v18n2_final Version

____ THE ______ MATHEMATICS____ _________ EDUCATOR _____ Volume 18 Number 2

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Fall 2008

MATHEMATICS EDUCATION STUDENT ASSOCIATION THE UNIVERSITY OF GEORGIA

Editorial Staff

A Note from the Editor

Editors Ryan Fox Diana May

Dear TME Reader,

Along with my co-editor Diana May, I welcome you to the second, and final, issue of the 18th volume of The Mathematics Educator (TME). It is my hope that the articles found in this issue engage and sustain the discussion among members of our audience and the larger mathematics education community.

Associate Editors Tonya Brooks Allyson Hallman Catherine Ulrich Advisor Dorothy Y. White

As has been our mission at TME, we aim to provide a variety of perspectives on issues within the mathematics education community. For this issue’s invited editorial, Azita Manouchehri presents a study on using tasks to understand how individuals preparing to become mathematics teachers learn mathematics content. In this issue there are three articles that provide a variety of perspectives on issues within mathematics education. In our first piece, Alison Castro Superfine shows a model for teachers’ planning of instruction as teachers are using a reformbased curriculum. Denise Forrest’s piece discusses the connection between communication theory and mathematics instruction, using the theory to help explain teachers’ beliefs. Finally, Elliott Ostler, Neal Grandgenett, and Carol Mitchell show us a new approach for analyzing assessment instruments, one that takes a critical use at the use of rubrics.

MESA Officers 2008-2009 President Brian Gleason Vice-President Sharon O’Kelley Secretary Allyson Hallman

There are many people involved here at MESA that have helped in making this issue possible, and I would like to thank them at this time. I want to thank my colleagues who worked as Co-Editor and Associate Editors for this issue; their names appear at the top of the column to the left. Additionally, there are many individuals who have provided additional assistance this semester in a unique capacity. These individuals have been part of a seminar that introduces them to the work of our journal. They have done great work in whatever task we have given them: Zandra de Araujo, Eric Gold, Erik Jacobson, Hee Jung Kim, Hulya Kilic, Ana Kuzle, Laura Lowe, Anne Marie Marshall, and Laura Singletary. I cannot thank you enough for all that you did!

Treasurer Richard Francisco Colloquium Chair Dana TeCroney NCTM Representative Ryan Fox Undergraduate Representative Emily Ferris Cynthia Thomas

Ryan Fox 105 Aderhold Hall The University of Georgia Athens, GA 30602-7124

tme@uga.edu www.coe.uga.edu/tme

About the Cover On the front cover of this issue, we include sketches from Elliott Ostler, Neal Grandgenett, and Carol Mitchell’s piece on new forms of assessment. In their article, readers are asked to reflect upon ways to assess that students can verify the area of a circle: one such way is presented here on the front cover.

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

Fall 2008

Volume 18 Number 2

Table of Contents 3 Guest Editorial… Motivating Growth of Mathematics Knowledge for Teaching: A Case for Secondary Mathematics Teacher Education AZITA MANOUCHEHRI 11 Planning for Mathematics Instruction: A Model of Experienced Teachers’ Planning Processes in the Context of a Reform Mathematics Curriculum ALISON M. CASTRO SUPERFINE 23 Communication Theory: Another Perspective to Think About for Mathematics Teachers’ Talk DENISE B. FORREST 33 Rethinking mathematics and science assessment: Some reflections on Solution Dynamics as a way to enhance quality indicators ELLIOTT OSTLER, NEAL GRANDGENETT, CAROL MITCHELL 40 Submissions information 41 Subscription form

© 2008 Mathematics Education Student Association All Rights Reserved

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The Mathematics Educator 2008, Vol. 18, No. 2, 3–10

Guest Editorial…

Motivating Growth of Mathematics Knowledge for Teaching: A Case for Secondary Mathematics Teacher Education Azita Manouchehri There is consensus within the teacher education community that effective teaching hinges upon several factors. Those factors include the teacher’s knowledge of the subject matter, ways the subject matter could be manipulated to be made meaningful and accessible to learners, a deep understanding of learners and their developmental trajectories, and a perspective on short and long term trajectory of curriculum. Teachers need to learn how to select appropriate strategies by reflecting on what factors influence the adaptation of particular approaches when teaching specific concepts.

They also need to develop a disposition of inquiry and a professional attitude that allows them to continue to learn from practice (Hiebert et al., 2003). A major challenge in mathematics teacher education is fostering prospective teachers’ knowledge base in all these domains. As a means to meet this challenge, scholars have proposed that case-based tasks can serve as a powerful vehicle for advancing teacher learning and nurturing the desired dispositions (Richardson, 1996).

It is suggested that as teachers examine dilemma driven tasks and analyze teaching actions they not only learn about teaching but also develop conditional knowledge that is crucial to effective practice (Kishner & Whitson, 1997). In light of these perceived benefits, the use of written, video, or animated case studies in methods courses designed for teachers has gained considerable momentum in the past decade (Merseth, 2003). Certainly, sound analysis of teaching actions calls for deep reflection on the subject matter, the structure of the discipline, and its associated ontological and epistemological obstacles and issues.

This specialized body of teaching knowledge can be better nurtured when the contexts for learning are presented to students of teaching at the appropriate time and juncture. As such, content courses for teachers present an ideal environment for raising teachers’ awareness of the complexities of teaching the subject matter to children. When used in a mathematics Azita Manouchehri is a professor of mathematics education at Ohio State University. Her research has focused on the study of classroom interactions and evolution of mathematical discourse in presence of problem based instruction.

Azita Manouchehri

content course however, the tasks need to be crafted carefully so as to ensure that mathematics is treated soundly while allowing for the development of insight in both areas. Our research was an attempt at first developing and then examining the potential of the type of case-based tasks that could be used in a content course designed for teachers. One research question guided our research efforts: What impact do case-based tasks have on prospective teachers’ mathematics learning when used as instructional tools in a content course required for prospective secondary mathematics teachers?

In this article, we will first describe the task we designed and used as the research instrument in our study. Drawing from data collected from two teaching experiments, we will outline ways in which the task seemingly enhanced mathematical and pedagogical development of the participants.

Task Design Issues: Goals and Considerations Our task design was guided by two prominent scholarly voices as they pertain to the scope, goals, and the audience of this project: recommendations of literature for development of contexts for learning in mathematics teacher education and a situated cognition perspective on task design. In listening to the voice of mathematics teacher education scholars, both mathematical and pedagogical goals were considered.

Mathematically, we wanted the task to motivate reflection on connections among various (seemingly disjointed) mathematical ideas, engaging them in mathematical problem solving and critical mathematical analysis. Pedagogically, we wanted to increase teachers’ awareness of ways in which children’s work could impact instruction and curriculum decision making. The content of the task was chosen, bearing in mind these recommendations.

The structure of the task was chosen so as to align with the constitutive elements of situated learning. We wanted it to be authentic, dilemma driven, in order to be conducive for the development of discourse, collaboration, reflection, and critical thinking.

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In search of the center: Analyzing construction algorithms 7. Examine the three attached chapters from three The following problem was assigned to a group of different textbooks on triangles and circles. How geometry students who had used GSP in exploring does the content of these chapters differ from the geometry concepts in class. The students had spent five mathematics content that the group is addressing? instructional periods learning about and working on How do you account for these differences? What explorations concerning centers of triangle. assumptions can you make about the teacher of this Additionally, the group had learned in the previous group and her choice of curriculum? What is your session that in a circle, if a chord is the perpendicular assessment of these assumptions? bisector of another chord, then it is a diameter of the 8. Study the article: “A journey with circumscribable circle. The teacher of this group posed the following quadrilaterals” by Charles Worrall in the October problem in order to see whether students could use 2004 issue of Mathematics Teacher Journal their knowledge of diameter in a novel context.

