v13n1

____ THE _____________ _____ MATHEMATICS ___ ________ EDUCATOR _____ Volume 13 Number 1

Spring 2003

MATHEMATICS EDUCATION STUDENT ASSOCIATION THE UNIVERSITY OF GEORGIA

Editorial Staff

A Note from the Editor

Editor Brian R. Lawler

Dear TME readers, An interesting collection of papers has come together for Volume 14 of The Mathematics Educator. Each article tells the story of mathematics educators wrestling with efforts to make meaning of their work. In his final piece of a three part series of case studies begun in Volume 13, Andy Norton peers into a mathematician’s spiritual beliefs. In contrast to earlier findings, this mathematician’s spiritual beliefs are in harmony with his attitudes toward and ways of knowing mathematics.

Rasmus Nielsen presents his personal theoretical difficulties in wrestling with the goals of conducting research in mathematics education. What is our commitment to society – reproduction or renewal? At what grain-level is our commitment to those who are our society?

Carmen Latterell; Norene Lowery; and Amy McDuffie, Valarie Akerson, & Judith Morrison provide a synergistic triad of reports of working with teachers implementing new classroom practices. As mathematics classrooms change and as what we value as mathematical learning evolves, new dilemmas arise for both teachers and researchers.

Each of these authors works to investigate sensible and meaningful assessment practices among the manifestation of these new dilemmas. Their findings begin to elaborate a research agenda in the era of larger scale adoption of reform curricula.

And finally, a graduating senior from The University of Georgia kicks off the issue. She reflects on the tensions of meeting the needs of her students in the face of an “alphabet soup” of curriculum standards, standardized tests, and the other stressors of student teaching.

I hope I have sufficiently sparked your interest to read our journal. Next, I hope to tempt you to consider an academic contribution to our efforts. TME is entirely a student run and student funded, peer-reviewed journal dedicated to sharing theory, research, and practices within the mathematics education community. In its thirteenth year of publication, the journal has proven to be a resource for serving, connecting, and learning for the writers, editors, and readers. Please contribute to TME by sharing articles with students and peers, linking to our web site, submitting manuscripts, or helping to review. You will find information on pp. 47-48 of this issue on how you can be involved.

Associate Editors Holly Garrett Anthony Dennis Hembree Zelha Tunç-Pekkan Publication Laurel Bleich Advisors Denise S. Mewborn Nicholas Oppong James W. Wilson

MESA Officers 2003-2004 President Dennis Hembree Vice-President Erik Tillema Secretary R. Judith Reed Treasurer Angel Abney NCTM Representative Holly Garrett Anthony Undergraduate Representative Tiffany Goodwin

Brian R. Lawler 105 Aderhold Hall The University of Georgia Athens, GA 30602-7124

tme@coe.uga.edu www.ugamesa.org

About the cover The cover art was inspired by a presentation given by Harold Asturias while he worked on the New Standards Project, a collaboration of the University of Pittsburgh and the National Center on Education and the Economy. In the mid 1990’s, these partners built an assessment system to measure student progress toward meeting national standards. In his presentation, Harold reminded the audience that any single tool that we peer through to attempt to measure student understanding has important limitations. Some provide reasonable clarity but offer too narrow a view; others open the lens wider but blur what we see. His conclusion was that we as educators are bound to consider multiple and varied assessments when making significant decisions about children. But more importantly, he reminds us that learners are more complex than any combination of measurement tools could ever report.

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

Spring 2003

Volume 13 Number 1

Table of Contents 2

Guest Editorial… A Learning Environment Crippled by Testing: A Student Teacher’s Perspective AMANDA AVERY

5

Testing the Problem-Solving Skills of Students in an NCTM-Oriented Curriculum CARMEN M. LATTERELL

15 Assessment Insights from the Classroom NORENE VAIL LOWERY 22 Designing and Implementing Meaningful Field-Based Experiences for Mathematics Methods Courses: A Framework and Program Description AMY ROTH MCDUFFIE, VALARIE L. AKERSON, & JUDITH A. MORRISON 33 How to do Educational Research in University Mathematics?

RASMUS HEDEGAARD NIELSEN 41 Mathematicians' Religious Affiliations and Professional Practices: The Case of Bo ANDERSON NORTON III 46 Upcoming conferences 47 Subscription form 48 Submissions information

© 2003 Mathematics Education Student Association All Rights Reserved

The Mathematics Educator 2003, Vol. 13, No. 1, 2–4

Guest Editorial…

A Learning Environment Crippled by Testing: A Student Teacher’s Experience Amanda Avery My student teaching experience was to begin in January 2003. In November prior to this, I was assigned to a mentor teacher who taught Applied Problem Solving (APS) and Advanced Algebra/Trigonometry. Because the state of Georgia has adopted (criterion referenced) End of Course Tests (EOCT) for Algebra I and Geometry, I will focus a majority of this commentary on my experiences with my APS class. This course and its complement, Applied Algebra, satisfy the requirement for Algebra I for technical career students (as opposed to those that are college prep students). The intention is for the technical career students to get more of a “hands on” approach to Algebra, with phrases like “contextual teaching and learning” being strongly emphasized and encouraged.1 My intention is to illustrate to you what I was up against concerning standardized testing in my APS classes, to express to you how I feel about these pressures, and to demonstrate to you how my students wanted me to assess them in other methods that aren’t necessarily measurable.

Because I was unfamiliar with the curriculum for the APS course, I carefully reviewed the state of Georgia’s educational standards, called the Quality Core Curriculum (QCC) and compared them to the CORE-Bridges APS textbook used in my school. I was relieved to discover that I didn’t need to teach the breadth of the entire textbook, but only about five key concepts in depth. I e-mailed my mentor teacher about his experience the previous year with this textbook: “I was looking at the QCCs for the APS class and discovered that we can most likely skip chapters 4 and 8 (along with a few sections from chapter 2) from their textbook. Let me know what you think.”

My mentor teacher’s response: “Due to the EOCT that follows the Applied Algebra, we do not follow the Amanda Avery is a mathematics education student at the University of Georgia. Upon completing her student teaching in a rural public high school, she was recognized as the outstanding graduating senior with the department’s Hooten Award. She will begin her teaching career this fall at John McEachern High School in Powder Springs, Georgia. She enjoys sailing, dabbles in desktop publishing, and collects black & white photographs of dogs. Her email address is ImagineALA@hotmail.com.

2

QCCs in this course as closely as we do others. We really don’t skip those sections at all, but modify and enrich as allowed by the students of the class. … Our two APS classes are just 23 and 17 students each. They range from special ed students to seniors who have algebra one, algebra two, and just need a third math to graduate under the Technical Career diploma.

However, the course is based on the slower students (bottom 25th percentile of the nation), and the others are used as tutors and mentors. Most like being the smartest in the class, as they are used to struggling in CP [College Prep] and failing.”

Just so you have a little more taste of what my first impressions were, I asked my mentor teacher for some more feedback: “I was hoping to skip a majority of chapter one. Of course, I really have no idea how appropriate this is because I don’t know what the students understand and can apply. I do feel, though, that I should give the students a little more credit than what their textbook implies.”

My mentor teacher’s feedback: “Most of the students need the reinforcement of the first chapter, but there are things we can do to not approach it the same way as the book. I agree that it seems elementary, but these students are not like the ones you observed in block four classes in the fall [Honors Advanced Algebra/Trigonometry]. They have struggled in math their whole life, and most math is forgotten between courses. Part of the hope is to let some find success if they have the skills, but look at it in an applied sense.

We may get lucky, but students have failed the first unit test (46% last spring).”

Now, for a challenge from the state legislature…

The state of Georgia, as part of its A+ Educational Reform Act of 2000 passed a law, O.C.G.A. §20-2-281 that “mandates that the State Board of Education adopt end-of-course-assessments in grades nine through twelve.” Each EOCT is “directly aligned with the standards in the QCC and will consist of multiple choice questions.” According to its 2001 Information Brochure, the purpose of EOCT is to “improve student achievement through effective instruction and assessment of the standards in the QCC and to ensure A Crippled Learning Environment

that all Georgia students have access to a rigorous curriculum that meets high-performance standards.” If students take the EOCT ; what, then, will teachers, schools, school boards, and the state board of education do with the data that is collected? “The results of the EOCT will be used for diagnostic, remedial, and accountability purposes to gauge the quality of education in the state.” The interesting part of all this is that all Georgia high school students take the Georgia High School Graduation Tests (GHSGT) as exit tests for getting a high school diploma: IB, college prep, tech prep, and general ed are all included (special ed seals allow for modifications, as appropriate). In addition to a comprehensive test that covers “up through Algebra I,” as a requirement for graduation, legislators also subject students to EOCTs. Let’s not forget, also, that legislators are submitting taxpayers to the costs of creating, administering, and scoring these tests.

Now you have the general idea of what I was to encounter, through my mentor teacher’s expectations and the state’s mandates. I taught in block scheduling, where students complete four entire courses in a single semester, sitting in four 90-minute classes each day.

Despite my mentor teacher’s outlook on keeping an iron fist on the APS textbook, I looked for ways to satisfy the National Council of Teachers of Mathematics (NCTM’s) vision from the Principles and Standards, the Georgia QCCs, the EOCTs, the GHSGTs, the PSAT, the SAT, the ACT, my APS students and ME.2 I don’t want to completely give you wrong impression—While I was very overwhelmed by the pull of all these outside forces, I did enjoy my student teaching. How? Because of my students—the interactions I’ve had with them, and the few sparks that I could witness igniting if they encountered something new and challenging.

Early on I was confronted with wondering how I can arrange a pacing guide that requires my students to be tested comprehensively as juniors in March for the GHSGT, tested cumulatively in April (or November) for Algebra I for the EOCT, tested state-wide for the PSAT as sophomores in October, make time for standardized testing “prep sessions,” athletic pep rallies, student organization meetings, special seminars for students, fire & tornado drills, and for, oh yea, teaching. I understand that assessment is a part of teaching, but within my classroom I also must place into the pacing guide room for formal assessments for finals, midterms, projects, presentations, quizzes, and reviewing homework. I hope you are beginning to understand that I feel as though my student teaching

Amanda Avery

experience was one big assessment. Sometimes I felt as though I never got to teach, and more importantly, my students rarely had the opportunity to learn significant mathematics, to struggle, to dabble—they were always being evaluated, whether informally or formally.

It is with this point that I am most disturbed by Georgia’s A+ Education Reform Act of 2000. I didn’t understand how multiple choice standardized tests could “improve student achievement through effective instruction and assessment of the standards in the QCC,” especially if the end results were to be used for “diagnostic, remedial, and accountability purposes to gauge the quality of education in the state.” It seems to me that standardized multiple choice questions are inadequate for assessing the “quality of education,” especially if you consider quality mathematics education as the vision put forth by NCTM’s Principles and Standards. Even more importantly, I was also vexed with what the results will do for my students—they sort, rank, and stigmatize my students against other students, other schools, other districts, and other states who have widely varying curriculum standards, resources, administrators, parents, and students than mine, especially if “holding schools accountable” means that my school district may not get additional funding. Ethically, this seems as a step backward if my community truly believes that ALL students can be successful and where ALL students can and must learn mathematics, where “no child is left behind.” Even though the idea that students must be able to illustrate proficiency in skills x, y, and z before moving on to the next course is quite valid, I don’t believe that standardized testing is helping my students, their parents, me, my administrators, my school board electorates, and my state board legislatures to provide a complete picture, even if my students excel at multiple choice standardized testing.

Lack of providing a complete picture is only the “tip of the iceberg” towards the argument that standardized testing is not the way to “improve student achievement.” While I don’t believe that assessment isn’t a part of the teaching process, I believe that students’ attitudes, behaviors, and oral and written communication skills must be also considered.

Yet, I am unconvinced that students should be continually assessed, every moment in every class. It was an obstacle for me to realize that my students were afraid to participate in class because they have learned the game—they know that the teacher is always listening, the teacher is always judging, the teacher is always thinking, “right or wrong.” No, actually, the obstacle was in trying to change the game, to let my

3

students experience different rules, where they weren’t judged on everything they thought, on everything they tried to start. This experience is not centered on my actions alone; in fact, the interactions that my students had with each other was the most crucial part of changing the rules. It took my students about six weeks to realize that when I ask for ideas that I (or other students) wouldn’t ridicule or thwart each other’s efforts. It was almost over night that my students started trusting me and trusting each other with their ideas and suggestions.

What happened? I was able to teach; they were allowed to struggle, they were given opportunities to test ideas, and they weren’t being graded. Correction: it wasn’t that I was suddenly able to teach; my students now taught each other. Through a problem solving day, my students demonstrated to me and to each other that they could effectively provide valid inferences, evaluate each other’s ideas constructively (not critically), and more importantly, they started questioning their own thought processes. The most significant comment I made that day is that I “will not grade what you do today. Rather, I want you to think about how you can evaluate how you think you understand what you are doing.”

My students informed me that I could understand what they were thinking if I watched them do problems, and if I listened to how they explained the concepts to each other. Even though I did these things informally, somehow this discourse allowed my students to become more relaxed with each other; it’s

4

almost as if I needed their “okay” for them to finally trust me and each other. The judging environment had been eased; the iron fist relaxed its grip. They understood that I didn’t like lecturing and being the only one doing the mathematics while they passively copied my notes, and I understood that they didn’t like this either—they want the challenge, they want to think. And standardized testing cannot ever measure that.

REFERENCES A+ Educational Reform Act of 2000 O.C.G.A:§20-2-281 (n.d.).

Retrieved May 16, 2003, from http://www.ganet.org/cgi-bin /pub/ocode/ocgsearch?docname=OCode/G/20/2/281 Georgia Department of Education. (n.d.). End of Course Test algebra one content description guide. Retrieved May 16, 2003, from http://www.doe.k12.ga.us/_documents/curriculum /testing/al1.pdf Georgia Department of Education. (n.d.). Information on End of Course Tests. Retrieved May 16, 2003, from http://www.doe.k12.ga.us/_documents/curriculum/testing/eoct -broc.pdf National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author.

1

You might understand why I’ve added the emphasis, but that is not the point of this editorial.

2 I wanted to emphasize that my students and I are also important in this alphabet soup of curriculum standards and standardized tests.

A Crippled Learning Environment

The Mathematics Educator 2003, Vol. 13, No. 1, 5–14

Testing the Problem-Solving Skills of Students in an NCTM-oriented Curriculum Carmen M. Latterell An interesting question concerns how well NCTM-oriented students do on standardized mathematics tests.

Another important question that has received less attention is: Are standardized tests truly measuring the skills that NCTM-oriented students have? Would other tests reveal skills that differentiate NCTM-oriented students from traditional students? Moreover, what are these skills? This paper contributes to the answers to these questions, and finds that students in one NCTM-oriented curriculum displayed such qualities as engagement, eagerness, communication, flexibility, and curiosity to a much higher degree than traditional students did. The implication given is that not only should we use standardized tests, but we should revise them and/or supplement them to measure the qualities that are not currently being measured.

The National Council of Teachers of Mathematics (NCTM) has been promoting standards for teaching secondary mathematics (NCTM, 2000; NRC, 1989). At the same time, many curriculum projects have been developed for the purpose of providing NCTMoriented curricula for teachers (e.g., the NSF-funded curricula). However, the implementation of these NCTM-oriented curricula has not been without problems (Curcio, 1999). In fact, the term “math wars” has been used to describe the controversies (Senk & Thompson, 2003b; Schoen, Fey, Hirsch, & Coxford, 1999). Some of these controversies surround the issue of standardized testing (Hancock & Kilpatrick, 1993).

In fact, some parents have indicated concern that if their children are in NCTM-orientated curricula, they will be at a disadvantage on standardized tests (Senk & Thompson, 2003a). Of course, standardized tests might change in time, but it is an interesting question whether NCTM-oriented students actually are at a disadvantage on standardized tests. Perhaps it is an interesting question because standardized testing should change in time. In other words, the question of interest is not simply “Are students in NCTM-oriented curricula at a disadvantage in standardized testing?” but rather “Are standardized tests the best measure of everything that students in NCTM-oriented curricula can do?” And then if the answer is that other tests can reveal abilities that standardized tests do not, what are these other abilities?

Carmen M. Latterell is an assistant professor of mathematics at the University of Minnesota Duluth. Her research interests include the testing of mathematics. This interest includes the types of questions seen in this article as well as such issues as placement testing into undergraduate mathematics courses.

Carmen M. Latterell

Standardized achievement tests are viewed as "blunt" measuring instruments by some (Kilpatrick, 2003, p. 479). Certainly issues are more complex when standardized tests attempt to measure students' problem-solving ability. Since problem solving is regarded as a process (NRC, 1989; NCTM 2000) and not a product, standardized tests must be well constructed to measure problem solving adequately.

While researchers say that standardized tests cannot measure problem solving, it is important to remember that testing is "value laden and socially constructed" (Gipps, 1999, p. 370). Are standardized tests measuring the abilities that we value for students in NCTM-oriented curricula?

Now that NCTM-oriented curricula have been put into action for several years, researchers are able to examine these issues with research studies. This study contributes to the literature by examining some of the issues involved in the testing of students in a particular NCTM-oriented curriculum, the Core Plus Mathematics Project Curriculum (CPMP). The overarching research question is how to better assess students who are in a NCTM-oriented curriculum to measure as fully as possible their problem-solving abilities.

To answer this, some comparisons between NCTM-oriented students and traditional students are made, as well as some comparisons between NCTMoriented students' results on different types of tests.

The reader is cautioned that the intent of this study is not to compare NCTM-oriented students to traditional students, per se. Many studies (Senk & Thompson, 2003a) have already been done (this paper will review some of them as applies to CPMP in a later section), giving evidence that students in NCTM-oriented

5

curricula can perform as well as students in traditional curricula on standardized tests.

The current study attempts to examine the subtleties of testing students in a NCTM-oriented curriculum. For example, although these students might do as well as others on standardized tests, are there other tests that are a better measure of these students' problem solving abilities? Or even, are there other tests that will measure abilities that standardized tests do not and that traditional students may not have?

And, perhaps most important, what is the nature of these abilities that are not (if in fact they are not) being measured by standardized testing?

The research questions are: 1. How do the scores of students in CPMP compare to students in a traditional curriculum on a standardized problem-solving test?

2. How do the scores of students in CPMP compare to students in a traditional curriculum on a parallel constructed-response problem-solving test?

3. Are there differences that qualitative data can illustrate between the manners in which pairs of students in CPMP and pairs of students in a traditional curriculum solve non-routine constructed-response problem-solving items?

Method The basic method was to give a standardized problem-solving test to students in CPMP and to comparable students in a traditional curriculum. Effort was made to have similar ability students in both sets of curricula before entering the curriculum, using Iowa Test of Educational Development (ITED; Feldt, Forsyth, Ansley, & Alnot, 1993) data. In addition, a test parallel to the standardized test but with constructed-response items was also given to both sets of students. Finally, both sets of students were given a performance assessment. This section describes the curricula, the students, a survey of the classroom environments, and the tests.

The CPMP Curriculum As stated earlier, the NCTM-oriented curriculum used CPMP, the Core Plus Mathematics Curriculum.

The author had access to students in the state of Iowa, and a large number of schools in Iowa had implemented CPMP for several years. Therefore, CPMP was chosen as the curriculum to represent NCTM-oriented curricula. Yet, no claim is made that CPMP is better or worse than other NCTM-oriented curricula, or that the results, therefore, would be the same under other curricula. Regardless, the researcher 6

judged it would be better not to have more than one NCTM-oriented curriculum and thus possibly confound the results.

CPMP has developed student and teacher materials for a three-year high school mathematics curriculum for all students and a fourth-year course for college bound students. The main theme of CPMP is mathematics as sense making. Students investigate problems set in real-life contexts within an integrated curriculum that includes algebra and functions, geometry and trigonometry, statistics and probability, and discrete mathematics. The curriculum for each year is seven units with a capstone section which is “a thematic two-week, project-oriented activity that enables students to pull together and apply the important mathematical concepts and methods developed in the entire course” (Schoen & Ziebarth, 1998b, p. 153). Mathematical modeling is emphasized throughout the curriculum. Graphing calculators are used. Additional characteristics of CPMP are that the curriculum is designed to be accessible to all students, to engage the students in active learning, and to provide multidimensional assessment (Hirsch & Coxford, 1997). Assessment opportunities are embedded within the curriculum and include students’ answers to questions in class, groupwork, student journals, quizzes, in-class and take-home end-of-unit assessments, cumulative written assessments, and extended projects (Hirsch, Coxford, Fey, & Schoen, 1995).

