____ THE _____ MATHEMATICS ___ ________ EDUCATOR _____ Volume 20 Number 2
Winter 2011
MATHEMATICS EDUCATION STUDENT ASSOCIATION THE UNIVERSITY OF GEORGIA
Editorial Staff
A Note from the Editor
Editor Catherine Ulrich Allyson Hallman
Dear TME readers, This issue closes out the twentieth volume of The Mathematics Educator. In the first issue of TME, on the inside front cover, the editorial panel laid out their motivation for starting a new mathematics education journal:
Associate Editors Zandra deAraujo Erik D. Jacobson Laura Lowe Laura Singletary Patty Wagner Advisors Dorothy Y. White MESA Officers 2003-2004 President Zandra deAraujo Vice-President Tonya DeGeorge Secretary Laura Lowe Treasurer Anne Marie Marshall NCTM Representative Allyson Hallman Undergraduate Representative Derek Reeves Hannah Channel Derek Reeves
The purpose of our journal is to fill a perceived need for providing students, faculty, alumni, and the broader mathematics education community a medium for more localized communication. We can foresee other journals improving as a result of our publication which provides current and future contributing authors and editors with additional experience in communicating ideas.
This issue’s table of contents reveals important qualities of the role The Mathematics Educator currently plays in the mathematics education community 20 years after its inception. Namely, in this issue, a broad array of the mathematics education community is represented. In fact, the vast majority of our contributions are now from outside of the UGA community. In this issue only one contributor has any direct tie to UGA; Sybilla Beckmann is a member of UGA’s Department of Mathematics, but her editorial is a call for action to all mathematics educators as the Common Core Standards are rolled out across the nation. In addition, the first authors on all of the other four articles are all emerging researchers. That is, they are graduate students or relatively recent graduates of mathematics education programs. Also, this issue communicates across a range of issues; elementary education (Inoue & Buczynski; Mueller, Yankelewitz, & Maher) to secondary education (Evans) to postsecondary mathematics education (Smith & Powell). In addition, some articles report on research studies focusing on student learning (Mueller, Yankelewitz, & Maher), others on teacher education (Evans), while one article is about a classroom experience, not a research project (Smith & Powell). In the end, there is no clear pattern to which issues of mathematics education that TME articles address. Instead, TME sets itself apart as a platform for emerging researchers to communicate about current issues in mathematics education, while also providing them experience in all facets of publication: submitting articles, reviewing articles, editing for the journal, and publishing the journal. As TME moves forward into its next decade of publication, we will strive to continue this service to the mathematics education community. Many thanks to all the people who have made The Mathematics Educator possible over the years. In particular, thank you to all of the contributors, reviewers, and editors who have helped shape the current issue. We hope you enjoy the results of our efforts! Sincerely, Catherine Ulrich Allyson Hallman 105 Aderhold Hall The University of Georgia Athens, GA 30602-7124
tme@uga.edu www.math.coe.uga.edu/TME/TMEonline.html
About the cover The cover art shows a student representation exploring combinatorial patterns. To learn more about this, please see the article by Mueller, Yankelewitz, and Maher.
This publication is supported by the College of Education at The University of Georgia
____________THE ________________ ___________ MATHEMATICS ________ ______________ EDUCATOR ____________ An Official Publication of The Mathematics Education Student Association The University of Georgia
2011
Volume 20 Number 2
Table of Contents 3
Guest Editorial… From the Common Core to a Community of All Mathematics Teachers SYBILLA BECKMANN
10 You Asked Open-ended Questions, Now What? Understanding the Nature of Stumbling Blocks in Teaching Inquiry Lessons NORIYUKI INOUE & SANDY BUCZYNSKI 24 Secondary Mathematics Teacher Differences: Teacher Quality and Preparation in a New York City Alternative Certification Program BRIAN R. EVANS 33 Sense Making as Motivation in Doing Mathematics: Results From Two Studies MARY MUELLER, DINA YANKELEWITZ, & CAROLYN MAHER 44 An Alternative Method to Gauss-Jordan Elimination: Minimizing
Fraction Arithmetic LUKE SMITH & JOAN POWELL
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© 2011 Mathematics Education Student Association All Rights Reserved
The Mathematics Educator 2011, Vol. 20, No. 2, 3–9
Guest Editorial… From the Common Core to a Community of All Mathematics Teachers Sybilla Beckmann As I write now, early in 2011, over 40 states have adopted the Common Core State Standards in Mathematics (National Governors Association Center for Best Practices and the Council of Chief State School Officers, 2010). This is a strong, coherent set of standards that asks students to understand and explain mathematical ideas and lines of reasoning. These standards should act as a framework to support vibrant teaching and learning of mathematics, in which students actively make sense of mathematics, discuss their reasoning, explore and develop ideas, solve problems, and develop fluency with important skills. Calls for vibrant mathematics teaching and learning and improved student proficiency in mathematics have been steady for a number of years (e.g., National Commission on Excellence in Education, 1983; National Council of Teachers of Mathematics [NCTM], 2000; National Commission on Mathematics and Science Teaching for the 21st Century [NCMST], 2000; National Mathematics Advisory Panel [NMAP], 2008). This new set of standards is one of many initiatives and projects that answer this call. But as strong as the Common Core standards are, they cannot improve students’ understanding of mathematics on their own—the standards will not teach themselves. Teachers are certainly key to enacting the standards as they are intended. They need to know the mathematics well, and they need to how to teach it in engaging and effective ways. Thinking about how to improve mathematics teaching and learning has led me to consider the larger environment in which this teaching and learning takes place. This, in turn, has led me to think about several interconnected groups and communities that are related to PreK-12 mathematics: the group of all mathematics teachers from pre-kindergarten through the college Sybilla Beckmann is a Professor of Mathematics at The University of Georgia. She is the author of a textbook for preservice elementary school teachers of mathematics, and her research interests are in the mathematical education of teachers and arithmetic geometry/algebraic number theory.
level; the community of mathematics researchers; and the community of mathematics educators, which includes teacher educators and mathematics education researchers. I am a member of all three groups and as I write I am drawing on my own experience as a mathematics researcher and member of a mathematics department; my experience teaching a variety of college-level mathematics courses, in particular, courses for prospective teachers; and my one year of teaching sixth grade mathematics. In this editorial, I want to make the case for the group of all mathematics teachers—from early childhood, to the elementary, middle, and high school grades, through the college and graduate levels, and including mathematics educators who teach teachers— to form a cohesive community that works together with the common goal of improving mathematics teaching at all levels. Although all parts of this community work individually towards improvement, I believe this community should take collective responsibility for improving the quality of all mathematics teaching. In making the case for the community of all mathematics teachers, I will draw on my knowledge of the mathematics research community and how it is set up to work towards excellence in mathematics research. I will also contrast research in mathematics and teaching of college-level mathematics, much of which is done by the same group of people. What Can Mathematics Research Tell Us About Mathematics Teaching? Why is it that at no level of mathematics teaching—from elementary school, to middle and high school, to the college level—do we have widespread excellence in mathematics teaching in this country? Of course, there are many examples of outstanding mathematics teaching and mathematics teachers, but, on the whole, there is cause for concern. At the K-12 level, mathematics teaching in the US is widely regarded as needing improvement (NCTM, 2000; NCMST, 2000; NMAP, 2008). Nor does it compare favorably with teaching in other countries, such as in Japan, where students perform well on international 3
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comparisons of mathematics achievement (Hiebert et al., 2003). At the college level, strong students who decide to leave the fields of mathematics, science, technology, and engineering often cite the quality of instruction as a key factor in their decision (Undergraduate Science, 2006; Seymore & Hewitt, 1997). The state of mathematics teaching in the US is especially perplexing in light of the strong state of mathematics research. The mathematics research community in this country is vibrant and active; it attracts students and researchers from all over the world. Unlike in the case of mathematics teaching, there are no calls for improving the quality of mathematics research. Yet the vibrant mathematics research community is also heavily involved in teaching: For many mathematics researchers, 50% of their job consists of teaching. It is perhaps surprising that mathematicians’ excellence in mathematics research has generally not translated into excellence in teaching. Could the differences in the way mathematics researchers conduct their research and their teaching shed light on why mathematics research is so full of vitality yet mathematics teaching seems to be suffering from malaise? If the conditions that lead to vibrancy in mathematics research could be adapted for and applied to mathematics teaching, could this lead to a similar vibrancy in mathematics teaching? This may seem like a preposterous question to ask, but there are some good reasons to believe the answer may be yes. What Conditions Make Mathematics Research Strong? Mathematics research is done within a cohesive community in which members share their work and build on each other’s ideas. Five factors strike me as key in making mathematics research so strong. First, mathematics researchers share their work, they discuss it in depth, and they built upon each other’s work. Second, the quality of a community member’s work is judged from within the community based on peer recognition and admiration, not from outside the community. Third, the mathematics research community is a meritocracy. Leaders in the community are active, enthusiastic community members whose work is admired within the community. Fourth, mathematics researchers have sufficient time to think about their research. And fifth, entry into the mathematics research community requires a high level of education and accomplishment. These five factors combine to create a highly motivating professional environment. Peer admiration within a cohesive, 4
meritocratic community of accomplished professionals provides a strong incentive for developing creative new approaches, sharing good ideas, and building upon each other’s work. In such a community, mathematics researchers are motivated to work in an especially deliberate and focused way. The mathematics research environment helps mathematics researchers to do more than just put in long hours of work; the very nature of the environment fosters an intense kind of work, a deliberate practice of honing and refining, of building on what others have done, and of looking for gaps and weaknesses. According to research on the development of expertise, it is precisely such a deliberate practice, done over a period of ten or more years, which is required for expertise (Ericsson, Krampe, & Tesch-Roemer, 1993; popularized by Colvin, 2008). Motivation research done over several decades and validated repeatedly in a variety of settings has shown that systems that fulfill people’s basic psychological needs for competence, autonomy, and relatedness lead to more internalized forms of motivation, which lead to more successful outcomes. In contrast, systems that people experience as externally controlling by such means as external evaluations, rewards, or punishments, lead to less internalized motivation and less successful outcomes (Deci & Ryan, 2008a, 2008b; Greene & Lepper, 1974; popularized by Pink, 2009)1. The mathematics research community fosters relatedness, namely the feeling of being involved with and related to others, because mathematicians share and discuss their work and build on each other’s ideas. In the process, the mathematics research community forms opinions about the quality of work, and community members attain a certain standing based on the community’s views about the quality of the work. The mathematics research community fosters competence because the quality of work matters in the community. The community fosters autonomy because finding innovative ideas and lines of reasoning leads to peer admiration. For mathematicians, the possibility of raising one’s standing within one’s community through the judgment of one’s peers—as opposed to through evaluation from outside of the community—may contribute to internalized motivation and a strong drive and desire to excel. Comparing Mathematics Teaching With Mathematics Research Now consider mathematics teaching with respect to the five factors— collaboration, internal evaluation, internal leadership, time, and high standards for entry—which make mathematics research so strong.
Sybilla Beckmann
First, mathematics teaching is often an isolated activity: Most teachers in the US do not share or discuss their practice in depth and do not have systematic ways of learning from each other. In the US, at the K-12 level, there is a “low intensity of teacher collaboration in most schools” and “the kind of job-embedded collaborative learning that has been found to be important in promoting instructional improvement and student achievement is not a common feature of professional development across many schools” (Darling-Hammond, Wei, Andree, Richardson, & Orphanos, 2009, pp. 23, 25). In contrast, the tradition of Lesson Study in Japan, in which groups of teachers collaborate to create, teach, revise, and publish research lessons, is an important factor in the high quality of teaching in Japan (Stigler & Hiebert, 1999; Lewis, 2002). Lesson Study has been specifically recommended for the new Common Core State Standards (Lewis, 2010). In the current system, at both the K-12 and college levels there is not a culture of looking for and using mathematical and pedagogical knowledge that has been developed by others to help improve mathematical understanding and teaching. Such knowledge does exist (although, of course, we still need more), but the lack of intellectual vigor concerning teaching sometimes makes mathematics teachers at all levels uninterested in considering new ideas. I have heard prospective elementary teachers claim that they do not need to know some mathematical concepts that directly relate to the school mathematics they will teach because the mathematical ideas are unfamiliar to them. Similarly, I have heard mathematicians express disdain for all mathematics education research. Second, because mathematics teachers do not routinely have opportunities to share or discuss findings about their teaching with any depth, they cannot develop good judgments about each other’s teaching. Also, teaching is usually evaluated from outside of the mathematics teaching community. At the college level, student evaluations are commonly used to evaluate teaching; K-12 teachers are evaluated by administrators, who are typically not active mathematics teachers and may have limited knowledge about mathematics teaching. Soon K-12 teachers may be evaluated and rewarded or rated based on their students’ performance on standardized tests (Duncan, 2009; Hearing on FY 2011, 2010). Third, it is not clear who the leaders are in mathematics teaching. Textbook authors and professional developers are sources of leadership; individuals may also think of a favorite teacher to
emulate. But, in the US, we do not seem to have a detailed and widely shared view of what constitutes effective teaching (Jacobs & Morita, 2002). In contrast, there is evidence that Japanese teachers do have a refined, shared conception of high-quality mathematics instruction (Corey, Peterson, Lewis, & Bukarau, 2010). Highly accomplished teachers in Japan become known through the public research lessons they teach during Lesson Study, thereby becoming leaders in teaching (Lewis, 2002). Fourth, teachers at all levels have many demands on their time. Most K-12 teachers do not have much time built into their demanding schedules for collaborative planning and thinking, for learning from and with outside experts, and for sharing, testing, and refining lessons or teaching ideas. According to Darling-Hammond et al. (2010, p. 20), “few of the nation’s teachers have access to regular opportunities for intensive learning” and “mathematics teachers averaged 8 hours of professional development on how to teach mathematics and 5 hours on the ‘in-depth study’ of topics in the subject area during 2003-04.” At the college level, the requirement to publish, the prestige of publishing research findings, and the dearth of opportunities to write in a scholarly way about teaching leave little or no time for serious, deliberate work that is devoted to teaching improvement. Fifth, as I will discuss below, the mathematical preparation of teachers is often weak. In sum, at both the college and K-12 levels, mathematics teachers are often not part of a strong professional community that promotes sharing and refining their practices or thinking deeply about mathematics teaching. Mathematics teaching is simply not set up to foster the development of internal motivation and deliberate practice towards expertise in the same way that mathematics research is. Entry Into the Mathematics Teaching Community A strength of the mathematics research community is the high standard for entry, namely, a PhD in mathematics, which involves intensive mathematics coursework, rigorous qualifying exams, and original research. In contrast, entry into the mathematics teaching profession is currently varied and often inadequate. Although some teachers receive excellent preparation for teaching mathematics, others are allowed to teach with very little mathematical preparation. The problem is especially severe for elementary teachers. The importance of mathematically knowledgeable teachers has been emphasized (NMAP, 2008), and there are recommendations that teachers take sufficient 5
Community of Math Educators
coursework to examine the mathematics they will teach in detail, with depth, and from the perspective of a teacher (Conference Board of the Mathematical Sciences, 2001; Greenberg & Walsh, 2008). But, in practice, the number and nature of the courses that are required often deviate considerably from these recommendations, as documented by Lutzer, Rodi, Kirkman, and Maxwell (2007, tables SP.5 and SP.6). Their research does not even take into account alternative routes to certification, which could require fewer courses still. The Common Core Standards in Mathematics are rigorous and will put a high demand on teachers. Many of us who teach teachers believe that most will need a much stronger preparation than they are currently getting to be ready to teach these new standards. What constitutes sufficient preparation? Based on my many years as a teacher of mathematics content courses for elementary teachers, I know that it takes far more work than most people realize to be ready to teach mathematics to children. My students (prospective teachers) are bright, hard working, and dedicated; I am not dealing with unmotivated or dull students. Yet it takes a full three semesters of courses (nine semester hours total) for us to discuss with adequate depth the ideas of PreK through grade five mathematics. In addition, I think that further content-heavy mathematics methods courses are necessary for such activities as examining curriculum materials used in elementary school, for studying how children solve mathematics problems, which may include examining videos and written work and interviewing children, and for learning how to question and lead discussions. It may seem surprising that so much coursework is needed in preparation for teaching elementary school. Yet even the mathematics that the very youngest children learn is surprisingly deep and intricate, and much is known about how children learn this mathematics (see Cross, Woods, & Schweingruber, 2009, for a summary about early childhood mathematics). Even mathematically well-educated people who have not specifically studied early childhood and elementary school mathematics from the perspective of teaching are unlikely to know it well enough to teach it. For example, if a child can count to five, and is shown five blocks in a row, will she necessarily be able to determine how many blocks there are, and, if not, what else does she need to know to do so? Why do we multiply numerators and denominators to multiply fractions, but we do not add numerators and denominators when we add fractions? Where do the formulas for areas and volumes come 6
from? Where does the formula for the mean come from? To teach the Common Core State Standards for Mathematics adequately, teachers will need to have studied all these details and many more. Children deserve to be taught by teachers who have studied such intricacies, inner workings, and subtle points that are involved in teaching and learning mathematics. If we think of other important professions, such as those in medicine, it is hard to imagine that doctors or nurses would be allowed to enter their professions without taking required coursework that focuses specifically on the knowledge these professionals rely on in their work. Yet in mathematics teaching, there are not such requirements. Would we be comfortable with doctors who had not had courses in chemistry and human anatomy, which underlie their work? Similarly, we should not be comfortable with teachers who have not studied the essential ideas they will need in their work. These essential ideas involve much more than being able to carry out procedures and solve problems in elementary mathematics or even in advanced mathematics. Governing boards and agencies set a bare minimum of coursework that is required for certification, but currently, the requirements do not ensure adequate coursework in mathematics before teaching. In my experience, without requirements from governing boards or agencies, it is difficult to ensure that individual certification programs will require prospective teachers to complete a sufficient amount of suitable mathematics coursework. Without changes, I believe that many teachers will not be ready to teach the Common Core State Standards when they begin teaching. A Community of All Mathematics Teachers Working Together Towards Excellence I have argued that mathematics research is strong along five factors—collaboration, internal evaluation, internal leadership, time, and high standards for entry—and that research in psychology indicates that these factors may play an important role in the success of mathematics research. I have also argued that mathematics teaching has considerable weaknesses in the five factors. Therefore it seems that mathematics teaching could benefit from an environment more like the environment of mathematics research. How could we create such an environment? First, suppose that all of us who teach mathematics could work within collaborative communities in which we share ideas and learn from each other about mathematics and about teaching. A number of small professional learning communities (including Lesson
Sybilla Beckmann
Study groups and Teacher Circles) exist. But, such smaller professional communities should also band together into a larger community—the community of all elementary, middle grades, high school, and college mathematics teachers and teachers of mathematics teachers. Why should the group of all mathematics teachers view itself as a cohesive community? One reason is the interconnectedness of mathematics teaching. At each grade level, mathematics teaching is intertwined with the teaching at all other grade levels. The mathematics teaching that students experience in elementary school influences what those students learn, which influences what the students will be ready to learn in later grades, which in turn influences the teaching that is possible and appropriate at those higher grade levels. In addition, the mathematics teaching that teachers experience in college surely influences their own understanding of mathematics and their subsequent mathematics teaching. Suppose that we—the community of all mathematics teachers—were to take collective responsibility for the quality of all mathematics teaching. The judgments we form about each other through the process of sharing our insights, ideas, and successes in improving our students’ performance could create a viable system of internal evaluation, so that, as with mathematics research, we might not need to be evaluated from outside the community. Sharing our knowledge within a strong professional community may motivate us to work deliberately, intensively, and continuously over the long term towards excellence in mathematics teaching. Given the electronic means of communication that are now available, we may have opportunities for sharing our work in teaching that were not available in the past. There may be new ways of organizing ourselves and working together that would help us learn useful information from each other and join together as we think about specific areas we are trying to improve in our teaching. Suppose that leadership within the community of all mathematics teachers were to evolve internally by peer recognition and admiration. Some intriguing research indicates that successful teaching communities that lead to improvements in student outcomes depend on certain kinds of leadership (Bryk, Sebring, Allensworth, Luppescu, & Easton 2010; Penuel, Riel, Krause, & Frank, 2009). So developing appropriate leadership could be important to developing effective communities of teachers. Suppose that all mathematics teachers had time built into their schedules to work together and to learn from each other and from outside experts, as
envisioned by Collins (2010, pp. 27, 36), in which teaching improvement is driven by “the kind of deep focus on content knowledge and innovations in delivery to all students that can only come when teachers are given opportunities to learn from experts and one another, and to pursue teaching as a scientific process in which new approaches are shared, tested, and continually refined across a far-flung professional community.” Suppose that the community of all mathematics teachers were to set professional standards for entry into the community. Although the relationships among teachers’ mathematical knowledge and skill, instructional quality, and student learning are not yet well understood and are a matter for research (NMAP, 2008), the mathematics teaching profession has the responsibility of setting reasonable standards for entry that fit with the duties of the profession. We should separate the need for research that can inform and guide us in making improvements in the preparation of teachers from making reasonable demands for entry into the profession, as is common in other professions. Doctors are required to study chemistry and biology because a certain level of knowledge of these subjects is a foundation for practicing medicine. Such a requirement is reasonable even though there may not be research evidence linking the study of biology and chemistry to good practice in medicine. To become a cosmetologist in Georgia requires at least 1500 credit hours of coursework in addition to passing written and practical exams (O.C.G.A., 2011). If we care about mathematics and about students, and if we want mathematics teaching to be treated as the serious profession it is, then we need to insist on higher minimum required coursework for entry into the profession even as we continue to study how to improve teacher preparation. We must also insist that agencies and boards in positions of responsibility for teachers honor our standards. One final thought about the evaluation of work from within a community by one’s peers: Albert Einstein supposedly had a sign outside his office saying, “Not everything that counts can be counted, and not everything that can be counted counts.” Although mathematicians do care about numbers of papers published and numbers of presentations, standing within the community is not determined purely by the numbers. An important component is the judgment of quality by one’s peers. Similarly, although it makes sense to find out how a teacher’s students do on common tests compared to other teachers’ students,
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Community of Math Educators
evaluating teachers purely in this way, without peer judgment in the mix, is counterproductive. The judgment of one’s peers is, of course, subjective and far from perfect, but it might be just what makes us try harder and look more closely at what other people have done. The process of looking closely at what others have done, trying to make improvements upon prior work, and bringing new ideas and insights to this work is precisely the process by which a field advances. Concluding Remarks The Common Core State Standards provide all of us with an opportunity for renewal, revision, and transition, and an opportunity to address the call for improving mathematics education that has been loud and clear over many years. But, in this process, two things seem certain: the first is that it will be tempting to make only superficial changes that merely repackage what we are already doing; the second is that we cannot create a top-notch system of mathematics education immediately and in one fell swoop. To create substantive improvements we must be in a system that helps us develop an authentic desire to improve and that promotes our internal motivation to do the hard work it will take to move towards excellence over the long term. I have argued that a key component in the success of the Common Core State Standards in Mathematics will be teaching and that in order to improve mathematics teaching, we must band together to form a cohesive community of mathematics teachers. Such a community should set standards for entry into the community, as do other important professions. I have argued that the possibility of raising one’s standing within the community through the judgment of one’s peers is likely to be a key driver of excellence. A stronger sense of community among all mathematics teachers, in which we challenge and support each other as we work together towards excellence in teaching, seems like a wonderful and exciting possibility. It is a vision for enlivening mathematics teaching from within through peer interactions rather than from without through external evaluations that will pit us against each other and sap our motivation. With apologies to John Lennon, you may say I’m a dreamer, but I hope I’m not the only one. Acknowledgements I would like to thank Kelly Edenfield, Francis (Skip) Fennell, Christine Franklin, and Tad Watanabe, for helpful comments on a draft of this paper. 8
References Bryk, A. S., Sebring, P. B., Allensworth, E., Luppescu, S., & Easton, J. Q. (2010). Organizing schools for improvement: Lessons from Chicago. Chicago: The University of Chicago Press. Collins, A. (2010). The Science of Teacher Development. Education Week, 30(13), 27, 36. Colvin, G. (2008). Talent is overrated. New York, NY: Penguin. Conference Board of the Mathematical Sciences (2001). The mathematical education of teachers (In “Issues in Mathematics Education” series, Vol. 11). Washington, DC: Author. (Available at http://www.cbmsweb.org/MET_Document/) Corey, D. L., Peterson, B. E., Lewis, B. M., & Bukarau, J. (2010). Are there any places that students use their heads? Principles of high-quality Japanese mathematics instruction. Journal for Research in Mathematics Education, 41, 438-478. Cross, C. T., Woods, T. A., & Schweingruber, H. (Eds.). (2009). Mathematics learning in early childhood, paths toward excellence and equity. Washington, DC: The National Academies Press. Darling-Hammond, L., Wei, R. C., Andree, A., Richardson, N., & Orphanos, S. (2009). Professional Learning in the Learning Profession: A Status Report on Teacher Development in the United States and Abroad. Dallas, TX: National Staff Development Council and The School Redesign Network at Stanford University. Deci, E. L., & Ryan, R. M. (2008a). Facilitating optimal motivation and psychological well-being across life's domains. Canadian Psychology, 49, 14-23. Deci, E. L., & Ryan, R. M. (2008b). Self-determination theory: A macrotheory of human motivation, development, and health. Canadian Psychology, 49, 182-185. Duncan, A. (June, 2009). Robust data gives us the roadmap to reform. Address by the Secretary of Education to The Fourth Annual Institute of Education Sciences Research Conference, Washington, DC. (Available at http://www2.ed.gov/news/speeches/2009/06/06082009.pdf) Ericsson, K. A., Krampe, R. T., & Tesch-Roemer, C. (1993). The role of deliberate practice in the acquisition of expert performance. Psychological Review, 100, 363-406. Greenberg, J., & Walsh, K. (2008). No common denominator: The preparation of elementary teachers in mathematics by America’s education schools Washington, DC: National Council on Teacher Quality. Greene, D., & Lepper, M. R. (1974). Effects of extrinsic rewards on children's subsequent intrinsic interest. Child Development, 45, 1141-1111. Hearing on FY 2011 Dept. of Education Budget: Hearing before the Subcommitee on Labor, Health and Human Services, Education, and Related Agencies, of the Senate Committee on Appropriations, 111th Cong. (2010) (testimony of Arne Duncan). (Available at http://appropriations.senate.gov/sclabor.cfm) Hiebert, J., Gallimore, R., Garnier, H., Givvin, K. B., Hollingsworth, H., Jacobs, J., … Stigler, J. (2003). Teaching mathematics in seven countries: Results from the TIMSS 1999 Video Study. Washington, DC: U.S. Department of Education, National Center for Education Statistics.