[Volume 98(3)]. How would you compare the Problem: investigations that the author described in his article with those of students in this class? If you The center of a circle was accidentally erased. had to use an activity from that article to use with Define a procedure for locating the center. this group of students, which would it be and why?

9. The NCTM Professional Standards (1991) suggest A list of responses offered by several students that the teachers of mathematics must provide during a whole group sharing time of strategies is opportunities for students to engage in building presented below. conjectures, verifying their conjectures and debating the accuracy of those conjectures. In such 1. Test each of the methods presented below, and a setting, the role of the teacher is to facilitate the decide whether the suggested procedure leads to students’ development by orchestrating tasks and locating the center. Make a list of “mathematical assignments that extend the students’ assumptions” that each child has seemingly made understanding of the concepts, while helping them and issues that might need to be resolved in each realize the efficiency and elegance of ideas. case.

Imagine that you are the teacher of this group.

2. How are the students’ responses similar? How are Write an outline of a whole group classroom they different? Which of these responses draw on discussion that you might lead, along with any the same mathematical concepts? Justify your tasks that you will use to structure their work. responses.

Justify why the exploration would be helpful in 3. Rank the student responses along a continuum addressing the mathematical needs of the group. ranging from Irrelevant to Sophisticated (you may choose your own ranking categories if you find this Student 1:Inscribe a triangle ABC in the circle. range inappropriate). Explain the basis for your Then construct the perpendicular bisectors of the sides. ranking (as well as your categories, if you have The point of intersection of these perpendicular chosen to use a different ranking scheme). bisectors is the center of the circle (the circumcenter of 4. Hypothesize about the mathematical issues that the the triangle). teacher needs to address with the group. That is, Student 2:First inscribe an equilateral triangle in what are the central mathematical topics that she the circle. Since in an equilateral triangle incenter, needs to bring up and synthesize? What is the basis orthocenter, circumcenter, and centroid coincide, once for your choice? we find one of them then that point is the center of the 5. Decide the type of feedback the teacher might circle. So, construct the medians and mark the point of provide to each child based on his/her suggested intersection of the medians. That point is the center of method. How could the teacher expand the the circle. thinking of each student and help him/her justify Student 3:Locate three points on the circumference her/his approach? For instance, what questions of the circle. From those points construct tangent lines could the teacher ask? What extensions could the to the circle. In this way the circle becomes the incircle teacher offer? What example could the teacher use of the triangle we drew. Now, if we construct angle to counter the false assumptions? bisectors of triangle ABC, the point of intersection is 6. What other methods could be used to find the the center of the circle we started with. center of the circle?

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Motivating Growth of Mathematics Knowledge for Teaching

Student 4:Consider circle c. Draw chord AB of circle c. From either B or A, construct a line perpendicular to AB to form an inscribed 90-degree angle. Mark the point of intersection of the perpendicular line and the circle, label it as C. Now, we know the side opposite an inscribed right angle is the diameter of the circle that circumscribes the angle. All we need now is another diameter which we can find using the same procedure as before. The point of intersection of these two diameters is the center of the circle.

Student 5:First fold the circle in half. Then fold it again. Now we have two diameters that intersect at the center.

Student 6:Consider circle j and chord AB of j. Find the midpoint of segment AB, label it as M. Construct a circle centered at M with radius MB. What we have now is two circles that intersect at two points. The segment connecting the centers of the two circles is the perpendicular bisector of the line segment that connects the points of intersection of the two circles.

So, if we construct a perpendicular to AB from M, we know that line contains the center of j. If we connect A and B, to the point of intersection of perpendicular line and circle j, we get two right triangles (or one isosceles triangle). If we construct the perpendicular bisectors of the sides, we get the center (their point of intersection).

Student 7:Draw a regular polygon in the circle.

Actually draw a rectangle. Construct the diagonals, and the point of intersection of diagonals is the center.

Figure 1. Description of case-based task.

In light of these considerations, we designed “Locating the center,” a case-based task to be used as the research instrument in our study (see Figure 1). The case contained a range of student responses to a geometry problem. The reader was then asked to analyze the children’s work and to hypothesize about how the teacher of the group could proceed with her lesson in the presence of the children’s diverse ideas.

Mathematical Goals of the Task The mathematical focus of the task is on two topics central to the study of Euclidean and NonEuclidean geometries: triangles and circles. These topics are usually addressed as separate chapters in standards textbooks. Frequently, students leave a geometry course without realizing the connections between them. The goal of the task is to help students develop a sense for how concepts that make up the field are closely related to each other and are sufficiently self-contained. Usually if not always, this kind of conceptual unity is not nurtured in undergraduate mathematics programs. The following list summarizes specific goals addressed by each guiding question.

Question 1 Question 2 Question 4 Question 6 Question 8

Mathematical problem solving Mathematical connections Content coherence and unity Mathematical analysis Extending mathematical inquiry and content specific pedagogical reasoning

The task contains a deliberate range of learner responses. Each of these responses could lead to a

Azita Manouchehri

series of important mathematical explorations as listed below.

• Exploring properties of centers of a triangle (Response #1) • Relationship between the lengths of inradii/circumradii and the area of triangle (Response #1, #2) • Trisecting an arc (Response #3) • Construction of tangents to circles and tangent circles (Response #3) • Relationship between the size of the equilateral triangles and their respective inradius and circumradius (Response #1,2, 3) • Inversions on circles (Response #4) • Angular measures of chords of circle (Response #4, 5) • Properties of tangent circles (Response #6) • Circumscribed quadrilaterals and their properties (Response #7) Pedagogical Goals of the Task Pedagogical goals of the tasks include assisting future teachers to develop an understanding of the connections between student learning and instructional decision making. The task was structured to allow for pedagogical problem solving. The reader is asked to analyze learners’ responses, hypothesize teaching actions, design assessment tasks, and develop activities. Furthermore, by asking them to explain, justify, and defend their choice of representations, assessment, and intervention, we envisioned that the task would provide opportunities for the readers to engage in pedagogical reasoning.

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Lastly, the task introduces teachers to professional journals and national professional standards (in the US this includes documents published by the National Council of Teachers of Mathematics). The following list summarizes the pedagogical goals addressed by each guiding question.

Question 3 Question 5 Question 7 Question 9

Content specific pedagogical decision making Content specific pedagogical analysis Curricular analysis and connections to student learning Content specific pedagogical reasoning and decision making

Context and Research Design The primary goal of our exploratory study was to investigate teacher learning in the presence of casebased tasks when used in a mathematics content course. Using a teaching experiment methodology (Steffe & Thompson, 2000), data was collected in two different sections of a course titled Modern Geometry, an advanced mathematics course required of all undergraduate and graduate mathematics majors pursuing a degree in secondary mathematics teaching.

Each teaching experiment lasted two and half weeks (five, 75-minute long sessions) and involved 40 prospective secondary teachers. The students enrolled in this course were either of junior or senior academic standing. All students had completed a minimum of 12 hours of coursework in general pedagogy, and a minimum of 21 hours of mathematics coursework prior to taking this class. The teaching experiment commenced during the second month of instruction, after the participants had examined both topics involved in the task.