Inclusion of topics in the CPMP curriculum is based on the merits of the topics themselves; that is, the topics must be important in their own right (Schoen et al., 1999). The instructional sequence follows a fourstep process labeled as launch, explore, share and summarize, and apply. The “launch” sets the context for what is to follow and consists of a class discussion of a problem. The “explore” is usually a group or pair activity in which students investigate the problems and questions. “Share and summarize” brings the class back together to discuss key concepts and methods.

“Apply” is time in which individual students practice what they learned (Hirsch et al., 1995).

CPMP authors have conducted numerous and extensive research into the effectiveness of CPMP. At the end of Course 1, CPMP students' average score on the Ability to Do Quantitative Thinking (a subtest of the nationally standardized Iowa Tests of Educational Development) was significantly higher (p < .05) than algebra students in traditional curricula (Schoen & Hirsch, 2003). At the end of Course 3, CPMP students performed significantly better (p < .05) on concept and

Testing Problem-Solving Skills

application tasks but significantly poorer on algebraic manipulation tasks when compared with Algebra II students in traditional curricula (Huntely, Rasmussen, Villarubi, Sangtong, & Fey, 2001). Using SAT 1, mathematics scores of CPMP III students versus Algebra II students showed no significant differences (Schoen, Cebulla, and Winsor, 2001). When American College Testing (ACT) Assessment Mathematics Test means were used, the Algebra II students had significantly higher scores (Schoen, Cebulla, and Winsor, 2001). Using placement tests constructed with items from a test bank from the Mathematical Association of America, CPMP Course 4 (N=164) versus Precalculus (N=177) students showed no significant differences on algebraic symbol manipulation skill but a significant difference in favor of CPMP on concepts and methods needed for the study of calculus (Schoen & Hirsch, 2003). Another study examined proof competence, as well as perceived need for proof using CPMP III students versus traditional Algebra II students; no significant differences were found (Kahan, 1999).

This research suggests CPMP is a curriculum that is at least as effective as a traditional curriculum except in some by-hand manipulation skills when these skills are needed outside of application context and graphing calculators are not allowed. In many areas (such as problem solving in context and with graphing calculators), CPMP students outperform traditional students. The design of the CPMP curriculum suggests a curriculum that is indeed in alignment with NCTM recommendations. The research indicates CPMP is effective in promoting the achievement of student reasoning, communication, problem solving, and representation. In traditional skills (such as by-hand manipulation skills), the CPMP curriculum might be less effective than traditional curricula. In less traditional skills (such as problem solving in multiple representations), the CPMP curriculum might be more effective than traditional curricula (Schoen et al., 1999; Schoen & Ziebarth, 1998a).

The Traditional Curriculum There is, of course, no such curriculum as “traditional curriculum” and as NCTM’s standards become increasingly respected, even the most traditional of curriculum begins to look like NCTMoriented curriculum.

The researcher fully acknowledges that there is no true dichotomy between NCTM-oriented curricula and traditional curricula any longer, but rather a continuum.

Carmen M. Latterell

For the purposes of this study, I chose to define the traditional curriculum to not be one of the NSF-funded, nor identify itself in any manner as reform. The traditional curriculum should lack an emphasis on groupwork and graphing calculators, while emphasizing hand symbolic manipulations as the only meaning of algebra. This approach moved the researcher to the traditional end of the continuum, but once there, could traditional curriculum be described in and of itself?

The traditional curricula in this study have an emphasis on separate units of mathematical content (in our case, algebra), which includes an emphasis on procedures (although conceptual understanding is present as well). The teacher most often serves as the "teller" of information (i.e., the students were not engaged in discovery work). Students most often work individually. Testing is usually easily accomplished through short-answer or even multiple-choice items.

Computation is important. In our case, since these were traditional algebra classes, the curriculum included solving (by hand) algebraic equations with solution processes not dependent on technology.

Students A sample of five Iowa schools using CPMP agreed to participate in this study. As previously mentioned, within the state of Iowa, there were many schools that had implemented CPMP. The researcher had access to a list of all Iowa schools that had CPMP in place for several years. A subset of these schools was identified and contacted, so that the subset were schools spread throughout Iowa. Five of these schools agreed to participate. This resulted in 230 students.

The researcher also had access to existing ITED data. This data was gathered for the schools that had already agreed to participate. Another sample of Iowa schools was gathered according to two characteristics: The size and location of the schools had to match one of the five already in the sample, and the mathematics curricula had to be of a traditional nature. The existing ITED data was used to narrow this sample to schools whose average school score was the same as those already in the sample. This resulted in a much smaller set of schools, but out of this set, five did agree to participate. This resulted in 320 students.

The Classroom Environment Form At this point, the teachers from each classroom completed a Classroom Environment Form (written by the researcher) to provide data on such interests as the textbook, availability of graphing calculators,

7

groupwork practices, and assessment practices (see Table 1). The purpose of gathering this data was twofold.

One purpose was to supply background and context for the discussion of the results. Although the purpose of this study was not to compare CPMP students to traditional students, it is through that comparison that the researcher is able to distinguish differences in testing results that may be created as a consequence of NCTM-oriented curricula. However, if the intended curriculum (whether that is CPMP or traditional) is not the enacted curriculum, then the worth of this comparison is questionable. (The researcher acknowledges that there is also the achieved curriculum.)

The Classroom Environment Form was also intended to supply possible explanations for results. An example might illustrate this. CPMP students work in groups as part of the curriculum. If it turns out that CPMP students work better under testing practices that include groups, perhaps that is reasonable. However, just because CPMP authors call for students to learn in groups, do the students actually spend time in groups?

Further, perhaps the traditional students also learn in groups, and thus groupwork is not really a difference

between the two curricula. Clearly the Classroom Environment Form will not necessarily give a full disclosure of a classroom. It will only give what the teacher chooses to say.

The Standardized Test A standardized problem-solving test, the subtest from the Iowa Test of Educational Development (ITED) titled Test Q: Ability to Do Quantitative Thinking, was used. The stated purpose of the ITED is to “provide objective, norm-referenced information about high school students’ development in the skills that are the long-term goals of secondary education—skills that constitute a major part of the foundation for continued learning” (Feldt et al., 1993, p. 4). Test Q at level 15 (grade nine) consists of 40 multiple-choice items with five response options.

Students are given 40 minutes to complete the test.

The questions, based on realistic situations, are “practical problems that require basic arithmetic and measurement, estimation, data interpretation, and logical thinking” ( Feldt, et al., 1993, p. 13). In addition, some of the questions test more abstract concepts. “The primary objective of the test is to

Table 1 Comparison of the Two Types of Classrooms NCTM-oriented Textbook

CPMP

Traditional Algebra I Explorations and Applications, McDougal Littell, 1997 (3 classrooms) Algebra I Applications and Connections, Macmillan/McGraw Hill, 1992 (2 classrooms)

Graphing calculators

constantly available

One teacher answered constantly available. The remaining teachers stated that they were available sometimes during classes and on tests, but less frequently during homework.

Mean percent estimate for the amount of class time that students are in groups

86%

22%

Groupwork outside of class

Four teachers encouraged students to work in groups outside of class.

Three teachers encouraged their students to work in groups outside of class.

Multiple-choice tests used

Never used

2 teachers sometimes used them.

Students receive partial credit for work and explanations.

Students receive credit for writing out reasons on graded assessments.

Students are encouraged to give reasons and not just an answer.

All teachers said “always”.

All teachers said “frequently”.

All teachers said “always”.

All teachers said “frequently”.

All teachers said “always”.

All teachers said “frequently”.

8

Testing Problem-Solving Skills

measure students’ ability to use appropriate mathematical reasoning, not to test computational facility under pressure” (Feldt et al., 1993, p. 13).

The Constructed-Response Test ITED has two parallel forms. One of the forms was given intact as the previous test. To create the second test, the remaining form was modified in the following manner: The researcher converted each item to a constructed-response item. For most items, this simply meant eliminating the choices and keeping the stem.

For items in which the choices completed a sentence, the item was changed to form a question. For a small number of items, the item was completely reworded, but the purpose of the item and the context remained the same. The researcher scored the constructedresponse tests using a scoring key in which each problem is worked including numerical calculations and a verbal explanation.

The Performance Assessment Due to time constraints, a subset of the sample consisting of two classrooms of CPMP students and two classrooms of traditional students solved problems in pairs using non-routine items while the researcher observed. This resulted in approximately 100 students.

The following two problems, each written out on separate pieces of paper, were given to pairs of students.

1. How many keystrokes are needed to put page numbers on a paper of length 124 pages?

2. Three friends, returning from the movie Friday the 13th Part 65, stopped to eat at a restaurant. After dinner, they paid their bill and noticed a bowl of mints at the front counter. Sean took 1/3 of the mints, but returned four because he had a momentary pang of guilt. Faizah then took 1/4 of what was left but returned three for similar reasons. Eugene then took half of the remainder but threw two that looked like they had been slobbered on back into the bowl. (He felt no pangs of guilt—he just didn’t want slobbered-on mints.)

The bowl had only 17 mints left when the raid was over. How many mints were originally in the bowl? (Herr & Johnson, 1994, p. 303) There are various solution processes for each of the questions. For example, on the first problem, the student could count. Another method would be to reason as follows. The page numbers 1 through 9 are one digit each. Thus, nine keystrokes are needed. The page numbers 10 through 99 are two digits each; thus 2 x 90 =180 keystrokes are needed. The page numbers Carmen M. Latterell

100 through 124 are three digits each; thus 3 x 25 =75 keystrokes are needed. The sum of 9, 180 and 75 is 264, which is the answer.

For the second problem, one could use algebra, producing a rather complicated, linear equation to be solved. An easier approach is to begin at the end. There were 17 mints left in the bowl. Just before that, Eugene threw two back in, so there were 15. Before that, Eugene took half of the mints and left 15 in the bowl.

There were 30 mints in the bowl before Eugene began.

Before that, Faizah put three mints back, leaving 30, so that was 27. Faizah took 1/4, leaving 27. So, there must have been 36 mints in the bowl before Faizah. Right before Faizah, Sean returned four to the bowl. So, there must have been 32. Sean took 1/3 of the mints, leaving 32, so there must have been 48. This brings us to the beginning of the problem, in which there were 48 mints.

The CPMP students had been working in pairs throughout the school year (this assessment was conducted in May). The same pairs were used for this assessment. The classroom teachers paired the traditional students. The students were asked to work on these problems and think out loud with each other on how to solve the problems. In addition, the students were asked to write out their solution process after they were happy with their solution. It was emphasized to the students that the researcher was not interested in a numeric answer, but interested in the process that they used when solving the problem. The students were observed with the researcher taking notes, but not intervening, and videotaped while solving these problems. The field notes, annotated transcripts, and student work were then analyzed for the quality of the problem-solving strategies and processes, the overall success of the students in solving the problems, the quality of the cooperative work, and other emergent themes in the process of problem solving.

Results with Discussion Given that the students were members of intact classes, it was determined that the individual student should not be the unit of analysis in analyzing the results from the standardized test and the constructed response test. So, schools were used as the unit of analysis. In comparing CPMP to traditional schools on the standardized test, the traditional schools had a mean proportion correct score of .52 (.06 standard deviation) and the CPMP schools had a mean proportion correct score of .52 (.05 standard deviation).

Clearly there is no significant difference between whether a school was CPMP or traditional.1

9

The standardized test in the format of constructed responses faired about the same. The mean proportion correct score for CPMP schools was .65 (.13 standard deviation) and traditional schools had a mean proportion correct score of .67 (.14 standard deviation).

Again, there is no significant difference between whether a school was CPMP or traditional.

On the performance assessment, considering only the correctness of a solution, there again was no difference between the CPMP students and the traditional students. Most of the students, whether from CPMP or traditional, were able to solve the typewriter problem. A slightly higher number of CPMP students than traditional were able to solve the mints problem.

This may be due to the fact that although the problem was much easier solved just working backwards, traditional students attempted to use algebra to solve the mints problem. However, if one counts only a numeric answer, on both problems, there was close to equivalence between the two groups of students.

A performance assessment allows one to look at other issues than just correctness of an answer. Five themes emerged from the analysis of the performance data (see Table 2).

For ease of reporting, the discussion of the themes will be given in Polya's framework for viewing the problem-solving process (Polya, 1945/1973). Polya outlined these stages as: (1) getting to know the problem, (2) forming a solution plan, (3) carrying out the solution plan, and (4) looking back.

The first two themes occurred in Polya’s getting to know the problem stage. CPMP students were more engaged in the problems than the traditional students were and CPMP students were more eager to work in pairs than the traditional students were.

CPMP students immediately became engaged in the problems. This was evidence in the words that CPMP students used. “Do we get to work on these?" was actually a question a CPMP student shouted with excitement toward the researcher as soon as the researcher handed her the problem sheets. "I bet I can solve this" and "we are good at problems" were two other comments that the researcher interpreted as positive.

In addition to expressing interests and enthusiasm for solving problems, the CPMP students talked with each other about what the problems meant. For example, on the typewriter problem, the majority of CPMP pairs immediately began to act out what it might mean to type numbers on pages of a book. One CPMP student said to her partner, "See, I know, it is that… well, 35, that would be a 2. See?", meaning that the number 35 has 2 digits in it.

In analyzing the annotated transcripts for students in the traditional curriculum, the researcher could not find a single positive comment toward problem solving. In addition, one traditional student told the researcher, "We are really bad at math" which promoted another to say, "He [the teacher] hates us."

Several pairs of traditional students stated that they did not want to work the typewriter problem or that they could not work the typewriter problem because they did not understand what the problem was asking.

Several traditional pairs read the problem and then stated, "I don't get it." However, on the mints problem, many of the traditional students stated that they probably could solve it, as it required algebra and they were in an algebra course.

As students were getting to know the problems, the cooperation aspect was the second theme that emerged.

Table 2 Five Themes of Student Problem Solving Activity Theme

Description

Students’ engagement in problems

CPMP students were more engaged in the problems than the traditional students.

CPMP students engaged in the problem-solving process beyond a numeric answer.

CPMP students asked to discuss the problems after they were done solving them.

Students’ enthusiasm for working in pairs

CPMP students were more eager and able to work in pairs than traditional students.

Students’ use of symbol manipulation for their solutions

The traditional students considered and actually used algebraic techniques.

Students’ ability to communicate mathematically

CPMP students were able to write about their mathematical processes.

Students’ flexibility in their solutions

CPMP students looked for more than one solution path.

10

Testing Problem-Solving Skills

In terms of the cooperation aspect, the CPMP students began cooperating as they were getting to know the problem. The traditional students didn't cooperate until possibly the looking back stage when they compared answers. Traditional students became familiar with the problems individually, and mostly inside their heads (not readily observable behavior). Since, CPMP students cooperated while getting to know the problem, the researcher could observe some of the ways that they interacted with the problem. For example, sometimes they tried to act out the situation. Other times they described the situations to each other and tried to clarify by talking to each other what the situation really meant. So, “getting to know the problem” is trying to understand the situation, and CPMP students did this cooperatively, while traditional students did not.

To summarize, while the CPMP students discussed the problem, the traditional students worked separately.

The only time the traditional students engaged together as a pair was when they compared answers with each other. This point was made obvious with many of the traditional pairs literally telling each other to work on the problem and "we will check our answers at the end." A minority of traditional students was unwilling to compare answers, however, saying, "Solve it yourself."

The first two themes (CPMP students were more engaged in the problems than the traditional students were and CPMP students were more enthusiastic to work in pairs than the traditional students were.) continued during Polya’s (3) carry out the plan stage.

The CPMP students engaged in conversation with each other about the problems, and actually worked on the problems together. An excerpt from the annotated transcripts might illuminate this. For the sake of this example, the students will be labeled simply Student A and Student B. The students are working on the typewriter problem.

Student A: How many two-digit numbers are there?

Student B: I don't know. 11, 12, 13, … [mumbles] … let's …

Student A: It will take too long to count them. 10, too.

Student B: Oh, yes. There are 10 through 100. 99.

There are 10 through 99.

Student A: So there are 89.

Student B: Yes, 89.

At this point, the students continued in the problem and started to examine the number of 3-digit numbers that existed. The reader might be worried about the 89 (knowing that the correct number is 90), but the reader

Carmen M. Latterell

is asked to be patient, as we will return to this same pair later in this section.

In addition to the first two themes continuing, another theme (Students’ ability to communicate mathematically) emerged. The CPMP students were able to write about their mathematical processes, demonstrating an ability to communicate mathematically. Only one-fourth of the traditional students showed anything other than a numeric answer.

All of the CPMP students showed something other than their answer, with most showing the steps and thought processes of their solution. For example, on the typewriter problem, many of the CPMP students wrote out the number of 1-digit numbers, 2-digit numbers, and 3-digit numbers, labeling each respectively. On the mints problem, several CPMP students wrote the words "work backwards" or just "backwards" on their sheet. Other CPMP students showed the backward progression of numbers, but did not use the word "backwards." On the mints problem, the traditional students who did write something down (again this was about one-fourth of them) wrote out the algebraic equation that they had developed. Although some traditional students did work backwards and solve the problem, they simply showed the answer in a box. When the researcher asked them to show their solution process, they said that there was nothing to show.

A theme (Students’ use of symbol manipulation for their solutions) emerged during (2) Polya’s forming a plan stage. The traditional students were much more likely than the CPMP students to think of using symbol manipulation. In fact, the traditional students at times seemed to skip the (1) getting to know the problem stage and enter right into (2) forming a plan, with that plan being to use algebraic symbols. The traditional students felt that they could work the mints problem, because they thought there would be manipulations of symbols for doing so. The traditional students, unlike the CPMP students, were very quick on the mints problem to use symbol manipulation.

Polya’s final stage, (4) looking back, revealed the final theme that emerged from the analysis of students problem solving activity: student’s flexibility to have different solutions. The CPMP students looked for more than one solution path and they stayed engaged in the process beyond a numeric answer. The fact that a numeric answer did not signify "done" to the CPMP students seemed important to the researcher. The CPMP students seemed to value finding a variety of solution paths, whether they lead to a numeric answer or not. Yet, the numeric answer was viewed to be

11

valuable as well. In particular, the numeric answer was viewed as a method of checking the various solution paths.

So, the researcher was actually amazed to find that after the CPMP students had written down a solution strategy, the vast majority of them asked each other if there was another way to solve the problem, and then continued to work on the problem. Although some wrote down more than one solution process, most did not. Although the alternative solution paths did not show up in writing, most CPMP students talked about alternative solution paths. There was no evidence any of the traditional students searched for more than one solution. This process of finding a second solution path on the part of the CPMP students seemed to serve two purposes. It appeared to be a check on the numeric answer, but it also appeared to be an aid for helping each other if one of the pair did not understand the previous process. In this manner, the numeric answer was important, but not the end goal. CPMP students repeatedly asked each other if they were in agreement on a process.

An example returns us to the students who thought that there were 89 two-digit numbers. This pair actually decided to count the digits by starting with 1 and ending with 124. (Many traditional students counted the digits, as well, but this was their only solution process.) These students then discovered that their first answer did not match their second answer.

The students then tried to go back over their solution to find out where there might be a mistake.

Finally, when the CPMP students stated that they were done, every pair asked the researcher to discuss the problems with them. None of the traditional students wished to see the solution processes for either question even when the researcher offered to work the problems. The CPMP students seemed to view part of the problem-solving process as explaining to each other and the researcher what they were thinking.