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Jacobs, J. K., & Morita, E. (2002). Japanese and American Teachers' Evaluation of Videotaped Mathematics Lessons. Journal for Research in Mathematics Education, 33, 154-175. Lewis, C. C. (2002). Lesson study: A handbook of teacher-led instructional change. Philadelphia, PA: Research for Better Schools. Lewis, C. C. (2010). A public proving ground for standards-based practice: Why we need it, what it might look like. Education Week, 30(3), 28–30. Lutzer, D. J., Rodi, S. B., Kirkman, E. E., & Maxwell, J. W. (2007). Statistical abstract of undergraduate programs in the mathematicalsSciences in the United States, Fall 2005 CBMS Survey. Washington, DC: Conference Board of the Mathematical Sciences . National Commission on Excellence in Education. (1983). A nation at risk: The imperative educational reform (Report No. 065000-00177-2). Washington, DC: U.S. Department of Education. National Commission on Mathematics and Science Teaching for the 21st Century. (2000). Before it’s too late: A report to the nation from the National Commission on Mathematics and Science Teaching for the 21st Century. Washington, DC: U.S. Department of Education. National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author. National Governors Association Center for Best Practices and the Council of Chief State School Officers. (2010). Common core state standards for mathematics. Retrieved July, 2010, from http://www.corestandards.org/the-standards/mathematics
National Mathematics Advisory Panel. (2008). Foundations for success: The final report of the National Mathematics Advisory Panel. Washington, DC: U.S. Department of Education. O.C.G.A. § 43-10-9 (LexisNexis, 2011). Penuel, W. R., Riel, M., Krause, A. E., & Frank, K. A. (2009). Analyzing teachers' professional interactions in a school as social capital: A social network approach. Teachers College Record, 111, 124–163. Pink, D. H. (2009). Drive, the surprising truth about what motivates us. New York, NY: Penguin. Seymour, E., & Hewitt, N. M. (1997). Talking about leaving: Why undergraduates leave the sciences. Boulder, CO: Westview Press. Stigler, J. W., & Hiebert, J. (1999). The teaching gap: Best ideas from the world’s teachers for improving education in the classroom. New York, NY: Free Press. Undergraduate science, math, and engineering education: What's working?, House of Representatives, 109th Cong. 14 (2006) (testimony of Dr. Elaine Seymore).
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Additional references can be found at the website http://www.psych.rochester.edu/SDT/index.php.
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The Mathematics Educator 2011, Vol. 20, No. 2, 10–23
You Asked Open-Ended Questions, Now What? Understanding the Nature of Stumbling Blocks in Teaching Inquiry Lessons Noriyuki Inoue & Sandy Buczynski Undergraduate preservice teachers face many challenges implementing inquiry pedagogy in mathematics lessons. This study provides a step-by-step case analysis of an undergraduate preservice teacher’s actions and responses while teaching an inquiry lesson during a summer math camp for grade 3-6 students conducted at a university. Stumbling blocks that hindered achievement of the overall goals of the inquiry lesson emerged when the preservice teacher asked open-ended questions and learners gave diverse, unexpected responses. Because no prior thought was given to possible student answers, the preservice teacher was not equipped to give pedagogically meaningful responses to her students. Often, the preservice teacher simply ignored the unanticipated responses, impeding the students’ meaning-making attempts. Based on emergent stumbling blocks observed, this study recommends that teacher educators focus novice teacher preparation in the areas of a) anticipating possibilities in students’ diverse responses, b) giving pedagogically meaningful explanations that bridge mathematical content to students’ thinking, and c) in-depth, structured reflection of teacher performance and teacher response to students’ thinking.
The things we have to learn before we do them, we learn by doing them. -Aristotle Many school reform efforts confirm the importance of inquiry-based learning activities in which students serve as active agents of learning, capable of constructing meaning from information, rather than as passive recipients of content matter (Gephard, 2006; Green & Gredler, 2002; National Council of Teachers of Mathematics, 1989, 2000; National Research Council [NRC], 2000). In inquirybased mathematics lessons, students are guided to engage in socially and personally meaningful constructions of knowledge as they solve mathematically rich, open-ended problems. Van de Walle (2004) emphasizes that conjecturing, inventing, and problem solving are at the heart of inquiry-based mathematics instruction. In inquirybased lessons, students develop, carry out, and reflect Noriyuki Inoue is an Associate Professor of Educational Psychology and Mathematics Education at the University of San Diego. His recent work focuses on inquiry pedagogy, Japanese lesson study, action research methodology, and cultural epistemology and learning. Sandy Buczynski is an Associate Professor in the Math, Science and Technology Education Program at the University of San Diego. She is the co-author of recently published: Story starters and science notebooking: Developing children’s thinking through literacy and inquiry. Her research interests include professional development, inquiry pedagogy, and international education.
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on their own multiple solution strategies to arrive at a correct answer that makes sense to them, rather than following the teacher’s prescribed series of steps to arrive at the correct answer (Davis, Maher, & Noddings, 1990; Foss & Kleinsasser, 1996; Klein, 1997). Inquiry-based lessons can be structured on a continuum from guided inquiry, with more direction from the teacher and a small amount of learner selfdirection, to open inquiry, where sole responsibility for problem solving lies with learner. In order to deliver an effective inquiry lesson, a set of general principles typically suggested in pedagogy textbooks are (a) to start the lesson from a meaningful formulation of a problem or question that is relevant to students’ interests and everyday experiences; (b) to ask open-ended questions, thus providing students with an opportunity to blend new knowledge with their prior knowledge; (c) to guide students to decide what answers are best by giving priority to evidence in responding to their questions; (d) to promote exchanges of different perspectives while encouraging students to formulate explanations from evidence; and (e) to provide opportunities for learners to connect explanations to conceptual understanding (e.g., NRC, 2000; Ormrod, 2003; Parsons, Hinson, & SardoBrown, 2000; Woolfolk, 2006). In effective mathematics inquiry lessons, students are supported in reflecting on what they encounter in the environment and relating this thinking to their personal understanding of the world (Clements, 1997).
Noriyuki Inoue & Sandy Buczynski
Preservice Teachers’ Difficulties with InquiryBased Lessons Though research indicates the importance of students’ construction of knowledge, multiple research reports show that preservice teachers are poor facilitators of knowledge construction in inquiry-based lessons, and that this persists even when they have gone through teacher-training programs focused on inquiry-centered pedagogy (Foss & Kleinsasser, 1996; Tillema & Knol, 1997). These research reports suggest that preservice teachers have a tendency to duplicate traditional methods, rather than implement the inquirybased pedagogy they experienced in their teacher education programs. Traditional pedagogy is typically associated with a style of direct instruction that is teacher-centered and front-loaded with subject matter. It is characterized by the teacher reviewing previously learned material, stating objectives for the lesson, presenting new content with minimal input from students, and modeling procedures for students to imitate. Throughout the lesson, the teacher periodically checks for learners’ understanding by assessing answers to closed-ended tasks and providing corrective feedback. In contrast, inquiry pedagogy is studentcentered and allows time for metacognitive development. In an inquiry classroom, the teacher presents an open-ended problem, and the learners explore solutions by defining a process, gathering data, analyzing the data and the process, and developing an evidence-supported claim or conclusion. Preservice teachers’ tendency to duplicate traditional methods has been attributed to a lack of a sound understanding of the mathematics content that they teach (Kinach, 2002a; Knuth, 2002), an inability to consider various ways students construct mathematical knowledge during instruction (Inoue, 2009), and a failure to consider how the content, curriculum map, and classroom situations contribute to students’ understanding (Davis & Simmt, 2006). Other researchers report that preservice teachers’ reluctance to stray from traditional methods is originates in the difficulty that they feel in conceptualizing their teaching in terms of the classroom culture and its social dynamics (Cobb, Stephan, McCain, & Gravemeijer, 2001; Cobb & Bausersfeld, 1995). These researchers suggest that preparing a non-traditional lesson requires the teacher to predict the possibilities of classroom interactions and carefully consider ways to shape the social norms of the classroom to facilitate studentcentered thinking. However, many preservice teachers go into teaching believing that knowledge transmission and teacher authority take precedence over students
constructing ideas (Klein, 2004). Even if preservice teachers learn about inquiry lessons in their teachertraining programs and believe students’ construction of ideas should take priority, they struggle to consider the multiple issues that are key for a successful inquiry lesson, limiting their ability to implement effective inquiry lessons. Current literature on inquiry learning focuses on identifying and theorizing various psycho-social factors that contribute to teachers’ ability to deliver an effective mathematics inquiry lesson in the classroom. Some researchers stress the importance of transforming teachers’ perceptions and understanding of inquiry teaching (Bramwell-Rejskind, Halliday, & McBride, 2008; Manconi, Aulls, & Shore, 2008; Stonewater, 2005) and transforming teachers’ beliefs (Robinson & Hall, 2008; Wallace & Kang, 2004). Others examine teachers’ personally constructed pedagogical content knowledge (PCK) that stems from their experiences as learners and their perceptions of students’ needs (Chen & Ennis, 1995). Wang and Lin (2008) add that students’ conception and understanding of inquiry lessons needs attention as well. Though some of these research findings are based on studies of inservice teachers’ struggles with implementing inquiry lessons, we believe that a majority of these research findings are applicable to preservice teachers as well. Rationale for Study Though the literature provides many insights on preservice teachers’ struggles in implementing inquirybased lessons, it is also essential to obtain a practicelinked understanding of why and how preservice teachers, particularly those who are motivated to teach mathematical inquiry lessons, encounter difficulty in authentic teaching contexts. This approach, taken together with the theoretical knowledge the literature provides, strengthens our understanding of how preservice teacher training should be improved. In this paper we address this identified need by presenting the results of one representative case study in which we analyzed a preservice teacher’s inquiry-based lesson taught in a mathematics classroom. Obtaining a practice-linked understanding of the nature of the difficulties that a preservice teacher might encounter in an inquiry lesson provides detailed insight into how specific contexts affect inquiry pedagogy. Research Questions In the process of implementing inquiry lessons, many interactions can serve as stumbling blocks to the inquiry process. Here, a stumbling block refers to instances where a teacher poses an open-ended 11
Stumbling Blocks
question, the students respond (or fail to respond), and the teacher does not know how to reply to students’ comments or questions and, therefore, fails to guide the learning activity towards the rich inquiry investigation initially envisioned. With this in mind, the questions guiding this investigation are: 1) What instances serve as stumbling blocks for preservice teachers motivated to teach inquiry lessons? 2) How do preservice teachers respond to stumbling blocks and how do those responses influence the direction of the lesson? Any preservice teacher who crafts an inquiry lesson could encounter these types of stumbling blocks. Therefore, the knowledge gained from this study can inform preservice teacher education in two ways: It can increase teacher educators’ awareness of preservice teachers’ issues in implementing inquirybased lessons, and it can guide teacher educators in helping preservice teachers deliver effective mathematics lessons that are characterized by meaningful construction of knowledge through mathematics inquiry activities. Methodology Context University faculty from the Mathematics Department in the School of Arts and Science were joined by faculty from the Learning and Teaching Department in the School of Leadership and Education Sciences to conduct a summer mathematics camp for third- through sixth-grade students. This cross-campus collaboration provided an opportunity for the faculty to mentor undergraduate preservice teachers to help them bridge mathematical content with pedagogical practice and knowledge of context. Preservice teachers were offered the opportunity to serve as camp instructors in order to gain experience teaching inquiry lessons. We then observed their inquiry-based lessons in order to answer our research questions. The summer mathematics camp served as an ideal environment for this investigation since the camp’s novice teachers could practice implementing inquiry lessons free from the pressure of supervisor evaluation and externally imposed state standards or tests. The camp also created an environment where learners were given time to be curious and to develop positive attitudes toward learning mathematics. The mission of math camp was two-fold: to provide mathematical enrichment for a diverse group of children and to support the mathematical and pedagogical development of preservice elementary school teachers. The summer math camp had unique contextual constraints that distinguished it from a traditional 12
classroom. The mathematics instruction was embedded in a thematic context of Greek mathematicians. Each class included combined grade levels; one for rising second through fourth graders and one for rising fifth through sixth graders. Students from across the city attended the camp. While this context diverged from a typical classroom, some features of the camp provided a context similar to a typical mathematics class: both classes had a heterogeneous mix of diverse students and class periods lasting 90 minutes. We believe that the educational context also highlighted opportunities for a preservice teacher to implement a quality inquirybased lesson because the students attended voluntarily and were not pressured to perform on tests or homework. Similarly, there was little pressure on the instructors to cover certain material or deliver inquiry lessons with the goal of students’ performing well on tests. Camp instructors (preservice teachers) University mathematics professors recruited camp instructors from an undergraduate elementary mathematics methods course. The professors informed preservice teachers enrolled in the course about the opportunity to practice inquiry-based lessons in this summer camp, and a number of them applied to be camp instructors. As part of the recruitment process, the candidates were informally interviewed about their interests and goals in mathematics teaching. Eight preservice teachers were selected to serve as camp instructors based on their enthusiasm and willingness to work in the team. All the eight camp instructors were female undergraduates working towards a bachelor’s degree in liberal studies combined with an elementary teacher credential. During the interview, all of the camp instructors professed an interest in developing their teaching skills and math content knowledge in an activity-rich environment and were willing to commit to one week of camp preparation mentoring and one week of classroom teaching during camp. Each camp instructor’s experience working with children varied, as did their time in the teacher education program. Two were sophomores, three were juniors, and three were seniors. Though they were at different points in the program, half of the camp instructors had completed foundation courses in education, and all had completed the mathematics teaching methods course. Camp students Because the camp was advertised in the local newspaper, children from across the city, as well as faculty and university-neighborhood children, applied
Noriyuki Inoue & Sandy Buczynski
and were accepted on a first-come, first-served basis. The price of the camp for each child was approximately $300. The university helped cover the operational cost of the camp with a $7,490 academic strategic priority fund award which applied to the camp instructors’ salaries, classroom resources, and tuition reduction for eligible children. Each of the two classes enrolled 30 students with approximately ten each of rising second, third, and fourth graders in the lower grade class and approximately 15 each of rising fifth and sixth graders in the upper grade class. Caucasian, Latino, and Asian students made up approximately 60%, 30%, and 10% of the student campers respectively. Because of the age range in each class, a wide range of skill levels was observed. Undergraduate preservice teacher preparation For entering the undergraduate elementary teacher education program preservice teachers must be in the university’s Bachelor’s degree program in a content area of their choice. To become a licensed elementary teacher they must then complete the 33-credit hour multiple-subject education program and pass a standardized state content exam. Most of the students who enroll in the undergraduate credential program are liberal studies majors with a concentration in one of the content areas. The credential program includes coursework in educational psychology, content pedagogy (including elementary mathematics teaching methods taught by mathematics faculty with expertise in pedagogy), educational theory, and courses on children’s learning. Through this coursework, the students gain field experience through a series of practicum placements in K-6 schools. In these placements they observe classroom instruction and teach inquiry lessons under the guidance of a schoolbased and a university-based supervisor. Camp instructor preparation Before the math camp program began, the camp instructors attended a required week-long preparation program focused on deepening their mathematics content knowledge, as well as mathematics pedagogy. Camp instructors learned about key developmental and learning theories and were exposed to current research on K-12 learners’ social and personal construction of meaning. They also learned how to develop lesson plans using a wide variety of instructional approaches that focused on helping students construct knowledge. Because exposure to inquiry-based lesson development differed across camp instructors, faculty mentors provided both group and one-on-one instruction and mentorship in this pedagogy.