The teaching sequence in each class consisted of a particular routine. The participants were assigned the task as a homework activity. The follow-up discussion in each teaching experiment included a large group discussion at the beginning of the session. During this time the participants were encouraged to offer their initial reactions to the case, share their responses to the guiding questions, ask any questions that they might have regarding the content and/or expectations of the task, their impressions of the children’s work and ways in which the teacher of the group could proceed with her instruction. A small group activity followed the large group discussion. A final whole group discussion allowed for observable evidence of learning including, synthesizing and formalizing processes that could determine a shared level of mathematical and 6

pedagogical analysis by the group. We used the initial sharing time as an opportunity to collect base line data on the participants’ initial approach to task analysis and used that data as a means to trace changes in their work as their interactions with the task intensified.

The decision to rely on two different teaching experiments was to verify our interpretations of the participants’ work. By collecting data from two different groups we would be in a better position to account for multiple variables that could impact the participants’ interactions with the task including: their mathematical background and experiences, interest levels, classroom routines and instructional practices to which they might have been accustomed.

In each class, two video cameras were set to capture both large group interactions as well as two targeted small groups. The small groups were selected randomly and the same groups were videotaped throughout the teaching experiment. All videotapes were transcribed and used in the data analysis.

Data Analysis In analyzing the impact of the task on the participants’ activities and learning, we considered two intertwined aspects of their work, including: (1) Interactions with the task—Issues that the participants raised about and/or extracted from the activity; (2) Mathematical activities of the participants—Ideas and problems the participants explored. Hence, data analysis was organized around these two key categories.

Participants’ Interactions with and Reactions to the Task In determining the participants’ particular approach to task analysis, we focused on their verbal exchanges during the small and large group discussions. We considered whether the participants showed an interest in learning about teaching, the learners, curriculum and mathematics by the type of questions they asked the facilitator or each other. Their comments regarding children’s work were also coded in order to trace sensitivity, or lack thereof, to relevant mathematical and pedagogical issues.

Participants Mathematical Analysis and Learning In seeking evidence of learning, we considered the participants’ modes of production (Balacheff, 2000) during the discussions as an indicator of learning.

Accordingly, we sought instances of mathematical action, conjecture formulation, and validation processes during each session. We considered situations of action to include instances of problem Motivating Growth of Mathematics Knowledge for Teaching

solving, problem posing, attempts at theoretical constructions, testing a method, or judging merit of ideas by reference to mathematical knowledge.

Instances of conjecture formulation included articulation of relationships among mathematical ideas, children’s solutions, and suggestions that peers offered.

Additionally, in seeking evidence of validation processes, we considered whether the participants referenced mathematical theory when analyzing the problem and its extensions, examining children’s work or judging the quality of mathematical arguments offered in groups.

Results In both sites, the classroom activities followed four phases: Primitive pedagogical theorizing, facilitator modeling, mathematical problem solving and curriculum analysis, and pedagogical inquiry.

Phase I (Approximately 35 Minutes) During the first phase of the activity, the participants were reluctant to engage in discussions about mathematics, showing a tendency to focus on pedagogical theorizing. Their suggestions, however, were not grounded in evidence or supported by a rationale for choice. They seemed confident in their assessment of children’s strategies and rarely elicited explanations and/or guidance from each other or the facilitator. They made brief and trivial references to children’s solutions and characterized them as right or wrong without offering a rationale for their choice.

None of the participants tried to formalize or justify their assessment beyond stating their personal preferences regarding the classroom environment they felt were conducive to building children’s confidence.

When asked to comment on how the teacher might decide which of the students’ responses to pursue in class, only one participant in both cohorts was willing to commit to a particular method. The facilitators’ comments during this phase were aimed at confronting the participants’ assessment of children’s work and the suggestions they offered for how the teacher might organize subsequent classroom instruction.

Phase II (Approximately 40 Minutes) In structuring individual and group analysis, the second phase consisted of facilitator modeling.

Choosing one of the children’s solutions the participants had labeled as incorrect (student 5) as an example, the facilitator spent approximately 40 minutes of the first session in each class describing her interpretation of one child’s method, and ways in which it connected to other solutions as well as Azita Manouchehri

different mathematical concepts. She listed additional questions the teacher could ask the child either to gain additional insight into his thinking or to advance his work.

Phase III (Approximately 220 Minutes) Following the modeling episode, the participants were instructed to examine children’s solutions again in small collaborative groups. A structure for group deliberation was also set to reach consensus on their assessment of children’s work as well as their hypothesis concerning pedagogy. They stated and explored several extension problems that could be shared with children. They also examined different textbook chapters to find places where specific topics could be shared with children in instruction. Three class sessions (210 minutes) were devoted to working on specific problems that different individuals had proposed as extensions to be used with children. The facilitator guided the discussions, offered explanations when asked, and continued to challenge over generalizations that the participants made.

Phase IV (Approximately 75 Minutes) The last phase of the participants’ activities focused on synthesizing and formalizing pedagogical and mathematical analysis of the case. The participants began to ask each other and the facilitator questions about curricular guides that could inform their practice.

During the last cycle of the activity, a major component of the participants’ discourse included articulation of concerns about their own knowledge of mathematics.

Tables 1 and 2 summarize the different types of comments that the participants made during each phase of their case analysis experience. The major categories of comment types included: Pedagogical, mathematical, eliciting support and feedback, and declarative.

Pedagogical themes included instances of: (1) hypotheses about instructional moves (i.e. the teacher should ask that each child go to the board and explain his method to the group), (2) references to the impact of instruction on children’s work (i.e. “Maybe the teacher should have told them to use only one method for solving the problem”), (3) references to the type of evidence they used to support their pedagogical decision making (i.e. The drawing is not accurate so the teacher should be sure to help them draw it right) and, (4) references to the impact of children’s work on instruction (i.e. children are not ready to move on to a

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Table 1 Typology of Comments: Pedagogical and Mathematical Comment Type Episodes of mathematical question posing (Is there a way to locate the incircle of a quadrilateral?)

Group 1 61

Group 2 54

Episodes of conjecturing about new mathematical relationships (I think there is a relationship between the area of inscribed regular polygon and its circumradius)

82

96

Episodes of eliciting explanations concerning connections among mathematical concepts (Can we connect the study of triangles to other polygons?)

75

82

Episodes of eliciting explanations concerning relevant mathematical theorems they could use (Is there a way to find the area of pentagon?)

234

186

Episodes of confronting peer’s analysis when discussing children’s work or extension problems (But this works for only one case! You need to generalize it)

187

198

Referencing evidence from children’s work when hypothesizing about teaching actions (It would be useful to start with the informal approach first, the folding paper part and then connect it to S7’s suggestion before moving on to 1st and 2nd methods)

56

61

Offering mathematical explanations on children’s solutions (When he says fold it twice he is finding two diameters of the circle, so he is right)

78

72

Statements indicating having gained new mathematical insights—Aha! (I know better why triangles are so important to geometry)

83

72

Number of mathematical problems on which participants worked*

28

25

Table 2

Declarative Statements

Eliciting Guidance

Typology of Comments: Eliciting Support and Declarative Statements

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Statement of need for additional guidance on pedagogical decision making Statement of need for additional information on curriculum Statement of need for additional guidance on mathematics Statement of need for additional information on learners and how they learn Statements of concern about the ability to teach

Phase I & II M=2 SD = 4.6 M = 12 SD = 4.2 M=3 SD = 9.08 M=0 M=0

Statements of concern about the ability to make sense of children’s work

M=0

Statements of concern for knowledge about appropriate decision making

M=0

Statements of concern about finding appropriate resources

M=0

Statement of concern about the quality of their teacher training

M=0

Phase III M = 32 SD = 4.2 M = 21 SD = 0.04 M = 89 SD = 0.09 M = 13 SD = 0.19 M = 12 SD = 0.16 M=0 M = 24 SD = 2.8 M = 18 SD = 1.88 M = 19 SD = 0.92

Phase IV M = 48 SD = 0.92 M = 52 SD = 4.8 M = 22 SD = 0.87 M = 76 SD = 2.11 M = 43 SD = 5.03 M = 76 SD = 6.4 M = 32 SD = 4.11 M = 29 SD = 2.01 M = 81 SD = 3.3

Motivating Growth of Mathematics Knowledge for Teaching

different topic, the teacher should go back and review her lesson).