Thus, the problem continued even when the CPMP students had exhausted their solution paths. One pair of CPMP students asked the researcher if she knew whether class time would be spent on the problems on the following day. The student stated that she wanted to show the class her solution and ask class members what their solutions were. When the researcher told her that actually she really did not know if more class time would be spent on these problems, the student ran to the chalkboard and began to act out what she would tell the class if she was given the time. Her partner watched patiently throughout this process, and even put in comments at appropriate places. Then the

12

partner turned to the researcher and asked, "Should I go to the chalkboard, too?" This enthusiasm for the problem-solving process was present in all the CPMP students to a more or lesser degree, and appeared to be virtually absent in the traditional students.

While the paper-and-pencil tests show CPMP students and traditional students being equivalent in their problem-solving abilities, the performance assessment paints a picture of CPMP students excelling at problem-solving characteristics and traditional students lagging considerably behind. Obviously, the researcher is not suggesting that the results (especially the performance assessment results, which were with 100 students and 4 classrooms, while the paper-andpencil results had considerably more students and from 10 schools) generalize to all CPMP students and all traditional students. However, the reader is reminded that the goal of this study is really not to compare these two types of students, but rather to comment on what aspects of studentsâ&#x20AC;&#x2122; â&#x20AC;&#x153;problem solving abilityâ&#x20AC;? may not be seen in testing. Clearly, the performance test revealed aspects of problem solving that were present in at least this sample of CPMP students that the other tests did not reveal with the CPMP students. None of the tests revealed these aspects of problem solving in the traditional students. It could be that the traditional students lacked these aspects of problem solving. Or, it is possible that the traditional students had these aspects of problem solving, but were unable to demonstrate them under any of these testing situations.

This last case, however, seems unlikely to the researcher.

This study suggests that it is time to examine how we are testing. If indeed we test what we value, do we value the problem-solving skills that are tested by the paper-and-pencil tests in this study, for example? Or do we value the problem-solving skills that became apparent in the performance test? Do we value engaging in the problem-solving process beyond a numeric solution? Do we value solving a problem more than one way? These abilities may indeed be present while taking paper-and-pencil tests, but they are difficult to measure. Of course, it is possible to give a paper-and-pencil test with constructed-response items, and tell students to find more than one solution process. It is, however, an interesting observation that without being told to find more than one solution process, CPMP students were inclined to do so. An implication to this study, then, is to call for an adjustment to how researchers and even classroom teachers evaluate NCTM-oriented programs.

Standardized testing is not the best method when

Testing Problem-Solving Skills

seeking to document types of studentsâ&#x20AC;&#x2122; mathematical features (e.g., engagement) seen in this study.

However, the implications can take researchers in the other direction, as well. Rather than a call to stop using standardized tests, perhaps this study calls for a revision of standardized tests. As stated in an earlier section, there have been numerous studies on the CPMP curriculum. Many of these use standardized testing materials, perhaps because this area concerns the public. Recall the mention of parental concerns in the introduction to this paper. However, it might be that it is time for testing and measurement experts to create alternatives to the standardized tests, so that researchers could attempt to evaluate CPMP and other NCTM-oriented curricula from a new perspective. Of course, researchers can write their own testing instruments, but there are advantages (including validity, reliability, and the standardization of the testing itself) to using a standardized test. An implication to this study, then, is to call for a massive revision of standardized tests.

Although it may be an obvious point, it probably should not go without saying that there is no question the CPMP students had more experience with aspects of problem solving than the traditional students. The CPMP students were more used to working together, writing about mathematics, discussing mathematics, not necessarily valuing a numeric answer, and looking for more than one solution. The CPMP students had had more opportunity to work with a variety of problems. If the traditional students had these experiences, would they too have these skills? This may be true, although it might be argued that if traditional students had these experiences they would not be referred to as traditional students. The traditional students were quick to think of algebraic techniques, and indeed the traditional students had more experience with algebra. Perhaps this returns us to a previous point: what it is that we value in problem solving? This is not to suggest that we have an either/or situation, as there is nothing to prevent us from including more traditional in reform, or including more reform processes in traditional.

Summary The problem-solving ability measured on the standardized tests did not show differences between CPMP and traditional schools. In spite of the absence of statistically significant differences, there were qualitative differences between CPMP students and traditional students observed in the performance assessment. For example, CPMP students appeared to

Carmen M. Latterell

be more engaged in the problem-solving process. They looked for alternative solutions processes; they worked together as a group; and they showed more steps on the written work.

With this rather large sample, the CPMP students were not at a disadvantage on standardized problemsolving tests, in the sense that they scored as well as the traditional students. However, the alternative assessment did reveal some aspects of the CPMP students that the standardized tests did not. Regardless of oneâ&#x20AC;&#x2122;s interest in the qualities (such as studentsâ&#x20AC;&#x2122; engagement, flexibility etc. discussed in emergent themes) that the standardized test did not measure, it appears to be the case that one can succeed on a standardized test while being in the CPMP curriculum.

And this study leaves us with the implication that perhaps it is time to reconceptualize standardized testing, and what, as researchers and classroom teachers, we really want to test.

REFERENCES Curcio, F. R. (1999). Dispelling myths about reform in school mathematics. Mathematics Teaching in the Middle School, 4, 282-284.

Feldt, L. S., Forsyth, R. A., Ansley, T. N., & Alnot, S. D. (1993).

Iowa Tests of Educational Development (Forms K & L).

Chicago: The Riverside Publishing Company.

Gipps, C. (1999). Socio-cultural aspects of assessment. In A. IranNejad, & P. D. Pearson (Eds.), Review of Research in Education, (pp. 355-392). Washington, DC: The American Educational Research Association.

Hancock, L., & Kilpatrick, J. (1993). Effects of mandated testing on instruction. In Mathematical Sciences Education Board, Measuring what counts: A conceptual guide for mathematics assessment (pp. 149-174). Washington, DC: National Academy Press.

Herr, T., & Johnson, K. (1994). Problem solving strategies: Crossing the river with dogs and other mathematical adventures. Emeryville, CA: Key Curriculum Press.

Hirsch, C. R., & Coxford, A. F. (1997). Mathematics for all: Perspectives and promising practices. School Science and Mathematics, 97, 232-241.

Hirsch, C. R., Coxford, A. F., Fey, J. T, & Schoen, H. L. (1995).

Teaching sensible mathematics in sense-making ways with the CPMP. Mathematics Teacher, 88, 694-700.

Huntely, M.A., Rasmussen, C. L., Villarubi, R. S., Sangtong, J., & Fey, J T. (2001). Effects of standards-based mathematics education: A study of the Core-Plus Mathematics Project algebra and functions strand. Journal for Research in Mathematics Education, 31, 328-361.

Kahan, J. A. (1999). Relationships among mathematical proof, high-school students and reform curriculum. Unpublished doctoral dissertation. University of Maryland.

Kilpatrick, J. (2003). What works? In S. Senk, & D. Thompson (Eds.), Standards-based School Mathematics Curricula (pp.

471-488). Mahwah, NJ: Erlbaum.

13

National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author.

National Research Council (1989). Everybody counts: A report to the nation on the future of mathematics education.

Washington, DC: National Academy Press.

Polya, G. (1945/1973). How to solve it. Princeton, NJ: Princeton University Press. (Original work published 1945).

Schoen, H. L., Cebulla, K. J., & Winsor, M. S. (2001). Preparation of students in a standards-oriented mathematics curriculum for college entrance tests, placements tests, and beginning mathematics courses. Paper presented at the Annual Meeting of the American Educational Research Association. Seattle, WA. Schoen, H. L., Fey, J. T., Hirsch, C. R., & Coxford, A. F. (1999).

Issues and options in the math wars. Phi Delta Kappan, 80(6), 444-453 Schoen, H. L., & Hirsch, C. R. (2003). Responding to calls for change in high school mathematics: Implications for collegiate mathematics. American Mathematical Monthly, 110, 109-123.

Schoen, H. L., & Ziebarth, S. W. (1998a). Assessments of studentsâ&#x20AC;&#x2122; mathematical performance. A Core-Plus Mathematics Project field test progress report. Unpublished manuscript, University of Iowa.

recent developments. In P. S. Hlebowitsh & W. G. Wraga (Eds.), Annual review of research for school leaders (pp. 141191). New York: Macmillan.

Senk, S., & Thompson, D. (2003a). Standards-based school mathematics curricula. Mahwah, NJ: Erlbaum.

Senk, S., & Thompson, D. (2003b). School mathematics curricula: Recommendations and issues. In S. Senk & D. Thompson (Eds.), Standards-based School Mathematics Curricula (pp. 327). Mahwah, NJ: Erlbaum.

Swafford, J. (2003). Reaction to the high school curriculum projects research. In S. Senk & D. Thompson (Eds.), Standards-based School Mathematics Curricula (pp. 457468). Mahwah, NJ: Erlbaum.

1 The use of school as the unit of analysis resulted in an enormous loss of power. The question may be asked as to whether the lack of significance observed here is due to this lack of power.

Two factors suggest that this is probably not the case. First, the values of the means analyzed here are very similar across groups.

Second, the analysis using the student as the unit of analysis showed a very similar pattern of nonsignificant results.

Schoen, H. L., & Ziebarth, S. W. (1998b). High school mathematics curriculum reform: Rationale, research, and

14

Testing Problem-Solving Skills

The Mathematics Educator 2003, Vol. 13, No. 1, 15â&#x20AC;&#x201C;21

Assessment Insights from the Classroom Norene Vail Lowery Reform efforts in mathematics education challenge teachers to assess traditional forms of assessment and to explore and implement alternative forms of assessment. Empowering all students with mathematical literacy demands methods of assessment that reflect and enhance the present state of knowledge about learning, about teaching, about mathematics, and about assessment. This discussion highlights insightful perspectives on assessment strategies and techniques currently being addressed and implemented. A cohort of middle school mathematics teachers reveal their experiences and reflections in addressing current assessment practices and ventures in innovative and alternative approaches to assessment.

The activity which is the subject of this publication was produced under a grant from the Texas Higher Education Coordinating Board and the U.S. Education Department under the auspices of the Eisenhower Grants Program (ID# 97053UH), titled the University of Houston/Southeast District (HISD) Mathematics Project for Teachers (Grades 6-8).

Assessment is the central aspect of classroom practice that links curriculum, teaching, and learning.

(NCTM, 1995). In the Principles and Standards for School Mathematics (NCTM, 2000) assessment is designated as one of the six underlying principles of mathematics education. The Assessment Principle states: "Assessment should support the learning of important mathematics and furnish useful information to both teachers and students" (NCTM, 2000, p. 22).

The emerging theme in assessment reform is to do more assessment than evaluation; to become assessors rather than evaluators.1 The aim is better assessment, not more. Standards were created to provide guidelines to improve mathematics education and to value the importance of alternative 2, as well as authentic3 assessment procedures and protocols.

Traditional forms of assessment have been utilized in mathematics classrooms for many years. However, reform efforts in mathematics education challenge teachers to reconsider traditional forms of assessment and to explore and implement alternative approaches.

Assessment of school mathematics is addressed in some manner in all of the NCTM documents (1989, 1991, 1995, 2000). It is essential that mathematics teachers be informed and proactive in addressing issues of assessment in mathematics classrooms. In response to the call for changes, a cohort of middle school Norene Vail Lowery, Ph.D., is an Assistant Professor of Mathematics Education in the Curriculum & Instruction Department of the College of Education at the University of Houston, Houston, Texas. Research interests include elementary and middle school mathematics education, preservice and inservice teacher education, assessment, and the integration of literature and mathematics. Her email address is nlowery@uh.edu.

Norene Vail Lowery

mathematics teachers in a large metropolitan area in Texas reflected on their current assessment practices and ventures in alternative forms of assessment in the classroom. These teachers were participants in a grant focusing on the strengthening of mathematical content knowledge, the improvement of instructional strategies, and the implementation of new curricula fulfilling national standards and state-mandated guidelines. In the light of education reform along with the looming accountability of state-mandated guidelines, these teachers began to realize the vision of achieving mathematical power for all students. This discussion highlights middle school mathematics teacher's new perspectives as they implemented alternative assessment strategies and techniques.

Research Methodology I have chosen to present my experiences with these teachers as a case study, as it is better suited for this context-specific inquiry (Lincoln & Guba, 1985). Data collection from the grant's workshop participants spanned an academic year. These participants were fifteen middle school mathematics teachers from five different middle schools in the same urban school district. Grant workshops were held on an urban university campus with site visits and observations conducted at individual teacher campuses throughout the year. The workshops were conducted as a graduate level course with each teacher receiving three semester hours of credit upon completion.

A hands-on/minds-on, standards-based (NCTM, 2000) approach to learning and teaching (model, observe, discuss) was implemented in the workshops.

The goals of the course were to explore mathematical representations through content, instructional

15

strategies, and authentic assessment. In order to initiate change in assessment strategies, workshop instructors focused the middle school teachers on three objectives while examining the following questions:

below were validated through multiple sources through the data collection and analysis that occurred throughout the year-long workshops.

1. Set goals for student learning: What are the important learner goals? What types of problems are students able to solve? What concepts and principles should students be able to apply?

2. Build an instructional program that reflects NCTMâ&#x20AC;&#x2122;s Principles and Standards (2000) and appropriate state-mandated objectives, include alternative and multiple assessments, and include a system for documentation and reporting: What strategies best assess student understanding and achievement?

3. Continually review and share classroom assessment and its effects: What collaborative supportive system in the mathematics learning community and within their school administrative personnel must be developed to change, alter, and improve the assessment in the classroom?

Assessment is sometimes viewed as simply a numerical value, a scale of student achievement.

Values of grades may vary from state to state, from district to district, and, yes, even from classroom to classroom. What measures of learning are really represented in the assigned and recorded values?

Reform efforts such as the Principles and Standards for School Mathematics (NCTM, 2000) provide needed guidance and direction in changing the face of classroom assessment.

Implementing new ways of doing assessment is not an easy task, but no longer can mathematics teachers afford to rely strictly on traditional formats. Alternative forms of assessment offer more opportunities to reveal a student's perceptions and conceptions of mathematical knowledge. Most forms of alternative assessment ask students to perform, create, produce, or do something; tap high-level thinking; and involve problem-solving skills (see Table 1). These forms use tasks that represent meaningful instructional activities, involve real-world application, are scored qualitatively, and require new instructional assessment roles of teachers (Herman, Aschbacher, & Winters, 1992).

As the workshop instructor modeled new assessment strategies during monthly workshops, teachers had opportunities to explore the assessments as learner and as teacher. Group activities involved planning new classroom assessment and practicing with peers before implementation in their own classrooms. In these sessions, teachers reported that change could be successfully implemented as they appreciated the collaborative efforts involved in their experiences.

The use of models and manipulatives in teaching, learning, and assessment encourages alternative forms of assessment. Teachers in this program learned that alternative assessment encourages the use of active hands-on learning. Learning experiences such as these create classroom activities and learning environments that are accessible to all students. The teachers in this project investigated and implemented innovative approaches to alternative assessment in their own classrooms.

Participants engaged in tasks such as developing mathematics, teaching mathematics, and designing and implementing instructional materials as well as alternative forms of assessment. As the workshop instructor and researcher, the author developed multiple perspectives through participant observations, random interviews, journal entries, reflections, and course artifacts. Required products from the workshops were used as data sources, including written assignments, anonymous session evaluations, assessment projects and surveys, classroom student examples, and a presentation project as a final assessment. Various workshop tasks included: evaluation, synthesis, and implementation of teaching strategies, learning strategies, and national standards; participation in inquiry and discovery activities (as learner and as teacher); and implementation of alternative forms of assessment in participantsâ&#x20AC;&#x2122; classrooms.

Here, as instructor and lead researcher of the grant program, I report findings of the research agenda as a narrative to weave together the responses from the cohort of teachers in a collective manner. This narrative draws most directly on data taken from the teacher responses in individual and small group discussions on assessment topics. The trends reported

16

The Workshop Design

Assessment Insights

Table 1 Workshop Participants' Identification Of Alternative Forms Of Assessment Product / Process Journal writing / writing prompts Projects Performance assessments / use of manipulatives

Purpose to assess development of mathematical concepts; writing activities show more of how the students are thinking to develop and apply concept/scoring rubrics to demonstrate concept attainment through the use and mastery of manipulatives

to provide insight into students’ mathematical thinking Problem solving to motivate interest; to promote critical thinking Diagnostic activities to determine student readiness for learning a particular concept Class discussion to assess learning informally Student conferences / to assess student’s ability to relate subject to conversations areas outside classroom Classroom challenges to motivate interest and assess learning Integration with other to use projects to integrate other subjects and subjects involve a variety of math concepts Rubrics to customize assessment to individual needs of tasks Questioning to understand depth of students’ understanding combined with through questioning during activities instruction Cooperative learning groups

Workshop Participant Insights Journals are quite informative, but very time consuming. … [Yet] more informative than just a number on a piece of paper. Another method, which I have not used, is self-evaluation.

Turning in performance assessments to mathematics department (in our school district, this is a regular procedure). [This is done] so they can analyze concepts being taught and relate them to curriculum standards.

Manipulatives, exploration through questions, make them work on becoming independent thinkers, bring outside experiences into the classroom. Show the application or necessity. Try to assure students they can do the work if they just try. Not looking for 100% accuracy, but risk taking. Talking to and challenging students, competitions – mini games.

Concept mapping

Portfolios Warm-ups Homework

Tests: Term – Cumulative Short answer Standardized & state-mandated Practice Quizzes – pre & post

Notebooks

Norene Vail Lowery

to represent learning through samples of student work to assess problem solving strategies; to get students into a mathematical framework to assess transfer of learning; to demonstrate application; to look for understanding and comprehension of objectives

to compare for growth from previous tests to develop higher order thinking skills to assess mastery/transfer of concept and skills; to track yearly growth as a diagnostic to diagnose areas that cause difficulty

… allows the student and teacher to see the objectives mastered and objectives not mastered, particularly TAAS objectives.

I believe a rubric is most informative, but as a new teacher, I find it very difficult to manage that in a classroom of students vying for attention.

Cooperative grouping/learning encourage students to help each other, games for competition, discovery learning/exploration, problem solving, and logic thinking activities.

Cooperative groups [are being used]. When student do projects, they really enjoy working in groups and like to do their best.

Problems [that I have] encountered [include] students do not study, do not do homework, do not ask questions. These are addressed in parent conferences and student-teacher conferences. [However], some kids realize objectives, concepts or skills they need to work on. For “lower” level kids, it [TAAS] may serve as a motivator, but for especially bright students, TAAS tends to limit what a teacher teaches. It tends to bring the high lower, and may frustrate the low into apathy.

We displayed TAAS benchmark results visually [bulletin board displays] so students could see where they stand related to other students.

[I] try to align TAAS objectives with the TEKS. Post results of TAAS benchmarks…the students love to compete with each other. Since I work in a low-performing school, TAAS gets emphasized over preparing the students for algebra, something which is hard to resolve. Sometimes students don’t see any correlation between what they are learning in class and what is tested. I think in a way it has discouraged studying.

to assess student’s study habits, completion of classwork, and concept understanding

17

The Results: Insights from Teachers The series of workshops was conducted over the academic year and provided teachers opportunities to implement, observe, and revise many assessment strategies. The workshop design created assessment strategy experiences for the teachers in planning, practicing with peers, implementing in their own classrooms, and collaborating through thoughtful mathematical discourse â&#x20AC;&#x201C; both positive and negative experiences were shared. As the instructor, I worked to create a community of mathematics learners and leaders by emphasizing these communication opportunities. This interactive professional dialogue was created and supported by access to peers and the instructor during monthly grant sessions as well as via a website between sessions. At the end of the grant program, each teacher responded individually and in small groups to a variety of assessment topic issues.

These teacherâ&#x20AC;&#x2122;s responses are presented in this section in italics, enclosed and in boxes. These reflections-inthe-moment are direct citations from the teachers' responses.

What Assessment Tasks were Explored Implemented by the Middle School Teachers?

and

Teachers used a variety of traditional and nontraditional approaches to student assessment. The teacher-developed chart (Table 1) represents the span of assessment strategies explored and currently in practice in the classrooms of these middle school teachers. The third column indicates the teacherâ&#x20AC;&#x2122;s reflections on the uses of these assessment strategies.