Four faculty mentors led seminars on the general principles of inquiry lessons. These faculty members also taught in the university’s regular preservice credential program, therefore, the seminars were highly comparable to the university’s regular preservice program. Constructivist philosophy influenced the design of the seminars. Preservice teachers were taught to encourage children to actively make sense of mathematics instead of teachers presenting and modeling procedures for solving problems. In other words, giving authoritarian feedback to students was not a pedagogical strategy valued by the math camp faculty mentors. The camp instructors were also taught lesson planning based on detailed task analyses of instructional goals called “backward design” (Wiggins & McTighe, 2005). In backward design, the teacher begins with the end in mind, deciding how learners will provide evidence of their understanding, and then designs instructional activities to help students learn what is needed to meet the goals of the lesson. Based on this model, the camp instructors started designing a camp lesson with an initial mathematical idea and then discussed with their peers how students’ understanding of this idea could be gauged. During the process, camp instructors were introduced to strategies including cooperative learning, active learning, mathematical modeling, and the use of graphic organizers. The instruction in these strategies emphasized inquiry pedagogy with the goal of learners developing understanding beyond rote knowledge. Faculty members also guided camp instructors in how to navigate the disequilibrium between what children want to do versus what they can do. Though the camp preparation lasted only one week, students instructors reviewed the basic principles of learning and designed a camp lesson based on pragmatic instructional fundamentals. They learned what to include in a lesson plan, how to pace activities within the 90-minute class period, how to pose appropriate questions, how to make use of wait time, how to manage the classroom, and what to consider in a thoughtful reflection on teaching experience. Camp instructors’ lessons were required to (a) provide a mathematically rich problem allowing for open-ended inquires of mathematical ideas, (b) ask open-ended questions, (c) encourage students to determine answers with rationales in their responses for problem solving activities, (d) and elicit exchanges of different ideas. Faculty mentorship Though the camp instructors had a theoretical understanding of how students make sense of 13
Stumbling Blocks
mathematical ideas and lesson planning, they did not have any practical experience in planning appropriate inquiry-based mathematics lessons for students. To guide and support them through this process, faculty mentors were available to provide generous assistance and offer advice. Two mathematics professors and two education professors, one specializing in educational psychology and the other in curriculum design and STEM education, served as mentors. During the precamp training session, the eight student instructors were paired into four teams of two instructors each. All four mentor professors worked with each team. Mentors met individually with each team to discuss their proposed lesson activities in terms of developmental appropriateness, mathematics content, and pedagogy. At the end of the preparation week, a survey developed by the education faculty members (see Appendix A) was administered to get a sense of teachers’ beliefs and attitudes toward inquiry learning after the camp instructor training program. According to this survey, all eight camp instructors had positive views about inquiry-based lessons and were motivated to deliver effective inquiry-based, activity-rich lessons in the camp. Each camp instructor team member designed one inquiry lesson for the lower grade class and then one for the upper grade class, or vice versa. These two lessons focused on the same content, but were modified to be appropriate for each age range. For instance, one camp instructor of each team-taught her lesson for the lower grade class during the morning session and the other taught her lesson for this class in the afternoon session. The teams then presented the upper grade lessons in the same manner later in the week. The camp instructors were completely responsible for classroom instruction, however, mentor professors were present in the classroom for additional support as needed. When camp instructors were not teaching, they were observing their peer camp instructors’ lessons. At the end of each day, all camp instructors met as whole group with all of the faculty mentors. These whole group meetings included discussions of how the day went and what aspects of the lesson were effective or ineffective, what revisions could be made, and what concepts should be revisited. Following this schedule, the camp instructors taught each lesson variation during the camp week and had a chance for individual feedback and advice from a faculty member after each presentation of their lesson. A large part of the camp instructors’ experiential learning arose from their reflection on their daily
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teaching experience and the mentors’ input about their classroom performance. Data collection and analysis During the camp session, the authors observed a total of 12 of the camp instructors’ inquiry lessons: three randomly chosen pairs of lower and upper grade lessons and six other randomly chosen lessons. These observations allowed the researchers to gain a conceptual understanding of the inquiry process that these novice teachers enacted from their lesson plans. Researchers made field notes and video-taped lessons as video cameras and audio-visual staff were available. Camp instructors also completed a post-lesson questionnaire (Appendix B) that probed their perceptions of their effectiveness as math teachers and their success with inquiry pedagogy. The 12 observed lessons offered a wide range of information about the camp instructors’ approach to inquiry learning in elementary mathematics. The crosscase analyses of observed lessons led us to believe that the camp instructors followed the design principles of an inquiry lesson. However, camp instructors had moments of difficulty that we have termed stumbling blocks. As described earlier, in these moments, the camp teacher responded to an instructional situation in such a way that derailed the inquiry-based goals of the lesson and created moments that significantly undermined the quality of the inquiry lesson. There were many different kinds of stumbling blocks. When we looked into the cases more closely, we found that the nature of the stumbling blocks was highly contextual and content specific. In each case, stumbling blocks emerged in math camp lessons, one after another, in ways that were nested. By nested we mean that once one stumbling block appeared in the lesson, it had the potential to contribute to the emergence of a subsequent stumbling block. For example, when a preservice teacher was faced with no student response to a question she posed, she resorted to guiding students with leading questions without giving ample opportunity for students to make sense of the concept. In this case, the initial problem that was created from the first stumbling block (i.e. not knowing how to respond when students have no input) served as a foundation for another stumbling block to emerge (i.e., guiding students with leading questions). These in-depth case study analyses revealed that each inquiry component of the lesson depended on other components of that lesson that developed from previous actions and interactions in the lesson. The only way to evaluate the inquiry process and conduct meaningful analyses of the stumbling blocks in inquiry
Noriyuki Inoue & Sandy Buczynski
pedagogy appeared to be step-by-step deconstructions of the camp instructors’ actions and utterances within each lesson. We reasoned that presenting a representative individual camp instructor as a case study was the most effective way to capture the nature of stumbling blocks that the camp instructors encountered during the presentation of their lessons. An analysis of one camp instructor’s performance provided the best insight into strengths and weaknesses of the inquiry teaching process. The following section describes the findings of this study based on this methodological framework. Findings The case analyses of the observed lessons indicate that all the teachers were not successful in giving mathematically and pedagogically meaningful explanations, ignored creative responses from the students, or switched the nature of instruction to the
direct transmission model where the teacher simply gave answers to students as an authority with little attention to students’ thinking about mathematics. A variety of kinds of stumbling blocks were identified, and each type of stumbling block was found in multiple cases. The type of stumbling blocks depended on the mathematical content covered in the lessons, the students, and the particular dynamics of the interactions in the classroom. We analyzed and identified different stumbling blocks that the camp instructors encountered when teaching a mathematics inquiry-based lesson. Based on the cross-case analyses of the observed lessons, we identified a total of thirteen stumbling blocks, summarized in Table 1. To exemplify these stumbling blocks, the following section describes an in-depth case study that illustrates the ways a preservice teacher actually encountered the stumbling blocks during the
Table 1 Stumbling Blocks Location of Stumbling Block
Type of Stumbling Block
Teacher Response
Planning the Inquiry Lesson
1. Problematic problem design
The teacher uses a poor or developmentally inappropriate set up of an inquiry problem or question for the lesson.
2. Insufficient time allocation
In the interest of time, the teacher moves on to the next planned activity scheduled in the lesson plan in spite of students’ confusion or teaching opportunities created by students’ responses.
3. Unanticipated student response
The teacher fails to anticipate students’ input and cannot give a pedagogically and mathematically meaningful response to the students.
4. No student response
The teacher fails to give a meaningful response to students’ silence or lack of input in reply to the teacher’s question.
5. Disconnection from prior knowledge
The teacher’s response severs connections between the lesson and students’ prior knowledge or their attempt to make sense of the concept using their experiential knowledge.
6. Lack of attention to student input
The teacher ignores the students’ input in reply to the teacher’s open-ended questions.
7. Devaluing of student input
The teacher diminishes student input by rejecting their suggestions and shuts down their attempts at making sense of a problem.
8. Mishandling of diverse responses
The teacher does not know how to effectively manage or give meaningful traffic controls to diverse responses that the students gave for open-ended questions.
9. Leading questions
The teacher’s questions directly guide learners to the answer without creating enough opportunities for learners to make sense of the concept.
10. Premature introduction of material
The teacher introduces a new concept or symbol without giving enough opportunity for students to make sense of previous content.
11. Failure to build bridges
The teacher misses important opportunities to effectively connect his or her question to the problem solving activity or the ideas that the students formulated during problem solving.
12. Use of teacher authority
The teacher uses his or her authority to impose the answer or strategy or judge the students’ answer or strategy as right or wrong.
13. Pre-empting of student discovery
The teacher provides the main conclusion that students were supposed to discover.
Teacher Response to Student Input
Teacher Delivery of Inquiry Lesson
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Stumbling Blocks
presentation of an upper grade lesson. This descriptive case study (Yin, 2003) illustrates a thick description of some of the issues faced in mathematics inquiry pedagogy. We chose this particular case among all the observed cases since it most vividly informs us of the nature of stumbling blocks that the camp instructors typically encountered in the inquiry lessons observed in the study. We labeled each stumbling block that the preservice teacher encountered at various points of the lesson in reference to the above table. Case study Jessica (pseudonym) was a university senior majoring in liberal studies and enrolled in the university’s elementary school teaching credential program. She had successfully completed an educational psychology class and other credential courses, but did not have any formal mathematics teaching experience. In the pre-survey Jessica described effective teaching as, “The teacher needs to prepare the students for what they will learn by getting them interested and providing a foundation to build on (pre-teach if necessary). Also the lesson/activity must be engaging (hands-on, collaborative).” This comment is representative of all the camp instructors’ responses to this survey item; many indicated their belief in the importance of using activities meaningful to children, eliciting children’s interest, and scaffolding students’ personal construction of knowledge that is grounded in their prior experiences. Even though camp instructors’ comments did not encompass the entirety of inquirybased learning principles, they did show understanding of the key ideas. Jessica, in particular, showed an understanding of her intention and plan to deliver an inquiry lesson in the summer camp. Jessica’s instruction contained a wide variety of stumbling blocks and can inform us of the nature of the difficulties that preservice teachers can encounter in teaching inquiry lessons. As discussed before, Jessica prepared her lesson plan in the pre-camp session with guidance from the faculty mentors. The objectives of Jessica’s lesson were to help children (a) understand the concept of ratio and (b) understand π as a constant ratio for any circle. As was true with the other camp instructors, Jessica was friendly and made personal contact with children very well. In the upper grade classroom, the children were divided into six groups sitting at different tables. First, with a picture of trail mix containing M&Ms projected, Jessica asked her students if they liked M&Ms. After hearing a positive response from most of the children, she indicated that she had three brands of trail mix, each containing M&Ms, nuts, and raisins. 16
She said, “We need to find out which brand we should buy if we would like to get the most M&Ms.” With this problem statement, she has started with an interesting story and formulated an open-ended question relevant to students’ everyday experiences, a key component of an inquiry-based lesson. Jessica then explained that each brand of trail mix advertised that it contained two scoops of M&Ms. She showed ladles of varying sizes and said that she was not sure which ladle each brand used to measure their two scoops. She asked the children how they might determine which brand of trail mix to purchase to maximize the amount of M&Ms. The children were listening to her attentively and appeared to be thinking about this question. Then one child answered, “What about finding how much sugar that they have on the box?” This child knew that the package should indicate its amount of sugar on the nutrition label and that this would vary directly with the amount of M&Ms. She had not anticipated the direction of this response that overall sugar content would indicate quantity of M&Ms nor had she anticipated this particular question from one of the children. Jessica did not know how to respond. If she simply said no, her inquiry lesson would have lost its real life meaningfulness and stumble just as it was starting. After a pause, Jessica responded, “But the raisins also have sugar, so we cannot compare trail mixes based on sugar [to determine amount of M&Ms in each brand].” With this clever response, the child who asked the question seemed convinced and began to consider other approaches. In responding to the child’s unexpected answer, Jessica managed to avoid using her authority as a teacher to silence the child. This child came up with a creative solution which she responded to by acknowledging his creativity while re-directing his thinking. While the children were still considering solutions, Jessica suggested using actual trail mixes as stimuli and distributed three plastic bags that contained different brands of trail mix along with a worksheet to each group of students. She asked the children to collaborate at each table to record 1) the number of M&Ms, 2) the number of nuts, 3) and number of raisins. First through her failure to elicit additional solution strategies from students to connect their thinking to the problem and second through her imposing a particular strategy to count M&Ms for problem solving, two stumbling blocks (SB11: Failure to build bridges & SB12: Use of teacher authority) emerged. In other words, this strategy of counting pieces of trail mix did not come from the students, and
Noriyuki Inoue & Sandy Buczynski
Jessica did not help the children make sense of what they were asked to do. One thing that needs to be pointed out here is that these stumbling blocks emerged even though a) she was trying to follow some aspects of the inquiry teaching principles by having students gather evidence and by giving priority to this evidence in responding to questions (NRC, 2000) and, b) the students were given the opportunity to connect the process of problem solving with the concrete experience of counting M&Ms and comparing their results for the different brands. After receiving the bags, the children immediately started collaborating and using various strategies to count the pieces in the trail mixes. When they finished, Jessica recorded and displayed their results to discuss with the class (Figure 1). Total pieces in trail mix
Brand of Trail Mix
M&Ms
Nuts
Raisins
Crunch Beans
66, 67
110, 117, 126, 111
32, 35, 36, 34
220
Sweet & Salty
30
69, 70
11
110
Snick Snack
71
167
91
329
Figure 1. Results of each group’s counting Note: Each cell displays the counting results from the groups. If the groups’ counting results are the same, the same number was not added to the table to avoid repetition.
It was not until this point in the lesson that we realized that each group’s bag of a particular brand of trail mix had the same number of M&Ms, nuts, and raisins; Jessica had set up the brands to have no counting variations among groups. Of course, the children made minor counting mistakes and this resulted in the variations shown in Figure 1. After the completing the chart, she suggested the correct number of pieces for each brand and totaled them in the table for the children. In other words, she told them the right answers as an authority (SB12: Use of teacher authority). After the counting activity, she asked the class, “Which one [brand] has more M&Ms compared to the whole package?” When no child responded to the question (SB4: No student response), Jessica pointed out the numbers in the table (Crunch Beans brand: 67 M&Ms in 220 pieces and Snick Snack brand: 71 M&Ms in 329 pieces). Again, she asked the question, “Which brand had more M&Ms compared to the total
number of trail mix pieces in the package?” Jessica attempted to assist children in finding the answers to her close-ended question by directing them to relevant evidence. However, the children remained confused because her explanation did not clarify that she was asking about the proportion of M&Ms compared to the total amount of trail mix. Still, with no child answering, she then asked “67 over 220 or 71 over 329?” (SB9: Leading questions). A child asked, “You mean, if the price of the packages is the same?” Again, Jessica clearly did not anticipate this question (SB3: Unanticipated student response), and responded by saying, “It's a good question,” but went on to say that price was not important here since the price of three packages of one brand could be the same as one package of another brand; she pointed out that price comparison can be very complicated, and is not what they should consider in the problem solving. Jessica’s reply indicated she did not understand the issue the student raised. The student was questioning a tacit assumption that Jessica did not address: if the prices were different then the comparison was invalid (SB5: Disconnect from prior knowledge). Jessica’s response confused this student and many students began interjecting comments about the price and taste of various trail mixes they liked. Finding out which trail mix to buy by holding the price constant is a meaningful assumption for the children since it is what shoppers (and parents) do in choosing a brand of trail mix in everyday life. However, this line of thinking was different from how Jessica’s problem set up: Her assumption was to hold the number of pieces constant, not a very meaningful set-up in everyday life. This discrepancy in interpretation of the problem served as another stumbling block for the inquiry process (SB1: Problematic problem design). She responded, “Let’s not think about the price; let’s explore this problem” (SB7: Devaluing student input). No one resisted this suggestion or asked why they needed to make such an assumption. Jessica began to subordinate children’s meaning construction with her response loaded with authority (SB12: Use of teacher authority). Then she asked the children if they knew what a ratio was, and wrote on the board, “Ratio = The relationship between quantities” (SB10: Premature introduction of material). At this point, the children began to be increasingly quiet. Without explaining why she was introducing the concept of ratio here, Jessica indicated that the children could use calculators to divide numbers and compare the ratios. She asked, “Does anyone know why divide?” No one answered the question, but some of the children were silently 17
Stumbling Blocks
taking a note of the formula on their notebooks. Here, Jessica did not follow up on her question or support learners’ meaning-making in the lesson (SB4: No student response and SB5: Disconnect from prior knowledge). This created another stumbling block that seems to have led the children to gradually shut down their personal construction of meaning in the following lesson segments. She pointed out to the children, “67/220 is like dividing a pizza. If you divide, you can compare, right?” (SB5: Disconnect from prior knowledge and SB9: Leading questions) Then she told the children that she could use her calculator to execute the division. She input two numbers to the calculator and wrote on the board (67/220) ≈ .30454 .1 Here, she did not take the time to explain what she was doing or why she was doing this procedure prematurely assuming that students knew the meaning of the mathematical symbols (SB5: Disconnect from prior knowledge and SB10: Premature introduction of material). The children became increasingly confused because she failed to give an effective explanation of the meaningfulness of the assumption (i.e., holding the number of pieces constant for ratio comparison), why they needed to divide the numbers, or why this relates to the action of dividing a pizza. It was apparent that her failure to provide a meaningful rationale for the new mathematical idea created another stumbling block in the lesson. It was no surprise that, at this point, most of the children became quiet and watched her actions rather than participating in a discussion about the mathematics, which created the atmosphere of a traditional mathematics classroom. Once she introduced the concept of ratio she began moving forward in her lesson plan despite student confusion (SB 2: Insufficient time allocation). Jessica hesitated for a while, but, in the interest of completing her planned lesson, she proceeded to introduce a new concept, a constant. She wrote Constant on the board and said, “Let's think about a constant. What is the quantity that does not change?” (SB10: Premature introduction of material) Jessica did not relate this question about constants to the M&M problem (SB11: Failure to build bridges). However, many of the children suddenly became engaged and raised their hands. They actively responded, “speed of light”, “fingers”, “gravity.” The sudden increase in participation was possibly because they knew that they could answer the question and project personal meaning in the activity. Jessica smiled and nodded in response to each of the children’s responses, but did
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not give any other reply (SB8: Mishandling diverse responses). Next, Jessica suddenly introduced a story where Romans killed Archimedes while he was thinking about a circle he drew on beach sands. Without providing a rationale for the story (SB11: Failure to build bridges), she asked the children, “So… what's so interesting about circles? Again, given this opportunity to participate in the open-ended question-and-answer activity, children presented many different responses: “The circle is round,” and “It looks like a hole.” She responded with nodding and smiling (SB8: Mishandling diverse responses). Then one child answered, “Unlimited angle, no end, no beginning.” Jessica looked a little puzzled by this child’s answer. Clearly, she did not expect this response, and did not know how to react (SB3: Unanticipated student response). She missed this educational opportunity to discuss central angles of a circle (SB5: Disconnect from prior knowledge). The child’s creative, yet unexpected, response served as another stumbling block in the lesson. She told the child that it was an interesting idea, and asked other children for more ideas. Without clarifying the link with her original activity (SB11: Failure to build bridges), she then distributed objects that contained circles (cans, lids, duct tape, etc.). She explained what circumference and diameter of a circle are, and asked each group to measure them on their object and explore the relationship. However, she did not give any instruction about how to measure these accurately (SB1: Problematic problem design). After some exploration, most of the groups could reason that the ratio is a little more than three (though some children already knew the ratio to be 3.14 from school mathematics classes). Then Jessica wrote on the board the symbol π and mentioned that this ratio is a constant for any circle, pointing out that the relationship values calculated were almost the same across the various groups (SB13: Pre-empting student discovery). At this point, a child raised her hand to say that their ratio was “a little less than three” in her group. Jessica approached this group and, while other groups were waiting, realized they were saying that three times diameter is a little less than circumference and therefore the ratio of circumference to diameter is a little less than three. She needed to spend a significant amount of the time for this particular group since she could not understand the logic underlying their claim, which served as another stumbling block in the lessons (SB3: Unanticipated student response). Essentially the
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group claimed that 3d < C implied that (C/d) < 3. Jessica missed another opportunity to compare these different ideas, address the misconception, and help the children construct their own meaning of ratio. Without sharing this group’s information with the entire class, Jessica began to explain the next activity (SB2: Insufficient time allocation). Then she was stopped when one child suddenly asked how to find π of an oval. Again, Jessica did not expect this question and did not know how to respond. This moment served as another stumbling block in the lesson (SB3: Unanticipated student response). She simply told the child that she would think about it. She continued her lesson by drawing examples of inscribed and circumscribed triangles on the board and asked if the circumference of the circle or perimeter of the triangles are bigger. At this point, she did not have enough time to finish the lesson (SB 2: Insufficient time allocation). She thanked all the children for their interesting ideas, and the lesson ended within the class’s allotted time. As shown in the above case study, the ways the stumbling blocks appeared and influenced the lessons and student instructor’s responses were complex. No simple descriptions seem to be able to capture the complexity and dynamics of these factors that were intertwined with each other. Please note that the camp instructors chose to teach in the inquiry-centered math camp and were highly motivated to teach inquiry lessons. This makes this group an unlikely representative of preservice teachers across the nation. However, we do not believe that this weakens our argument, but strengthens it. It highlights one of the key points of the study: Even if pre-teachers are motivated to teach inquiry lessons, they encounter stumbling blocks and often do not know how to overcome them. The following examples illustrate stumbling blocks that the camp instructors encountered in other lessons observed in the study. 1. When students explored how to expand a 2’ x 3’ picture of a face into a larger dimension without distorting the image, one of the students responded 5’ x 6’ since 2 + 3 = 5 and 3 + 3 = 6. The teacher simply responded, “That’s not quite right,” in front of all the students without explaining or examining this. (SB6: Lack of Attention to Student Input and SB7: Devaluing Student Input) 2. In a lesson to understand the effects of volume and mass on water displacement, the teacher started the lesson by asking very broad questions: “Have you ever heard the term volume?”, “How does it relate to math?”, and “What are some ways to find
volume?” When the students gave diverse responses to these broad questions, the camp teacher merely listened to them without giving any sort of meaningful response and proceeded on to the planned water displacement activity. This lack of validation or even acknowledgement of students’ responses quieted their eagerness to answer, as after that the students spoke up much less in the lesson. (SB8: Mishandling Diverse Responses and SB6: Lack of Attention to Student Input) 3. In a lesson to find the height of a pyramid, the camp teacher asked students to measure their shadows and compare the measures with their actual heights. Though the children made the connection between this activity and finding the height of a pyramid, the teacher did not make the relationship between the two activities explicit. (SB5: Disconnect from prior knowledge) Most of the camp instructors expressed a sense of failure in their first round of teaching, but did not clearly know the reasons why their inquiry lessons did not work very well. After the first lesson, each teacher had an individual meeting with the faculty mentor(s) who observed the lesson to go over their reflection and receive suggestions for improvement. This opportunity to discuss the lesson presentation with faculty helped the camp instructors determine reasons why these stumbling blocks were encountered and provided them expert advice on what to do next.