Eliciting support included instances of statements of need for additional information on: (1) classroom conditions (i.e. How did the teacher organize classroom activities? What is the teacher’s curriculum?), (2) mathematics (i.e. Is this mathematically sound?), (3) learners’ thinking (i.e. Is this method common to all children this age?) and (4) pedagogical decision making (i.e. Is this approach developmentally appropriate? Is this question adequate to be posed to this group?).

Mathematical themes included instances of mathematical analysis consisting of: (1) references to mathematical theory in analyzing children’s work (i.e.

What this student is suggesting is related to the theorem that says the diameter is the perpendicular bisector of a chord), (2) references to the impact of children’s work on instruction with a focus on identifying mathematical significance of the ideas presented by a child (i.e. I think the teacher should ask the students to study this solution first cause they can see there are several things that need to be resolved), (3) references to mathematical connections among ideas (i.e. Circumscribing a triangle is the same as finding the perpendicular bisector of three chords, so the sides of the triangle are actually three chords of the circle that intersect at vertices).

Declarative statements included instances of volunteered remarks regarding self knowledge (i.e. I am not sure I know the subject well enough now; how do we decide which textbook to use?), learning or lack thereof (i.e. I used the last student’s method to solve this problem), particular needs (i.e. Why can’t we do this type of activity more often?), and projected plans for professional development either in mathematical or pedagogical domains (i.e. Maybe we should start building a resource book; I really should work on learning the software more.)

Following the modeling episode by the facilitator (Phase II), mathematical references that the participants made when they analyzed children’s work increased significantly. The participants asked more questions about theories they could use or ways in which children’s methods related to other mathematical ideas. Indeed, the participants’ declarative statements were indicative of the impact of the task on raising the participants’ sensitivity to the quality of their own knowledge and ways in which they could improve their capacity to teach.

Azita Manouchehri

The Participants’ Mathematical Work The number of mathematical problems on which the participants worked averaged 21 per group. The sheer number of problem posing, conjecturing, and explaining episodes is remarkable considering that the participants had rarely practiced such processes in their regular classroom. The children’s solutions were used as a springboard for extending the study of triangles to circumscribed and inscribed polygons. The following is a partial list of common problems on which the participants worked during the extended problem solving episode.

1. Construction of tangent line to a circle, and tangent circles 2. Constructing three kissing circles 3. Relationship among the radii of kissing circles 4. Properties of quadrilaterals

circumscribed

and

inscribed

5. Finding areas of regular polygons using the measure of inradius and circumradius 6. Star Trek Lemma (formulated by the group) 7. Bow-tie theorem (formulated by the group) 8. Determining the interior angle sum of polygons using the Star Trek lemma 9. Properties of Pedal and Orthic triangles 10. Derivation of the extended law of sines using circumcircles 11. Describing the area of triangle and quadrilateral using the law of cosines Discussion Our data indicate that the case-based task successfully engaged teacher candidates in doing mathematical inquiry and pedagogical analysis.

Evidence of mathematical learning from the task was manifested not only in the number of problems that the participants explored in the course of their case analysis sessions, but also in the amount of mathematical information shared as they made and verified conjectures. Further evidence was evident in how the participants justified their assessment of children’s work. They elicited and articulated connections among solutions and detailed ways in which these connections could be made public in instruction. They elicited information about, and also identified mathematical structures that could be used when discussing specific problems with children.

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The participants’ comments revealed that their experience with the task helped them to realize connections between a teacher’s own mathematical knowledge and his or her pedagogical choices. The significant number of participants’ requests for theoretical guidance on both mathematics and pedagogy is a strong indication of their interest in learning. Additionally, the large number of statements of concerns they raised about their own knowledge, the quality of their preparation, and their ability to teach further support our perspective that the goal of the task in raising teachers’ sensitivity to complexities associated with pedagogical decision making.

Lastly, a careful analysis of case-based tasks in order to maintain the integrity of both mathematics and pedagogy is a time intensive process. Discussion of the center activity took five class sessions. Indeed, if we had the opportunity to pursue each of the mathematical and pedagogical questions that the participants raised, we could have easily doubled the length of time spent on each case. In making decisions about what to pursue with the teacher candidates, we focused mainly on mathematical objectives of the course. Considering the amount of learning developed as the result of exposure to only one case-based task, we are prepared to conjecture that if used systemically facilitators will be

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in a better position to strike a balance in instruction when simultaneously addressing mathematical and pedagogical issues in a course.

References Hiebert, J. , Morris, A. K., & Glass, B. (2003). Learning to learn to teach: An “experiment” model for teaching and teacher preparation in mathematics. Journal of Mathematics Teacher Education, 6(3), 201–222.

Kirshner, D. & Whitson, J. A. (1997). Situated cognition: Social, semiotic, and psychological perspectives. Mahwah, NJ: Erlbaum.

Merseth, K.K. (2003). Windows on teaching math: Cases of middle and secondary classrooms. New York: Teachers College Press.

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

Richardson, V. (1996). The case for education: Contemporary approaches for using case methods. Boston: Allyn & Bacon.

Steffe, L. P. and Thompson, P.W. (2000). Teaching experiment methodology: Underlying principles and essential elements in Research design in mathematics and science education. In R. Lesh & A. E. Kelly (Eds.), Research design in mathematics and science education (pp. 267-307), Hillsdale, NJ: Erlbaum.

Motivating Growth of Mathematics Knowledge for Teaching

The Mathematics Educator 2008, Vol. 18, No. 2, 11–22

Planning for Mathematics Instruction: A Model of Experienced Teachers’ Planning Processes in the Context of a Reform Mathematics Curriculum Alison Castro Superfine Planning is an important phase of teaching, during which teachers make decisions about various aspects of instruction that ultimately shape students’ opportunities to learn. Prior research on teacher planning, however, fails to adequately describe experienced teachers’ planning decisions, and is unclear about the extent to which teachers use curriculum materials to inform their decisions. Using data from 6th grade mathematics teachers’ use of curriculum materials, this study presents a discipline-specific model of experienced mathematics teachers’ planning. The proposed model provides a lens for understanding the nature of teachers’ planning decisions, and the conditions under which such decisions change over time.

Planning is an important and often underappreciated aspect of teaching practice, when teachers make decisions that ultimately impact students’ opportunities to learn (Clark & Peterson, 1986; Floden, Porter, Schmidt, Freeman, & Schwille, 1980; Stigler & Hiebert, 1999). Planning commonly refers to the time teachers spend preparing and designing activities for students. From tasks and activities to instructional practices employed during lessons, teachers need to consider a variety of aspects of their instruction before students even enter the classroom. Teachers need to pay careful attention to designing their lessons; “effective teachers understand that teaching requires a considerable effort at design.