Teachers found that short answer tests, journal writing, manipulatives, projects, concept mapping, and performance assessments revealed a broad range of capability, understanding, and communication of mathematical concepts. Many different tasks were used to create a complete picture of the students' mathematical knowledge. Strategies for evaluating performance on assessment activities also varied.

Teachers used rubrics quite extensively, as they became comfortable with this system through the workshops. In addition, concept maps, journal entries, textbook assignments, and worksheets were very informative. Teachers identified sources of feedback such as group grades, participation grades, praise, peer evaluation, and self-evaluation.

18

What Mathematics was Assessed? How Did State and National Guidelines and Accountability Affect Assessment Strategies?

The mathematical skills and concepts assessed by the middle school teachers in the workshop were typical for grades 6-8. As with many other states, Texas has state-mandated curricula objectives, as do many districts. Texas guidelines are called the Texas Essential Knowledge and Skills (TEKS). For each grade level and each subject area, there are specific learning objectives and goals for Pre-K through 12th grade. These curricula guidelines are correlated to the statewide student test, the Texas Assessment of Academic Skills (TAAS).4 The TAAS test is taken based on grade level and subject matter. The final TAAS is an exit test that must be passed as a prerequisite for high school graduation. Within this framework of curricula are thirteen TAAS objectives that are assessed in mathematics. These have been determined by the state, but are also related to the national standards identified by the NCTM (cf. 2000).

These objectives were created to help ensure quality and consistency. Learning accountability, in some school districts, is even more defined by specific objectives and goals for the grade levels.

[I use a] mastery tracking sheet, standardized tests, TAAS, computer programs, independent practice manipulatives, projects, and worksheets. We also review and practice test-taking strategies. I feel this has helped students become a little more confident because they at least know what to expect.

Direct test preparation for the TAAS is widespread. Many teachers used the item analysis from the previous year's TAAS test to determine the areas of strengths and weaknesses to improve on the objectives that were deficient. Practice tests, six-week tests, quizzes, and a section of the student's daily homework are formatted so students can practice on how the questions are structured as well as practicing and applying the objectives. Teachers and students review and practice test-taking strategies to develop more confidence. Many of the workshop teachers felt that too much focus was placed on the standardized test, thus limiting the time available for alternative assessments. Even so, teachers valued the need for change and explored the potential of other forms of assessment. Tutoring, motivation techniques, and parental involvement were common efforts.

Assessment Insights

How Can Information from Alternative Assessment be Integrated into Grading and Reporting Progress?

Weekly reports to parents; scheduled progress reports; promoting ways parents can help students at home; tutorials after school; TAAS data used to group students by abilities work with parents on skills students need; math make-and-take sessions [as a] parent workshop; parent conferences; phone conferences/conversations; parent involvement in schools; display example work for the school; student/teacher conferences; Saturday school; students are able to track themselves by objective using TAAS data; award certificates; honor pictures taken and put on the wall; and, TAAS classes. â&#x20AC;Ś kids grade in groups. [I use] class participation grades.

The teachers shared strategies to integrate information from alternative assessment into grading policy. Alternative assessments were sometimes counted as a test grade and sometimes as a daily grade, depending on how much time was required. For example, some teachers used notebooks as test grades.

It was common for teachers to offer extra credit opportunities when implementing new forms of assessment. Extra points were given for creativity and originality, hoping to build student confidence. Most of these teachers used homework to determine the depth of student understanding and which concepts needed re-teaching. Projects and journals offered students opportunities to express their ideas, understanding, and concerns. Some students worked better with manipulatives; others with pen and paper.

The teachers reported a creative variety of alternative forms of assessment implemented into traditional protocol. Each type of assessment determined a certain percentage of the grade.

Discussion of the variety of assessment practices and grade recording encouraged all teachers to try more alternative forms of assessment as well as developed increase confidence in this endeavor.

The teachers communicated the types and importance of assessment strategies and approaches to students and parents through many venues. Some of the ways used by the teachers include weekly reports to parents, scheduled progress reports, promoting ways parents can help students at home, tutorials after school, and Saturday school. As a result, parents and teachers participated in workshops, conferences, and conversations to encourage and support student learning. Positive reinforcements included special privileges at school and at home, award certificates, and other classroom and school acknowledgements.

Norene Vail Lowery

Teachers reported that sometimes students do not see any correlation between what they are learning in class and what is tested. Teachers tried to address these issues by using real-world problems and scenarios.

Typical problems encountered involved students that do not study or complete homework, or that do not ask questions. These were addressed in parent conferences and student-teacher conferences. Through these many approaches, students were able to ask questions about concepts they had not mastered.

What Results Did the Teachers See as they Used Assessment to Improve Curriculum And Instructional Practices?

I use concept maps, journal entries, textbook assignments, and worksheets. These methods are very informative [One method I use is to have the] whole class solve their problems, [but] only take one solution from [the entire] class on chalkboard. This encourages total class collaboration, a step beyond small group work.

I try to celebrate different learning styles. Let students explain to me what was just taught, if they are having difficulties this means that I have to use another strategy.

Assessment does alter instruction.

The teachers studied and shared strategies to improve mathematics curriculum and instructional practices. They found that different assessment instruments helped to take the focus off the "computation and accuracy" aspect of mathematics, and helped to encourage mathematical thinking. New sorts of tasks in classrooms created a more complete picture of the students' mathematical knowledge. The workshop teachers reported that assessment informed re-teaching, addressed students with math anxiety, and identified students' need for more instruction and/or reinforcement. Students were able to see the objectives mastered and not mastered, as well as their own strengths and weaknesses. Alternative assessment took the emphasis away from right/wrong answers and concentrated students and teachers on thought processes.

What Assessment Encouraged Mathematics Learning?

While addressing curricula objectives, the teachers made high priority of planning relevant activities that connect mathematics with the real world and creating a rich learning environment. The teachers tried innovative approaches and teaching strategies to address the mathematical content in a hands-on, mindson manner. Teachers used a variety of assessment approaches in a traditional and non-traditional manner

19

for student assessment. Different learning styles were more easily addressed by alternative assessment.

These teachers developed and implemented some effective approaches to alternative assessment that fostered student learning and helped to address motivation concerns. Some strategies that encouraged students to learn math were: doing extra credit assignments, using peer tutoring, valuing classroom discourse, and finding ways to justify their answer.

Teachers used manipulatives and exploration through questioning to assist students in developing as independent thinkers. Showing the application and necessity of mathematics while bringing in real-world scenarios was also an effective and valuable strategy.

Teachers reassured students that they can do the work and encouraged risk taking. Students developed selfconfidence as they were asked to provide their opinion on problems in classroom discourse and in writing.

This created a safe learning environment more conducive to learning. Motivation appeared to be the ultimate goal for ensuring student encouragement and interests. Challenging students with competitions and games was a good motivator for the middle school student. Teachers also reported that cooperative grouping encouraged students in problem solving and logic while they learned to help each other. Teachers encouraged students to justify why they did what they did, focusing on the thinking processes rather than just the answer.

Teachers confronted their own perspective of the nature of mathematics by participating in learning activities that encouraged deep reflection and discourse. Davis, Maher, & Noddings (1990) believe that this perspective has a direct bearing on the ways reform can be approached. Unveiling or developing one's own conception of the nature of mathematics was an enlightening experience that promoted a deeper understanding of reflective teaching and learning mathematics. Teachers developed a better conceptual understanding as they explored mathematics topics as learners and teachers to better inform instruction and assessment. Teachers examined and explored reasons for evaluating and assessing student achievement.

Being aware that teachers evaluate and assess in order to enable decision-making about mathematics instruction and classroom climate was a critical aspect of these teachers' learning. The protocols presented above communicate important tensions for the middle school mathematics teachers among testing expectations, assessment of student understanding, and the need to assign grades.

Appreciating the need for reform was another area of study for the teachers. For the teachers, this meant acknowledging that current testing procedures are inadequate and realizing the need for further research.

Through the workshop experiences and the teachers' own personal classroom action research, teachers discovered why there is a need for reform in assessment. It was apparent that using multiple assessment strategies was a significant step toward creating a more complete picture of the student's mathematical understanding and achievement. New evaluation models and technologies that utilize assessment procedures that reflect the changes in school mathematics are needed. Ultimately, the middle school teachers demonstrated a belief that classrooms should be active learning environments where instruction is interactive and multiple forms of assessment are interwoven with teaching.

Conclusion

Looking Forward

The teachersâ&#x20AC;&#x2122; learning experiences focused on developing and promoting better classroom assessment. Initially, the teachers explored the recent trends in changes from behavioral to cognitive views of learning and assessment, as well as changes to authentic, multi-dimensional, and collaborative assessment. Teachers learned about the constructivist perspective of teaching and learning school mathematics that is predominant in the NCTM Standards documents (NCTM, 1989, 1991, 1995, 2000).

Multiple forms of assessment are being advocated as we come to understand that traditional means of assessment have not addressed the needs of all learners. Richard Stiggins estimates that educators spend about a third of their time involved in assessment-related activities that guide the instructional and classroom decisions which directly affect learning (1993). A time investment such as this demands that teachers examine their current assessment practices. Simply testing student

[Using] assessments that make them have a feeling of selffulfillment, to develop confidence... [have] students write their opinion on a problem, ... [and] let students show different ways of solving a given problem, let them justify why they did it.

It is a slow process to get our students to do in-depth work. I have not quite figured out the right formula to motivate them. Motivation is a real challenge for me.

Students don't seem to have the confidence to try. I'm working at it...

20

Assessment Insights

achievement with traditional instruments and protocols is insufficient. Empowering all students with mathematical literacy demands methods of assessment that reflect and enhance the present state of knowledge about learning, teaching, mathematics, and assessment.

Implementing improved assessment in the mathematics classroom begins with combining instruction with assessment to better meet the needs of the learner.

In order to plan and implement new strategies for assessment, mathematics teachers should have opportunities for professional development, as did these middle school teachers. It is crucial that a support system in the mathematics learning community be developed along with any efforts to change, alter, and improve assessment in the classroom. Mathematics teachers must personally explore alternative assessment strategies. They should be involved in creating and implementing tasks that are exemplars of mathematics instruction as envisioned by the NCTM. As part of this effort to develop tasks, teachers should have opportunities to observe students doing mathematics and to examine the their products. A solid basis for mathematics teaching, learning, and assessment is created when teachers value and comprehend recent trends, perspectives towards mathematics teaching and learning, evaluation and assessment, and the need for reform. The informed mathematics teacher has the ability and the tools to offer the best learning environment for improving student achievement and understanding.

In this paper, I have attempted to present a multiperspective approach toward understanding and implementing assessment reform. The middle school mathematics teachers encountered many problems on this journey from traditional classroom assessment to implementing alternative assessment strategies. Some problems were unique, but many were common among all teachers. Some problems were collectively resolved, while others, such as student motivation, remain as ongoing obstacles to address. These teachers learned about assessment and implementing innovative strategies in a collaborative environment. As a result, the need for a strong support system to implement change was revealed and valued. The experiences and insights of these teachers may promote and encourage other middle school mathematics teachers to move outside the comfort zone of traditional assessment protocols and begin implementing innovative and alternative approaches to assessment.

Norene Vail Lowery

REFERENCES Davis, R., Maher, C., & Noddings, N. (1990). Introduction. In R. Davis, C. Maher, & N. Noddings (Eds.), Constructivist views of the teaching and learning of mathematics. Journal for Research in Mathematics Education Monograph no. 4 (1-3).

Reston, VA: National Council of Teachers of Mathematics.

Herman, J., Aschbacher, P., & Winters, L. (1992). A practical guide to alternative assessment. Alexandria, VA: Association for Supervision and Curriculum Development.

Lincoln, Y., & Guba, E. (1985). Naturalistic inquiry. Newbury Park, CA: Sage.

National Council of Teachers of Mathematics (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: Author.

National Council of Teachers of Mathematics (1991). Professional standards for teachers of mathematics. Reston, VA: Author.

National Council of Teachers of Mathematics (1995). Assessment standards for school mathematics. Reston, VA: Author.

National Council of Teachers of Mathematics (2000). Principles and standards for school mathematics. Reston, VA: Author.

Stiggins, R. (1993). Teacher training in assessment: Overcoming the neglect. In S. Wise (Ed.), Teacher training in assessment and measurement skills (27-40). [Buros-Nebraska Series on Measurement and Testing]. Lincoln, NE: Buros Institute.

1 Drawing upon NCTM's Assessment Standards for School Mathematics (1995), I make the following distinction between assessment and evaluation. Evaluation is the process of determining the worth of, or assigning a value to, something based on careful examination and judgment. Assessment (as a noun) is used to emphasize understanding and description of both qualitative and quantitative evidence in making judgments and decisions.

Assessment (as a verb) is the process of gathering evidence about a student’s knowledge of, ability to use, and dispassion toward mathematics and making inferences from that evidence for a variety of purposes. Evaluation is one use of assessment information.

2 “Alternative”, as well as “traditional”, forms of assessment may be less clearly demarked. There are common characteristics in alternative assessment, most ask students to perform, create, produce, or do something; tap higher-level thinking and problemsolving skills; use tasks that represent meaningful instructional activities; involve real-world application; are scored qualitatively; and, require new instructional and assessment roles of teachers.

Traditional assessment efforts seem skill or process-oriented, such as common practices of end-of-the-unit testing. These efforts present a clear line of distinction between instruction and assessment.

3 Authentic assessment is a type of alternative assessment, emphasizing practices that are relevant, real-world and focused on meaningful learning.

4 Beginning in 2003, this test has been renamed the Texas Assessment of Knowledge and Skills (TAKS).

21

The Mathematics Educator 2003, Vol. 13, No. 1, 22–32

Designing and Implementing Meaningful Field-Based Experiences for Mathematics Methods Courses: A Framework and Program Description Amy Roth McDuffie, Valarie L. Akerson, & Judith A. Morrison Performance-based approaches to learning and assessment are consistent with goals for standards-based instruction and show promise as a vehicle for teacher change. Performance assessment involves students participating in an extended, worthwhile mathematical task while teachers facilitate and assess their learning.

We designed and implemented a project in an elementary mathematics methods course in which preservice teachers developed performance assessment tasks and then administered these tasks in K-8 classrooms. We present our guiding framework for this project, the project design, and the teaching and learning experiences for project leaders and preservice teachers. Recommendations and reflections are included for others intending to implement similar projects.

This article is based on a paper presented at the Association of Mathematics Teacher Educators Annual Conference, Costa Mesa, CA, January 18 – 20, 2001.

The current reform movement in mathematics education is based on national and state standards for students’ learning (e.g., National Council of Teachers of Mathematics [NCTM], 1989, 2000) and on the perspective that students learn by actively constructing their own knowledge and understandings. Within this context, educators and researchers have identified two different but complementary needs for preservice teacher education in mathematics. First, preservice teachers need to learn to use performance assessment strategies to effectively meet and assess standardsbased learning objectives. Second, preservice teacher learning should be situated in classroom practice to facilitate their pedagogical knowledge constructions and their enculturation into a community of practice.

Below we discuss each of these needs for preservice teacher education, and then we describe the program we have developed to meet these needs.

Call for New Forms of Assessment Following the release of NCTM’s Curriculum and Evaluation Standards (1989), many states and local Amy Roth McDuffie is an assistant professor of mathematics education at Washington State University Tri-Cities. Her research focuses on inservice and preservice professional development toward Standards-based practices. Her email address is mcduffie@tricity.wsu.edu.

Valarie Akerson is an assistant professor of science education at Indiana University. Her research focuses on elementary teacher and student conceptions of nature of science.

Judith Morrison is an assistant professor of science education at Washington State University Tri-Cities. Her research interests focus on teachers' diagnosis of students' science conceptions.

22

school districts have developed standards for students’ learning in mathematics. Included in these standards and in NCTM’s updated standards, the Principles and Standards for School Mathematics (PSSM) (NCTM, 2000), are greater emphases on the processes of doing mathematics (e.g., problem solving and reasoning) and on communicating thinking and solution strategies (NCTM, 1989, 2000).

Also included in these standards is a call for new forms of assessment. Traditional paper and pencil classroom tests and standardized multiple-choice tests focused on recall of facts and basic procedures do not effectively measure what is valued for standards-based learning (Darling-Hammond & Falk, 1997; Shepard, 2000). While traditional measurement approaches to assessment were once aligned with the instructional practices of a century past, these approaches are not consistent with current teaching and learning goals from a social constructivist perspective (Shepard, 2000). This incongruity has resulted in an emerging paradigm for assessment that involves teachers’ assessment of students’ understandings and students’ self-assessments as part of the social process of knowledge construction (Shepard, 2000). Educators and researchers argue that to align assessment with standards-based learning, the following changes are needed: (a) the form and content of assessments must represent higher order thinking, reasoning, communication, problem solving skills, as well as a conceptual understanding of subject matter; and (b) the focus of assessment policy needs to shift to using assessment for learning (Borko, Mayfield, Marion, Meaningful Field-Based Experiences

Flexer, & Cumbro, 1997; Darling-Hammond & Falk, 1997; Shepard, 2000).

Consistent with these views, in mathematics education the PSSM state that the primary purpose of assessment should be to “support the learning of important mathematics and furnish useful information to both teachers and students…. Assessment should be more than merely a test at the end of instruction to see how students perform under special conditions” (NCTM, 2000, p. 22). To achieve this goal, the Standards call for embedding assessment in instruction, rather than keeping assessment as separate from learning (NCTM, 1995, 2000). Indeed, this call is supported by research that indicates use of formative assessments, the continual assessment of learning throughout an instructional sequence, in instruction enhances student learning (Black & Wiliam, 1998).

Performance Assessment to Improve Teaching and Learning As a result of this call, attention has been directed to more authentic forms of assessment, including performance assessment (PA). Indeed, well-designed PA tasks can assess student understanding as well as teach concepts as a formative assessment (DarlingHammond & Falk, 1997; Shepard, 2000). While a single definition for PA does not exist, Stenmark’s (1991) definition for PA in mathematics education seems to capture the important aspects of this approach. Stenmark states, “A performance assessment in mathematics involves presenting students with a mathematical task, project, or investigation, then observing, interviewing, and looking at their products to assess what they actually know and can do” (1991, p. 13).

Educators and researchers argue that the advantages of classroom based performance assessment are that they provide the opportunity to: 1. Examine the process as well as the product and represent a full range of learning outcomes by assessing students’ writing, products, and behavior (Danielson, 1997; Shepard, Flexer, Hiebert, Marion, Mayfield, & Weston, 1996).

2. Situate tasks in authentic, worthwhile, and/or realworld contexts (Stenmark, 1991).

3. Preserve the complexity of content knowledge and skills (Shepard et al., 1996).

4. Assess higher-order thinking skills and deeper understandings (Firestone, Mayrowtz, & Fairman, 1998).

5. Embed assessment in instruction, rather than separating it from learning (Stenmark, 1991).

Amy Roth McDuffie, Valarie L. Akerson, & Judith A. Morrison

6. Apply criterion referenced as sessment approaches based on important learning outcomes, rather than norm-referenced (Stenmark, 1991).

Early research indicated that using performance assessment in instruction can improve student learning and teaching. Fuchs, Fuchs, Karns, Hamlett, and Katzaroff (1999) studied the effects of classroom based performance-assessment-driven instruction. They found that students in PA-driven instruction classes demonstrated stronger problem solving skills than comparison groups that were not PA-driven. Shepard et al. (1996) found that the teachers involved in their study were beginning to show substantial changes in practice. The changes included: greater use of manipulatives; increased emphasis on the teaching and learning of problem solving strategies; and increased class time and focus on written explanations in mathematics. Similarly, in Borko et al.’s (1997) study of a professional development program on using performance assessment strategies in mathematics instruction, they found that their teachers changed their instructional practices to incorporate more problem solving activities, student explanations of strategies as a central component of their programs, and scoring rubrics for assessing students’ solutions of open-ended tasks. These changes all represent a shift towards standards-based instruction. Given that these studies indicated that work with PA served as a vehicle for change for inservice teachers, we pursued a program for preservice teachers that focused on PA as a means of building their understanding of standards-based practices.