Discussion Teachers are known to possess personalized understanding of how to support children’s construction of knowledge based on their own learning experiences (Chen & Ennis, 1995; Segall, 2004). The weaknesses observed in the student instructors’ inquiry lessons could be seen to stem from a novice teacher’s immature understanding of how elementary school students think and understand mathematics. Ample literature supports this point that teachers’ beliefs and understanding about how children learn significantly impact the effectiveness of teaching (e.g., Kinach, 2002b; Warfied, Wood, & Lehman, 2005). The camp instructors knew that helping children make connections between abstract concepts and the material representations of those concepts is critical to a meaningful inquiry-based lesson. However, there were many instances in our observations where such connections were not made, in spite of the camp instructors knowledge about inquiry lesson principles 19
Stumbling Blocks
and their willingness to deliver an inquiry lesson. Through the study, we found that asking preservice teachers to take command of a full classroom of students with only a crash preparation course was insufficient to circumvent problems. In a way, it is not a surprise that the camp instructors had so much room for improvement in lesson planning, time management, and transitioning from one activity to the next. However, we believe that Jessica’s lesson was informative for any teacher, novice or veteran, who attempts to deliver an inquiry lesson because it addresses several important stumbling blocks that anyone could encounter in delivering an inquiry lesson, yet not recognize in the moment of teaching. There are several lessons that could be learned from this case study. First, Jessica often asked openended questions to help the children personally construct meaning. But it was often the case that she had not considered possible learner responses, and caught off guard, did not know how to respond, as she admitted in the post-teaching interview. As a result, in the face of these rich educational opportunities provided by the diverse learner responses, Jessica was unprepared, inflexible, and unable to make use her knowledge of content and children’s thinking to improvise within the parameters of the lesson. Consequently, the children who contributed these thought-provoking answers did not receive any meaningful response or validation of their ideas from the teacher or their peers. We observed many such instances throughout the math camp and suspect this is true in many classrooms where teachers attempt to deliver an inquiry lesson. To be fair, in preparing these camp instructors to teach a math concept, anticipating student response was not a part of their lesson plan template. Inoue (2011) points out that this should be a key component of lesson design; in his cross-cultural lesson study research, “failure to anticipate students’ diverse responses” was one of the reasons that an inquiry lesson was ineffective and deviated from the initially planned instructional goal. One solution for this could be adding a section to the lesson plan template that includes thinking through possible student answers to questions, as Japanese educators are known to include in their lesson plans (Fernandez & Yoshida, 2004). This would help them prepare for conceptual conversations in the classroom and help them evaluate the lesson by envisioning students’ diverse perspectives. Furthermore, we discovered that preservice teachers were more focused on their own performance
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than on their students’ performance in these classroom experiences. Berliner (1994) reports similar finding from his research that inexperienced teachers had a tendency to focus on teachers’ actions, rather than students’ actions, and lacked the ability to identify meaningful sub-activities integrated within a larger lesson. The camp instructors’ tendency to focus on their own performance could work in favor of their learning from their pedagogical mistakes, strengthening their content delivery, and gaining insight into the inquiry process. However, it could do little to help them learn to consider each action in the lesson in reference to the goals of student learning, a necessity for successful inquiry instruction. Passing over or ignoring a response that has merit in the conceptual framework of the lesson could not only lower the learner’s inclination to participate in the lesson but also invalidate or devalue the learner’s prior knowledge (Cooper, 1994, 1998). If a learner’s response falls outside the realm of anticipated responses, yet presents an opportunity to expose the class to a different facet of understanding of a mathematical concept, the teacher needs to first validate a student’s legitimate response and then use that response to navigate to the instructional goals. In addition, the teacher needs to have the flexibility and confidence in content matter to build a consensus among the students and achieve the instructional objective within the allocated time. More importantly, expecting diverse and high-quality responses and knowing how to incorporate a learner’s prior knowledge in the lesson is an important skill for teachers to have when delivering an inquiry lesson. What holds the key seems to be a deep understanding of how children think and might react to concepts. For example, Lubienski (2007) points out that lower socioeconomic students are more likely to use “solid common sense” (p.54) than they are to use a sophisticated mathematical concept. Researchers point out that mathematical word problems are often written without accurately reflecting the experiences described in the problems (Greer, 1997; Inoue, 2005; Verschaffel, Greer, & De Corte, 2000). In Jessica’s episode, this was evident when students tried to compare prices of brands of trail mix rather than use ratios of ingredients. Being aware that students often become engaged with the real-world aspects of math problems rather than focusing on the mathematical concept intended by the problem would help teachers anticipate students’ responses and prepare a means to incorporate that line of thinking into the math concept being studied.
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Second, in spite of the camp instructors’ attempt to explain abstract concepts in ways that were grounded in students’ prior experiences and concrete models, they often failed to explain mathematical concepts in pedagogically meaningful ways. Jessica simply did not know how to explain the concept of ratios effectively, and gave irrelevant, misleading, and disconnected instructions to the students. This issue could be due to her lack of deep knowledge on how to deliberately unpack a mathematical concept, as seen with her treatment of ratio, constant, and π. As Jessica herself pointed out in the post-teaching interview, it is important for teachers to possess multi-layered content knowledge in order to utilize it in a pedagogical meaningful way to make connections among concepts and to a learner’s prior knowledge. In this sense, teaching inquiry lessons effectively requires going beyond merely following the principles of inquiry lessons to developing a deep pedagogical understanding of how one could construct each mathematical concept in a meaningful way (Ball, Hill, & Bass, 2005; Ma, 1999; Mapolelo, 1998). This point is emphasized by Shulman (1987) who claimed “the key to distinguishing the knowledge base of teaching lies at the intersection of content and pedagogy” (p. 15). Shulman (1986, 1987) described the construct of pedagogical content knowledge as an integrated synthesis of subject matter content knowledge and pedagogical knowledge that is specific to education and separates teachers from mere content experts. For instance, when the student says a circle is interesting because it has “unlimited angles, no end, no beginning,” the teacher needs to be able to confidently respond to this mathematical statement in a pedagogical meaningful way without losing the scope of the planned lesson. For instance, the teacher could have responded by first pointing out that central angles are an important concept to explore in understanding a circle. She could have instructed the students on drawing central angles and challenged them to draw a 180° central angle, the diameter, before starting the activity to discover π, the ratio of diameter to the circumference. Preservice teachers must have meaningful criteria for suitable open-ended questions that are supported by deep pedagogical content knowledge. This will enable them to anticipate probable responses and have sufficient confidence in their content knowledge to determine which avenues are worth exploring and how best to follow up on diverse student input. Finally, we learned that the evaluation of an inquiry lesson for teacher training requires step-by-step
analyses of the preservice teacher’s actions and utterances linked with prior actions, appropriateness of content, and students’ understanding. Instructional dialogues that teachers engage in to support students’ understanding are highly complex and do not allow linear, simplistic formulation (Inoue, 2009; Leinhardt 1989, 2001). We found this to be the case with the inquiry lessons that we observed. It is also true that we cannot expect epistemological enlightenment to arise spontaneously through two weeks of mentoring, no matter how strong the mentoring or the mentees. However, we infer that experience, combined with consistent, constructive step-by-step analysis of teacher performance, curricular materials, and learner interaction with both, are needed to support the teachers in order to build an effective teaching practice. Likewise, what is helpful to any teacher is to plan a lesson, deliver the lesson, and reflect on their step-bystep actions in the classroom. From this careful scrutiny of the meaningfulness of their every action and reaction, the teacher can become aware of possible stumbling blocks in their instructional path and use this awareness to strengthen future performances. We do not deny the importance of learning the guidelines for delivering effective inquiry lessons. However, we also learned that actually teaching an inquiry lesson based on the guidelines had many possible pitfalls for teachers. We learned the importance of improving teachers’ ability to explain content, anticipate children’s responses, respond appropriately to children's answers, and link new content to appropriate models and experiences. For meaningful knowledge construction to occur, implementing an inquiry lesson is not enough. It is more important to effectively negotiate the topic’s meaning as different perspectives and interpretations emerge at each moment in the classroom’s instructional dialogue (Cobb & Yackel, 1998; Voigt, 1996). Without such micro-level support for students’ thinking, any attempt to deliver inquiry lessons will encounter many serious stumbling blocks.
Implications for Teacher Training These stumbling blocks of inquiry-based lessons are not bumps to be ignored. In designing professional development for teachers or coursework for preservice teachers, highlighting the role of teacher awareness on teacher actions and re-actions to learners is critical to developing practice (Buczynski & Hansen, 2010). For example, a teacher may not spend enough time acknowledging or validating students’ responses. If, from careful examination of a teaching event, the teacher is made aware of this behavior, then this 21
Stumbling Blocks
awareness creates a heightened sensitivity to the issue and a potential for change in the teacher’s future behavior. Teacher practice then, goes beyond principled pedagogy to conscious responsiveness. We found that teachers asking open-ended questions instead of giving answers provided learners with an opportunity to blend new knowledge with prior knowledge. However, this approach also presented a stumbling block. The teacher opens herself up to the unexpected nuances of the mathematical concept. By being aware that posing open-ended questions can lead to uncharted territory and take extra instructional time, teachers can design a lesson plan that includes consideration of strategies for anticipating responses and allowing contingency time for the subsequent discussion that might arise. A planned approach to the student comment would allow validation of the student’s ideas and integration of student’s prior knowledge with the topic at hand, two essential components of an inquiry learning activity.
Conclusion This close examination of a preservice teacher’s performance in math camp resulted in valuable information about potential stumbling blocks that stand in the way of effectively executing a well designed inquiry lesson. This study points teacher educators to focus teacher preparation in the areas of (a) anticipating possibilities in children’s diverse responses, (b) developing deep pedagogical content knowledge that allows them to give pedagogically meaningful responses and explanations of the content, and (c) step-by-step analysis of a teacher’s actions and responses in the classroom. Although the case study described in this article provides only a snapshot of one novice teacher’s practice, we believe that uncovering these stumbling blocks across all camp instructors overcomes this limitation. To truly transform traditional teaching toward the inquiry model, we need to make every effort to help teachers become aware of potential missteps so that they may avoid these stumbling blocks in future inquiry lessons.
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Bramwell-Rejskind, F. G., Halliday, F., & McBride, J. B. (2008). Creating change: Teachers’ reflections on introducing inquiry teaching strategies. In B. M. Shore, M. W. Aulls, M. A. B., Delcourt (Eds.), Inquiry in education, volume 2: Overcoming barriers to successful implementation (pp. 207–234). New York: Erlbaum. Buczynski, S., & Hansen, B. (2010). Impact of professional development on teacher practice: Uncovering connections. Teaching and Teacher Education, 26, 599–607. Chen, A., & Ennis, C. D. (1995). Content knowledge transformation: An examination of the relationship between content knowledge and curricula. Teaching and Teacher Education, 4, 389–401. Clements, D. H. (1997). In my opinion: (Mis?) Constructing constructivism. Teaching Children Mathematics, 4, 198–200. Cobb, P., & Bausersfeld, H. (1995). Introduction: The coordination of psychological and sociological perspectives in mathematics education. In P. Cobb & H. Bausersfeld (Eds.), The emergence of mathematical meaning (pp.1-16). Hillsdale, NJ: Erlbaum. Cobb, P., Stephan, M., McClain, K., & Gravemijer, K. (2001). Participating in classroom mathematical practices. The Journal of the Learning Sciences, 10, 113–163. Cobb, P., & Yackel, E. (1998). A constructivist perspective on the culture of the mathematics classroom. In F. Seeger, J. Voigt, & U. Waschescio (Eds.), The culture of the mathematics classroom. City Needed: Cambridge University Press. Cooper, B. (1994). Authentic testing in mathematics? The boundary between everyday and mathematical knowledge in national curriculum testing in English schools. Assessment in Education: Principles, Policy & Practice, 11, 143–166. Cooper, B. (1998). Using Bernstein and Bourdieu to understand children's difficulties with “realistic” mathematics testing: An exploratory study. International Journal of Qualitative Studies in Education, 11, 511–532. Davis, R. B., Maher, C. A., & Noddings, N. (1990) Constructivist views on the teaching and learning of mathematics. Journal for Research in Mathematics Education, 1, 1–6. Fernandez, C. & Yoshida, M. (2004). Lesson study: A Japanese approach to improving mathematics teaching and learning. Mahwah, NJ: Lawrence Erlbaum. Foss, D., & Kleinsasser, R. (1996) Preservice elementary teachers’ views of pedagogical and mathematical content knowledge, Teaching and Teacher Education, 12, 429–442. Greer, B. (1997). Modeling reality in mathematics classrooms: The case of word problems. Learning & Instruction, 7, 293–307. Green, S. K., & Gredler, M. E. (2002). A review and analysis of constructivism for school-based practice. School Psychology Review, 31, 53–71. Inoue, N. (2005). The realistic reasons behind unrealistic solutions: The role of interpretive activity in word problem solving. Learning and Instruction, 15, 69– 83. Inoue, N. (2009). Rehearsing to teach: Content-specific deconstruction of instructional explanations in preservice teacher trainings. Journal of Education for Teaching, 35, 47– 60. Inoue, N. (2011). Consensus building for negotiation of meaning: Zen and the art of neriage in mathematical inquiry lessons through lesson study. Journal of Mathematics Teacher Education, 14, 5–23.
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Kinach, B. M. (2002a). A cognitive strategy for developing pedagogical content knowledge in the secondary mathematics methods course: Toward a model of effective practice. Teaching and Teacher Education, 18, 51–71. Kinach, B. M. (2002b). Understanding and learning-to-explain by representing mathematics: Epistemological dilemmas facing teacher educators in the secondary mathematics “methods” course. Journal of Mathematics Teacher Education, 5, 153– 186. Klein, M. (1997) Looking again at the ‘supportive’ environment of constructivist pedagogy. Journal of Education for Teaching, 23, 277–292. Klein, M. (2004). The premise and promise of inquiry based mathematics in preservice teacher education: A poststructuralist analysis. Asia-Pacific Journal of Teacher Education, 32, 35–47. Knuth, E. J. (2002). Teachers’ conceptions of proof in the context of secondary school mathematics. Journal of Mathematics Teacher Education, 5, 61–88. Leinhardt, G. (1989). Math lessons: A contrast of novice and expert competence. Journal for Research in Mathematics Education, 20, 52–75. Leinhardt, G. (2001). Instructional explanations: A commonplace for teaching and location for contrast. In V. Richardson (Ed.), Handbook of research on teaching (4th ed., pp.333–357). Washington, DC: American Educational Research Association. Lubienski, S. (2007). What we can do about achievement disparities? Educational Leadership, 65, 54–59. Ma, L. (1999). Knowing and teaching elementary mathematics: Teachers' understanding of fundamental mathematics in China and the United States. Mahwah, NJ: Lawrence Erlbaum. Mapolelo, D. C. (1998). Do pre-service primary teachers who excel in mathematics become good mathematics teachers? Teaching and Teacher Education, 15, 715–725. Manconi, L., Aulls, M. W., & Shore, B. M. (2008). Teachers’ use and understanding of strategy in inquiry instruction. In B. M. Shore, M. W. Aulls, & M. A. B., Delcourt (Eds.), Inquiry in education, volume 2: Overcoming barriers to successful implementation (pp. 247–269). New York: Lawrence Erlbaum. National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: Author. National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author. National Research Council (NRC). (2000). Inquiry and the national science education standards: A guide for teaching and learning. Washington, D.C.: National Academy Press. Ormrod, J. R. (2003). Educational psychology: Developing learners (5th Ed.). Upper Saddle River, NJ: Merrill. Parsons, R., Hinson, S. L., & Sardo-Brown, D. (2000). Educational psychology: A practitioner-researcher model for teaching. Belmont, CA: Wadsworth.
Robinson, A., & Hall, J. (2008). Teacher models of teaching inquiry. In B. M. Shore, M. W. Aulls, & M. A. B., Delcourt (Eds.), Inquiry in education, volume 2: Overcoming barriers to successful implementation (pp. 235–246). New York: Lawrence Erlbaum. Stonewater, J. K, (2005). Inquiry teaching and learning: The best math class study. School Science and Mathematics, 105, 36– 47. Segall, A. (2004). Revisiting pedagogical content knowledge: The pedagogy of content/the content of pedagogy. Teaching and Teacher Education, 20, 489–504. Shulman, L. S. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 15, 4–14. Shulman, L. S. (1987). Knowledge and teaching: Foundations of the new reform. Harvard Educational Review, 57, 1–22. Tillema, M., & Knol, W. (1997). Collaborative planning by teacher educators to promote belief changes in their students. Teachers and Teaching: Theory and Practice, 3, 29–46. Van de Walle, J. (2004). Elementary and middle school mathematics: Teaching developmentally. Boston, MA: Pearson. Verschaffel, L., Greer, B., & De Corte, E. (2000). Making sense of word problems. Lisse, The Netherlands: Swets & Seitlinger. Voigt, J. (1996). Negotiation of mathematical meaning in classroom processes: Social interaction and learning mathematics. In L. P. Steffe, P. Nesher, P. Cobb, G. A. Goldin & B. Greer. (Eds.), Theories of mathematical learning. Mahwah, NJ: Erlbaum. Wallace C. S., & Kang, N. H. (2004). An investigation of experienced secondary science teachers’ beliefs about inquiry: An examination of competing belief sets. Journal of Research in Science Teaching, 41, 936–960. Wang, J-R, & Lin, S-W. (2008). Examining reflective thinking: A study of changes in methods students’ conceptions and understandings of inquiry teaching. International Journal of Science and Mathematics Education, 6, 459–479. Warfield, J., Wood, T., & Lehman, J. D. (2005). Autonomy beliefs and the learning of elementary mathematics teachers. Teaching and Teacher Education, 21, 439–456. Wiggins, G., & McTighe, J. (2005). Understanding by design. Alexandria, VA: Association for Supervision and Curriculum Development. Woolfolk, A. E. (2006). Educational psychology (10th Ed.). Boston, MA: Allyn & Bacon. Yin, R. (2003). Case study research: Design and methods (3rd ed.). Thousand Oaks, CA: Sage. Mathematics Curricula (pp. 457468). Mahwah, NJ: Erlbaum.
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Though “≈” was incorrectly used, and likely unfamiliar to the students, we do not classify this as a significant stumbling block given the context of the on-going issues in the lesson.
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The Mathematics Educator 2011, Vol. 20, No. 2, 24–32
Secondary Mathematics Teacher Differences: Teacher Quality and Preparation in a New York City Alternative Certification Program Brian R. Evans Providing students in urban settings with quality teachers is important for student achievement. This study examined the differences in content knowledge, attitudes toward mathematics, and teacher efficacy among several different types of alternatively certified teachers in a sample from the New York City Teaching Fellows program in order to determine teacher quality. Findings revealed that high school teachers had significantly higher content knowledge than middle school teachers; teachers with strong mathematics backgrounds had significantly higher content knowledge than teachers who did not have strong mathematics backgrounds; and mathematics and science majors had significantly higher content knowledge than other majors. Further, it was found that mathematics content knowledge was not related to attitudes toward mathematics and teacher efficacy; thus, teachers had the same high positive attitudes toward mathematics and same high teacher efficacy, regardless of content ability.
In fall 2000, New York City faced a predicted shortage of 7,000 teachers and the possibility of a shortage of up to 25,000 teachers over the following several years (Stein, 2002). In response to these shortages the New Teacher Project and the New York City Department of Education formed the New York City Teaching Fellows (NYCTF) program (Boyd, Lankford, Loeb, Rockoff, & Wyckoff, 2007; NYCTF, 2008). The program, commonly referred to as Teaching Fellows, was developed to recruit professionals from other fields to fill the large teacher shortages in New York City’s public schools with quality teachers. The Teaching Fellows program allows careerchangers, who have not studied education as undergraduate students, to quickly receive provisional teacher certification while taking graduate courses in education and teaching in their own classrooms. Teaching Fellows begin graduate coursework at one of several New York universities and begin student teaching in the summer before they start independently teaching in September. Those who lack the 30 required mathematics course credits are labeled Mathematics Immersion, and must complete the credits within three years, while those with the minimum 30 required credits are labeled Mathematics Teaching Fellows. Brian R. Evans is an Assistant Professor of mathematics education in the School of Education at Pace University in New York. His primary research interests are in teacher knowledge and beliefs, social justice, and urban mathematics education.