Such design is often termed planning, which many teachers think of as a core routine of teaching.”

(Kilpatrick, Swafford, & Findell, 2001, p. 337).

Reviews of teacher planning and decision-making further emphasize the centrality of planning processes in teachers’ practice (Clark & Peterson, 1986; Clark & Yinger, 1977; Shavelson & Stern, 1981). Despite this general agreement about the importance of planning, few researchers have explicitly examined the precise ways in which teachers plan for mathematics instruction.

Prior research related to teacher planning presented a “linear” or “rational” model of teacher planning by delineating the various lesson elements teachers Alison Castro Superfine is Assistant Professor of Mathematics Education and Learning Sciences at the University of Illinois at Chicago. Her current work focuses on teacher-curriculum interactions and elementary preservice teacher education..

The research reported in this article is based upon the author’s doctoral dissertation under the direction of Dr. Edward A. Silver at the University of Michigan.

Alison Castro Superfine

generally considered when planning their lessons (Popham & Baker, 1970; Taylor, 1970; Tyler, 1950).

Under this model, teachers first consider the learning activities that take into account students’ interests and abilities, then the learning goals and objectives of the lesson, and finally the evaluation procedures to be used during the lesson. Some researchers later argued that linear models of teacher planning do not adequately describe experienced teachers’ planning processes and do not account for the complexities inherent in mathematics teaching. Rather, a variety of additional factors, such as teachers’ experiences and conceptions of mathematics teaching and learning, also influence the ways in which teachers plan their lessons (Kilpatrick et al., 2001; Leinhardt & Greeno, 1986; Yinger, 1980).

More recent research on teachers’ planning does not clearly indicate the extent to which teachers draw from curricular resources when making planning decisions. Moreover, there is even less research that focuses explicitly on teachers’ planning in the context of the reform mathematics curricula that provide much of the instructional design for teachers (Kilpatrick et al., 2001; Trafton et al., 2001). Such reform curricula are increasingly prevalent in classrooms in the United States, embodying new modes of instruction (Reys, 2002). The challenges of planning lessons using such curricula may be somewhat different from the challenges of planning lessons with more conventional mathematics curricula. Thus, exploring how teachers plan in the particular context of reform curricula is critical if mathematics educators want to understand this important phase of teaching.

In order to explore the theoretical considerations presented in this article, the author has selected 11

examples of teachers’ planning routines taken from a larger study examining experienced 6th grade teachers’ use of the Connected Mathematics Project (CMP) materials. CMP is a middle school reform curriculum developed in response to the National Council of Teachers of Mathematics (NCTM, 1989) Curriculum and Evaluation Standards. By emphasizing the discovery of mathematical ideas through tasks, CMP encourages students to make connections between topics and important mathematical ideas in order to help them apply their learning to real-world contexts.

The larger study focused on how four teachers used the CMP teacher’s guide in both the planning and enactment of their lessons. The teacher’s guide includes summaries of the mathematical content, specific questions to ask students throughout a lesson, and examples of student errors.

Teacher Planning Past research on teacher planning focused on the broad features and order of teachers’ planning decisions and considerations, with minimal attention given to the particular ways that teachers considered engaging students with the content. Adhering to a linear model of teacher planning, Tyler (1950) and Popham and Baker (1970) found that teachers specified ordered objectives, selected learning activities, organized learning activities, and specified evaluation procedures. Similarly, Taylor (1970) found that teachers sequentially considered four aspects of a given lesson when planning: materials and resources, students’ interests, the aims and purposes of teaching, and evaluation. Implicit in these studies is the notion that teachers create their own objectives and activities for students, which may reflect the design of the types of curriculum materials available to teachers at the time in which these studies were conducted.

In a later study on teacher planning, Brown (1988) examined the extent to which 12 teachers adhered to a linear model of planning. Focusing on teachers’ planning in different subject areas, Brown found that teachers tend to use curriculum materials and the objectives expressly stated in these resources as a starting point for their planning. She noted, “teachers operate as curriculum implementers and not curriculum planners as they consider objectives already written in curriculum guides” (p. 79). Yackel and Cobb (1996) noted that planning decisions about ways of facilitating students’ activity in a history or English classroom are considerably different from those in a mathematics classroom. Nevertheless, Brown’s (1988) study points to the integral role of curriculum materials in the 12

process of teachers’ planning, which was not clearly addressed by proponents of the linear planning model.

Some researchers have focused on the role of curriculum materials as a resource for teachers to draw upon when making planning decisions. For example, McCutcheon (1981) found that when planning for daily lessons, teachers tend to rely heavily on suggestions in the teacher’s guide. In a study of one teacher’s planning throughout the school year, Clark and Elmore (1981) found that curriculum materials are primary resources in the teacher’s planning. Similarly, Smith and Sendelbach (1979) studied this issue at the level of teachers’ unit planning. They found that teachers tend to construct a mental image or plan of the unit and then supplement their plan with notes based on the suggestions in the teacher’s guide. Additional research has highlighted various factors that influence how teachers use these curricular resources in the planning and instructional processes (Ben-Peretz, 1990; Cohen et al., 1990).

Research points to experience as a potential factor that influences teachers’ planning. Such research suggests that experienced teachers have more extensive and well-organized knowledge of both pedagogy and student learning, making them more flexible and attentive to the nature of the students’ learning opportunities that they create (Borko & Shavelson, 1990; Leinhardt & Greeno, 1986; Livingston & Borko, 1989). For example, researchers found that when planning, experienced teachers make more extensive mental plans than written plans and rely less on curriculum materials than their less experienced counterparts (Bush, 1986; Leinhardt, 1983; Livingston & Borko, 1990).

As teachers’ experience with a particular curriculum program increases, they become more familiar with the details, nuances, and presentation of the specific mathematics content in the curriculum.

Thus, they may have developed daily routines in planning and engaging with the curriculum in particular ways. For the purposes of this study, experienced teachers are identified as having at least five years of teaching experience and at least three years experience using a certain curriculum program.

Other possible factors influencing teachers’ planning decisions are the various conceptions teachers bring to bear on their practice. These conceptions contribute to what Remillard and Bryans (2004) refer to as teachers’ orientation toward the curriculum.

Experienced teachers have refined their conceptions of mathematics teaching, learning, and curricula because they have spent considerable time formulating and Teacher Planning

applying these conceptions in the classroom. Teachers’ conceptions of mathematics content are also important for understanding how teachers engage with the curriculum (Lloyd, 1999; Remillard, 1999; Remillard & Bryans, 2004). Moreover, it is important to understand the extent to which teachers’ conceptions of mathematics teaching and learning align with the ideas about teaching and learning underlying the curriculum (Manouchehri & Goodman, 1998; Remillard & Bryans, 2004).

While linear models are useful for capturing certain basic elements of teacher planning, these models fail to account for an array of factors that have been identified as influencing teachers’ planning processes, such as curriculum materials, teaching experience, and the various conceptions teachers have about teaching and learning. Although teachers have a variety of conceptions, this article will focus on teachers’ conceptions of mathematics teaching, learning, and curricula for the purposes of this study.

The Work of Reform-Oriented Mathematics Teaching Although the various factors highlighted by researchers as influencing teachers’ planning are essential to consider when developing a new planning model, it is also important to consider the demands and characteristics of the particular discipline in which such planning occurs. Most of the research discussed previously does not explicitly focus on planning or instruction in the context of a specific discipline.

Mathematics teaching, specifically in the context of reform mathematics curricula, involves particular demands and challenges that may shape teachers’ planning processes.