While it is possible to derive many instructional benefits from PA strategies, it is not clear that teachers can easily or quickly learn to implement these strategies in practice. Firestone, Mayrowetz, & Fairman (1998) studied teachers where state testing programs included PA tasks, and therefore teachers felt compelled to use PA in instruction, however, little change in instructional strategies resulted. Firestone, Mayrowetz, & Fairman identified two major barriers to change: a lack of the sophisticated content knowledge required in implementing PA approaches, and a lack of rich tasks and problems in curricular materials to support this approach to instruction. Firestone, Mayrowetz, & Fairman concluded that to effectively implement performance assessment and thereby realize the potential for improved student learning, teachers needed substantive training opportunities (not just new policies requiring new assessment approaches) and new curricular materials that are aligned with

23

performance assessment strategies and a standardsbased vision for teaching and learning.

In accordance with Firestone, Mayrowetz, and Fairman’s (1998) research, Borko et al. (1997) found that substantive and sustained professional development is needed for teachers to effectively use and realize the benefits for PA approaches. Their research also indicated that time was a major obstacle to implementing PA approaches. In particular, time served as a barrier in planning for the implementation of new strategies; applying more complex scoring rubrics in assessment; administering the assessment tasks; recording observations of students’ working and thinking as part of the assessment; and interviewing students before, during, and after the assessment. For successful change to occur, teachers need time to implement new assessment approaches.

Recognizing the value of PA and the complexity of using these strategies, we decided to make PA a focus of our mathematics methods course. This decision was part of our effort to prepare our preservice teachers from the beginning of their careers to use these approaches and to implement standards-based teaching and learning in their own instructional practice. While we view performance assessment as one form of alternative assessments (cf., Stenmark, 1991), it allows the opportunity for preservice teachers to implement other forms of alternative assessment (e.g., brief interviews with students and systematically observing students) while students perform a task. Additionally, the nature of performance assessment (focusing on the process and product of doing mathematics), pushes preservice teachers to think deeply about how students think about and do mathematics. Performance assessment also provides an approach for preservice teachers to use in which assessment is part of instruction, a primary focus of the PSSM. That is, tasks facilitate students’ learning of content and processes through meaningful problems while teachers assess their work and products. Moreover, as is described throughout this paper, the process of designing and implementing a performance assessment task provided us, as teacher educators, the opportunity to assess the performance of the preservice teachers; consequently the preservice teachers experienced performance assessment as students while they designed and conducted performance assessment with their students.

Situated and Constructivist Perspectives on Teacher Learning With the goal of developing preservice teachers’ abilities to implement PA in their classrooms, we

24

considered a second need identified in teacher education literature: a need to situate preservice teacher learning in classroom practice. Borko et al (1997) emphasized the importance of this approach for professional growth. They found that a key component of their program was their teachers’ ability to experiment with and implement the ideas of the professional development workshops in their own classroom practice and then to reflect on these efforts in follow-up workshops.

This finding is consistent with the perspective of teacher learning put forth by Putnam and Borko (2000). They argue that for teachers to construct new knowledge about their practice the learning needs to be situated in authentic contexts. First, learning needs to be situated in authentic activities in classrooms to support transfer to practice. For preservice teachers, a combination of university learning for theoretical foundations and school-based learning for a situated perspective is needed (Putnam & Borko, 2000).

Second, preservice and inservice teachers should participate in discourse communities as part of learning and enculturation in the profession. In particular, preservice teachers need to learn about and contribute to a community’s way of thinking (Putnam & Borko, 2000). This process of enculturation is especially important to future teachers of mathematics because many come to their education program with limited views of teaching, learning, and doing mathematics (Roth McDuffie, McGinnis, & Graeber, 2000).

Putnam and Borko (2000) recognize that implementing this perspective in teacher preparation programs can be problematic. While we want to place preservice teachers in schools to experience the activities of teaching as part of their learning, K-12 placement classrooms may not embody the kind of teaching and learning advocated in university classrooms and/or these kinds of classrooms may not be available. Moreover, the pull of traditional school culture is strong, and these traditions make it difficult for student teachers to implement different approaches and views (Putnam & Borko, 2000).

Smith (2001) discusses specific approaches for situating teachers’ learning in practice based on a synthesis of the literature. We incorporated two of the approaches she recommends: using “samples of authentic practice” (p. 9), and designing our project around “the cycle of teachers work” (p. 10). The first approach involved selecting an example of a mathematical task with a set of carefully chosen student responses. Teachers complete the task and engage in doing mathematics as learners. Next,

Meaningful Field-Based Experiences

teachers analyze the task and a range of students’ responses, examining understandings, approaches, and misconceptions in students’ thinking and work. The second approach is intended to mirror the nature and cycle of teachers’ work. This cycle begins with planning for instruction by targeting learning goals, considering students’ prior knowledge, and selecting and/or designing experiences that will promote students’ construction of knowledge and understandings. The cycle continues as teachers enact the plan, making adjustments in the plan and instructional decisions to meet students’ needs while formally and informally assessing students’ learning.

Teachers complete the cycle as they reflect on the teaching and learning experience, and use their reflections to guide future instruction. In the next section we describe how these ideas were incorporated in our program.

Program Description We first implemented our PA program in an undergraduate mathematics methods course at Washington State University Tri-Cities in Spring 2000 and have continued the program in 2001 and 2002.

This description focuses on the initial implementation.

While the program has changed slightly each year with changes in university faculty, most of the core elements have remained the same, and the revisions and adjustments made over the two years will be discussed at the end of the article. This methods course focused on mathematics teaching and learning at the K-8 level. The PA program was included as part of a one-semester mathematics methods course that met for three hours, once each week of the semester. Twenty-

two preservice teachers were enrolled in the methods course, with 18 being between the ages of 20 and 24 and the remaining 4 being second-career students. The PA program aimed to provide a learning experience with both a university component to build theoretical foundations and a field-based component to situate learning in the authentic context of the school classroom, as recommended by Putnam and Borko (2000) and Smith (2001). The primary goals for preservice teacher learning in this program were: 1. To develop skills and habits of mind for assessing and diagnosing students’ mathematical thinking, skills, understandings, and lack of understandings; 2. To understand issues of and strategies for implementing classroom-based performance assessment; 3. To have a meaningful field-based experience including an opportunity to collaborate with expert inservice teachers and work with students.

A brief timeline of the PA program is provided in Table 1, and a description of these activities is provided below.

Planning the Program A collaborative team planned the performance assessment program prior to the beginning of the semester, and continued to meet and adjust the program as needed during the semester. The planning team was composed of a mathematics educator (first author), a science educator (second author), four middle school mathematics teachers, a middle school social studies teacher (for inter-disciplinary

Table 1 Performance Assessment Program Timeline.

Week of Semester 3 3-5 5 6 7 8 9 9-12 13 14

Activity Introductory PA workshop conducted during regular class meeting (3 hours).

Preservice teachers began to research PA task topics and plan task outside of class.

Preservice teachers submitted their PA task planning guides and their journal article reviews on their selected PA task topics.

Collaborative team met to match mentors with preservice-teacher-groups.

Mentors met with their assigned preservice-teacher-groups during class to provide advice and feedback on preservice teachers’ initial plans for their PA tasks (1 hour).

Preservice teachers submitted first draft of PA tasks to their mathematics methods professor (first author) and to their mentor teachers.

Preservice teachers received written feedback from their mathematics methods professor (first author) and from their mentor teachers.

Preservice teachers revised their tasks and field-test tasks in their mentor teachers’ class.

Preservice teachers submitted their report of their PA tasks Preservice teachers submitted a follow-up lesson plan based on PA findings.

Amy Roth McDuffie, Valarie L. Akerson, & Judith A. Morrison

25

connections), and a secondary program administrator from a Washington State Educational Service District.

The middle school teachers were recognized regionally as teacher-leaders for their expertise in performance assessment strategies, and more generally, for implementing standards-based approaches to teaching and learning. The team worked together to develop the preservice teachers’ understanding of PA, match preservice teachers to mentors, and to support the preservice teachers in their PA task design and implementation. These efforts were aimed at ensuring that our program was providing for meaningful interaction between preservice teachers and inservice teachers, as called for by Putnam and Borko (2000) (Goal 3), and thereby facilitating the preservice teachers’ growth in understanding students’ thinking and learning (Goal 1) and in implementing the PA strategies (Goal 2). It should be noted that a practicum field component was not built into the semester for the preservice teachers, and thus this field experience was initiated and arranged entirely by the planning team.

Introductory Assessment Workshop This workshop was conducted during the regular methods class meeting time for a three hour period.

The collaborative team planned and facilitated the workshop with team members leading different parts of the workshop. It was conducted to address our second goal by briefly discussing general assessment issues, providing an overview of the standards-based assessment program in Washington State (e.g., see Washington Commission on Student Learning, 1998), and introducing the preservice teachers to performance assessment issues and strategies.

To introduce the preservice teachers to performance assessment, we asked them to work in groups on a sample performance assessment task that was written and field-tested as part of an assessment program in Washington State. The task required the preservice teachers to design a cereal box that would reduce the amount of cardboard needed and still maintain a specific volume, and then to write a letter to the cereal company describing and defending their design. While we only provided approximately twenty minutes for the preservice teachers to work on the task, they had enough time to identify key issues of the task and key components of task-design. Next, we discussed some of the features and purposes of the task. Consistent with our framing of the features and purposes of PA, we examined the authentic context of the tasks, the open-ended questions involved, the descriptive and persuasive writing components, the

multiple entry points and various solution methods possible in performing the task, and opportunities for assessing higher order thinking. After this discussion, we gave the groups scoring rubrics and samples of ninth grade students’ work on the task at various performance levels. Using the scoring rubrics, the groups assigned scores to their sample students’ work.

Following this group work, we discussed the scoring process, the rubrics, and the task as a class. This component was designed to attend further to our first goal regarding students’ thinking and understandings by exploring a sample of “authentic practice” (Smith, 2001, p. 9) in that the task selected was used in local classrooms and students’ work (in their own hand) on this task was analyzed for understandings and approaches.

Next, we worked to formalize their knowledge of performance assessment (Goal 2) by discussing defining characteristics of performance assessment, as well as advantages and limitations. Additionally, a middle grades language arts teacher-leader facilitated a brief discussion of types of writing used in performance assessment (e.g., descriptive, expository, and persuasive). We concluded the workshop with an introduction of the planning guide (described in detail below) and provided a few minutes for generating ideas for the preservice teachers’ PA projects.

Researching Topics and Generating a Plan for the PA Task The preservice teachers formed groups of two to three to collaborate on the PA task project. Each group chose a mathematics topic for the focus of their task.

The groups were restricted to middle school mathematics topics because all of the mentor teachers selected were teaching at the middle school level. This restriction was due to the planning team’s decision to select mentor teachers with experience in PA, and we had difficulties finding such teachers at the elementary level. Once the topic was chosen, each group member found a minimum of two journal articles discussing teaching and/or learning issues for that topic. The preservice teachers submitted a brief summary of each of their articles and an explanation on how the information in the article contributed to their thinking and plans for their PA project. The purpose of this component of the project was to lay a foundation for understanding students’ thinking and learning (Goal 2) by ensuring that the preservice teachers had some awareness of the pedagogical issues surrounding their topic as reported in mathematics education literature.

Additionally, each group used a planning guide to outline major features of their task and keep them focused on goals and purposes of performance assessment (versus other types of projects or assessments). To show a clear and mathematically important purpose for the task, the preservice teachers described the concepts and processes targeted for assessment. To demonstrate how the task would engage learners, the preservice teachers explained the task’s authentic and/or worthwhile context, the role the learner would play in performing the task (other than a student doing math for a class), and the audience for the product (other than a teacher grading a project). To ensure alignment with selected goals and define criteria for quality performances, the preservice teachers created a table showing connections among learning standards, task products and/or performances, and criteria for measuring whether learning goals had been met. Because the Washington State Essential Academic Learning Requirements (EALRs; Washington Commission on Student Learning, 1998) were emphasized in this course, our students identified appropriate EALRs for their task. However, PSSM could have been used in lieu of the EALRs. Regardless of which standards were applied, this component focused preservice teachers’ thinking on the notion that assessments need to be aligned with important instructional goals (part of Goal 2).

From this point, the groups continued developing their tasks outside of class time. While many groups created original tasks, the preservice teachers were permitted to use outside resources (e.g., activity books, journal articles, their Van de Walle (1998) textbook, etc.) in developing their tasks. We did not require that their work be entirely original because we wanted the process to mirror that of teachers’ planning (cf., Smith’s (2001) recommendations), and teachers often draw from existing resources, rather than write their own tasks. Even in the cases where a problem, activity, or task was used from an outside source, significant work was required to develop the problem into a performance assessment task and meet the assignment requirements.

Collaboration Teachers

Between

Mentors

and

Preservice

Using the information provided in the preservice teachers’ planning guides (i.e., grade level and topic targeted), we matched each preservice-teacher-group to one of four mentor teachers. Each mentor teacher was responsible for advising two groups of preservice teachers.

Amy Roth McDuffie, Valarie L. Akerson, & Judith A. Morrison

After the mentor teachers had been assigned to groups of preservice teachers, the mentors attended one hour of a methods class. The preservice teachers brought their planning guides and drafts of their PA tasks to this meeting. During this hour, the mentor teachers met with each of their groups to discuss their ideas and plans for implementing the PA tasks. We provided the mentors and preservice teachers with specific discussion prompts including individual students’ learning needs, mentor’s typical teaching practices, and classroom norms. The members of the planning team circulated to assist groups in designing their tasks and keep groups focused on objectives.

These meetings were planned to address our third goal of facilitating meaningful collaboration with teachers, and consequently, more authentically engage in the planning phase of teachers’ work as recommended by Smith (2001).

Submitting the First Draft and Field-Testing the PA Task Continuing on the theme of experiencing the cycle of teachers’ work (Smith, 2001), in the eighth week of the semester, the groups submitted their first drafts of their PA tasks to their mathematics methods professor and to their mentor teacher. This draft included a brief overview of the task, a table showing alignment between task items and the EALRs (revised and developed further from their initial plans), instructions for administering the task and a list of materials needed, the task as it would be administered to students, and rubrics for scoring the task. Within a week, both parties provided written feedback and comments for the groups to consider before administering their tasks to middle school students.

As part of our third goal of situating the project in the schools, each group arranged a time to field-test their PA tasks in their mentor’s class. The tasks were designed to be completed in two to three 50-minute class periods. Each mentor teacher decided with his or her groups who would facilitate the task. In some cases the mentor teacher was the primary facilitator and in other cases the groups facilitated the task administration. However, in all cases, the preservice teachers observed throughout the task administration, talked with students, and in some cases, interviewed students about their thinking, and recorded notes on the process.

Analyzing Results and Reporting on the PA Task Following the field-test, the preservice-teachergroups scored the students’ work and analyzed selected

27

students’ work in greater depth for the purpose of understanding students’ thinking and learning (Goal 1).

Finally, they prepared a written report of their analysis of students’ work and their reflections on the performance assessment process and project to examine the strengths and limitations of PA, as part of Goal 3.

Writing a Follow-up Lesson Plan To help preservice teachers understand the teaching and learning cycle of using assessment to inform instruction (Goal 2), the preservice teachers were required to write a lesson plan based on their findings in the performance assessment task administration. In some cases the lesson plans were on a topic closely related to their PA task topic, and in other cases the preservice teachers identified weaknesses in underlying skills and thinking through the PA, and correspondingly chose topics that were less obviously related to their PA topic. As part of the lesson plan, the preservice teachers explained how the lesson was motivated by their findings in the PA task administration.

Providing Release Time and Compensation for the Mentor Teachers Throughout the semester inservice teachers played a key role in the project. They attended two class meetings during the school day, an evening meeting, and provided written comments on the first draft of each of their two groups tasks. For this project, we were able to provide substitute teachers to release the mentor teachers from their teaching responsibilities on the days they attended the methods class. Additionally, the mentor teachers were compensated for their time during the evening meeting and for their reading of the projects. This funding was available through the Washington State Educational Service District. We believe that this support enhanced the extent to which the mentor teachers were committed to the program, and contributed to our efforts to meet Goal 3, creating a meaningful collaboration with inservice teachers.

Reflections on the PA Program We found that all of our goals were achieved in that students began to develop understanding in our areas of focus (Goals 1 and 2) through careful facilitation of field-based experiences (Goal 3), and indeed we experienced some unanticipated benefits.

However, these achievements were not gained without some significant challenges. In the process of implementing this project, we also recognized areas to preserve and to change, and have made changes in our 28

program in semesters following the initial implementation. These reflections and changes are described below.

Benefits of the Program Our first goal of developing skills in assessing and diagnosing student thinking was met in that the preservice teachers provided substantive analysis and interpretations of students’ thinking, understanding, and lack of understanding in their reports on their PA tasks and follow-up lesson plans. For example, in Karen’s (all names used are pseudonyms) final report, she reflected on her students’ work and remarked, Although [the group’s] worksheets were not… complete, … [they] added new insights to the final group discussion by introducing conjectures to the problem…they exhibited a higher level of reasoning. … They argued various points and brought up ideas that even [we] had not considered.

Their inferences and thought processes led others to question their own conclusions.

These comments demonstrate how the preservice teachers were observing and analyzing their students’ work on a deeper level than simply looking for correct answers.

In regard to our second and third goals, we believe that our preservice teachers cannot truly come to understand performance assessment, its complexities, its benefits for understanding students’ thinking and learning, and its benefits for informing teaching without experiencing the process of designing and implementing performance assessment tasks first hand.

At the end of the semester, the preservice teachers demonstrated their understandings of PA in their reports and comments. Sarah’s explanation of PA was typical of preservice teachers’ understandings when she described PA as: A task which has a real world problem to assess students’ understanding of a topic. …[It can be used] to assess what someone already knows, like at the beginning [of a unit], … or at the end to evaluate what they have learned and how your teaching has helped them to understand that concept.

While we intended for the preservice teachers to consider worthwhile or meaningful contexts, not just “real world,” it was clear that Sarah understood the primary purposes and approaches of PA. Our experience in this project and their work in designing and implementing PA tasks suggest that the preservice teachers meaningfully constructed ideas as to what constitutes performance assessment. For Meaningful Field-Based Experiences

example, one student designed a PA task entitled, “City Park” in which middle school students worked as landscape architects (the role) to design a park with playground equipment and a sprinkler system (the context). In this task the students had to construct a budget, calculate the area of their design, and satisfy various design criteria established by the city council, represent their design visually with a scale, and write a letter persuading the city council (the audience) that their design proposal should be accepted. This task exemplifies how the preservice teachers were able to incorporate key elements of performance assessment in tasks that involve several important mathematical concepts and processes.

Perhaps an even greater benefit was that the preservice teachers began to understand assessment as a formative process, rather than merely a grade in the grade book. They began to generalize the ideas from performance assessment to understand and be interested in other forms of authentic and alternative assessment such as interviewing and observational record keeping. Dora exemplified these understandings for assessing in multiple ways when she said, [This type of assessment] engages the students in real-world problems, capitalizes on their prior knowledge, requires them to think critically, and allows the teacher to assess by observation.… As I circulated throughout the room listening to students, making mental notes about what was going well and what changes need to be made, it was obvious that the students were using their prior knowledge.

Moreover, as is evident in the earlier example of the “City Park” task, designing and administering a performance assessment task also seemed to help the preservice teachers construct a more sophisticated notion of problem solving in mathematics and more fully understand what is meant by an open-ended task, consistent with Shepard et al.’s (1996) findings for inservice teachers. Focusing on our third goal specifically, the situated nature of the project (i.e., designing an open-ended task for actual students, working with an experienced teacher, and administering the task in a school classroom) seemed to be the most important factor in bringing about the preservice teachers’ interest in the project and learning from the project. Robert’s reflections represented what we heard from virtually all of the preservice teachers in their final reports and/or course evaluations. He stated, “The project was an excellent opportunity to work with an actual math class. It gave me a good picture of what the students know and how they can learn.” Thus, we Amy Roth McDuffie, Valarie L. Akerson, & Judith A. Morrison

found that following Borko et al.’s (1997), Putnam and Borko’s (2000), and Smith’s (2001) recommendations for situating tasks in actual classrooms were an important part of our program.