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Prior to teaching in September, Teaching Fellows must pass the Liberal Arts and Sciences Test (LAST) and the mathematics Content Specialty Test (CST) required by the New York State Education Department (NYSED) for teaching certification. Teaching Fellows receive subsidized tuition, earn a one-year summer stipend in their first summer, and are eligible to receive full teacher salaries when they begin teaching. Over the next several years Teaching Fellows continue taking graduate coursework while teaching in their classrooms with a Transitional B license from the NYSED that allows them to teach for a maximum of three years before earning Initial Certification. The Teaching Fellows program has grown very quickly since its inception in 2000. According to Boyd et al. (2007), Teaching Fellows “grew from about 1% of newly hired teachers in 2000 to 33% of all new teachers in 2005” (p. 10). Currently, Teaching Fellows account for 26% of all New York City mathematics teachers and a total of about 8,800 teachers in the state of New York (NYCTF, 2010). Of all alternative certification programs in New York, the Teaching Fellows program is the largest (Kane, Rockoff, & Staiger, 2006). There has been concern that teachers prepared in alternative certification programs are lower in quality than those prepared in traditional teacher preparation programs (Darling-Hammond, 1994, 1997; DarlingHammond, Holtzman, Gatlin, & Heilig, 2005; LaczkoKerr & Berliner, 2002); thus measures of teacher quality are of particular concern to the Teaching
Brian R. Evans
Fellows Program, New York State policymakers, and other states implementing and evaluating alternative certification programs. Teacher Quality Teacher quality is one of the most important variables for student success (Angle & Moseley, 2009; Eide, Goldhaber, & Brewer, 2004). In this study three variables that indicate teacher quality were analyzed: content knowledge, attitudes toward mathematics, and teacher efficacy. The National Council of Teachers of Mathematics (NCTM, 2000) defined highly qualified mathematics teachers as teachers who, in addition to possessing at least a bachelor’s degree and full state certification, “have an extensive knowledge of mathematics, including the specialized content knowledge specific to the work of teaching, as well as a knowledge of the mathematics curriculum and how students learn” (p. 1). NCTM recommends that high school mathematics teachers have the equivalent of a major in mathematics, commonly understood in New York to be at least 30 credits of calculus and higher. For middle school teachers NCTM recommends that mathematics teachers have at least the equivalent of a minor in mathematics. The NYSED requires both high school and middle school mathematics teachers to have at least 30 credits in mathematics. Researchers have supported the notion that strong mathematical content knowledge is essential for quality teaching (Ball, Hill, & Bass, 2005; Ma, 1999; NCTM, 2000). Teachers prepared in alternative certification programs, such as the Teaching Fellows program, have on average higher content test scores than other teachers (Boyd, Grossman, Lankford, Loeb, & Wyckoff, 2006; Boyd et al., 2007). While these findings are encouraging, there has been a lack of concentrated focus on the content knowledge of secondary mathematics teachers specifically. Building on this position, this study examined the content knowledge of the Teaching Fellows with teacher content knowledge defined for this study to be the combination of knowledge, skills, and understanding of mathematical concepts held by teachers. Despite strong academic credentials (Kane et al., 2006), few differences are found between the mathematics achievement levels of students of Teaching Fellows and traditionally prepared teachers in grades 3 to 8 (Boyd, Grossman, Lankford, Loeb, Michelli, & Wyckoff , 2006; Kane et al., 2006), but, after several years of teaching experience, the students of Teaching Fellows outperform the students of traditionally prepared teachers in academic
achievement (Boyd, Grossman, Lankford, Loeb, Michelli, & Wyckoff , 2006). However, very few studies have focused on Teaching Fellows who teach mathematics in particular, and an emphasis on secondary mathematics Teaching Fellows is needed because much of the existing research has focused on teachers in elementary schools only. Teacher quality typically addresses content and pedagogical knowledge, but examining teacher attitudes is also important. Previous studies have shown that attitudes in mathematics have a positive relationship with achievement in mathematics for students (Aiken, 1970, 1974, 1976; Ma & Kishor, 1997), which may translate to teachers as well. Attitudes toward mathematics are defined for this study as the sum of positive and negative feelings toward mathematics in terms of self-confidence, value, enjoyment, and motivation held by teachers. Amato (2004) found that negative teacher attitudes can affect student attitudes. Trice and Ogden (1986) found that teachers who had negative attitudes toward mathematics often avoided planning mathematics lessons. Charalambous, Panaoura, and Philippou (2009) called for teacher educators to actively work to improve teachers’ attitudes. Like teacher attitudes, teacher efficacy is a strong indicator of quality teaching (Bandura, 1986; Ernest, 1989). Teachers with high efficacy, defined as a teacher’s belief in his or her ability to teach well and belief in the ability to affect student learning outcomes (Bandura, 1986), are more student-centered, innovative, and exhibit more effort in their teaching (Angle & Moseley, 2009). Additionally, teachers with high efficacy are more likely to teach from an inquiry and student-centered perspective (Czerniak & Schriver, 1994), devote more time to instruction (Gibson & Dembo, 1984; Soodak & Podell, 1997), and are more likely to foster student success and motivation (Angle & Moseley, 2009; Ashton & Webb, 1986; Haney, Lumpe, Czerniak, & Egan, 2002). Mathematics anxiety is one hurdle in building efficacy in teachers: Teachers with higher mathematics anxiety were found to believe themselves to be less effective (Swars, Daane, & Giesen, 2006). Research in Alternative Certification Concern about alternative teacher certification programs has led to an interest in studying the effects of these programs in U.S. classrooms, particularly in terms of teacher quality issues (Darling-Hammond, 1994, 1997; Darling-Hammond et al., 2005; Evans, 2009, in press; Humphrey & Wechsler, 2007; LaczkoKerr & Berliner, 2002; Raymond, Fletcher, & Luque, 25
Mathematics Teacher Differences
2001; Xu, Hannaway, & Taylor, 2008). Many recent studies examining the Teaching Fellows in New York schools focus on teacher retention and student achievement as variables to determine success. Though these variables are important (Boyd, Grossman, Lankford, Loeb, Michelli, & Wyckoff, 2006; Boyd, Grossman, Lankford, Loeb, & Wyckoff, 2006; Boyd et al., 2007; Kane, et al., 2006; Stein, 2002), there is also a need to investigate other variables related to success, such as teacher content knowledge, attitudes toward mathematics, and teacher efficacy because these variables can affect student learning outcomes (Angle & Moseley, 2009; Ball et al., 2005; Bandura, 1986; Ernest, 1989). Few studies have examined the relationship between mathematical content knowledge and teacher efficacy. Those that exist have examined preservice teacher content knowledge and efficacy for traditionally prepared teachers (i.e. Swars et al., 2006; Swars, Hart, Smith, Smith, & Tolar, 2007). Researchers have called for a strong academic coursework component for alternative certification teachers (Suell & Piotrowski, 2007), yet little is known about the knowledge and skills that these teachers already possess on entering the program. In order to most effectively use limited teacher training resources, policymakers need more research in this area. Humphrey and Wechsler (2007) noted, “Clearly, much more needs to be known about alternative certification participants and programs and about how alternative certification can best prepare highly effective teachers” (p. 512). Theoretical Framework The theoretical framework of this study is based upon the positive relationship between mathematical achievement and attitudes found in students (Aiken, 1970, 1974, 1976; Ma & Kishor, 1997), the need for strong teacher content knowledge (Ball et al., 2005), and teaching efficacy theory (Bandura, 1986). Bandura found that teacher efficacy can be subdivided into a teacher’s belief in his or her ability to teach well and his or her belief in a student’s capacity to learn well from the teacher. Teachers who feel that they cannot effectively teach mathematics and affect student learning are more likely to avoid teaching from an inquiry and student-centered approach (Angle & Moseley, 2009; Swars et al., 2006). Purpose of the Study and Research Questions This study is a continuation of a previous study (Evans, in press) that examined changes in content knowledge, attitudes toward mathematics, and the teacher efficacy over time of new teachers in the 26
Teaching Fellows program. The previous study found that Teaching Fellows increased their mathematical content knowledge and attitudes over the course of the semester-long mathematics methods course while teaching in their own classroom. They also held positive attitudes toward mathematics and had high teacher efficacy both in terms of their ability to teach well and their ability to positively affect student outcomes. The focus of the present study is finding differences in the various categories of Teaching Fellows across these three variables. Teacher quality is an important concern in teacher preparation (Eide et al., 2004), and particularly for mathematics teachers of high-need urban students (Ball et al., 2005). The purpose of this study was to determine differences in these variables among different categories of alternative certification teachers in New York City. Determining these differences is important for two reasons. First, it is important for teacher recruitment. If policy makers, administrators, and teacher educators know which teacher characteristics lead to the highest levels of content knowledge, attitudes, and efficacy, recruitment can be better focused. Second, it is important for teacher preparation. Knowing which teachers need the most support, and in which areas, can lead to increased teacher quality through better preparation and focused professional development. This study addresses the following research questions: 1. Are there differences in mathematical content knowledge, attitudes toward mathematics, and teacher efficacy between middle and high school Teaching Fellows? 2. Are there differences in mathematical content knowledge, attitudes toward mathematics, and teacher efficacy between Mathematics and Mathematics Immersion Teaching Fellows? 3. Are there differences in mathematical content knowledge, attitudes toward mathematics, and teacher efficacy between undergraduate college majors among the Teaching Fellows? 4. Is mathematical content knowledge related to attitudes toward mathematics and teacher efficacy? The first three research questions addressed the differences that existed among types of teachers in content knowledge, attitudes toward mathematics, and teacher efficacy. These questions are important because it is imperative that policy makers, administrators, and teacher educators determine teacher quality for those who will be teaching mostly
Brian R. Evans
high-need urban students. In this study “high-need” refers to urban schools in which students are of lower socio-economic status, have low teacher retention, and lack adequate resources. The fourth research question involved synthesizing the results of the first three questions to generate further implications. Methodology This study employed a quantitative methodology. The sample consisted of 42 new teachers in the Teaching Fellows program (N = 30 Mathematics Immersion and N = 12 Mathematics Teaching Fellows) with approximately one third of the participants male and two thirds of the participants female. The teachers in this study were selected due to availability and thus represented a convenience sample with limited generalizability. The Teaching Fellows in this study were enrolled in two sections of a mathematics methods course, which involved both pedagogical and content instruction in the first semester of their program. These sections, taught by the author, focused on constructivist methods with an emphasis on problem solving and real-world connections in line with NCTM Standards (2000). Teaching Fellows completed a mathematics content test and two questionnaires at the beginning and end of the semester. The mathematics content test consisted of 25 free-response items ranging from algebra to calculus and was designed to measure general content knowledge. The mathematics content test taken at the end of the semester was similar in form and content to the one taken at the beginning. Prior to their coursework and teaching, the Teaching Fellows take the Content Specialty Test (CST). CST scores were recorded as another measure of mathematical content knowledge. The scores range from 100 to 300, with a minimum state-mandated passing score of 220. The CST consists of multiplechoice items and a written assignment and has six subareas: Mathematical Reasoning and Communication; Algebra; Trigonometry and Calculus; Measurement and Geometry; Data Analysis, Probability, Statistics and Discrete Mathematics; and Algebra Constructed Response. Data from the CST were analyzed to validate findings suggested by the mathematics content test. Attitudes toward mathematics were measured by a questionnaire designed by Tapia (1996) that has 40 items measuring characteristics such as selfconfidence, value, enjoyment, and motivation in mathematics. The instrument uses a 5-point Likert scale of strongly agree, agree, neutral, disagree, to strongly disagree. Teacher efficacy was measured by a
questionnaire adapted from the Mathematics Teaching Efficacy Beliefs Instrument (MTEBI) developed by Enochs, Smith, and Huinker (2000). The MTEBI is a 21-item 5-point Likert scale instrument with the same choices as the attitudinal questionnaire. It is grounded in the theoretical framework of Bandura’s efficacy theory (1986). Based on the Science Teaching Efficacy Belief Instrument (STEBI-B) developed by Enochs and Riggs (1990), the MTEBI contains two subscales: Personal Mathematics Teaching Efficacy (PMTE) and Mathematics Teaching Outcome Expectancy (MTOE) with 13 and 8 items, respectively. Possible scores range from 13 to 65 on the PMTE, and 8 to 40 on the MTOE. Higher scores indicated better teacher efficacy. The PMTE specifically measures a teacher’s concept of his or her ability to effectively teach mathematics. The MTOE specifically measures a teacher’s belief in his or her ability to directly affect student-learning outcomes. Enochs et al. (2000) found the PMTE and MTOE had Cronbach α coefficients of 0.88 and 0.77, respectively. Research questions one and two were answered using independent samples t-tests on data collected from the 25-item mathematics content test, CST, 40item attitudinal test, and 21-item MTEBI with two subscales. Research question three was answered using one-way ANOVA on data also collected from the same instruments. In this study there was a mix of middle school and high school teachers in the Mathematics and Mathematics Immersion programs. For the third research question Teaching Fellows were divided into three categories based upon their undergraduate college majors: liberal arts, business, and mathematics and science majors. Liberal arts majors consisted of majors such as English, history, Italian, philosophy, political science, psychology, sociology, Spanish, and women studies. Business majors consisted of majors such as accounting, business administration and management, commerce, economics, and finance. Mathematics and science majors consisted of majors such as mathematics, engineering, and the sciences (biology and chemistry). Research question four was answered through Pearson correlations with the same instruments used in the other research questions. The data were analyzed using the Statistical Package for the Social Sciences (SPSS), and all significance levels were at the 0.05 level. Teachers were separated by teaching level (middle and high school), mathematics credits earned (Mathematics and Mathematics Immersion), and undergraduate major (liberal arts, business, and mathematics and science majors) in order to determine differences between the
27
Mathematics Teacher Differences
different types of mathematics teachers sampled to determine teacher quality. Results To determine internal reliability of the attitudinal instruments, it was found that the Cronbach α coefficient was 0.93 on the pretest and 0.94 on the posttest for the 40-item attitudinal test. For the efficacy pretest, α = 0.80 for the PMTE α = 0.77 for the MTOE. For the efficacy posttest, α = 0.82 for the PMTE and α = 0.83 for the MTOE, respectively. These values are fairly consistent with the literature (Enochs et al., 2000; Tapia, 1996). The first research question was answered using independent samples t-tests comparing middle and high school teacher data using responses on the mathematics content test, CST, attitudinal test, and MTEBI with two subscales: PMTE and MTOE. There was a statistically significant difference between middle school teacher scores and high school teacher scores for the mathematics content pretest, posttest, and CST (see Table 1). Thus, high school teachers had higher content test scores than middle school teachers, and the effect sizes were large. There were no statistically significant differences found between middle and high school teachers on both pre- and posttests measuring attitudes toward mathematics and teacher efficacy beliefs. Table 1 Independent Samples t-Test Results on Mathematics Content Tests by Level Assessment
Mean
SD
t-value
d-value
Mathematics Content Pre-Test 68.42
15.600
High School (N = 16)
85.13
16.041
-3.334**
1.056
Mathematics Content Post-Test Middle School (N = 26)
79.46
15.402
High School (N = 16)
92.63
6.582
Middle School (N = 26)
255.31
20.372
High School (N = 16)
269.25
17.133
-3.230**
1.112
Mathematics CST
28
Table 2 Independent Samples t-Test Results on Mathematics Content Tests by Background Assessment
Mean
SD
t-value
d-value
Mathematics Teaching Fellows (N = 12)
89.50
7.868
-4.005**
1.555
Mathematics Immersion (N = 30)
68.90
17.008
Mathematics Teaching Fellows (N = 12)
94.33
7.390
-3.130**
1.202
Mathematics Immersion (N = 30)
80.53
14.460
276.33
16.104
-3.636**
1.277
Mathematics 254.33 Immersion (N = 30) N = 42, df = 40, two-tailed ** p < 0.01
18.291
Mathematics Content Pre-Test
Mathematics Content Post-Test
Mathematics CST
Middle School (N = 26)
N = 42, df = 40, two-tailed * p < 0.05 ** p < 0.01
The second research question was answered using independent samples t-tests comparing Mathematics Immersion and Mathematics Teaching Fellows data using the mathematics content test, CST, attitudinal test, and MTEBI with two subscales: PMTE and MTOE. There was a statistically significant difference between Mathematics Immersion Teaching Fellows’ scores and Mathematics Teaching Fellows’ scores for the mathematics content pretest, posttest, and CST (see Table 2). Thus, Mathematics Teaching Fellows had higher content test scores than Mathematics Immersion Teaching Fellows, and the effect sizes were large. There were no statistically significant differences found between Mathematics and Mathematics Immersion Teaching Fellows on both pre- and posttests measuring attitudes toward mathematics and teacher efficacy beliefs.
-2.283*
0.741
Mathematics Teaching Fellows (N = 12)
The third research question was answered using one-way ANOVA comparing different undergraduate college majors using the mathematics content test, CST, attitudinal test, and MTEBI with two subscales: PMTE and MTOE. Teaching Fellows were grouped according to their undergraduate college major. Three categories were used to group teachers: liberal arts (N = 16), business (N = 11), and mathematics and science (N = 15) majors. The results of the one-way ANOVA revealed statistically significant differences between
Brian R. Evans
undergraduate major area for the mathematics content pretest, posttest, and CST, with large effect sizes in each case (see Tables 3, 4, 5, and 6). A post hoc test (Tukey HSD) revealed that mathematics and science majors had significantly higher content knowledge than business majors with p < 0.01 (pretest, posttest, and CST) and liberal arts majors with p < 0.01 (pretest) and p < 0.05 (posttest and CST). There were no other
statistically significant differences. In summary, in this study mathematics and science majors had higher content knowledge scores than non-mathematics and non-science majors. No statistically significant differences were found between the undergraduate college majors on both pre- and posttests in attitudes toward mathematics and teacher efficacy.