The model developed in this study is grounded in a specific conception of mathematics teaching, drawing from the works of Stein, Smith, Henningsen, and Silver (2000), Lampert (1992, 2001), Clark and Elmore (1981), and Lampert and Ball (1998). These researchers describe in detail the nuances and complexities of mathematics teaching in a way typically embodied in reform curricula. Teachers need to consider the mathematical content and ways to engage students in discussion about the content, while simultaneously guiding students towards a particular goal. For example, during planning and instruction teachers modify tasks and ask high-level questions in order to promote students’ understanding of the underlying ideas and concepts.

To support students’ understanding, teachers need a variety of pedagogical skills. For example, teachers Alison Castro Superfine

need to be able to resist the urge to tell students how to work on the content so that they provide students with adequate time to think through what they are asked to do (Donovan, Bransford, & Pellegrino, 2000).

Anticipating student responses and having an awareness of common errors can also help teachers effectively respond to and redirect students’ discussions (Chazan & Ball, 1999; Fennema, Franke, Carpenter, & Carey, 1993). In addition, modifying tasks based on students’ current knowledge and abilities may help teachers to be mindful of the cognitive activity in which students should be engaged (Stein et al., 2000). Employing such instructional practices to facilitate student learning in accordance with the principles of reform mathematics, however, requires extensive and demanding work on the part of teachers. Therefore, teachers face several challenges supporting students at such a high level of mathematical activity.

These challenges to teachers’ work are considered the “problems” in mathematics teaching (Lampert, 1992, 2001). “Problems” in mathematics teaching refer to the work teachers do to further students’ understanding of mathematics. This includes facilitating students’ discussion around the content, continuously pressing students to explain their ideas and to communicate with each other, posing questions, and selecting solution strategies to present to the class.

Teachers need to make important and often simultaneous decisions in ways that do not undermine students’ thinking or the mathematical opportunities afforded by the content in reform curricula. Hereafter, “problems” will be used to refer to the challenges and decisions teachers face during mathematics teaching, as described in Lampert (1992, 2001).

Teaching includes not only the physical act of teaching, during which teachers interact with students, but also includes the time teachers spend preparing for these interactions (Jackson, 1966, 1968). Planning for the demands and challenges of mathematics instruction requires teachers to engage in a planning process that involves the development of skeletal frameworks rather than detailed scripts for teaching lessons (Rosebery, 2005). In particular, teachers must identify a particular mathematical topic to discuss and the means necessary to cover that topic, without necessarily delineating the precise steps needed to teach that topic. Therefore, planning for reformoriented instruction requires teachers to select specific topics or concepts and to identify particular activities, instructional strategies, and suitable materials for discussing and engaging students with the topics or 13

concepts. In addition, during planning, teachers must anticipate potential problems that may arise during instruction and then make decisions regarding how to manage these problems. Planning problems refer to the considerations and decisions teachers face when both planning for and anticipating what will happen during a specific lesson.

A Model of Teacher Planning There are several elements that a model of teacher planning in the specific context of reform-oriented mathematics instruction must capture. These elements include the approaches to mathematics teaching and learning embodied in reform curricula and teachers’ various experiences and conceptions they bring to their planning decisions. Such a model must particularly capture how these elements interact with each other and ultimately influence teachers’ planning decisions.

The concept of planning problems is well suited for developing a conceptualization of planning because it incorporates the influence of these different elements.

The model developed in this study draws heavily on the notion of planning problems and highlights the various elements that drive the emergence of such planning problems in teachers’ practice.

As discussed previously, the work of reformoriented mathematics teaching includes facilitating and supporting students’ understanding in ways that will neither constrain students’ opportunities to learn nor undermine students’ thinking. Consequently, teachers need to plan for engaging in this sort of work during instruction. For example, teachers must plan questions they will ask students that will guide students’ thinking about the content without giving them too much information, while also encouraging students to explain their ideas (Hiebert & Wearne, 1993; Maher & Martino, 1992; Moyer & Milewicz, 2002). Teachers need to anticipate different solutions students may offer, as well as alternative ways of thinking about a task, in order to facilitate students’ learning and discussion of these strategies in ways that foster a shared understanding of the ideas (Kilpatrick, 2003; NCTM, 1991). Teachers also need to anticipate potential errors in order to respond appropriately and help students learn from incorrect solutions. Finally, teachers should be prepared to change or modify a task, in the case that students are struggling with a

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concept, in ways that both preserve the task’s complexity and help students learn from working on the task (Stein et al., 2000). Though their intended plans often differ from their enacted plans, teachers need to carefully plan their lessons and anticipate how students will interact with the content during implementation in order to further students’ understanding of different mathematical ideas. In this way, planning problems can be considered to be the anticipation of instructional problems.

Planning problems are inherently different for each teacher depending on the teacher’s experiences, ideas, and conceptions, as well as the curriculum being used.

For instance, asking higher-order questions that press students to justify and explain their thinking is also only a planning problem for teachers who view the use of such questions as contributing to student understanding. Planning problems also may be quite different for a teacher who adheres to a more conventional conception of mathematics teaching.

Such a teacher may need to determine how to incorporate opportunities for students to practice the application of certain skills and procedures within a curriculum the teacher perceives as deficient.

Furthermore, a teacher with experience implementing multiple curriculum programs must consider how to apply what they know of other mathematics curricula to their planning with a specific Standards-based curriculum. Therefore, planning problems appear to be a useful lens for understanding the relationship between teachers’ experiences, conceptions of mathematics teaching and learning, and the curriculum used in the planning process.

As the Planning for Mathematics Instruction (PMI) Model in Figure 1 illustrates, teachers’ various conceptions influence their engagement with curriculum materials during planning. Additionally, the conceptions influence the type of planning problems teachers encounter in the course of their work. These planning decisions then influence teachers’ lesson enactment and the types of learning opportunities they create with students during instruction. This lesson enactment then informs experiences and shapes the information teachers have to draw upon when they plan for and enact the lesson in subsequent classes or school years.

Teacher Planning

Figure 1. Planning for Mathematics Instruction Model.

As teachers’ conceptions help to frame the planning problems they encounter, their various conceptions also serve as a resource for managing planning problems that arise in the course of their planning. When confronted with planning problems, teachers draw upon their previous experiences with the task and their ideas about what it means to learn and teach mathematics in order to make decisions about ways of managing these different planning problems.

In some cases, teachers may also draw from the information and support provided within the actual mathematics curriculum materials. Notably, the CMP teacher guide provides the means for teachers to manage certain planning problems, such as anticipating solution methods students may generate, questions to ask students, and errors and misconceptions students may have in relation to a task. Though, the extent to which teachers use curriculum materials to inform their planning decisions is largely dependent on the nature of their conception of the curriculum.

Teachers’ Planning Practices The following teacher examples illustrate the various ways in which experienced teachers, with distinct conceptions towards the CMP curriculum, can engage with reform curriculum materials in the course of their planning and demonstrate how the PMI Model applies to actual teaching practices. Alicia, Richard, and Susan were selected for the present study because their planning decisions and considerations captured the range of variation in planning routines and problems encountered.1 All three teachers were teaching sixth grade at the same middle school at the time the study took place. As this study uses teaching and curricular experience to define experience, Alicia has been teaching 16 years, using CMP for 3 years; Richard has been teaching 17 years, using CMP for 10 Alison Castro Superfine

years; Susan has been teaching 6 years, using CMP for 3 years. These teachers were observed planning and enacting the same unit, Bits and Pieces III, which focuses on operations with rational numbers (Lappan et al., 2006). The data collected includes interviews with teachers prior to and immediately following classroom observations to understand teachers’ lesson plans and their reflections on their lesson enactments. Although post-hoc examinations of teachers’ planning were conducted, teachers were interviewed the same day of the observation in order to increase the accuracy of teachers’ responses. Field notes from classroom observations and artifacts from teachers’ lesson planning are additional data sources.