Additionally, we observed professional growth opportunities for the mentor teachers. All of the mentor teachers commented that they learned more about performance assessment strategies and gained ideas for their own teaching through their involvement in the project. For instance, one mentor stated, “[Working with a preservice teacher in this program] affirmed my strong belief in observable assessment for young learners. It gave me a chance to teach someone else techniques I have developed.”

Challenges of the Program Similar to the Borko et al. (1997) finding, time emerged as a primary challenge in implementing our project. Time was a challenge in the form of demands on the methods professor, mentor teachers, preservice teachers, and the mentor teachers’ class time. For the methods professor (first author), this project certainly demanded more time in planning and assessing. As a new endeavor, more time was required to plan the project, especially in regard to the time required to meet with the project team. While collaboration often produces better results for learning, it seems to take more time than working independently in teaching.

Additionally, assessing and providing feedback on the preservice teachers’ work throughout the project required more time than is typically spent assessing written work in a methods class.

As described earlier, the mentor teachers in this project were provided release time and compensation for their significant time committed to the project.

Certainly, we preferred to offer support to inservice teachers who took on this responsibility. However, this funding was not available to us after the initial implementation and we have found that the program is manageable without funding.

The preservice teachers also experienced significant time demands. While most preservice teachers commented (either orally or in course evaluations) that the project was worth the effort, they all seemed to feel that the workload for the class was heavier than other classes due to the PA project. This challenge is consistent with Borko et al.’s (1997) finding for the increased planning time required in using PA. In addition to the PA project, the preservice teachers had several additional course assignments and requirements. In semesters following the initial implementation, we reduced other assignments

29

recognizing the time this project requires and the multiple purposes it serves (i.e., we found that writing the PA assignment could serve in lieu of a lesson plan).

In addition to challenges with the magnitude of the project, some of the preservice teachers had difficulty arranging for administering their tasks in classrooms.

This mathematics methods course did not have a field experience as part of the course. As such, time to be in the schools was not allocated in their schedules.

Moreover, given that the timing had to meet the needs of the mentor teacher, scheduling was not simply a matter of finding time in the preservice teachers’ schedules. In some cases, not all group members were able to be present for each day of the task; however, everyone managed to be present for some part of it.

Our teacher education program soon will include a practicum experience as part of a methods block scheduling structure. As this practicum is instituted, we are hopeful that some of the logistical issues, particularly the scheduling problems associated with the field component will be mitigated.

Most tasks required more time than anticipated by preservice and mentor teachers, and correspondingly either the task was modified or the mentor teacher allowed the preservice teachers to use more than three days of class time. Consistent with Borko et al.’s (1997) findings, PA requires a substantial investment of class time, and it is not easy to predict how long the students will need to complete their work.

In addition to time demands, we faced a challenge identified by other researchers (Putnam & Borko, 2000; Sykes & Byrd, 1992): finding appropriate mentor teachers. We wanted the mentor teachers to have expertise in PA and to be able to provide the needed support to the preservice teachers. We had limited success in finding these candidates. The teachers involved with our planning team were well qualified and successful mentors; however, the other two teachers that were recruited were not as informed about PA strategies and did not seem to be as committed to supporting our preservice teachers. The preservice teacher groups working with these teachers commented that they provided limited support in designing and implementing the task, and it seemed that the mentors did not feel qualified to discuss PA strategies. While we initially perceived that all of the mentor teachers were interested in the project and had the necessary expertise to provide support to the preservice teachers, these teachers needed more experience with these approaches before they could adequately advice our preservice teachers.

30

Additionally, two groups of preservice teachers mentioned that they had difficulty communicating with their mentor teachers (e.g., emails and phone messages were not returned, minimal written comments on their PA task draft was provided, etc.), and these groups perceived that they did not receive the same level of support as their classmates. One mentor teacher had some health concerns during the semester, and the other teacher seemed to have pressing issues in her teaching that resulted in less time being devoted to the project. While these cases could be called exceptions, we believe these situations are to be expected when asking inservice teachers to take on another responsibility. Thus, accommodations for unexpected situations with mentor teachers should be expected and planned for as much as possible.

We have recruited more mentor teachers through referrals from participating teachers, and are adding teachers who have been involved in summer workshops and/or graduate courses focusing on assessment offered at our university. Even as we have expanded our pool of mentors, challenges remain. As with any field-based work, we have found that we need to be flexible with project due dates while still trying to structure the program through the three-part assignment (planning guide, task draft, final report) to keep the preservice teachers progressing during the semester.

Features of the PA Program to be Preserved In attempting a program for the first time, we found that we made several decisions along the way, some that were well conceived and others that were quick solutions. In this section we reflect on some of the key decisions that worked well for us. First, we were asked whether the mentor teacher or the preservice-teacher-group should facilitate the task.

Given that the preservice teachers did not necessarily have any experience in the mentor teachers’ classes prior to administering the PA task, we allowed the mentors and the preservice teachers to decide on the preservice teachers’ level of involvement in facilitating the task. The preservice teachers had various levels of classroom experience, and leaving this decision to the mentor-preservice teacher groups enabled everyone to make decisions within individual contexts. The primary purpose of the field-based component of this project was not to provide a student teaching experience as much as it was for preservice teachers to learn about performance assessment in a situated context of the middle school classroom. For preservice teachers and mentors that were not comfortable with

Meaningful Field-Based Experiences

the preservice teachers facilitating the tasks, this flexibility seemed to enhance the preservice teachers’ abilities to focus on performance assessment and diagnose students’ thinking and learning more than it might have if they had the added stress of teaching during the task.

In regard to the assignments of the project, two non-field-based components were important to preservice teacher learning: the initial research of the mathematics topic and the follow-up lesson plan. By requiring that the preservice teachers find journal articles examining the teaching and learning issues surrounding their topic, the preservice teachers gained in-depth expertise on the theoretical foundations of their topic beyond what is normally discussed in class.

While this type of assignment has been a part of our methods courses in the past, connecting it to the situated context of the PA project gave it more meaning for the preservice teachers. Additionally, the project provided some assurance that they were better informed about the pedagogical issues surrounding their topic as they designed tasks, and many preservice teachers commented that the project helped them anticipate and/or avoid potential problems in the classroom. Likewise, the follow-up lesson plan helped the preservice teachers to see what a classroom teacher would do with the information gained from the assessment. We found the preservice teachers to be more invested in these lesson plans than in others required for the course because they had their classroom experience and real students as their referent when they designed them.

Features of the PA Program to be Changed or Added First, in attempting to find qualified mentor teachers, we were able only to find middle school teachers who seemed to have appropriate experience.

Correspondingly, we limited our preservice teachers to writing PA tasks for middle school mathematics. Some of the preservice teachers were unhappy with this limitation because they intended to teach at the elementary level and were not interested in the middle school level. For these preservice teachers, the PA project seemed less authentic because it was not situated in the grades in which they intended to teach.

Another problem was that all of the mentor teachers were not selected prior to the start of the semester. It may not be a coincidence that the two less committed mentors were called upon part way into the semester and therefore were not included in early planning efforts. We believe that we would have been more successful if we had involved all of the mentors in the

Amy Roth McDuffie, Valarie L. Akerson, & Judith A. Morrison

PA project throughout the entire semester. Since this first implementation, we have been more successful in assembling a cadre of mentor teachers at all grade levels to draw from each semester and to better match the PA project requirements with our preservice teachers’ interests. However, we continue to struggle with having all of the mentors selected prior to the start of the semester. Some teachers and school districts are reluctant to commit to the program in advance, especially for the fall semester when schools are still organizing their own teaching assignments.

In addition to more mentors, we realize that our preservice teachers need more opportunities to interact with their mentors. For example, a final meeting between preservice teachers and mentor teachers would provide an opportunity for preservice teachers to review their analyses and report on the students’ performances. This meeting would provide the mentor teachers with a new perspective on their students’ thinking, learning, abilities, and skills. It also might serve to help the mentor teachers improve their mentoring skills by more carefully examining the products of the preservice teachers’ work. Moreover, this meeting would provide preservice teachers with feedback on their analyses based on the teachers’ knowledge of their students, and this feedback and perspective is not possible from their methods professor. However, logistics with scheduling and the need for substitutes have impeded these plans.

Next Steps We are continuing to implement our PA program in mathematics methods courses. Our current efforts include offering this PA program in both the mathematics and the science methods course with the students using PA to make connections between the disciplines. The benefits we have experienced compel us to continue to develop this program. A study is underway to empirically investigate the effects of our program on our preservice teachers’ learning of PA, and more generally, the teaching and learning of mathematics REFERENCES Black, P., & Wiliam, D. (1998). Inside the black box. Phi Delta Kappan, 80 (2). 139-148.

Borko, H., Mayfield, V., Marion, S., Flexer, R., & Cumbro, K. (1997). Teachers’ developing ideas and practices about mathematics performance assessment: Successes, stumbling blocks, and implications for professional development.

Teaching and Teacher Education, 13 (3), 259-278.

31

Danielson, C. (1997). A collection of performance tasks and rubrics: Upper elementary school mathematics. Larchmont, NY: Eye on Education.

Darling-Hammond, L., & Falk, B. (1997). Using standards and assessment to support student learning. Phi Delta Kappan, 79, 190-199.

Firestone, W., Mayrowetz, D., & Fairman, J. (1998). Performancebased assessment and instructional change: The effects of testing in Maine and Maryland. Education Evaluation and Policy Analysis, 20, 95-113.

Fuchs, L. S., Fuchs, D., Karns, K., Hamlett, C., & Katzaroff, M. (1999). Mathematics performance assessment in the classroom: Effects on teacher planning and student problem solving. American Educational Research Journal, 36, 609646.

National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: Author.

National Council of Teachers of Mathematics. (1995). Assessment standards for school mathematics. Reston, VA: Author.

National Council of Teachers of Mathematics (2000). Principles and standards for school mathematics. Reston, VA: Author.

Roth McDuffie, A., McGinnis, J.R., & Graeber, A. (2000).

Perceptions of reform-based teaching and learning in a college mathematics class. Journal of Mathematics Teacher Education, 3 (3), 225 â&#x20AC;&#x201C; 250.

Shepard, L., Flexer, R., Hiebert, E., Marion, S., Mayfield, V., & Weston, T. (1996). Effects of introducing classroom performance assessment on student learning. Educational Measurement: Issues and Practices, 15, 7-18.

Shepard, L. (2000). The role of assessment in a learning culture.

Educational Researcher, 29 (7), 4-14.

Smith, M. (2001). Practice-based professional development for teachers of mathematics. Reston, VA: NCTM. Stenmark, J. (1991). Mathematics assessment: Myths, models, good questions, and practical suggestions. Reston, VA: National Council of Teachers of Mathematics.

Sykes, G., & Bird, T. (1992). Teacher education and the case idea.

Review of Research in Education, 18, 457-521.

Van de Walle, J. (2001). Elementary and middle school mathematics: Teaching developmentally. New York: Addison Wesley Longman.

Washington Commission on Student Learning (1998). Essential Academic Learning Requirements. Olympia, WA: Author.

Putnam, R., & Borko, H. (2000). What do new views of knowledge and thinking have to say about research on teacher learning?

Educational Researcher, 29, 4-15.

32

Meaningful Field-Based Experiences

The Mathematics Educator 2003, Vol. 13, No. 1, 33–40

How to Do Educational Research in University Mathematics?

Rasmus Hedegaard Nielsen Situated as a Ph.D. student in university mathematics education, I present some of my considerations about my identity as a researcher in this field. I discuss the larger global and local societal issues and their connections to educational research in university mathematics. My discussion goes beyond personal considerations and touches upon the structures and ideas that are both internal and external to university mathematics education. I discuss the different political projects that I can identify from my personal experiences across the fields of educational research, practice, and policy in university mathematics. I place myself firmly within the tradition of critical education, but also draw on postmodern theories. The results of the discussion are the identification of challenges for a postmodern critical mathematics education, with a focus on university mathematics.

This paper is a modified version of a paper presented at the Third International Conference on Mathematics Education and Society held in Helsingør, Denmark in April 2002.

I live in Denmark, one of the richest countries in the world, a small democratic kingdom in the western world. I am a white male living in a welfare society with all the commodities of the western way of life. I have free access to education—even to all the universities. I have free access to medical care. I live in a peaceful neighborhood. I can walk on the streets in the middle of the night without any fear of being attacked in any part of my city. I will not have to sell my home or change my life drastically if I lose my job.

I have never felt starvation. I have democratic rights to vote and to participate in political life. I do not risk discrimination or being arrested at random. I live in a country where everybody gets what he or she needs and deserves. This is a brief glimpse of the Danish society from my perspective.

Mathematics Education and the Larger Society Is this short story from the Danish society in any way relevant for the question of how to do educational research? Does it make sense to ask what kind of educational research should be conducted in the context of this society? These questions I have asked myself as a Ph.D. student in university mathematics education. 1 I have done so because it is important for me to consider the role that my work and I will take in this society and it is important for me to contribute to my society. And my society is the one you just glimpsed. In the following analysis I try to sketch some of the answers that I have found.

Rasmus Hedegaard Nielsen is a Ph.D. student in the Department of Education at Aalborg University, Denmark. His thesis focuses on the social roles of mathematics and university mathematics education. His email address is riege@ofir.dk.

Rasmus Hedegaard Nielsen

The first answer I listened to when I began my work focused on the internal problems of university mathematics education, and thereby deemed my glimpse of the Danish society irrelevant. It suggested, for example, that I could look at how to get students to learn the concept of function space better, or how to understand how students actually learn concepts.2 Another group of issues I could look at was how to get the students motivated and how to get more students to pass the exams. This answer almost totally ignores the relation between the larger society and university mathematics education and thereby makes the context of this education irrelevant, but it certainly offers a paradigm of research in university mathematics education. This I call the conservative answer.

Another answer suggests that the institutions of university mathematics education are very important for the welfare of the Danish society, understood mainly in an economic sense. This makes my opening glimpse of the Danish society relevant in the sense that we need the institutions of university mathematics education for maintaining the things that we like which are represented by this glimpse. But this answer also suggests that these institutions are in crisis in the sense that the very important link between these institutions and the larger society (understood as the economic system) is not close enough. Furthermore, they often claim that the institutions of university mathematics education are also in crisis in the sense that they are inefficient. ‘We’, the taxpayers, do not get enough for the money ‘we’ pour into these educations. This means that the focus of educational research should be on the learning of mathematical concepts by the individual student, but also on what and how the students should learn. What and how they should learn should be 33

relevant for the economy, which in this context translates mainly to the industry. It also suggests that I could research into how to measure the efficiency of the institutions, in order to increase the efficiency of the institutions by benchmarking them. This I call the neoliberal answer.

Many additional answers that reflect moments of both the conservative and the neoliberal answers I’ve already offered can be found. For example, one answer takes the internal concerns from the conservative answer seriously, but also finds mathematics education to be a crucial institution in the Danish society, though these institutions are not seen to be in crisis because the link to the larger society is not tight enough. On the contrary, it sees mathematics education in crisis because it is being challenged by the neoliberal project.

Yet another answer tells me it is not certain that university mathematics education is one of the institutions that helps guarantee all the good things in Danish society. Also it is not certain that university mathematics education has no influence on the larger Danish society. It might be that these institutions are also the source of some of the bad things that I mention next. This I call the critical group of answers.

Another way to experience my country—an alternate glimpse—is to take a walk in the red-light district near the railroad station, or simply walk down the main shopping street. There you will notice the junkies and homeless people that nobody wants. They get pushed around between different places in the city.

Some of the homeless and alcoholics used to drink their strong beers near a traffic junction in the middle of Nørrebro where many people pass by. They used to sit there peacefully, or at least as peacefully as they could while being an alcoholic and having many other problems, until the authorities took their bench. They literally removed it. Now the homeless are at another place where they do not ‘disturb’ the people that have a job and ‘contribute’ to ‘our’ society. These people we often do not see, are near the bottom of our small peaceful society. They are not alone down there, others are just harder to notice. They are being rejected by the peaceful, democratic, and just Danish welfare society; they are not considered good for anything (other than trouble and someone to spend ‘our’ money on).

As with the first gaze into the Danish society, there are different opinions about the relevance of this glimpse for research in mathematics education. The conservatives would tell me that this picture has nothing to do with mathematics education and especially university mathematics education. The neoliberals would tell me that what ‘we’ need is more

34

and better mathematics education in order to address these problems (if they are seen as problems of society and not only individual problems). The critical educators try to tell something akin to what is mentioned in the Aims of the Third Mathematics Education and Society (MES) Conference: Mathematics qualifications remain an accepted gatekeeper to employment. Thus, managing success in mathematics becomes a way of controlling the employment market. Mathematics education also tends to contribute to the regeneration of an inequitable society through undemocratic and exclusive pedagogical practices which portray mathematics and mathematics education as absolute, authoritarian disciplines.

(Skovsmose & Valero, 2002, p. 3)

This means that this second glimpse becomes relevant.

But this is not all. Both the neoliberal and critical group tell me to look further than the Danish society, to the global world. They urge me to look at different parts of the global world and in different ways. Let me try to share one glimpse into the larger society.

Yet another way to experience the world (and not only Denmark) is to take a plane from inside to outside the Western World (or more correctly from inside to outside the industrialised world and its holiday resorts in the sunny or “exotic” parts of the not-soindustrialised world). Here you can meet hunger, war, serious environmental problems, cultures being destroyed, peoples and countries being plundered, torture, and disasters and crimes of any kind.3 (Well, not personally, it is likely you would have a nice room in a hotel and a return ticket to your home country in your pocket.4) The original voices, conservative and neoliberal, are again answering me, whispering different suggestions in my ear: “Mathematics education is the hero of civilization,” or “mathematics education is innocent.” They continue whispering seductively, “these problems that you see are only small errors in the system and if only people would not resist the system these errors would be easily fixed.” The critical group of answers would suggest that mathematics education might play a role in producing all the nightmares that haunt the world globally, and that mathematics education might play a role in the unequal distribution of wealth globally.

Mathematics and the Larger Society Not only can university mathematics education be seen as playing different roles in our local and global society, mathematics can also be seen as playing

How to do Educational Research?

different roles. And how we see and understand university mathematics education might depend quite a lot on the role played by mathematics in our societies, both globally and locally. The conservatives claim that mathematics is objective and neutral in itself, and it is only the use of mathematics that can lead to good or bad things. This means that university mathematics education is protected from considerations about the role of mathematics in society, and the only problem with mathematics education is that students have problems learning it. Mathematics is of major importance because it is an important part of our culture. The neoliberals are saying that mathematics is of major importance in the pursuit of economic growth and thereby the success of our societies, and in this way mathematics education is made important as well, namely as a producer of competencies in mathematics in the shape of a mathematically skilled work force.

But I can also hear other people, for example in the field of ethnomathematics5, that try to tell me a more critical story: The critical strand [of ethnomathematics] is not just interested in the mathematics of Angolan sand drawings and their use in story telling, but also in the politics of imperialism that arrested the development of this cultural tradition and in the politics of cultural imperialism that discounts the mathematical activity involved in creating Angolan sand drawings. (Powell, Knijnik, Gilmer, & Frankenstein, 1998, p. 45)

These voices say that mathematics and mathematics education might not be innocent and might not be our hero and problems might not just be errors, but mathematics and mathematics education might have something more substantial to do with all these problems.

Who is right? Which story should be believed and on what grounds should the different kinds of answers be judged?

The Larger Society and Hegemonic Projects I think it should be clear by now that what I have called the larger society or ‘our’ society are terms that are highly disputed and that these terms play a crucial role in the stories that I am offered when I ask about how to do educational research in university mathematics. Personally, I felt that I had no firm foundation that I could stand on when I was to judge the different ideas about ‘our’ society. I found that the discourse theory of Laclau and Mouffe (1987; see also Torfing, 1999) expressed theoretically just that feeling.

In discourse theory, society is not seen as something

Rasmus Hedegaard Nielsen

that is without conflicts or something that can be described from a neutral and objective standpoint.

Theoretically put, what I have sketched above is my experience of the struggle of different political projects that all try to dominate society and to that end give different interpretations of what is important in society.