Table 3 Means and Standard Deviations on Content Knowledge for Major Pre-, Post-, and CST Tests
Mean
Content Knowledge Pre Test; Total (N = 42)
Standard Deviation
74.79
17.605
Liberal Arts (N = 16)
70.13
16.382
Business (N = 11)
64.45
15.820
Math/Science (N = 15)
87.33
12.804
84.48
14.225
Liberal Arts (N = 16)
81.19
15.132
Business (N = 11)
76.82
14.034
Math/Science (N = 15)
93.60
7.679
CST Content Knowledge; Total (N = 42)
260.62
20.184
Content Knowledge Post Test; Total (N = 42)
Liberal Arts (N = 16)
255.81
18.784
Business (N = 11)
249.64
18.943
Math/Science (N = 15)
273.80
15.857
Table 4 ANOVA Results on Mathematics Content Pretest for Major Variation
Sum of Squares
df
Mean Square
F
Ρ2
Between Groups
3883.261
2
1941.630
8.582**
0.31
Within Groups
8823.811
39
226.252
Total
12707.071
41
** p < 0.01
Table 5 ANOVA Results on Mathematics Content Posttest for Major Variation
Sum of Squares
df
Mean Square
F
Ρ2
Between Groups
2066.802
2
1033.401
6.469**
0.25
Within Groups
6229.674
39
159.735
Total
8296.476
41
** p < 0.01
29
Mathematics Teacher Differences
Table 6 ANOVA Results on Mathematics Content Specialty Test (CST) for Major Variation
Sum of Squares
df
Mean Square
F
Ρ2
Between Groups
4302.522
2
2151.261
6.765**
0.26
Within Groups
12401.383
39
317.984
Total
16703.905
41
** p < 0.01
Research question four was analyzed using Pearson correlations to determine if there were any relationships between content knowledge and attitudes toward mathematics or efficacy. No significant relationships were found. This suggests that Teaching Fellowsâ&#x20AC;&#x2122; attitudes toward mathematics and efficacy are unrelated to how much content knowledge they possess. Discussion and Implications The results of the analyses on the data collected from this particular group of Teaching Fellows revealed that high school teachers had higher mathematics content knowledge than middle school teachers, Mathematics Teaching Fellows had higher mathematics content knowledge than Mathematics Immersion Teaching Fellows, and mathematics and science majors had higher mathematics content knowledge than non-mathematics and non-science majors. The sample size in this study was small, but effect sizes were found to be quite large. Moreover, no differences in attitudes toward mathematics and teacher efficacy were found between middle and high school teachers; between Mathematics and Mathematics Immersion Teaching Fellows; or among liberal arts, business, and mathematics and science majors. Surprisingly, no relationships were found between mathematical content knowledge and attitudes toward mathematics and teacher efficacy. The statistically significant differences in content knowledge found in this study led to further analysis to determine if there were differences in gain scores for content knowledge on the mathematics content test over the course of the semester for any group; however, no significant differences were found in gain scores between middle and high school teachers, between Mathematics Teaching Fellows and Mathematics Immersion Teaching Fellows, or among the different undergraduate college majors. In the first study (Evans, in press) the sampled teachers had positive attitudes toward mathematics and high teacher efficacy. The present study revealed that there were no differences between the different 30
categories (teaching level, immersion status, and major) of Teaching Fellows in attitudes toward mathematics and efficacy, and that content knowledge was unrelated to attitudes toward mathematics and efficacy. Combining the results of the first study (Evans, in press) with the results found in this present study, an interesting finding emerged. Teachers in this study had the same high level of positive attitudes toward mathematics and the same high level of teacher efficacy regardless of content ability. Thus, some of the teachers in this study believed they were just as effective at teaching mathematics, despite not having the high level of content knowledge that some of their colleagues possessed. This finding is significant because high content knowledge is a necessary condition for quality teaching (Ball et al., 2005). This finding also contradicts other research conducted that found a positive relationship between content knowledge and attitudes (Aiken, 1970, 1974, 1976; Ma & Kishor, 1997). It is possible that the unique sample of alternative certification teachers may have contributed to this difference, and this possibility should be further investigated. It should also be noted that the instructor in the mathematics methods course was also the researcher. Thus, consideration must be given for possible bias in participant reporting since the participants in this study knew that the instructor would be conducting the research. Participants were assured that their responses would not be used as an assessment measure in the methods course. Although New York State requires a minimum of 30 mathematics credits for both middle and high school teachers, high school teachers had higher content knowledge than middle school teachers. This may be due to their experience working with higher level mathematics in their teaching. However, this does not explain the reason that sampled high school teachers scored better on the CST and content pretest instruments: this study began at the beginning of their teaching careers, and the teachers did not yet have significant classroom experience. It is possible that teachers with stronger content knowledge may be drawn more to high school teaching, rather than middle
Brian R. Evans
school teaching, and the more rigorous content that comes with teaching high school mathematics. Because the participants in this study represent a convenience sample due to availability, which restricts the generalizability of this study, further research should extend to larger sample sizes. Many alternative certification teachers, such as the Teaching Fellows, teach in high-need urban schools in New York City (Boyd, Grossman, Lankford, Loeb, & Wyckoff, 2006) and throughout the United States. Therefore, it is imperative that policy makers, administrators, and teacher educators continually evaluate teacher quality in alternative certification programs. NCTM (2005) stated, “Every student has the right to be taught mathematics by a highly qualified teacher—a teacher who knows mathematics well and who can guide students' understanding and learning” (p. 1). New York State holds the same high standards for both high school and middle school teachers. Thus, educational stakeholders should investigate and implement strategies to better middle school teachers’ content knowledge. Based on the results of this study it is recommended that middle school teachers be given strong professional development in mathematics content knowledge by both the schools in which they teach and the schools of education in which they are enrolled. Future studies should examine this issue with References Aiken, L. R. (1970). Attitudes toward mathematics. Review of Educational Research, 40, 551–596. Aiken, L. R. (1974). Two scales of attitude toward mathematics. Journal for Research in Mathematics Education, 5, 67–71. Aiken, L. R. (1976). Update on attitudes and other affective variables in learning mathematics. Review of Educational Research, 46, 293–311. Amato, S. A. (2004). Improving student teachers’ attitudes to mathematics. Proceedings of the 28th Annual Meeting of the International Group for the Psychology of Mathematics Education (IGPME), 2, 25–32. Bergen, Norway: IGPME. Angle, J., & Moseley, C. (2009). Science teacher efficacy and outcome expectancy as predictors of students’ End-ofInstruction (EOI) Biology I test scores. School Science and Mathematics, 109, 473–483. Ashton, P., & Webb, R. (1986). Making a difference: Teachers’ sense of efficacy and student achievement. New York: Longman. Ball, D. L., Hill, H. C., & Bass, H. (2005). Knowing mathematics for teaching: Who knows mathematics well enough to teach third grade, and how can we decide? American Educator, 14– 17, 20–22, & 43–46. Bandura, A. (1986). Social foundations of thought and action: A social cognitive theory. Englewood Cliffs, NJ: Prentice Hall. Boyd, D. J., Grossman, P., Lankford, H., Loeb, S., Michelli, N. M., & Wyckoff, J. (2006). Complex by design: Investigating
larger samples of Teaching Fellows and teachers from other alternative certification programs to increase generalizability. It is imperative that future research address whether or not there are differences in actual teaching ability among the Mathematics and Mathematics Immersion Teaching Fellows and different college majors held by the teachers. One way to determine this would be to measure students’ mathematics performance to identify differences in student achievement among the variables examined in this study. As earlier stated, Teaching Fellows currently account for one-fourth of all New York mathematics teachers (NYCTF, 2008), and increasingly alternative certification programs account for more teachers coming to the profession throughout the United States (Humphrey & Wechsler, 2007). For the sake of students who have teachers in alternative certification programs, the certification of high quality teachers must continually be a priority for policy makers, administrators, and teacher educators. Considering the call for high quality teachers, high stakes examinations, and accountability, now more than ever we need to ensure that the teachers we certify are fully prepared in both content knowledge and dispositions to best teach our high-need students. pathways into teaching in New York City schools. Journal of Teacher Education, 57, 155–166. Boyd, D., Grossman, P., Lankford, H., Loeb, S., & Wyckoff, J. (2006). How changes in entry requirements alter the teacher workforce and affect student achievement. Education Finance and Policy, 1, 176–216. Boyd, D., Lankford, S., Loeb, S., Rockoff, J., & Wyckoff, J. (2007). The narrowing gap in New York City qualifications and its implications for student achievement in high poverty schools (CALDER Working Paper 10). Washington, DC: National Center for Analysis of Longitudinal Data in Education Research. Retrieved August 26, 2008, from http://www.caldercenter.org/PDF/1001103_Narrowing_Gap.p df. Charalambous, C. Y., Panaoura, A., & Philippou, G. (2009). Using the history of mathematics to induce changes in preservice teachers’ beliefs and attitudes: Insights from evaluating a teacher education program. Educational Studies in Mathematics, 71, 161–180. Czerniak, C. M., & Schriver, M. (1994). An examination of preservice science teachers’ beliefs. Journal of Science Teacher Education, 5, 77–86. Darling-Hammond, L. (1994). Who will speak for the children? How "Teach for America" hurts urban schools and students. Phi Delta Kappan, 76(1), 21–34. Darling-Hammond, L. (1997). The right to learn: A blueprint for creating schools that work. San Francisco, CA: Jossey-Bass. Darling-Hammond, L., Holtzman, D. J., Gatlin, S. J., & Heilig, J. V. (2005). Does teacher preparation matter? Evidence about
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teacher certification, Teach for America, and teacher effectiveness. Education Policy Analysis Archives, 13(42), 1– 32. Eide, E., Goldhaber, D., & Brewer, D. (2004). The teacher labour market and teacher quality. Oxford Review of Economic Policy, 20, 230–244. Enochs, L. G., & Riggs, I. M. (1990). Further development of an elementary science teaching efficacy belief instrument: A preservice elementary scale. School Science and Mathematics, 90, 695-706. Enochs, L. G., Smith, P. L., & Huinker, D. (2000). Establishing factorial validity of the Mathematics Teaching Efficacy Beliefs Instrument. School Science and Mathematics, 100, 194–202. Ernest, P. (1989). The knowledge, beliefs and attitudes of the mathematics teacher: A model. Journal of Education for Teaching, 15, 13–33. Evans, B. R. (2009). First year middle and high school teachers’ mathematical content proficiency and attitudes: Alternative certification in the Teach for America (TFA) program. Journal of the National Association for Alternative Certification (JNAAC), 4(1), 3–17. Evans, B. R. (in press). Content knowledge, attitudes, and selfefficacy in the mathematics New York City Teaching Fellows (NYCTF) program. School Science and Mathematics Journal. Gibson, S., & Dembo, M. H. (1984). Teacher efficacy: A construct validation. Journal of Educational Psychology, 76, 569–582. Haney, J. J., Lumpe, A.T., Czerniak, C.M., & Egan, V. (2002). From beliefs to actions: The beliefs and actions of teachers implementing change. Journal of Science Teacher Education, 13, 171–187. Humphrey, D. C., & Wechsler, M. E. (2007). Insights into alternative certification: Initial findings from a national study. Teachers College Record, 109, 483–530. Kane, T. J., Rockoff, J. E., & Staiger, D. O. (2006). What does certification tell us about teacher effectiveness? Evidence from New York City. Working Paper No. 12155, National Bureau of Economic Research. Laczko-Kerr, I., & Berliner, D. C. (2002). The effectiveness of “Teach for America” and other under-certified teachers on student academic achievement: A case of harmful public policy. Education Policy Analysis Archives, 10(37). Retrieved August 26, 2008, from http://epaa.asu.edu/epaa/v10n37/. Ma, L. (1999). Knowing and teaching elementary mathematics. Mahwah, NJ: Lawrence Erlbaum Associates Publishers.
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Ma, X., & Kishor, N. (1997). Assessing the relationship between attitude toward mathematics and achievement in mathematics: A meta-analysis. Journal for Research in Mathematics Education, 28, 26–47. National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author. National Council of Teachers of Mathematics. (2005). Highly qualified teachers. NCTM Position Statement. Retrieved February 18, 2009, from http://www.nctm.org/about/content.aspx?id=6364. New York City Teaching Fellows. (2008). Retrieved August 26, 2008, from http://www.nyctf.org/. New York City Teaching Fellows. (2010). Retrieved May 25, 2010, from http://www.nyctf.org/. Raymond, M., Fletcher, S. H., & Luque, J. (2001). Teach for America: An evaluation of teacher differences and student outcomes in Houston, Texas. Stanford, CA: The Hoover Institution, Center for Research on Education Outcomes. Soodak, L. C., & Podell, D. M. (1997). Efficacy and experience: Perceptions of efficacy among preservice and practicing teachers. Journal of Research and Development in Education, 30, 214–221. Stein, J. (2002). Evaluation of the NYCTF program as an alternative certification program. New York: New York City Board of Education. Suell, J. L., & Piotrowski, C. (2007). Alternative teacher education programs: A review of the literature and outcome studies. Journal of Instructional Psychology, 34, 54–58. Swars, S. L., Daane, C. J., & Giesen, J. (2006). Mathematics anxiety and mathematics teacher efficacy: What is the relationship in elementary preservice teachers? School Science and Mathematics, 106, 306–315. Swars, S., Hart, L. C., Smith, S. Z., Smith, M. E., & Tolar, T. (2007). A longitudinal study of elementary pre-service teachers’ mathematics beliefs and content knowledge. School Science and Mathematics, 107, 325–335. Tapia, M. (1996). The attitudes toward mathematics instrument. Paper presented at the Annual Meeting of the Mid-South Educational Research Association, Tuscaloosa, AL. Trice, A. D., & Ogden, E. D. (1986). Correlates of mathematics anxiety in first-year elementary school teachers. Educational Research Quarterly, 11(3), 3–4. Xu, Z., Hannaway, J., & Taylor, C. (2008). Making a difference? The effects of Teach for America in high school. Retrieved April 22, 2008, from http://www.urban.org/url.cfm?ID=411642.
The Mathematics Educator 2011, Vol. 20, No. 2, 33â&#x20AC;&#x201C;43
Sense Making as Motivation in Doing Mathematics: Results from Two Studies Mary Mueller, Dina Yankelewitz, & Carolyn Maher In this article, we present episodes from two qualitative research studies. The studies focus on students of different ages and populations and their work on different mathematical tasks. We examine the commonalities in environment, tools, and teacher-student interactions that are key influences on the positive dispositions engendered in the students and their interest and engagement in mathematics. In addition, we hypothesize that these positive dispositions in mathematics lead to student reasoning and, thus, mathematical understanding. The resulting framework is supported by other educational research and suggests ways that the standards can be implemented in diverse classrooms in order to achieve optimal student engagement and learning.
The National Council of Teachers of Mathematics (NCTM, 2000) describes a vision for mathematics education focusing on conceptual understanding. This vision includes students engaged in hands-on activities that incorporate problem solving, reasoning and proof, real-world connections, multiple representations, and mathematical communication. NCTM and others have prepared multiple documents and resources (e.g., Chambers, 2002; Germain-McCarthy, 2001; NCTM, 2000; Stiff & Curcio, 1999) to support teachers in achieving this vision and putting the standards into practice. However, differences in age, gender, ethnicity, and school culture often impede the implementation of successful teaching practice in mathematics classrooms and prevent students from taking ownership of mathematical ideas in the ways that have been envisioned. While NCTM addresses factors such as classroom environment and mathematical tasks, this provides an incomplete picture of how to build studentsâ&#x20AC;&#x2122; conceptual understanding. For example, motivation to Mary Mueller is an Associate Professor in the Department of Educational Studies at Seton Hall University. Her research interests include the development of mathematical ideas and reasoning over time. Dina Yankelewitz is an Assistant Professor in the School of General Studies at the Richard Stockton College of New Jersey. Her research interests include the development of mathematical thinking and the identification and development of mathematical reasoning in students and teachers of mathematics. Carolyn Maher is a Professor of mathematics education in the Graduate School of Education at Rutgers University. Her research interests include the development of mathematical thinking in students, mathematical reasoning, justification and proof making in mathematics, and the development of a model for analyzing videotape data.
learn is pivotal in studentsâ&#x20AC;&#x2122; attainment of understanding in all content areas (Middleton & Spanias, 1999), but the NCTM vision does not explicate how to help students experience motivation as they learn mathematics. We have developed a framework for mathematics teaching and learning that provides this missing link. It provides teachers and researchers with a conceptual tool that explains how students build the positive attitudes (motivation, autonomy, self-efficacy, and positive dispositions) towards mathematics that are necessary to engage in mathematical reasoning. We believe that this approach that can be implemented across the spectrum of mathematics classrooms in the US. Our research focuses on students who are working collaboratively as they engage in mathematical problem solving. We videotaped students as they engaged in mathematical tasks and then analyzed the reasoning that occurred as they worked to formulate strategies and defend their solutions. We have found that, although the demographics of the groups of students and the tasks may be different, the reasoning and subsequent understanding that occurs is quite similar. In this article, we present two episodes from our research, focusing on students of different ages and populations as they work on different mathematical tasks. We then examine the commonalities in environment, tools, and teacher-student interactions that are key influences both on the positive attitudes towards mathematics engendered in the students and on their engagement in mathematics. We hypothesize that these positive attitudes towards mathematics lead to student reasoning and, thus, mathematical understanding. Based on our research, we created a
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framework for teaching and learning that identifies the key factors in encouraging positive attitudes in the mathematics classroom as well as their role in enabling student reasoning and understanding. We support this framework using the extensive literature base centering on students’ motivation in the mathematics classroom. The resulting framework suggests ways that the standards can be implemented in diverse classrooms in order to achieve optimal student engagement and learning. Although the development of our framework began with our data and then was supported by the literature, we begin by presenting the supporting literature in order to give the readers a background for the framework. In our framework, there are four factors that mediate between elements in the classroom environment, such as tasks, and the development of conceptual understanding through mathematical reasoning. These four factors are autonomy, instrinsic motivation, self-efficacy and positive dispositions towards mathematics. Because the literature concerning all four of these factors is interrelated, we have picked one factor, intrinsic motivation, to organize our discussion around. All students must be motivated in some way to engage in mathematical activity, however, the nature of that motivation largely determines the success of their endeavor. In particular, students’ motivations can be divided into two distinct types: extrinsic motivation and intrinsic motivation. Extrinsically motivated students engage in learning for external rewards, such as teacher and peer approval and good grades. These students do not necessarily acquire a sense of ownership of the mathematics that they study; instead they focus on praise from teachers, parents and peers and avoiding punishment or negative feedback (Middleton & Spanias, 1999). In contrast, students who are intrinsically motivated to learn mathematics are driven by their own pursuit of knowledge and understanding (Middleton & Spanias, 1999). They engage in tasks due to a sense of accomplishment and enjoyment and view learning as impacting their selfimages (Middleton, 1995). Intrinsically motivated students, therefore, focus on understanding concepts. Thus, intrinsic, rather than extrinsic, motivation benefits students in the process and results of mathematical activities.
motivation include perceptions of autonomy, interests in given tasks, and the need for competence. Brophy (1999) concurs and notes that a supportive social context, challenging activities, and student interest and value in learning are crucial to the development of intrinsic motivation. Autonomous students, in attending to problem situations mathematically, rely on their own mathematical facilities and use their own resources to make decisions and make sense of their strategies (Kamii,1985; Yackel & Cobb, 1996). Autonomy promotes persistence on tasks and thus leads to higher levels of intrinsic motivation (Deci, Nezdik, & Sheinman, 1981; Deci & Ryan, 1987; Stefanou, Perencevich, DiCinti, & Turner, 2004). Furthermore, through participation in classroom activities, mathematically autonomous students begin to rely on their own reasoning rather than on that of the teacher (Cobb, Stephan, McClain, & Gravemeijer, 2001; Forman, 2003) and thus become arbitrators of what makes sense. Studies show that teacher support and classroom environments play a crucial role in the development of another source of intrinsic motivation, namely, positive (or negative) dispositions toward mathematics (Bransford, Hasselbring, Barron, Kulewicz, Littlefield, & Goin, 1988; Cobb, Wood, Yackel, & Perlwitz, 1992; Middleton, 1995; Middleton & Spanias, 1999). According to NCTM (2000), “More than just a physical setting … the classroom environment communicates subtle messages about what is valued in learning and doing mathematics (p. 18). The document then describes the implementation of challenging tasks that challenge students intellectually and motivate them through real-world connections and multiple solution paths (NCTM, 2000). Stein, Smith, Henningsen, and Silver (2000) stress that teachers need be thoughtful about the tasks that they present to students and use care to present and sustain cognitively complex tasks. They suggest that during the problem solving implementation phase, teachers often reduce the cognitive complexity of tasks. Overall, when students are presented with meaningful, relevant, and challenging tasks; offered the opportunity to act autonomously and develop self-control over learning; encouraged to focus on the process rather than the product; and provided with constructive feedback, they become intrinsically motivated to succeed (Urdan & Turner, 2005).
Sources of Intrinsic Motivation
Effects of Intrinsic Motivation
Researchers (Deci & Ryan, 1985; Hidi, 2000; Renninger, 2000) have found that sources of intrinsic
Intrinsic motivation leads to self-efficacy, an individual’s beliefs about their own ability to perform
The Role of Intrinsic Motivation
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specific tasks in specific situations (Bandura, 1986; Pajares, 1996). Students’ self-efficacy beliefs often predict their ability to succeed in a particular situation (Bandura, 1986). Specifically, in mathematics, research has shown that self-efficacy is a clear predictor of students’ academic performance (Mousoulides & Philippou, 2005; Pintrich & De Groot, 1990). Furthermore, studies suggest that students with highly developed self-efficacy beliefs utilize cognitive and metacognitive learning strategies more vigorously while being more aware of their own motivational beliefs (Mousoulides & Philippou, 2005; Pintrich, 1999). Unlike sources of extrinsic motivation, which need to be constantly reinforced, research shows that the common sources of intrinsic motivation are reinforced when students are encouraged to develop their selfefficacy (Urdan & Turner, 2005), For example, intrinsic motivation helps students succeed at a given learning objective, thereby further developing students’ self-efficacy.In general, students are more likely to engage and persist in tasks when they believe they have the ability to succeed (Urdan & Turner, 2005). Therefore, intrinsic motivation can lead to an increased willingness to engage in reasoning activities. In summary, research shows that when students are intrinsically motivated to learn mathematics, they spend more time on-task, tend to be more persistent, and are confident in using different, or more challenging, strategies to solve mathematical problems (Lepper, 1988; Lepper & Henderlong, 2000). These qualities of mathematical learners better enable them to actualize the recommendations put forth by NCTM (2000) and to master key mathematical processes in their pursuit of understanding mathematics. Intrinsic motivation, then, is correlated with self-efficacy and positive dispositions towards a conceptual understanding of mathematics, whereas extrinsic motivation results in merely a superficial grasp of the information presented. Results from Two Studies Through a combination of cross-cultural and longitudinal studies we have observed that a mixture of factors contribute to students’ motivation to participate in mathematics and their dispositions towards mathematics (for details on our methodologies, see Mueller, 2007; Mueller & Maher, 2010; Mueller, Yankelewitz, & Maher, 2010; Yankelewitz, 2009; Yankelewitz, Mueller, & Maher, 2010). These include classroom environment, teacher questioning that evokes meaningful support of conjectures, and welldesigned tasks. Together, these factors positively
influence the establishment of favorable dispositions towards learning mathematics. In their quest to make sense of appropriately challenging tasks, students enjoy the pursuit of meaning and thereby become intrinsically motivated to engage in mathematics. In this paper we present results from two research studies investigating students’ mathematics learning. In particular, we present specific examples of elementary and middle school students who demonstrated sense making and higher order reasoning when working on mathematical tasks. In these episodes, the students were engaged, motivated, and, importantly, confident in their ability to offer and defend mathematical solutions; they demonstrated positive dispositions towards mathematics. We identified student behaviors that indicated confidence in mathematics and a high level of engagement. These behaviors include perseverance; the ability to consider more challenging, alternative solutions; and the length of time spent of the task. In the discussion that follows, we analyze the commonalities in the two teaching experiments, and consider how these commonalities may have positively influenced the level of motivation and confidence that students exhibited as they worked on mathematical tasks. In the discussion, we use our findings to define a framework that can be used to inform a teaching practice that will motivate students and encourage student engagement and mathematical understanding. Data Analysis and Results The episodes presented below come from two data sets. Data from the first study is drawn from sessions during an informal after-school mathematics program in which 24 sixth-grade students from a low socioeconomic urban community worked on openended tasks involving fractions. The students represented a wide range of abilities and thus their mathematical levels ranged from those who were enrolled in remedial mathematics to those who were successful in regular mathematics classrooms. The present discussion focuses on one table of four students, two boys and two girls. The second source of data includes segments from sessions in which fourth and fifth grade students from a suburban school investigated problems in counting and combinatorics. This data is drawn from a longitudinal study of children’s mathematical thinking. As part of the students’ regular school day, researchers led the students in exploring open-ended tasks during which students were expected to justify their solutions to the satisfaction of their peers. These strands of tasks were separate from the school-mandated curriculum. Because of space limitations, we give examples of one 35
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task from each data set, one involving fractions and the other focusing on combinatorics. Episode 1: Reasoning about Fractions in the Sixth Grade The students in the first study worked collaboratively on tasks involving fraction relationships. Cuisenaire® rods (see Figure 1) were available and students were encouraged to build models. A set of Cuisenaire rods contains 10 colored wooden or plastic rods that increase in length by increments of one centimeter. For these activities, the rods have variable number names and fixed color names. The colors increased incrementally as follows: white, light green, purple, yellow dark green, black brown, blue, and orange.
staircase to name the rods and explained the dilemma of naming the orange rod, “See this is One-ninth, twoninths, three-ninths, four-ninths, five ninths, six-ninths, seven-ninths, eight-ninths, nine-ninths - what’s this one?” Dante replied, “That would be ten-ninths. Actually that should be one. That would start the new one (one-tenth)”. Chanel and Michael then named the blue rod “a whole”. The students worked for a few more minutes and then Dante explained that he had overhead another table naming the rods. Dante:
Why are they calling it ten-ninths and [it] ends at ninths?