Prior to using CMP, Alicia used a more conventional mathematics curriculum for 13 years and claims to strongly believe that CMP does not provide students with sufficient opportunities to practice skills and procedures. She views her role as a teacher as that of an intervener, providing direct guidance and explicit instructions for students, which is evident in her planning decisions. Alicia first reads through the entire lesson in the student book and solves the task: “And I do the whole [lesson] myself, you know without looking at the teacher’s guide or anything…And then I have an idea about what might be tricky and what might not be.” Alicia says she then looks through the accompanying teacher’s guide and decides the most appropriate course of action, taking into consideration both who her students are as learners and the constraints of class time. “And I kind of pick and choose what I think will work best with my students…

And most times I won’t use all of it.” In general, Alicia claims to regularly modify the suggested content of the lesson and the suggestions for how students should engage with the content during the lesson. Her view of CMP, and ultimately her conception of what students 15

should learn, is readily apparent in the nature of her lesson additions and deletions: “I’m usually thinking about what I need to add…Like word problems I usually skip…and substitute other practice problems that I feel like need to be emphasized more.”

Richard, on the other hand, claims to be a strong proponent of the instructional approach embodied by the program. Having taught CMP for over 10 years, he believes that CMP is a desirable alternative to a conventional mathematics curriculum because it allows him as a teacher to facilitate student discussion and to play less of a central role in the classroom discussion: “I would characterize it less as teacher-driven…and more kid-driven…I think the focus becomes--to me I’m giving up being the center of attention.” When planning, to make sure that he understands the content for himself, Richard says he first reads through the lesson and solves the task that he will use with students. He then tries to ascertain, from reading the student book, exactly what the lesson is about, the “big idea” students are to come away with, and also places where students may struggle or misconceptions students may have while working on the lesson: “…[J]ust so I get a sense…of what problems they’re going to struggle with. Just looking at the answer doesn’t help me out. If I actually work through the problem, that gives me a sense of where they’re going to struggle.” Richard says he uses the teacher’s guide during planning only when an idea or concept or even the wording of a particular question in the student book is unclear. The student book provides him with the necessary information he needs for enacting the task during instruction.

I think if I would read through the [teacher] manual and not read through the kid edition, I’d just feel like I’d be at a disadvantage to know what to expect from the kids…It tells me as a teacher what to know conceptually…but it doesn’t help me understand quite how I think the kids are going to react to what I’m asking them to do.

In general, Richard claims to plan for using most, if not all, of the lesson elements as described in the curriculum.

Although both Alicia and Richard use the curriculum materials for the content features of the lessons (albeit to a limited degree in Alicia’s case), both teachers appear to have different views of the curriculum, and thus use the materials in very different ways when planning. While Alicia “picks and chooses” from the lesson suggestions what she considers relevant and important for her students, Richard tends to plan for enacting the lesson as described in the 16

materials. Notably, both of these teachers do not follow the lesson suggestions in a prescriptive fashion, including Richard who seems to only rely on the materials for content features of the lessons and not for suggestions as to how to engage students with the content. By engaging with the materials in such an adaptive, or even modified fashion, Richard and Alicia leave the lesson open to interpretation, making room for their conceptions of mathematics teaching and learning to inform their lesson planning.

In contrast to the other two teachers, Susan, having taught for 6 years and used CMP for 3 years, appears to adhere closely to the lesson and corresponding suggestions in the curriculum. She believes the instructional approach espoused in CMP is an ideal way to support students’ mathematical development and their ability to communicate their understanding.

She views her role as that of a referee, mediating students’ discussions of their proposed solution strategies during instruction: And so sort of mediating that discussion is sort of the biggest part because…learning of the strategies is supposed to take part amongst themselves. So it’s like I have very little, this is how it works, you know, it’s more, ok what are your ideas? Let’s put them together.

When planning for a lesson, Susan follows the suggestions in the teacher’s guide almost as a script for the lesson. She will first read through and solve the task in order to think about how students will approach and solve the task, as well as to consider potential solution strategies that may arise. After working through the content, Susan says she reads through the longer lesson summary in the teacher’s guide to understand the overall direction and purpose of the lesson. In addition to reading the detailed teaching notes, she reads both the suggested questions and answers to the task, and then includes these elements in a slideshow presentation she uses during instruction.

Though, she only includes suggested questions in her lesson if she decides they are worthwhile and appropriate for her students: “So…whenever there are suggested questions, I see ok, is this a meaningful question for my students?” Susan says she reads through the lesson suggestions to ensure she does not extend the discussion of a concept further than what is expected or stated in the materials, and also to gain an overall sense of the lesson and the mathematical ideas embedded in the lesson: “So, it’s good to read that sometimes just to see ok, this is only where they need to get to at the end.”

Teacher Planning

Table 1 Summary of planning problems teachers encountered during unit Planning Problem

Alicia

Richard

Susan

Anticipating students’ work on task Treatment of content in curriculum

Recalled previous experience with lesson Read teacher guide to clarify content, but focused only on “important” aspects

Read student book, solved task himself Read teacher guide to clarify content, but focused only on “important” aspects

Planned for more teacherdirection of task Read teacher guide to clarify content, and planned to follow lesson suggestions

Susan also says the teacher’s guide provides her with an image of how students will engage with the task: “…it gets me ready for what they might say.

What’s the book going to be after? You know, what’s sort of the big idea that they want to come away with.”

She states that her purpose for using the student book is to understand the task for herself. She uses the teacher’s guide, on the other hand, to understand how students may approach and solve the task, including potential misconceptions students may have in relation to the task.

In contrast to Alicia, Susan appears to agree with the principles underlying CMP, and accordingly plans for enacting lessons in the unit largely as described in the materials. Moreover, unlike Richard, Susan uses the curriculum not only for the content features of the lesson, but also she uses the suggestions for how to engage students with the content during the lesson.

Thus, Susan, the teacher with less teaching experience as compared to Alicia and Richard, draws heavily from the suggestions in the curriculum when planning, and plans to enact the lesson largely as described in the materials. By adhering to the lesson suggestions in an almost prescriptive fashion, Susan leaves little room for her own interpretation of the lesson.

As these examples illustrate, the nature of teachers’ engagement with curriculum materials during planning is determined by a variety of factors. Although Richard agrees with CMP’s overall approach to teaching and learning, he largely relies on the curriculum materials solely for its content. It appears that he does not require much pedagogical support when planning; instead, he typically limits himself to reading and working through the student book. Susan, on the other hand, seems to rely on the materials for both content and pedagogical purposes during her planning for the unit, closely following the lesson suggestions. In contrast to Richard and Susan, Alicia does not seem to believe that CMP provides students with sufficient opportunities to practice basic skills and procedures, and therefore Alison Castro Superfine

modifies the lesson suggestions as needed when planning. In fact, all three teachers held varied conceptions of the curriculum – curriculum as a guide to varying extents (Alicia and Richard) and curriculum as a script (Susan). As these examples illustrate, teachers’ various conceptions influence the types of planning problems these teachers encounter and the ways in which teachers manage these problems as they arise during planning for the unit.