They all try to make their descriptions look neutral and objective—to look like the truth about our society. In this sense, such theories are in the same vein as the theory of Foucault. In the words of discourse theory, these efforts are called hegemonic projects and they are said to try to gain hegemony. Hegemony translates roughly into leadership, including cultural and political dimensions. My point is that these struggles also extend to the arena of university mathematics education, and that this arena is both used as a resource and as a stake in the struggles. It is not the case that a hegemonic project is always struggling in all arenas and it might be that the project takes different forms in different arenas. The hegemonic projects are not some overarching ideology that structures everything. The answers I sketched in the previous section are very different in scope, but they are all more or less entering the arena of university mathematics education.

It should also be clear that I could have chosen to give you, the reader, quite different glimpses of society. But as I am also situated in these struggles (on the side of the critical group, which you might have guessed) I want to obtain something, and to this end I have chosen these particular glimpses. I do not have hopes that I can show you that things necessarily must be like I see them; I only hope to show you something you might not have seen before. This might be seen as an answer of how to judge the different answers; yet there is no way to stand on a firm ground and be able to judge. You are always a part of these hegemonic projects; you always see the world from somewhere.

In my work I wanted to get a little closer to the hegemonic projects in order to know the terrain that I was entering. This implied that I took a closer look at how the different hegemonic projects are connected to university mathematics education. In the following, I concentrate on the Danish context, but I am sure that the discussion also extends to most Western countries, though maybe with different emphasis caused by the different contexts. The works of Michael Apple have inspired the following discussion of hegemonic projects.

The Field of Educational Research Different projects are present in the field of educational research, such as the conservative and

35

neoliberal. When it comes to general education, critical education can be said to constitute a project, but as far as I can see, critical education has mainly been interested in the primary and secondary educational system. I want to mention critical education anyway, because it is a project that I have sympathy for and because I think it is possible to extend it to university mathematics. In the field of educational research the conservative project is dominant when it comes to university mathematics. However, the field itself is quite young.

Critical Mathematics Education In the Aims of the MES Conference, I see concern for identifying structural problems that affect the people who are learning mathematics: “Mathematics qualifications remain an accepted gatekeeper to employment”, and “mathematics education also tends to contribute to the regeneration of an inequitable society through undemocratic and exclusive pedagogical practices” (Skovsmose & Valero, 2002, p.

3). Here we see a focus on democracy, and elsewhere on the idea of citizenship. Mathematics is not seen as unproblematic, but seen as a potential social actor that supports the production of risks in society.

Ethnomathematics is also interesting, since it for me has a completely different focus on mathematics than what is usual in the field of research in mathematics education.

Only in the very broadest sense can these concerns and focuses be said to be a part of a hegemonic project.

Critical mathematics education is a movement that is connected to practitioners of the teaching of mathematics but might have weak links to fields outside mathematics education. This should be seen as a challenge to critical mathematics education. It is a movement that has not entered the field of university mathematics education in a substantial way. This is something that I would like to change.

University Mathematics Education The conservative project dominates the field of educational research in university mathematics (Hart, 1999). Hart identified the dominating research tendencies in post-secondary mathematics education, “Except for a handful of studies, most research at this level has focused on the student or on various pedagogical methods…” (p. 3). She proposed a research agenda that can be characterized as a postconservative agenda. It retains conservative characteristics since it still sees mathematics and mathematics education as disconnected to the larger

36

society. It is post-conservative since it clearly goes beyond the conservative agenda by proposing constructionism (Hart refers to Gergen, 1992; and Phillips, 1995) as the epistemological foundation of research within what she calls post-secondary mathematics education.

The Field of Educational Practice At many departments of mathematics in the ‘old’ universities in Denmark, the teaching is centred on courses based on lectures and classes (where the student are supposed to solve problems) with typically large numbers of students attending. The pedagogy is often authoritative, picturing mathematics as an absolute discipline and teachers as holding the absolute truth about mathematics. Mathematics is seen as packages of knowledge that should be put into the heads of the students. The students are seen as individuals and their context is unimportant (unless to the degree that their motivation is of interest). The teaching of mathematics and mathematics itself are seen as unproblematic.

At some departments of mathematics, the teaching is centred on group-based project work, but also lectures. Some of the project work focuses on links to the larger society and the role played by mathematics in society. These universities have become the best suppliers of workers because they focus on project work in groups while the old universities more or less try to copy their ways of organising the educations.6 My impression of groups that are in these environments is that they both more or less represent different degrees of what I call the conservative project.

In the context of one of the old universities, some of the problems that I have heard talked about are economic problems and pedagogical problems. The first kind is caused by the decrease in the number of students studying mathematics 7 and the second by the fact that the student population is becoming more heterogeneous 8 and that the students lack motivation.

The first problem is understandable so, since economic problems will mean less funding for the researchers and teachers in the department. The second problem consists partly of increasing difficulties teaching at a level where as many as possible benefit, and partly of increasing exam failure.

What I think is characteristic of these problems is that they focus on economics and on the individual students—they are both more or less external problems being imposed on the departments. This naturally puts other problems on the sideline. For example, there is

How to do Educational Research?

not much attention on mathematics itself: No one asks why there is such a thing as university mathematics education, what kind of mathematics should be taught, or the relation between mathematics and the larger society. Or this view perceives such problems and issues in a particular way. The focus is on how mathematics should be taught so that more students want to study mathematics, so that they complete their study faster, and so that they become better mathematicians. It seems like the conservative ideas of mathematics education that prevail in the departments of mathematics are under attack from the neoliberal ideas, and that this attack comes mainly from the area of national university policy.

This identification of certain problems is not innocent. It has caused different actions to be taken.

That tells me a certain story of what research in university mathematics education should be and what I ought to do to be a ‘normal’ researcher. For example, there is a suggestion of making elite courses alongside a normal course to accommodate the problem of a heterogeneous student population. A Centre for Science Education 9 has been built to undertake research and development of mathematics education to make it more ‘sexy’ 10 and thereby attract and motivate more students.

The Field of University Education Policy National I have tried to understand the kinds of arguments and understandings of the universities and society. One thing that is striking in these debates on university and society is the use of a particular idea that is always connected to the role of the universities—the idea of a knowledge society. This idea is used to refer to the kind of society that we live in (at least in the Western World) and by connecting to the universities via the idea of knowledge, a certain perspective on universities is constructed that dominates the debates. The idea comes in different versions—for example, the concept of a “learning society” in Michael Young’s The Curriculum of the Future (1998, p. 137-155). I think of the knowledge society as a contested concept.11 This means that different groups in society (not necessarily political parties) with different interests try to gain the power to define the idea of a knowledge society and to connect it with other different ideas. This would help the groups gain the power to define facts, problems, and solutions concerning, among other things, the university and the role of the university in the knowledge society.12 In other words, they try to make their ideology hegemonic. Some groups try to connect Rasmus Hedegaard Nielsen

the idea of a knowledge society with the ideologies of business and management using ideas as production, competition, management, and markets. Other groups try to connect the knowledge society with the idea of democracy with emancipation, the risk society, and ethics.13 The typical dominant argument goes like this: We are in, or partly in, a knowledge society, therefore the role of the universities have changed in a certain way and we, as a society, have to react responsibly to these new conditions. This is the general form of the argument, and when it is presented like this it is obvious that defining the knowledge society to some extent determines the new conditions of the universities and thereby the kinds of reactions there are. What is also obvious is that in this form of argument there are reactions—not actions. This supposes that the universities have the role of reacting to the conditions in the society, and not the other way around. This makes the university a ‘service’ institution of society, making sure that the right amount and kind of knowledge is produced, and not an institution that can critically examine parts of the larger society, including itself! This idea of a ‘service’ institution nicely fits with the idea that research in university mathematics education is ‘efficiency’ research, that it never gets critical in any profound sense, but only makes sure that the ‘service’ institution is as efficient as possible.

As examples of different contested ideas of the knowledge society, I will examine articles from Universities for the Future14 (Maskell & Jensen, 2001) and from Education15. Sometimes there is a small description of what is meant by a knowledge society, normally focusing on economic. There are no discussions of the processes that have lead to this development or the adequacy of the concept itself. It is taken as fact that we live in a knowledge society and that this is a fact that we have to adjust to and, in particular, the universities have to adjust to. These kinds of description and this kind of construction of necessity are also found in the political policies on education of most of the political parties in Denmark.

After establishing the fact that we have to react to the emerging knowledge society, the writers draw conclusions about the role of the university. These writers agree that it is a very important institution and much more important than it used to be. They see it is an institution where knowledge is ‘produced’ mainly in two forms: as research results and as academic workers. Both are conceived as inputs to the private corporations that are so important for our welfare

37

system. This means that universities, as a knowledge society, are conceived only from an economic perspective and not from a cultural or political perspective.16 This means that changing the structure of the universities will have effects on the economy, and more importantly, it means that this is the only relation that is conceived in the relation between university and society.

International In this context, economy connects to competition, markets, and freedom. The economic description of the knowledge society is typically followed by some kind of description of a globalisation process, constructing a link between the success of ‘our’ welfare system and how competitive our country is. If knowledge is the most important factor for competition between countries, then the success of our welfare system is dependant on the success of our society as a knowledge society. The General Agreement on Trade and Services (GATS) that is a part of the World Trade Organization (WTO)17 contains a clear neoliberal approach to higher education, including university mathematics education.

It sees mathematics and mathematics education as a commodity that should be given the opportunity to be traded on a free market. The Bologna Declaration can be seen as an attempt to clear the way for a free market in university education in Europe.18 This declaration has also some “regionalistic” agendas, such as the building of a common European culture. The agreements and declarations are beginning to have effects on both the thinking and the everyday life of higher education, including university mathematics education, in most European countries.

How to do Educational Research in Mathematics Education?

I hope that I have sketched with some clarity the different answers that have been given to me in my search for an identity as a researcher in university mathematics education. These answers make different suggestions for a research paradigm. I have tried to sketch how these possible research paradigms are contained in different hegemonic projects with very different scopes and identities. I leave it to the reader to think about the names that I have given to the different answers: conservatives, neoliberals and critical.

The conservative suggestions are mainly focused on mathematics and try to ignore external relations, though the conservatives have been under pressure from the neoliberals. The neoliberals focus on the economic link between university mathematics

38

education and the larger society, which they understand in mainly economic terms.

One of my main points would be that whether you like it or not, deciding how to do educational research in university mathematics education makes you a part of these struggles in one way or the other. It is not a neutral realm that can refer to the pursuit of truth for the legitimisation of work being done. In this way, the three glimpses of society that I have given are relevant to consider. I have found it is not an easy thing to choose how to do research in this field (or any other field for that sake); there is no firm ground to stand on from where to make a neutral and necessary judgement.

Challenges for Postmodern Critical Educational Research in University Mathematics As mentioned before, I can identify with the concerns of critical education, though I also find some of the ideas problematic. Therefore, I have chosen the word “postmodern” from critical education; this signals my flirtation with discourse theory. As mentioned, both the ideas of discourse theory and those of critical mathematics education have not been especially interested in university mathematics, therefore there are a manifold of challenges and uncertainties for a research paradigm that is inspired by these two approaches. The challenges I focus on here are those that I find important. This does not mean that I see them as the most important or the only ones, but it means that they are those that I have found interesting and within my reach as a researcher.

Theoretical Challenges There is need of a theoretical framework that can help: 1. Conceptualise the multiplicity of educational research paradigms, practices, and policies; and the way that they internally compete and struggle.

2. Conceptualise the relations between the different fields. I have focused on educational practice, educational research, and educational politics in this paper, but many others exist.

3. Conceptualise key-concepts such as society, politics, and mathematics.

4. Conceptualise political implications as to what democracy, citizenship, and mathematics education should be like.

These are the challenges that I feel to be urgent. I have appropriated the theoretical framework of the Laclau and Mouffe’s discourse theory (1987) as an approach to the concepts of society and politics and as How to do Educational Research?

an approach to understand the struggle for power within the discipline of education research. I have also drawn on the work of critical mathematics education, especially the work of Ole Skovsmose (1994), to conceptualise mathematics in society.

The theoretical framework I call for should not only be descriptive; it should also provide directions and strategies for research in university mathematics education, but also in society and education at large.

Also, the framework should give us directions that we can explore as to theorize how we would like the educational institutions of university mathematics to be. I have myself focused on concepts such as democracy, citizenship, and the apparatus of reason.

Empirical Challenges Parallel to such a theoretical framework, there are also challenges that relate to the understanding of the actual state of the fields and their connections. How is educational research in university mathematics actually being done? What are the conceptualisations of students, of mathematics, of connections to the larger society, and so forth, that are implicit or explicit in the kinds of research that takes place? How are the educational practices in the universities and what are the connections to the larger society and other fields?

In other words, what kind of world is it that the critical strand is a part of and in which it finds itself?

But this is not all. There are also empirical challenges connected to the normative part of the theoretical framework. We need to explore empirically how the ideas such as democracy and citizenship can be realized in a university mathematics education.

I have personally concentrated on the first part of these empirical challenges, and I have done so by focusing on the three fields that I also mention above. I have looked at the educational practices at a certain department of mathematics at a university. I have looked at international policy on higher education. And I have looked at educational research in university mathematics.

Building a Hegemonic Project One of the most important features that a postmodern critical mathematics education should have is that it should be able to form an alliance of different groups in order to get enough momentum. It should be able to connect to other fields as the neoliberal project has done. In my opinion, some of the most important groups to connect to are the teachers at all levels of the educational system, the students, and others. It is not clear to me how these connections could be made, but

Rasmus Hedegaard Nielsen

the idea of a radical plural democracy19 and a democratic citizenship seem to be a concern that can be traced throughout many fields.

REFERENCES Apple, M.W. (2001). Educating the â&#x20AC;&#x2DC;rightâ&#x20AC;&#x2122; way. New York: Routledge Falmer.

Danish Ministry of Education. (n.d.). Uddannelse. Retrieved May 16, 2003, from http://udd.uvm.dk/?menuid=4515 Gergen, K. (1992, February). From construction in contest to reconstruction in education. Paper presented at the Conference on Constructivism in Education. University of Georgia, Athens, GA. Hart, L. C. (1999). The status of research on postsecondary mathematics education. Journal on Excellence in College Teaching, 10 (2), 3-26.

Kjersdam, F., & Enemark, S. (1994). The Aalborg experience.

Aalborg: Aalborg University Press.

Laclau, E., & Mouffe, C. (1987). Hegemony & socialist strategy: Towards a radical democratic politics. London: Verso.

Maskell, P., & Jensen, H. S. (Eds.) (2001). Universiteter for fremtiden. Copenhagen: Rektorkollegiet.

Phillips, D.C. (1995). The good, the bad and the ugly: The many faces of constructivism. Educational Researcher, 24 (7), 5-12.

Powell, A. B., Knijnik, G., Gilmer, G., & Frankenstein, M. (1998).

Critical mathematics. Proceedings of the First International Mathematics Education and Society Conference, 45-48.

Powell, A.B., & Frankenstein, M. (Eds.) (1997).

Ethnomathematics. Albany: SUNY Press.

Skovsmose, O. (1994). Towards a philosophy of critical mathematics education. Dordrecht: Kluwer.

Skovsmose, O., & Valero, P. (2002). Proceedings of the Third International Mathematics Education and Society Conference.

Copenhagen: Centre for Research in Learning Mathematics.

Torfing, J. (1999). New theories of discourse. Oxford: Blackwell Publishers Ltd. Vithal, R., & Skovsmose, O. (1997). The end of innocence: A critique of ethnomathematics. Educational Studies in Mathematics, 34, 131-157.

Young, M. F. D. (1998). The curriculum of the future. London: Falmer Press.

1 My focus is on educational research within the departments of pure or applied mathematics at the universities, not on the teacher training colleges. Although in Denmark, the departments of mathematics at the universities are educating the upper secondary school teachers (the high school/gymnasium level). These teachers are educated in just the same way as the students that choose to be a researcher in mathematics or choose to work in industry or elsewhere. In fact, at the university of Copenhagen no course in mathematics education is available for those who later want to be teachers in high school.

2

I.e., a focus on learning theory, and a certain kind of learning theory.

39

3

I would like to mention only two numbers, namely the number of 1,200,000,000 and 7,000,000. The first in the number of persons in the world today that daily have under 1 dollar to live on. The second is the number of children that die every year of hunger.

Compare this number to the 5,000 people that died when the World Trade Center was destroyed.

4 This comment could be applied to what I am doing here; in some sense this paper is also exploiting the people in hopeless situations.

5

See for example Powell & Frankenstein (1997), and for a discussion of ethnomathematics see for example Vithal & Skovsmose (1997).

6 In Kjersdam & Enemark (1994) there is a presentation of Aalborg University as a success in the sense that they supply the employment market with some of the best workers.

7 In Denmark some of the funding of the universities are partly dependent on the number of students.

8

This means that there is a group of about 20% that finds learning mathematics easy and a group of about 80% that has difficulties, at least according to some of the lecturers in the department.

9 Science includes mathematics in this context. The centre’s homepage is http://www.naturdidak.ku.dk.

10 As I heard one of the speakers say at the opening of the centre the 27th of March 2001.

40

11

More correctly I think the concept is partly contested because the dominant part of the debates actually agrees to a large extent, but there are also many disagreements.

12 I must admit I am a little uncertain about this formulation. I do not want to think of the idea of a knowledge society as something that one can apply or use like a tool to gain power. It is more like something that is a part of the construction of the way one perceives the society and one’s identity. I am not sure if I really end up doing what I do not want in this paper.

13

These thoughts on how ideas, concepts and power interact are to a large extend inspired by Michael Apple (e.g., 2001).

14 This is my translation of the title ‘Universiteter for fremtiden’.

This book consists of articles written by politicians and others.

15 This is my translation of the title of the Danish magazine ‘Uddannelse’ published by the Danish Ministry of Education (n.d.), 16 Young (1998, p. 156) has identified a similar trend in public education.

17

This agreement is being negotiated continuously, and has been the target of many protests especially for making education a commodity that can be traded as bananas are traded.

18 See http://www.unige.ch/cre/activities/Bologna%20Forum/ Bologne1999/bologna%20declaration.htm for the text of the declaration. There is resistance to this declaration from for example the Attac movement.

19 This is a hegemonic project proposed by Laclau and Mouffe.

See (Torfing, 1999, p. 247-261) for an introduction.

How to do Educational Research?

The Mathematics Educator 2003, Vol. 13, No. 1, 41–45

Mathematicians’ Religious Affiliations and Professional Practices: The Case of Bo Anderson Norton III Bo’s case is the third of three case studies exploring relationships between the domains of religious belief and mathematical practice among university research professors. As a Buddhist, Bo’s mathematics and religious views are integrated in a surprising epistemology. His epistemology and other relationships are contrasted by those presented in previous case studies of a Jewish professor and a Christian professor, at the same university.

While the previous cases highlighted the transfer of methods of practice across domains and the need to reconcile potentially conflicting aspects of the two domains, Bo’s case reminds educators that each student holds her own universe of thought and that mathematics plays a prominent role in developing that universe; or is it “the way of knowing the universe?”

This paper reports on the third of three case studies, all intended to investigate the implications of religious affiliation in the professional lives of mathematicians. These case studies offer contrasting perspectives in answer to my research question: How do strong religious convictions influence professional mathematicians’ practices and their views of mathematics? The previous cases revealed the need for reconciliation of mathematical truth and professional practice with religion in order to make mathematical practice meaningful. Reconciliation can be difficult because one realm may supercede the need for the other (Norton, 2002b). However, in the case of Bo, the two realms are fundamentally integrated so that, together, they provide an epistemology.

I selected the three participants for my study because they had reputations as devout representatives of three distinct religious groups—Judaism, Christianity, and Buddhism—among professors in the mathematics department of a large southern university.

Before conducting one-hour interviews with each of them I was not certain that I would be able to identify more than a superficial influence. In fact, the participants themselves were largely unaware of such a relation, but as they recounted their personal histories, evidence of significant connections emerged. For Joseph, the Jewish participant, religion helped to define and inform his professional practice of research and teaching as “meritorious activity;” on the other hand, Charles struggled through years of conflict before Andy Norton is currently working on his doctoral dissertation in mathematics education and master’s degree in mathematics at the University of Georgia. His research interests include students’ mathematical conjectures and their role in learning. His email address is anorton@coe.uga.edu.