Michael: Not the orange one. The orange one’s a whole. Dante:
But I’m hearing from the other group from over here, they calling it ten-ninths.
Michael: Don’t listen to them! The orange one is a whole because it takes ten of these to make one. Dante:
I’m hearing it because they speaking out loud. They’re calling it ten-ninths
Michael: They might be wrong! … Chanel: Let me tell you something, how can they call it ten-ninths if the denominator is smaller than the numerator? Dante:
Figure 1. “Staircase” Model of Cuisenaire Rods. Students were encouraged to build models to represent fraction tasks. For example, in one task, the blue rod was given the number name one and students initially worked on naming the red rod (two-ninths) and the light green rod (three-ninths or one-third). When the group completed this task, they initiated their own task of naming all of the rods in the set, given that the blue rod was named one. Chanel used the staircase model (shown in Figure 1) to incrementally name the remainder of the rods beginning with naming the white rod one-ninth. As she was working, she said the names of all of the rods, “One-ninth, two-ninths, three-ninths, four-ninths, fiveninths, six-ninths, seven-ninths, eight-ninths, nine ninths, ten..– wow, oh, I gotta think about that one ….. nine-tenths”. Disequilibrium. The teacher/researcher encouraged Chanel to share her problem with Dante. Chanel showed Dante her strategy of using the 36
Yeah, how is the numerator bigger than the denominator? It ends at the denominator and starts a new one. See you making me lose my brain.
A teacher/researcher joined the group and asked what the students were working on. Dante presented his argument of naming the orange rod one-tenth and explained that “it starts a new one”. The teacher/researcher reminded him that the white rod was named one-ninth and that this fact could not change. Again she asked him for the name of the blue rod and he stated, “It would probably be ten-ninths”. When prompted, Dante explained that the length of ten white rods was equivalent to the length of an orange rod. The teacher/researcher asked Dante to convince his partners that this was true. Chanel: No, because I don’t believe you because– Michael: I thought it was a whole. Dante:
But how can the numerator be bigger than the denominator?
T/R:
It can. It is. This is an example of where the numerator is bigger than the denominator.
Mary Mueller, Dina Yankelewitz, & Carolyn Maher
Chanel:
But the denominator can’t be bigger than the numerator, I thought.
Michael: That’s the law of facts. T/R:
Who told you that?
Chanel: My teacher. Dante:
One of our teachers.
Direct reasoning. The students continued working on the task. At the end of the session students were asked to share their work. Another student explained that she named the orange rod using a model of two yellow rods, “We found out the denominator doesn’t have to be larger than the numerator because we found out that two yellows [each] equal five-ninths so fiveninths plus five-ninths equals ten-ninths.” Another student explained that the orange rod could also be named one and one-ninth and used a model of a train of a blue rod and a white rod lined up next to the orange rod to explain (see Figure 2), “If you put them together then this means that it’s ten-ninths also known as one and one-ninth.”
Figure 2. A train of rods to show that
9 9
+ 19 = 109 or
1 19 Finally, Dante came to the front of the class and explained that he found a different way to name the orange rod. Building a model of an orange rod lined up next to two purple rods and a red rod (see Figure 3), he explained that the purple rods were each named fourninths and therefore together they were eight-ninths; the red rod was named two-ninths and therefore the total was eight-ninths plus two-ninths or ten-ninths, “four and four are eight so which will make it eightninths right here and then plus two to make it tenninths.”
In the beginning of the session described above, Dante and his partners were convinced that a fraction’s numerator could not be greater than its denominator. At some point it seems that they were taught about improper fractions and may have internalized this to mean incorrect fractions. The children referred to this rule as “the law of facts” and, when presented with the task, although they visually saw that the orange rod was equivalent to ten white rods (or ten-ninths), they resisted using this nomenclature. We highlight this episode to show that the students did not simply accept the rule that they recalled and move on to the next task. Instead they heard another group naming the orange rod ten-ninths and grappled with the discrepancy between this name and their rule. Remaining engaged in the task, the students focused on sense making; they were motivated to make sense of the models they built and in doing so exhibited confidence in their solutions. For over an hour, Dante attempted to make sense of his solution by building alternative models, sharing his ideas, conjectures, and solutions, questioning the teacher, and revisiting the problem. When faced with a discrepancy between what he had previously learned and the concrete model that he built, Dante relied on reasoning, rather than memorized facts, to convince himself and others of what made sense. In particular, he relied on his understanding of the model that he had constructed to make sense of the fraction relationships. This quest for sense making triggered the use of a variety of strategies, and the success of meaningbuilding led to persistence and flexibility in thinking, which, as described by Lepper and Henderlong (2000), are positively correlated with self-efficacy. Dante’s self-efficacy gave him the confidence and autonomy to move beyond his erroneous understanding that was based on previous memorized facts. Similarly to discussions about autonomy from Kamii (1985) and Yackel and Cobb (1996), this autonomy encouraged Dante to believe in his own mathematical ability and use his own resources to make sense of his model. This autonomy, coupled with his positive dispositions toward mathematics, allowed him to use reasoning to make sense of and fully understand the mathematics inherent in the problem.
Episode 2: Reasoning about Combinatorics in the Fourth and Fifth Grades Figure 3. A train of rods to show that 94 + 94 + 92 = 109 .
In the second study, fourth- and fifth-grade students were introduced to combinatorial tasks. The students were given Unifix cubes and were asked to find all combinations of towers that were four tall when selecting from cubes of two colors. Over the course of the two years, students revisited the task in 37
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various settings. This provided multiple opportunities for them to think about and refine their thinking about the problem. Stephanie, along with her partner, Dana, first constructed all possible towers four cubes tall by finding patterns of towers and searching for duplicates. After her first attempt to find all possible towers, Stephanie organized her groups of towers according to color categories (e.g., exactly one of a color and exactly two of a color adjacent to each other) in order to justify her count of 16 towers, thus she organized the towers by cases (see Figure 4). Stephanie then used this organization by cases to find all possible towers of heights three cubes tall, two cubes tall and one cube tall when selecting from two colors.
and 8 times 2 is 16. It goes like a pattern! You have the 2 times 2 equals the 4, the 4 times 2 equals the 8 and the 8 times 2 equals the 16.” A few minutes later, Stephanie gave a rule to describe a method for generating towers, “all you have to do is take the last number that you had and multiply by two.” Stephanie’s persistent attempts to make sense of the problem enabled her to think about the problem in flexible, yet durable, ways. She used multiple forms of reasoning to examine the problem from different angles and was confident in her findings. She was motivated by her own discoveries and the chance to create and share her own conjectures. Milin also used cases to organize towers five cubes tall. He then went back to the problem and used simpler problems of towers four cubes tall and three cubes tall to build on to towers five cubes tall. While his partners based their arguments on number patterns and cases, Milin explained his solution using an inductive argument. Milin’s explanation in each instance was based on adding on to a shorter tower to form exactly two towers that were one cube taller (see Figure 5). For example, when asked to explain why he created four towers from two towers, Milin explained: Milin: [pointing to his towers that were one cube high] Because – for each one of them, you could add … two more – because there’s … a blue, and a red- … for red you put a black on top and a red on top – I mean a blue on top instead of a black. And blue – you put a blue on top and a red on top – and you keep doing that.
Figure 4. Stephanie’s organization of towers by cases. During further investigation, Stephanie noticed a pattern in the sequence of total number of towers for each height classification: “Two, four, eight, sixteen… that’s weird! Look! Two times 2 is 4, 4 times 2 is 8,
Blue
Red
Figure 5. Milin’s inductive method of generating and organizing towers.
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Later in the year, four students participated in a group session, during which Stephanie and Milin presented their solutions to the towers problem. The next year, in the fifth grade, when the students again thought about this problem, Stephanie worked with Matt to find all tower combinations. Initially, they used trial and error to find as many combinations as they could. However, they only found twelve combinations. Stephanie remembered the pattern that she had discovered the year before. Stephanie:
Well a couple of us figured out a theory because we used to see a pattern forming. If you multiply the last problem by two, you get the answer for the next problem. But you have to get all the answers. See, this didn't work out because we don't have all the answers here.
Matt:
I thought we did.
Stephanie:
No. I mean all the answers, all the answers we can get . . . I don't know what happened! Because I am positive it works. I have my papers at home that say it works.
Persistence. Stephanie and Matt worked to find more tower combinations, but their search proved unsuccessful. Stephanie insisted that there were more combinations. Stephanie:
I don’t know how it worked. I know it worked. I just don’t know how to prove it because I’m stumped.
Matt:
Steph! Maybe it didn’t work!
Stephanie:
Oh no. No. Because I’m pretty sure it would… I think we goofed because I’m still sticking with my two thing. I’m convinced that I goofed, that I messed up because I know that…
Flexible Thinking. The teacher/researcher encouraged Stephanie and Matt to discuss the problem with other students. Stephanie and Matt approached other groups to see how they had solved the problem. They visited Milin and Michelle, who had been discussing the inductive method of finding all tower combinations. After hearing Michelle’s explanation of Milin’s method, Matt adopted that method and told other students about it. Stephanie attempted to explain Milin’s strategy to others, and, after the teacher/researcher questioned Stephanie about her explanation, she returned to her seat to work on refining her justification. Later in the session, the teacher/researcher again asked Stephanie to explain her
original prediction of the number of four-tall towers using the inductive method. This time she demonstrated a newfound understanding and enthusiastically presented the solution to the class. The motivation to make sense of the mathematical task and the confidence in the power of their own reasoning exhibited by this young group of students is evident from the transcripts and narrative above. In addition, the students exhibited characteristics that are correlated with intrinsic motivation (e.g., Lepper & Henderlong, 2000), including perseverance, the length of time spent on the task, and the students’ flexibility of thought as they considered and adopted the ideas of others. Stephanie’s investigations are especially interesting. Although she had previously solved the problem and was certain of her previous solution, Stephanie’s autonomy motivated her to continue to work on the problem until she was convinced that her strategy made sense. Rather than accept the solutions of her classmates, Stephanie persisted in verifying her model in order to make sense of the mathematics. The episode described took a full class period, during which the students were actively engaged in solving the task. Stephanie insisted on rethinking the problem, eventually learning from Milin’s explanation, and then she used her newfound knowledge to reason correctly about the task and verify her solution. Similar to Dante, she persisted in understanding why her solution worked and insisted on reasoning about the problem, thereby successfully solving and understanding the mathematical task.
Discussion Both highlighted tasks, one dealing with fraction ideas and the other with combinatorics, engaged students in sense making. The students described in the above episodes demonstrated confidence in their own understanding as they justified their solutions in the presence of their peers, even as their partners offered alternate representations. It is important to note that the episodes described above are exemplars of numerous similar incidents involving many of the students. Students developed this confidence as they were encouraged to defend their solutions first in their small groups and then in the whole class setting. They relied on their own models and justifications and did not seek approval from an authority or guidance from the teacher/researchers for validation of their ideas. These findings correspond with Francisco and Maher’s (2005) findings that certain classroom factors promote mathematical reasoning. The factors identified by Francisco and Maher include the posing of strands of challenging, open-ended tasks, establishing student 39
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ownership of their ideas and mathematical activity, inviting collaboration, and requiring justification of solutions to problems, all of which were present in the episodes above. On the surface, the two classroom episodes seem quite different from one another. Specifically, the two classrooms were comprised of students of different ages and demographics. In addition, one of the highlighted tasks focuses on fractional relationships and the other on combinatorics. However, despite these dissimilarities, they share many characteristics that encouraged students to be intrinsically motivated by the mathematics that they learned. In both episodes, an environment was created that facilitated an active, responsible, and engaged community of learners: Students were encouraged to share ideas and representations and to listen to, question, and convince one another of their solutions. The teacher/researchers facilitated learning while affording students the opportunity to create and defend their own justifications. The teacher/researchers employed careful questioning and support when needed, but the students were the arbitrators of what made sense, giving them a sense of autonomy. Students had opportunities to be successful in building understanding and in communicating that understanding through the arguments they constructed to support their solutions. The resulting discussions also required students to develop representations of their thinking in order to express their ideas with others. In both groups, students used rich and varied forms of direct and indirect reasoning. The reasoning that emerged during these tasks may be explained, at least in part, by the open-ended nature of the two tasks: The tasks lent themselves to multiple strategies, and, hence, they elicited various forms of reasoning. The behaviors that were observed and the depth of reasoning exhibited can also be explained as a byproduct of intrinsic motivation. The students in both groups strove for conceptual understanding, were persistent in their endeavors, and displayed confidence in their final solutions. Perhaps most importantly, in both episodes described, the students gained ownership of new mathematical ideas after being confronted with other studentsâ&#x20AC;&#x2122; differing understanding of challenging tasks. In accordance with other research (Deci, Nezdik, & Sheinman, 1981; Deci & Ryan, 1987; Stefanou et al., 2004), the studentsâ&#x20AC;&#x2122; autonomy led to their perseverance to find or defend their solutions and further increased their intrinsic motivation to make sense of the tasks at
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hand. Rather than accept the solutions of their classmates, both Stephanie and Dante verified their own strategies using the models they built and, thus, relied on their own reasoning to gain mathematical understanding. Dante and Stephanie were both motivated to rethink their understanding and justify their solutions after being exposed to the ideas of others and being challenged by the researchers to make sense of the task. Dante and Stephanie are representative of the other students we worked with, who displayed the ability to think about the solutions of others and use their own models to make sense of and acquire these solutions as their own. The consistency of these behaviors among our diverse sample suggests that, given the correct environment, all students can reason mathematically and succeed in engaging in mathematics. Based on our analysis, we hypothesize that motivation and positive dispositions toward mathematics lead to mathematical reasoning, which, in turn, leads to understanding. Furthermore, we constructed a framework to show the relationship between contextual factors and the chain of events leading to conceptual understanding (Figure 6). Our framework begins with the posing of an open-ended, engaging, and challenging task that the students have the ability to solve. The task is supported by a carefully crafted learning environment, carefully planned facilitator roles and interventions, student collaboration, and the availability of mathematical tools. In the episodes described above, both challenging tasks allowed students to deploy their own, personal solution strategy. Both tasks encouraged students to work collaboratively and utilize mathematical tools. In addition, the teacher/researcher adopted the role of facilitator and allowed the students to grapple with their own strategies as they listened to the strategies of their peers. Stephanie was given the opportunity to work on the problem independently and with a partner. She then listened to the strategies of others before refining her own solution strategy. Likewise, Dante was given the space and time to work through his misconception that the numerator of a fraction could not be larger than the denominator. Due to the nature of the task and the environment, Dante and his peers were motivated to resolve the discrepancy and find a solution. As with Stephanie, after listening to the ideas of others, Dante worked to make sense of the problem himself and create his own justification. Both students spent over an hour developing their solutions. Their positive dispositions,
Mary Mueller, Dina Yankelewitz, & Carolyn Maher
coupled with intrinsic motivation, gave them the confidence and desire to find a solution. This is apparent in the amount of time that they spent developing their solutions. Both students persevered even after a classmate had offered a viable solution. In both episodes, the studentsâ&#x20AC;&#x2122; motivation to succeed at the tasks at hand led to feelings of self-efficacy and autonomy. Both Stephanie and Dante took the
initiative to build several models and justifications in order to justify their solutions, first to themselves and then to the larger community. The students relied on reasoning, rather than memorized facts or the solutions of others, to convince themselves and others of what made sense. This reasoning led to their mathematical understanding. In
Mathematical Tools
Environment Tasks Open-ended Engaging Challenging yet attainable Student Collaboration
Teacher Variables
Positive Dispositions Toward Math
Intrinsic Motivation
Autonomy
Self-efficacy
Mathematical Reasoning
Understanding
Figure 6. The relationship between contextual factors, motivation and other events leading to conceptual understanding. 41
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particular, Dante proved to himself that 10/9 was a reasonable fraction and Stephanie was able to defend her doubling rule. In summary, in such a learning environment, students are encouraged to communicate their understandings of the task, and their ideas are valued and respected. This respect engenders students’ positive self-concepts in mathematics. At the same time, students become intrinsically motivated to succeed at mathematics. Intrinsic motivation fosters positive dispositions toward mathematics, which, in turn, encourage students to develop self-efficacy and mathematical autonomy as they discuss and share their understandings with their classmates. At the same time, students enjoy doing mathematics and develop ownership of their ideas. In such an environment and with such dispositions, students are more likely to engage in mathematical reasoning and, thus, acquire conceptual understanding. Our framework and research suggest that with careful attention to developing appropriate and engaging tasks, a supportive mathematical environment, and timely teacher questioning, students can be encouraged to build positive dispositions towards mathematics in all mathematics classrooms. These positive dispositions towards mathematics, in turn, form the ideal conditions for achieving conceptual understandings of mathematics.
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Mary Mueller, Dina Yankelewitz, & Carolyn Maher
Proceedings of the 29th Conference of the International Group for the Psychology of Mathematics Education (PME) (pp. 321-328). Melbourne, Australia: PME. Mueller, M. F. (2007). A study of the development of reasoning in sixth grade students (Unpublished doctoral dissertation). Rutgers, The State University of New Jersey, New Brunswick. Mueller, M., & Maher, C. A. (2009). Learning to reason in an informal math after-school program. Mathematics Education Research Journal, 21(3), 7–35. Mueller, M., Yankelewitz, D., & Maher, C. (2010). Promoting student reasoning through careful task design: A comparison of three studies. International Journal for Studies in Mathematics Education, 3(1), 135–156. National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author. Pajares, F. (1996). Self-efficacy beliefs in achievement settings. Review of Educational Research, 66, 543–578. Pintrich, P. R. (1999). The role of motivation in promoting and sustaining self regulated learning. International Journal of Educational Research, 31, 459–470. Pintrich, P. R., & De Groot, E. (1990). Motivational and selfregulated learning: Components of classroom academic performance. Journal of Educational Psychology, 82, 33–50. Renninger, K. A. (2000). Individual interest and its implications for understanding intrinsic motivation. In C. Sansone & J. Harackiewicz (Eds.), Intrinsic and extrinsic motivation: The search for optimal motivation and performance (pp. 373–404). New York, NY: Academic Press. Sansone, C., & Harackiewicz, J. M. (2000). Looking beyond extrinsic rewards: The problem and promise of intrinsic motivation. In C. Sansone & J. Harackiewicz (Eds.), Intrinsic and extrinsic motivation: The search for optimal motivation and performance (pp. 1–9). New York, NY: Academic Press. Stiff, L. V., & Curcio, F. R. (Eds.). (1999). Developing mathematical reasoning in grades K-12. Reston, VA: National Council of Teachers of Mathematics. Stefanou, C. R., Perencevich, K. C., DiCintio, M., & Turner, J. C. (2004). Supporting autonomy in the classroom: Ways teachers
encourage students’ decision making and ownership. Educational Psychologist, 39, 97–110. Stein, M. K., Smith, M.S., Henningsen, M. A., & Silver, E. A. (2000). Implementing standards-based mathematics instruction: A casebook for professional development. New York, NY: Teacher College Press. Urdan, T., & Turner, J. C. (2005). Competence motivation in the classroom. In A. J. Elliot & C. S. Dweck (Eds.), Handbook of competence and motivation (pp. 297–317). New York, NY: Guilford. Wigfield, A., & Eccles, J. S. (2002). The development of competence beliefs, expectancies for success, and achievement values from childhood through adolescence. In A. Wigfield & J. S. Eccles (Eds.), Development of achievement motivation (pp. 91–120). San Diego, CA: Academic Press. Yackel, E., & Cobb, P. (1996). Sociomathematical norms, argumentation, and autonomy in mathematics. Journal for Research in Mathematics Education, 27 , 458–477. Yankelewitz, D. (2009). The development of mathematical reasoning in elementary school students’ exploration of fraction ideas. Unpublished doctoral dissertation, Rutgers, The State University of New Jersey, New Brunswick. Yankelewitz, D., Mueller, M., & Maher, C. (2010). Tasks that elicit reasoning: A dual analysis. Journal of Mathematical Behavior, 29, 76–85.
1
This work was supported in part by grant REC0309062 (directed by Carolyn A. Maher, Arthur Powell and Keith Weber) from the National Science Foundation. The opinions expressed are not necessarily those of the sponsoring agency and no endorsements should be inferred. 2 The research was supported, in part, by National Science Foundation grants MDR9053597 and REC-9814846. The opinions expressed are not necessarily of the sponsoring agency and no endorsement should be inferred.
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The Mathematics Educator 2011, Vol. 20, No. 2, 44â&#x20AC;&#x201C;50
An Alternative Method to Gauss-Jordan Elimination: Minimizing Fraction Arithmetic Luke Smith & Joan Powell When solving systems of equations by using matrices, many teachers present a Gauss-Jordan elimination approach to row reducing matrices that can involve painfully tedious operations with fractions (which I will call the traditional method). In this essay, I present an alternative method to row reduce matrices that does not introduce additional fractions until the very last steps. The students in my classes seemed to appreciate the efficiency and accuracy that the alternative method offered. Freed from unnecessary computational demands, students were instead able to spend more time focusing on designing an appropriate system of equations for a given problem and interpreting the results of their calculations. I found that these students made relatively few arithmetic mistakes as compared to students I tutored in the traditional method, and many of these students who saw both approaches preferred the alternative method.