Anderson Norton III

reconciling his early desire to do research mathematics with his most fundamental Christian beliefs. Bo’s situation was different in that he developed his Buddhist beliefs and his mathematical career while simultaneously exploring other possibilities in both realms.

In my analysis of Bo’s interview I identified two major themes: his belief in cause and consequence, and his world of quantifiable objects with infinite coordinates. In this paper I report on these themes along with Bo’s background and relevant history, which I use to contrast Bo’s unique perspective with those of past mathematicians and with the other two cases. I also include a poetic transcription in order to give a flavor for Bo’s own language; though I employed artistic license in the order of phrases, the words are his (see Figure 1). A detailed account of my methods for developing both the narratives and the poetic transcription can be found in Norton (2002a).

Einstein and Bo Because of their similarities in practice and belief, I find it especially interesting to contrast Bo’s views with those of Albert Einstein. I begin here with a brief summary of Einstein’s philosophy on science and religion, as reported in his bibliographies. I return to these points in the discussion section following Bo’s narrative.

Like Bo, Einstein was a mathematician with Buddhist views. Though he was a Jew by heritage, he did not believe in a personal god and instead referred to a “cosmic religious feeling” (1990, I, p. 26). He claimed that Buddhism had a strong element of this feeling. Far from believing that science and religion were at odds with one another, he claimed, “in this materialistic age of ours the serious scientific workers 41

are the only profoundly religious people” (p. 28) because they are able to think abstractly and universally. In Out of My Later Years, Einstein noted that “the realms of religion and science are clearly marked off from each other” in that they answer different questions (1990, II, p. 26). Still, he proclaimed, “science without religion is lame; religion without science is blind” (p. 26). Much of this thinking is echoed in Bo’s story, though there are some notable differences of viewpoint and profound differences in background.

Bo’s Narrative Bo is a 30-year old Chinese man who has been living in the United States for about 10 years. He was raised in a family without religious beliefs, but began to explore his own beliefs as an undergraduate in Shanghai. There he studied philosophy, the Bible, Taoism, Buddhism, and other religions. He found that Buddhism fit his nature: It offered him a “home for his mind to rest.” His beliefs were strengthened when he met a group of Buddhists in graduate school in the United States.

When Bo was denied admittance for undergraduate study in physics at the Shanghai University of Science and Technology, he turned to his second choice: mathematics. He found that he was better suited for mathematical study because it offered him freedom that physics did not—there were no experiments or computer skills required in the study of pure mathematics. He went on to receive a Ph.D. from the State University of New York at Stony Brook.

His interests in mathematics were piqued even before college, when he learned about infinity. The infinite still plays a role in his post-doctoral research.

He studies operator theory, a branch of mathematics that examines behaviors of objects in infinitedimensional space. He feels that this research should occupy 80% of his time and energy, while the rest is reserved for teaching.

Cause and Consequence In Buddhism, there is no personal god controlling things: “Everything is just cause and consequence.” In fact, Bo believed in this universal phenomenon of cause and consequence before learning of Buddhism.

His belief in the phenomenon contributed to his natural inclination toward Buddhism. Since all is cause and consequence, he cannot expect someone else to save him, and this view countered a major tenet of many western religions he had explored.

42

If you do bad things, you are going to be suffering from that in the future. If you help other people, you will be helped eventually. So, it’s a cause and consequence kind of thing that I believe. And, I also believe that by purifying one’s own soul… you get rid of delusions to see your own nature.

You find a way to save yourself.

Bo refers to this purification as “a way to control your own thoughts.” This is the central theme of his religion, which provides him with a set of values.

Bo describes thoughts as clouds that come and go.

You use good thoughts to do good deeds and evil thoughts to do evil deeds. If an evil or bad thought enters your mind, you can just let it go. “Your mind is like the sky. A cloud is like thought. They go and pass.” This approach applies to mathematical study as well. Any thought that distracts Bo from his research is a bad thought. Letting go of distracting, bad thoughts allows him to focus on his research.

Bo emphasizes the importance of being oneself.

This value is based on the nature of life. He believes that he is defined as a mathematician because mathematical thoughts are the most frequent thoughts in his mind. In fact, on his failure to gain admission in the physics program of his university he says that “life made a correct decision for me.” Rather than ascribing this decision to a mindful deity, he refers to the natural consequence of his failure that suited his nature, embodied by his decision to study mathematics.

“Being a researcher is a value of [one’s own] spirit.” Bo finds freedom in mathematics, as he has in Buddhism. This openness is common to Bo’s nature as well. Perhaps his value of freedom offers further explanation of his affinity for both mathematics and Buddhism. While Buddhism offers him “a feeling of [being] at home,” mathematics makes him happy. “If it makes me happy, then I can make friends around me happy.”

Making others happy is another important religious value for Bo, and “teaching… is a happy thing to do.”

Bo describes teaching as “telling other people what you understand” so that they can appreciate your ideas.

He likes teaching because it allows him to interact with “vibrant students.” He calls teaching “a social value,” and feels that it is important to practice patience in the classroom. When students ask repetitive questions or criticize him in his teaching, Bo keeps a peaceful mind.

Rather than letting negative remarks aggravate him, he reminds himself “there is no target to be hit” by these remarks and lets them pass by. This orientation, then, is another influence of his religion upon his practice of teaching.

Mathematician’s Religion: Bo

It seems that many aspects of Bo’s profession make him happy. It is a part of his nature or Karma, which he knows through a seventh sense—the sense of being oneself (the sixth sense is consciousness). “Your Karma was a seed. So, for example, a person may develop talent for mathematics.” Karma is carried in the eighth sense—an ever-present and immense store of knowledge. Because the seed grows, the eighth sense is the knowledge that “grows out of the seed” and is like creation. Because the seed is eternal, attaining knowledge is like discovery. “Many times, we discover a thing that should be there…So I may think it is a discovery or I may think it brings back memory.”

In mathematics too, Bo found that it is difficult or even impossible to distinguish between the discovery and creation of knowledge. “Mathematics is like a tree.

It's already there, [but] grows different branches….

Only history can tell…. I don’t think it’s purely creation or purely discovery. It is in between.”

Infinite Sequences of Coordinates The strongest relation between Bo’s Buddhism and mathematics exists in the intertwining of the two realms resulting in an epistemology that stems from his belief in cause and consequence. In Buddhism, everything is an image in the mind that is given by objects that we cannot otherwise know. “We cannot say that the thing itself is ‘what what.’ We can only say that the image it gives us is ‘what what.’” In other words, we cannot know an object for what it is, independent of our own unique perspective. Thus there is already a strong epistemology embedded in Buddhism. It includes the belief that our knowledge of objects depends on the observer. “[Bo’s epistemology] is an association of objects with numbers, because ultimately we can process numbers in our minds – not an object itself.” So, in Buddhism, “everything is understood as a sequence of coordinates.” Bo explains that because objects can be seen from infinitely many perspectives, objects must be infinite sequences of coordinates. These are quantitatives, and it is through mathematics that people study quantitatives and their relationships. Thus mathematics is “the way of studying the universe.”

In mathematics objects are also viewed as sequences of coordinates. In fact, this aspect of mathematics is the central focus in Bo’s chosen branch of study, operator theory. In operator theory, mathematicians study objects and relations between pairs of objects in infinite-dimensional space. Bo’s decision to study operator theory may have risen from

Anderson Norton III

his initial high school interest in mathematics, infinity.

This initial interest in the infinite then may be a common cause to both his religious and professional pursuits.

Bo’s religion and mathematics seemed to grow together in many ways. He was drawn to mathematics and Buddhism for at least two common reasons: his nature and his value of freedom. In fact, given Bo’s theory of Karma, we can say that it was in his nature to become a Buddhist mathematician. Certainly many ideas and practices from one domain flow to the other.

In particular, Bo’s view of the world and his means of understanding it are intertwined with his profession and his religion.

Discussion In discussing each of the three cases from my study, I have used Charlotte Methuen’s four categories of historical relationships between mathematics and religion: conflict, independence, dialogue, and integration (1998). These categories provide general contexts from which to examine mathematicalreligious influence and to compare these influences within and across cases (both historical cases and those from my study.) Methuen recounted the life and philosophy of the 16th century philosopher, Philip Melanchthon. Melanchthon clearly fell into the last category, claiming, “the study of mathematics offers a vehicle by which the human mind may transcend its restrictions and reach God” (p. 83). Bo is another example of integration, where mathematics is the vehicle to which we are restricted in reaching the universe. Though some disciples of Buddha may have been able to transcend this restriction and “know without thought,” mathematics is his primary way of knowing.

Bo seems to share the cosmic religious feeling of which Einstein wrote. He might also agree with Einstein that religion provides an avenue for abstract thought that contributes to scientific study. However, it is not clear for Bo that “the realms of religion and science are clearly marked off from each other”. In fact they seem to coalesce into a single realm of thought that is uniquely mathematical and Buddhist, but can be neither of these alone.

Mathematics educators can learn from Bo’s example. Though they might stop short of promoting a mathematical religion, there is an element of Bo’s view that educators may want to instill in their students.

Mathematics may not be the way of knowing the universe, but it certainly provides ways of understanding it. Bo’s view is an admission that

43

A Feeling of Being at Home There’s no ultimate consciousness that creates this world And sets rules for other things to play.

Everything is just cause and consequence—the universal law.

Buddhism gives freedom and a home for my mind to rest.

I have a new feeling of being settled, and I am finding a way To save myself by purifying my own soul.

I am responsible for my deeds and thoughts, but to be honest Thoughts themselves are not distinguished by good and evil.

Fighting in battle, you use evil thoughts. Just be watchful of these thoughts.

My mind is like the sky. A cloud is like thought. They go and pass.

I am defined as a mathematician; I cling to mathematical thoughts.

Centering on mathematical problems, I am using a cloud in a drought.

To thoughts, there’s a deep part and a shallow part. The shallow part is Given by the object that stands in front of you. What is the deep part?

I don’t know. Ability of dealing with image & thoughts is immense.

Limit, infinity, derivative - mathematics is a very freestyle subject.

Between creation and discovery, it’s like a tree branching out.

Only history can tell, but life made a correct decision for me.

Being a researcher is a value of the spirit, and it makes me Happy. Teaching—interacting with energetic & vibrant Students—is a social value. It’s a happy thing to do.

Buddhists believe everything is image in our mind.

We cannot really say the thing itself is what what.

See the cup? A fly may see this cup in a different way.

Buddhism gave me another way to look at mathematics.

Mutually, mathematics deepened my understanding of Buddhism.

Everything is understood as a sequence of coordinates.

Every element is described in infinite-dimensional space Everything has the ability to be infinite: every particle, Every human, every social event, and mathematics…

It turns out to be the way of studying the universe. Or is Mathematics just one approach humans adopt to study this world?

Ultimately we can process numbers in our mind – not an object itself.

There was one disciple of Buddha who knew things without thought, Like when I’m thirsty I know I’m thirsty without thought.

He just expanded this capability. It brings back memory and is there forever.

There are things that exist beyond human sensation and we will never know.

But we should have a peaceful mind and remember that, ultimately, There is no self.

Figure 1: A poetic transcription of excerpts from Bo’s interviews.

44

Mathematician’s Religion: Bo

humans cannot know the universe for what it is (i.e., that an object is “what what”), but that mathematics offers a myriad of lenses for viewing it—perhaps for examining different subsets of the infinite coordinates within it.

This characteristic of mathematics is recognizable in its employment in the sciences. Chemistry, geology and economics (to name just a few fields) all use mathematics in order to explain the biological and sociological environments of humans. By accepting particular assumptions and adopting prescribed methods associated with a field, in a sense one reduces the study of the universe to a few measurable coordinates. After all, these presumptions enable ascription of a cause to a consequence and prediction of phenomena, yet this pattern of assuming and ascribing says nothing about truth except that humans cannot directly perceive it.

There is at least one more aspect of Bo’s view from which educators can learn: Each human being has a different view of the universe. Since mathematics is (at least in Bo’s view) the human way of understanding the universe, each person might infer that she develops her own mathematics. That is, people use mathematical thoughts as they occur in them to satisfy their own goals. The way people use those thoughts yields consequences that determine their direction in future development. In trying to foster development, teachers must first recognize their students’ universes of thoughts and then try to determine the causes and consequences associated with the use of those thoughts. Moreover, in teaching students, teachers must understand what motivates student thinking, else students may let pass the products of teachers’ best intentions as clouds through the sky.

Because it admits observer-dependent truths (or at least observer-dependent perceptions of Truth), Bo’s religious philosophy for mathematics may be the most desirable for establishing meaning for mathematical activity without conflicting with religious views.

Clearly Bo’s and Einstein’s mathematical philosophies were in harmony with their religions, but Einstein could also have carried on the faith of his Jewish

Anderson Norton III

heritage without abandoning his philosophy. In fact, he claimed that Judaism already had present in it an element of this view (1990, I). Certainly in the case of Joseph (the Jewish participant of my study), there was a strong religious respect for science and its role in humanity. In Charles’ Christianity, Charles made a distinction between God’s knowledge and our own and believed that man was capable only of “wavering toward” divine knowledge through trial and error; thus religious Truth and scientific thought need not conflict and often compliment one another.

Whatever their religions, in all three cases the mathematicians felt the necessity of making religious meaning for their practice and defining the role of mathematics in their spiritual lives. Charlotte Methuen’s four categories provide contrasting descriptors for the relationships between the two realms in establishing this role. Though Joseph’s Judaism stood independent of any mathematical truth, his practices within the two realms overlapped, and he held a religious value for mathematical study and teaching as meritorious activity. In that sense, the relationship between his two practices was also one of dialogue. For Charles, there was no built-in religious value for his mathematical practice, so he struggled to integrate the conflicting domains and find some religious value for his mathematical practice in serving God. Bo’s case provides the strongest example of integration – one that led to an essential meaning of mathematical study in understanding the Universe.

REFERENCES Einstein, A. (1990). The world as I see it; Out of my later years.

New York: Quality Paperback Books.

Methuen, C. (1998). Kepler's tübingen: Stimulus to a theological mathematics. Sydney, Australia: Ashgate.

Norton, A. (2000a). Mathematicians’ religious affiliations and professional practices: The case of Joseph. The Mathematics Educator, 12(1), 17-23.

Norton, A. (2000b). Mathematicians’ religious affiliations and professional practices: The case of Charles. The Mathematics Educator, 12(2), 28-33.

45

Conferences 2003 CMESG/GCEDM Canadian Mathematics Education Study Group http://plato.acadiau.ca/courses/educ/reid/cmesg/cmesg.html

Acadia University Nova Scotia, Canada

May 30–June 3

ICIAM International Congress on Industrial and Applied Mathematics http://www.iciam.org/iciamHome/iciamHome_tf.html

Sydney, Australia

July 7–11

CIAEM Eleventh Interamerican Conference on Mathematical Education http://www.furb.br/xi-ciaem

Blumenau-SC, Brazil

July 13–17

PME and PME-NA Joint International and North American Conference on the Psychology of Mathematics Education http://www.hawaii.edu/pme27

Honolulu, HI

July 13–18

JSM of the ASA Joint Statistical Meetings of the American Statistical Association http://www.amstat.org/meetings

San Francisco, CA

Aug. 3–7

ISA RC04 International Sociological Association Research Committee on Sociology of Education Midterm Conference Europe 2003 http://www.ucm.es/info/isa/rc04.htm

Lisbon, Portugal

Sep. 18-20

GCTM Georgia Council of Teachers of Mathematics http://www.gctm.org

Eatonton, GA

Oct. 16–18

SSMA School Science and Mathematics Association http://www.ssma.org

Columbus, OH

Oct. 23–25

RUME Research in Undergraduate Mathematics Education http://www.math.la.asu.edu/~hauk/arume

Phoenix, AZ

Oct. 23–26

AMTE Association of Mathematics Teacher Educators http://amte.net

San Diego, CA

Jan. 22-24

AERA American Education Research Association http://www.aera.net

San Diego, CA

Apr. 12-16

NCTM National Council of Teachers of Mathematics http://www.nctm.org

Philadelphia, PA

Apr. 22-24

PME-NA North American Conference on the Psychology of Mathematics Education http://www.pmena.org

Toronto Ontario, Canada

Oct. 21–24

Conferences 2004

46

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 Georgiaâ&#x20AC;&#x2122;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.

Visit MESA online at http://www.ugamesa.org

TME Subscriptions TME is published both online and in print form. The current issue as well as back issues are available online at http://www.ugamesa.org, then click TME. A paid subscription is required to receive the printed version of The Mathematics Educator. Subscribe now for Volume 14 Issues 1 & 2, to be published in the spring and fall of 2004.

If you would like to be notified by email when a new issue is available online, please send a request to tme@coe.uga.edu To subscribe, send a copy of this form, along with the requested information and the subscription fee to The Mathematics Educator 105 Aderhold Hall The University of Georgia Athens, GA 30602-7124

___ I wish to subscribe to The Mathematics Educator for Volume 14 (Numbers 1 & 2). ___ I would like a previous issue of TME sent. Please indicate Volume and issue number(s): ___________________

Name

Amount Enclosed ________________ subscription: $6/individual; $10/institutional each back issue: $3/individual; $5/institutional

Address

47

The Mathematics Educator

(ISSN 1062-9017) is a biannual publication of the Mathematics Education Student Association (MESA) at The University of Georgia. 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. The Mathematics Educator is abstracted in Zentralblatt für Didaktik der Mathematik (International Reviews on Mathematical Education).

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, philosophical studies, and historical studies), curriculum projects, or classroom experiences; commentaries on issues pertaining to research, classroom experiences, or public policies in mathematics education; literature reviews; theoretical analyses; critiques of general articles, research reports, books, or software; mathematical problems; 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.

The Mathematics Educator 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.

Submission by Mail: Submit five copies of each manuscript. Manuscripts should be typed and double-spaced, conform to the style specified in the Publication Manual of the American Psychological Association, 5th Edition, and not exceed 25 pages, including references and endnotes. Pictures, tables, and figures should be camera ready. The author’s name and affiliation should appear only on a separate title page to ensure anonymity during the reviewing process. If the manuscript is based on dissertation research, a funded project, or a paper presented at a professional meeting, a note on the title page should provide the relevant facts. Send manuscripts to the postal address below.

Electronic Submission: Submit an attachment of your manuscript saved as a Microsoft Word or Rich Text Format document. The manuscript should be double-spaced and written in a 12 point font. It must conform to the style specified in the Publication Manual of the American Psychological Association, 5th Edition, and not exceed 25 pages, including references and footnotes. Pictures, tables, and figures should be in a format compatible with Word 95 or later. The author’s name and affiliation should appear only on the e-mail message used to send the file to ensure anonymity during the reviewing process. If the manuscript is based on dissertation research, a funded project, or a paper presented at a professional meeting, the e-mail message should provide the relevant facts. Send manuscripts to the electronic address given below.

To Become a Reviewer: Contact the Editor at the postal or email address below. Please indicate if you have special interests in reviewing articles that address certain topics such as curriculum change, student learning, teacher education, or technology.

Postal Address: The Mathematics Educator 105 Aderhold Hall The University of Georgia Athens, GA 30602-7124

48

Electronic address: tme@coe.uga.edu

In this Issue, Guest Editorial… A Learning Environment Crippled by Testing: A Student Teacher’s Perspective AMANDA AVERY Testing the Problem-Solving Skills of Students in an NCTM-Oriented Curriculum CARMEN M. LATTERELL Assessment Insights from the Classroom NORENE VAIL LOWERY Designing and Implementing Meaningful Field-Based Experiences for Mathematics Methods Courses: A Framework and Program Description AMY ROTH MCDUFFIE, VALARIE L. AKERSON, & JUDITH A. MORRISON How to Do Educational Research in University Mathematics?

RASMUS HEDEGAARD NIELSEN Mathematicians’ Religious Affiliations and Professional Practices: The Case of Bo ANDERSON NORTON III