When solving systems of equations by using matrices, many teachers present a Gauss-Jordan elimination approach to row reducing matrices that can involve painfully tedious operations with fractions (which I will call the traditional method). In this essay, I present an alternative method to row reduce matrices that does not introduce additional fractions until the very last steps. As both a teacher using this alternative method and a tutor working with students instructed in the traditional method, I have some anecdotal experience with both. The students in my classes seemed to appreciate the efficiency and accuracy that the alternative method offered them. Since they were freed from unnecessary computational demands, they were instead able to spend more time focusing on designing an appropriate system of equations for a given problem and interpreting the results of their calculations. I found that these students made relatively few arithmetic mistakes as compared to students I tutored in the traditional method, and many of these students who saw both approaches preferred the alternative method. I find (and it is likely true for students) that it takes significantly less time to row reduce a matrix using the alternative approach than the traditional approach. Teachers are free to choose a preferred method (some may want to emphasize practice with fractions), but I believe this alternative method to be a strong alternative to the traditional Luke Smith has several years of experience teaching high school mathematics. He currently manages a math and science tutoring lab at Auburn University Montgomery. Joan Powell is a veteran professor with over 26 years of college teaching experience. 44
method since students will perform significantly fewer computations and teachers can extend the technique to finding the inverse of matrices. Many students are not proficient at solving problems involving fractions, and this lack of proficiency is not restricted to any one grade band. For example, when Brown and Quinn (2006) studied 143 ninth graders enrolled in an elementary algebra course at an upper middle-class school, they found that many of the students had a lack of experience with both fraction concepts and computations. In their study, 52% of the students could not find the sum of 5/12 and 3/8, and 58% of the students could not find the product of 1/2 and 1/4. Unfortunately, studentsâ&#x20AC;&#x2122; difficulty with fractions can persist into postsecondary education. When studying elementary education majors at the University of Arizona, Larson and Choroszy (1985) found that roughly 25% of the 391 college students incorrectly added and subtracted mixed numbers when regrouping was involved. Hanson and Hogan (2000) studied the computational estimation skills of 77 college students who were majoring in a variety of disciplines; many of the students in their study struggled with problems that involved fractions and became frustrated with the process of finding common denominators. They noted that a few students in the lower performing groups added (or subtracted) the numerators and denominators and did not find common denominators. Commenting on the lack of understanding commonly associated with fractions, Steen (2007) observed that even many adults become confused if a problem requires anything but the simplest of fractions.
Luke Smith & Joan Powell
The use of matrices to solve systems of equations has long been a topic in high school and college advanced algebra and precalculus algebra courses. An increasing number of colleges and high schools teach Finite Mathematics, sometimes as a core course option. This means that increasing numbers of college and college-bound students are introduced to solving systems of equations by converting them into matrices and then row reducing them. For example, at the university where I teach, childhood education majors see this topic in a required core course. Fraction skills may be a reasonable requirement for all of these students, but I believe this is not the best context for practicing numerous fraction computations, particularly for students who are not typically math or science majors. Indeed, students’ difficulties with fractions lead many instructors to carefully pick matrices that do not involve fractions during the intermediate steps of the traditional approach to rowreducing a matrix. However, the alternative method discussed below is similar to traditional Gauss-Jordan elimination but allows instructors to use any system of linear equations over the rational numbers because it prevents new fractions from appearing until the very last steps. Furthermore, the alternative method involves a similar number of computations as the traditional method, which decreases the likelihood of arithmetic mistakes. When deciding which approach students should learn in order to row reduce matrices, teachers need to consider their motivation for showing students how to row reduce matrices. Typically, we want our students to be able to solve resource allocation problems, geometric problems, or other types of applications by finding the values of the variables in a system of equations and then correctly interpreting the results of their findings. In other words, we are interested in showing our students how to solve problems where row reduction of matrices is an appropriate strategy. Therefore, if we have two mathematically sound approaches for finding the values of the variables, one whose computational demands may distract from the main concept and the other that involves fewer computations and is less distracting, it seems reasonable to show students the method that will free them to focus on setting up the problem and interpreting the results rather than being immersed in the intermediate calculations. Such an instructional decision aligns with the National Council of Teachers of Mathematics (2000) teaching principle (2000) that advocates the skillful selection of teaching strategies to communicate mathematics.
The alternative method is not a new approach, but after reviewing many Finite Mathematics and Linear Algebra textbooks from a variety of publishers, I found that the vast majority of the texts do not clearly present to students with a method of solving a system of equations without incurring fractions in the intermediate steps (Goldstein, Schneider, & Siegel, 1998; Poole, 2003; Rolf, 2002; Uhlig, 2002; Young, Lee, & Long, 2004). Even the texts used at my university (Barnett, Ziegler, & Byleen, 2005; Lay, 2006) do not demonstrate the alternative method. Warner and Costernoble (2007), Shifrin and Adams, (2002), and Lial, Greenwell, and Ritchey (2008) were the only texts that I found that clearly presented the alternative method. In all of the aforementioned books no characteristics seemed to predict whether or not the alternative method was presented and they all covered roughly the same concepts that are traditionally presented in Finite Mathematics and Linear Algebra courses. For the benefit of students and teachers who have only been exposed to the traditional Gaussian methods of row-reduction, the remaining portion of the article develops the alternative technique. The following paragraphs describe operations with matrices of the type provided below (Figure 1). a1,1 a 2,1 a 3,1
a1, 2
a1,3
a 2, 2 a 3, 2
a 2, 3 a 3,3
k1 k2 k 3
Figure 1. A typical 3× 3 augmented matrix. The most common method that students are taught Gauss-Jordan-elimination for solving systems of equations is first to establish a 1 in position a1,1 and then secondly to create 0s in the entries in the rest of the first column. The student then performs the same process in column 2, but first a 1 is established in position a2,2 followed secondly by creating 0s in the entries above and below. The process is repeated until the coefficient matrix (Figure 1) is transformed into the identity matrix, where 1s are along the main diagonal and 0s are in all other entries (Barnett, Ziegler & Byleen, 2005). Some teachers use a variation of GaussJordan elimination called back-substitution that simplifies the process somewhat for solving systems of equations; however, back-substitution can not be used to find inverses of matrices. The traditional approach of finding first the 1s for each of the diagonal entries and secondly finding the 0s for the remaining elements in each corresponding column becomes extremely cumbersome when 45
Gauss-Jordan Elimination
fractions are involved. Students who are not comfortable or proficient with fractions may become frustrated with these types of problems. Asking instructors to teach students a method that they are only able to use to solve a limited class of problems does those students a disservice. The alternative Gaussian approach where 1s on the diagonal are not obtained until the very end of the problem is a nice alternative to the traditional method. In my opinion, the strength of this approach is that (a) no new fractions are introduced until the very last steps and (b) this
process can still be implemented to find the inverse of a matrix (in contrast to the back-substitution method). To set up this method, I review an approach for solving a system of two equations in two variables. For this smaller system, teachers commonly teach the addition method, which relies on multiplying each equation by the (sometimes oppositely signed) coefficients in the other equation and then adding the two equations to eliminate the target variable. Consider the following problem (Example 1 in Figure 2).
Step 1: We can choose to eliminate either the x or y variable. For this example, we will eliminate the x variable. Step 2: To eliminate the x variable, we will multiply the top row (R1) by 2 and the bottom row (R2) by -3. Then we will add the two equations together to create a new equation. Note: We know that we are proceeding in the correct direction because we successfully eliminated the x variables when we added the equations together.
3x + 2 y = 8 2 x − 5 y = −1
3x + 2 y = 8 2 x − 5 y = −1
(2) (− 3)
6 x + 4 y =16 − 6 x +15 y = 3 19 y = 19
Step 3: At this point, we simply solve for y and substitute our solution back into either equation to solve for x, checking both in the other equation.
y =1 x=2
Figure 2. Solving Example 1, a 2×2 linear system. The process of eliminating the x variable in the above problem (Figure 2) by producing opposite coefficients of x is used in the alternative method for
row-reducing matrices. Next, I show how to use the above idea to solve a typical system of n equations with n variables without incurring any fractions (Example 2 in Figures 3a and b).
Step 1: Recopy from the original system of equations into augmented matrix form.
3x + 4 y + 2 z = 5 2x − 3y + z = − 7 4 x + y − 2 z = 12
3 4 2 5 2 − 3 1 − 7 4 1 − 2 12
Step 2: Multiply R1 and R2 in such a way that you create oppositely signed common multiples in entries a1,1 and a2,1 as shown below.
3 4 2 5 2 −3 1 −7
(−2) (3)
Adding and then substituting the sum for row 2 results in a 0 in entry a2,1.
−6 +
− 8 − 4 − 10
6 −9 0 − 17
3 4 2 5 0 − 17 − 1 − 31 4 1 −2 12
3 − 21 − 1 − 31
Figure 3a. Step-by-step process for solving Example 2 using the alternative Gaussian approach. Note: The process in the left column produces the matrix in the right column for each step. 46
Luke Smith & Joan Powell
Step 3: Multiply R1 and R3 in such a way that you create oppositely signed common multiples in entries a1,1 and a3,1.
3 4
2
( − 4)
5
4 1 − 2 12
(3)
Adding and then substituting the sum for row 3 results in a 0 in entry a3,1.
− 12 − 16 +
− 8 − 20
12 3 −6 0 − 13 − 14
3 4 2 5 0 − 17 − 1 − 31 0 − 13 − 14 16
36 16
Figure 3b. Step-by-step process for solving Example 2 using the alternative Gaussian approach.
It is not important what values are produced on the main diagonal until the last step of this process. So, I will not divide the top row by 3 to get a value of 1 in position a1,1 which would
produce fractions in this intermediate step. Now, I will must establish 0s in the entries above and below a2,2 (Figure 4).
Step 4: Multiply R1 and R2 in such a way that you create oppositely signed common multiples in entries a1,2 and a2,2.
3 4 2 5 0 − 17 − 1 − 31
(17) ( 4)
Adding and then substituting the sum for row 1 results in a 0 in entry a1,2.
51 +
68
34
0 30 − 39 51 0 − 17 − 1 − 31 0 − 13 − 14 16
85
0 − 68 − 4 − 124 51 0 30 − 39
Step 5: Multiply R2 and R3 in such a way that you create oppositely signed common multiples in entries a2,2 and a3,2.
0 − 17 − 1 − 3` 0 − 13 − 14 16
( −13) (17)
Adding and then substituting the sum for row 3 results in a 0 in entry a3,2.
0
221
13
85
0 30 − 39 51 0 − 17 − 1 − 31 0 0 − 225 675
+ 0 − 221 − 4 − 124 0 0 30 − 39 Figure 4. A continuation of the solution of Example 2 using the alternative Gaussian approach.
Having now established 0s in the appropriate positions in columns 1 and 2 (Figure 4), we repeat the process to establish 0s in column 3. However, it would
be useful at this point to reduce the numbers in row 3 before we establish the last set of 0s (See optional step in Figure 5).
47
Gauss-Jordan Elimination
Optional Step: Since 675 is a multiple of -225, simplifying R3 by dividing the entire row by “-225” (or multiplying by the reciprocal) will make the arithmetic easier from this point on:
( 225−1 )
0 0 − 225 675
→
0 0 1 −3
Note: Dividing a row by a common factor simplifies the arithmetic by producing smaller values for each entry.
0 30 − 39 51 0 − 17 − 1 − 31 0 0 1 − 3
Step 6: Multiply R1 and R3 in such a way that you create oppositely signed common multiples in entries a1,3 and a3,3.
51 0 30 − 39 0 0 1 −3
(1) ( −30)
Adding and then substituting the sum for R1 results in a 0 in entry a1,3.
51 0
+
30 − 39
0 0 − 30 51 0 0
0 0 51 51 0 − 17 − 1 − 31 0 0 1 − 3
90 51
Step 7: Multiply R2 and R3 in such a way that you create oppositely signed common multiples in entries a2,3 and a3,3
0 − 17 − 1 − 31 0 0 1 −3
(1) (1)
. And then substituting the answer in for R2 results in a 0 in entry a2,3.
0 − 17 − 1 − 31
+ 0 0 0 − 17
0 0 51 51 0 − 17 0 − 34 0 0 1 − 3
1 −3 0 − 34
Final Step: The last step in this process is to divide each row by its first non-zero entry (multiply by its reciprocal), in this case the values on the main diagonal.
51
(51−1 )
0 − 17 0 − 34 0 0 1 −3
(−17 −1 ) (1)
51
0 0
0 0 51 51 0 − 17 0 − 34 0 0 1 − 3
Thus, x = 1, y = 2, z = -3.
Figure 5. Concluding steps for solving Example 2 using the alternative Gaussian approach.
Showing students how to solve systems of linear equations using the alternative version of Gaussian elimination allows them to avoid becoming inundated with fraction computations. For Example 2, if the operation between any two integers counts as one computation, then using the traditional method to solve the system of equations results in 58 computations; the alternative method results in 46 computations. Because the alternative method produced 21% fewer computations than the traditional method, students are less likely to get lost in the intermediate computations
48
and are more able to focus on the overall purpose of the method. Note again that the alternative method can be used for systems of rational equations and can be followed fairly mechanically for rational systems containing n equations with n variables. In the event that the system of equations has infinitely many solutions or no solution, the idea behind the alternative method is the same: get 0’s for entries above and below the leading non-zero entry in each row, then divide each row by the value of this non-zero entry. The following example illustrates this point (Example 3 in Figure 6).
Luke Smith & Joan Powell
Step 1: Recopy from the original system of equations into augmented matrix form.
6 x + 4 y + 13 z = 5 9x + 6 y 4x + 8 y −
= 7 z =12
6 4 13 5 9 6 0 7 12 8 − 1 12
Step 2: Multiply R1 and R2 in such a way that you create oppositely signed common multiples in entries a1,1 and a2,1 as shown below.
6 4 13 5 9 6 0 7
( −3) ( 2)
Adding and then substituting the answer for R2 results in a 0 in entry a2,1.
− 18 − 12 − 39 − 15 + 18 12 0 14 0 0 − 39 − 1
13 5 6 4 0 0 − 39 − 1 12 8 − 1 12
Step 3: Multiply R1 and R3 in such a way that you create oppositely signed common multiples in positions a1,1 and a3,1.
(−2) (1)
6 4 13 5 12 8 − 1 12
Adding and then substituting the answer for R3 results in a 0 in entry a3,1.
− 12 − 8 − 26 − 10 + 12 8 − 1 12 0 0 − 27 2
13 5 6 4 0 0 − 39 − 1 2 0 0 − 27
Figure 6. Beginning steps of solution for Example 3.
Looking at the preceding matrix, we have a 0 in position a2,2, so I cannot use it to eliminate the 4 in position a1,2; and since I have a 0 in position a3,2, I do not benefit from switching row 2 and row 3. Thus, I can focus our attention on -39 in position a2,3. (I could
also focus our attention on -27, but the end result would not change). The objective is still the same: get “0’s” in the entries above and below -39 (Figures 7a and 7b).
Step 4: Multiply R1 and R2 in such a way that you create oppositely signed common multiples in positions a1,3 and a2,3.
6 4 13 5 0 0 − 39 − 1
(3) (1)
Adding and then substituting the answer in for R1 results in a 0 in position a1,3.
18 12
+
39
0 14 18 12 0 0 − 39 − 1 0 0 − 27 2
15
0 0 − 39 − 1 18 12 0 14
Figure 7a. Continuation of solution for Example 3.
49
Gauss-Jordan Elimination
Step 5: Multiply R2 and R3 in such a way that you create oppositely signed common multiples in positions a2,3 and a3,3.
0 0 − 39 − 1 0 0 − 27 2
(27) ( −39)
Adding and then substituting the answer in for R3 results in a 0 in position a3,3..
0 0 − 1053
+ 0 0 0 0
− 27
0 14 18 12 0 0 − 39 − 1 0 0 0 − 105
1053 − 78 0 − 105
Figure 7b. Continuation of solution for Example 3.
Based on the previous matrix (Figures 7a and 7b) we can see that the system of equations does not have a solution since row 3 states that 0 = -105 (clearly a false statement). If we wanted to finish simplifying the matrix, we would divide rows 1 and 2 by the values of their leading non-zero entries to get the following (Figure 8). Final Step: R1
÷ 18 → R1
R2
÷ −39 → R2
7 1 23 0 9 1 39 0 0 1 0 0 0 −105
Figure 8. Final steps of solution for Example 3.
I hope that those who have not considered this alternative method will see the possible advantages for themselves and their students. First, this method may increase the accessibility of matrix material for students with weaknesses in fractions. Next, the method has the potential to increase the speed and accuracy of computations for students and teachers alike by the substitution of integer computations for rational number computations. I have found that some students avoid fractions by using decimal approximations, sacrificing precision. However, with this method, teachers can still require the precision of fractional solutions without the excessive mire of fractions, potentially encouraging more student effort and success. Finally, teachers who are wary of requiring extensive fractional computations may be freed by this method to have a greater flexibility in problem selection.
50
REFERENCES Barnett, R., Ziegler, M., & Byleen, K. (2005). Applied mathematics for business and economics, life sciences, and social sciences. Upper Saddle River, NJ: Pearson Prentice Hall. Brown, G., & Quinn, R. (2006). Algebra students’ difficulty with fractions. Australian Mathematics Teacher, 62(4), 28–40. Goldstein, L., Schneider, D., & Siegel, M. (1998). Finite mathematics and its applications, 6th ed. Upper Saddle River, NJ: Prentice Hall. Hanson, S., & Hogan, T. (2000). Computational estimation skill of college students. Journal for Research in Mathematics Education, 31, 483–499. Larson, C., & Choroszy, M. (1985). Elementary education majors’ performance on a basic mathematics test. Retrieved from http://www.eric.ed.gov/. Lay, D. (2006). Linear algebra and its applications, 3rd ed. Boston, MA: Pearson Education. Lial, M., Greenwell, R., & Ritchey, N. (2008). Finite mathematics, 9th ed. Boston: Pearson Education. National Council of Teachers of Mathematics (2000). Principles and standards for school mathematics. Reston, VA.: Author. Poole, D. (2003). Linear algebra: A modern introduction. Pacific Grove, CA: Thompson Learning. Rolf, H. (2002). Finite mathematics, 5th ed. Toronto: Thompson Learning. Shifrin, T., & Adams, M. (2002). Linear algebra: A geometric approach. New York, NY: W. H. Freeman and Company. Steen, L. (2007). How mathematics counts. Educational Leadership, 65(3), 8–14. Warner, S., & Costernoble, S. (2007). Finite mathematics, 4th ed. Pacific Grove, CA: Thompson Learning. Young, P., Lee, T., Long, P., & Graening, J. (2004). Finite mathematics: An applied approach, 3rd ed. New York, NY: Pearson Education.
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(ISSN 1062-9017) is a biannual publication of the Mathematics Education Student Association (MESA) at The University of Georgia and is abstracted in Zentralblatt für Didaktik der Mathematik (International Reviews on Mathematical Education). The purpose of the journal is to promote the interchange of ideas among students, faculty, and alumni of The University of Georgia, as well as the broader mathematics education community. The Mathematics Educator presents a variety of viewpoints within a broad spectrum of issues related to mathematics education. Our editorial board strives to provide a forum for a developing collaboration of mathematics educators at varying levels of professional experience throughout the field. The work presented should be well conceptualized; should be theoretically grounded; and should promote the interchange of stimulating, exploratory, and innovative ideas among learners, teachers, and researchers. The Mathematics Educator encourages the submission of a variety of types of manuscripts from students and other professionals in mathematics education including: • • • • • • •
reports of research (including experiments, case studies, surveys, and historical studies), descriptions of curriculum projects, or classroom experiences; literature reviews; theoretical analyses; critiques of general articles, research reports, books, or software; commentaries on research methods in mathematics education; commentaries on public policies in mathematics education.
The work must not be previously published except in the case of: • •
translations of articles previously published in other languages; abstracts of or entire articles that have been published in journals or proceedings that may not be easily available.
Guidelines for Manuscript Specifications •
Manuscripts should be typed and double-spaced, 12-point Times New Roman font, and a maximum of 25 pages (including references and endnotes). An abstract (not exceeding 250 words) should be included and references should be listed at the end of the manuscript. The manuscript, abstract, references and any pictures, tables, or figures should conform to the style specified in the Publication Manual of the American Psychological Association, 6th Edition.
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An electronic copy is required. The electronic copy must be in Word format and should be submitted via an email attachment to tme@uga.edu. Pictures, tables, and figures should be embedded in the document and must be compatible with Word 2007 or later.
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The editors of TME use a blind review process. Therefore, to ensure anonymity during the reviewing process, no author identification should appear on the manuscript. A cover age should be submitted as a separate file and should include the author’s name, affiliation, work address, telephone number, fax number, and email address. If the manuscript is based on dissertation research, a funded project, or a paper presented at a professional meeting, a footnote on the title page should provide the relevant facts.
To Become a Reviewer Contact the Editors at tme@uga.edu. Please indicate if you have special interests in reviewing articles that address certain topics such as curriculum change, student learning, teacher education, or technology.
In this Issue, Guest Editorial… From the Common Core to a Community of All Mathematics Teachers SYBILLA BECKMANN You Asked Open-Ended Questions, Now What? Understanding the Nature of Stumbling Blocks in Teaching Inquiry Lessons NORIYUKI INOUE & SANDY BUCZYNSKI Secondary Mathematics Teacher Differences: Teacher Quality and Preparation in a New York City Alternative Certification Program BRIAN R. EVANS Sense Making as Motivation in Doing Mathematics: Results From Two Studies MARY MUELLER, DINA YANKELEWITZ, & CAROLYN MAHER An Alternative Method to Gauss-Jordan Elimination: Minimizing Fraction Artihmetic LUKE SMITH & JOAN POWELL
The Mathematics Education Student Association is an official affiliate of the National Council of Teachers of Mathematics. MESA is an integral part of The University of Georgia’s mathematics education community and is dedicated to serving all students. Membership is open to all UGA students, as well as other members of the mathematics education community.