Planning Problems As discussed previously, planning problems constitute a fundamental structural component of the PMI Model because they highlight the relationships between teachers’ experience, conceptions of mathematics teaching and learning, and the actual curriculum program used by teachers. Applying this model to teachers’ practices requires close analysis of the planning problems experienced by these teachers and the factors underlying the emergence of these planning problems. The teachers in this study primarily encountered two different planning problems – (1) anticipating potential errors and misconceptions students may have in relation to a task and (2) treatment of content in the curriculum. Although teachers encountered several planning problems throughout the unit, these two problems were selected for analysis because they illustrate how the PMI Model depicts teachers’ planning practices. Though the two types of planning problems teachers primarily encountered during their planning were quite similar, teachers varied considerably in the ways in which they managed these two planning problems. Table 1 summarizes the planning problems these three teachers encountered during their planning for the unit, and briefly describes the ways in which the teachers managed these different problems. Table 1 does not reflect the frequency in which participating teachers encountered planning problems during the unit.

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Anticipating Students’ Work on Tasks A specific planning problem all three teachers encountered during their planning for the unit was anticipating how students would work on the content of the lesson. Alicia, for example, relied on her experience from previous classes to anticipate how students would engage with the content of several lessons in the unit. Based on experiences with classes in previous years, Alicia anticipated that students would struggle with lining up the decimal points when adding and subtracting decimals in a certain lesson.

She therefore planned to enact the entire lesson as a whole-class activity to help students through the lesson, as opposed to providing opportunities for students to work collaboratively in groups during the lesson, as was the suggested organization. Similarly, she anticipated students would struggle with another task involving computing discounts by drawing upon her previous experience with that lesson, and again planned the lesson as a whole class activity. As she described in her planning, Alicia believed that by implementing lessons as either whole-class or individual activities, she was better able to address student difficulties and “guide them in the right direction.” Richard regularly encountered this same planning problem but managed it quite differently than Alicia. Rather than relying on his previous experiences, Richard anticipated how students would work on the different tasks by working through the lessons himself in the unit – he read the student book and solved the task while thinking about how students would approach the task and what potential aspects might confuse students. In doing so, Richard attempted to forecast the various ways in which students could engage with the content, which reflected his more nonconventional conception of mathematics learning.

Susan also anticipated how students would work on the task, primarily drawing upon what she knew of her students’ previous work throughout the unit, but also the information included in the teacher’s guide.

She became aware of this planning problem by not only reading the teacher’s guide, but also from her previous experience with a particular lesson involving decimal division in which students seemed to struggle with long division. Although the materials alerted her to this potential source of confusion for students, she did not seem to use the suggestions in the teacher’s guide to support students’ understanding of long division. Instead, she used her view of how students should learn in order to address students’ difficulty with the content and planned to enact the lesson in particular ways to lessen the likelihood that students 18

would struggle. This was also the case in Susan’s planning for a lesson involving computing discounts.

In both situations, Susan’s previous experience, her view of how struggling students should learn, and her proclivity to follow the curriculum suggestions closely, influenced how she managed the problem of anticipating how students would work on the lesson.

She often reduced the complexity of the tasks by telling students how to solve them, taking away students’ opportunities to wrestle with the central ideas, but still enacted the lesson largely as written.

Treatment of Content in Curriculum The treatment of the content within two lessons dealing with long division also emerged as a planning problem for these teachers. All three teachers considered long division as a particularly important concept for students to know and to be able to do.

However, the long division algorithm was not explicitly presented in the unit; it was presented as a set of two interrelated lessons in order for students to understand the underlying rationale of the algorithm and the role of place value when dividing decimals. All three teachers had to consider how to enact these two lessons in light of their conceptions towards the content and the curriculum. Alicia and Richard modified the lesson to focus on the procedural aspects of decimal division in these two lessons. This modification reflected their conceptions about what they considered to be the most important aspects of the content. Moreover, this modification comported with their conceptions of the curriculum as a guide rather than a script for their lesson planning. This particular conception of the curriculum left room for the teachers’ conceptions toward the content to dictate how teachers planned to enact the lessons.

While Susan also encountered this planning problem, she planned to enact the lesson largely as written in the curriculum despite her reluctance to do so. Her conceptions toward both the curriculum and the content influenced how she framed and managed this problem. Susan felt inclined to change the treatment of the content because her conception of the content clashed with the treatment of the content in the curriculum. However, her desire to plan her lessons largely in accordance with the lesson suggestions provided a push in the opposite direction to teach the content as written: “I don’t know about this lesson because students have always struggled with division….Though [the lesson] helps students understand, so I just have to be patient.” Susan had to consider what content to enact with students in light of Teacher Planning

these conflicting conceptions. Her conception to plan for lessons in accordance with the lesson suggestions ultimately outweighed her conception of the content.

The shape of Susan’s planning problem contrasts with that of Alicia’s and Richard’s in that they did not negotiate conflicting conceptions. In summary, the three teachers in this study encountered different problems in the course of their planning for the unit.

Despite the fact that the CMP materials provided the means to manage some potential planning problems, teachers seemed to rely largely on their previous experiences and particular conceptions to manage their planning problems. Therefore, in the case of all three teachers, the PMI Model highlights how teachers with diverse conceptions and experiences frame and manage particular planning problems.

Discussion and Conclusion The previous discussion highlights the interrelationship among curriculum materials, teachers’ various conceptions, and the types of and ways in which teachers frame and manage planning problems that arise in the course of their work. During planning, teachers often use curriculum materials as a starting point for their lesson planning. The nature and extent of teachers’ engagement with the curriculum materials, however, is determined primarily by their various conceptions. Teachers’ various conceptions then influence the type of planning problems they encounter, and also how teachers manage these planning problems.

Teachers’ lesson enactment also contributes to the pool of knowledge and information they have to draw from in subsequent years, thereby influencing their conceptions. As Figure 1 illustrates, the PMI Model represents an iterative process that is continuously shaped by teachers’ experiences over the course of their careers. With every lesson, teachers potentially encounter unanticipated questions or new strategies that contribute to the knowledge they can draw from when planning the same or related lessons in subsequent years. The ways in which teachers’ enact lessons with students over time can also inform how teachers conceive of what it means to teach and learn school mathematics. The proposed model provides a way to understand how teachers’ planning practices change, or fail to change, over the course of their careers.

The PMI model suggests a possible cause for the “experience problem,” perhaps one of the most significant problems teachers face as they advance through their careers. Unlike their less experienced Alison Castro Superfine

colleagues, experienced teachers have to consider how to make use of their prior knowledge and experience.

With regard to teachers’ various conceptions, the “experience problem” consists of how to utilize experienced teachers’ assumptions about and prior knowledge of mathematics curricula, and their ideas about what it means to learn and teach mathematics.

The planning routines of Alicia, Richard, and Susan reflect how the experience problem plays out in actual teaching practice. Experienced teachers do not face significantly fewer or different planning problems as compared to less experienced teachers. On the contrary, all three teachers anticipated that a group of students would struggle with a particular aspect of a lesson, or even struggled themselves with certain aspects of the mathematics content. Yet, these teachers encountered these planning problems differently. The differences among these teachers seem to be in their conceptions of the curriculum and content, the prevalence of their conceptions in their planning decisions, and ultimately their instructional decisions.

Regardless of their conceptions, teachers’ conceptions of curriculum and mathematics teaching and learning can become calcified over time (Leinhardt, 1983; Leinhardt & Greeno, 1986). As a result, teachers may become inattentive to how their planning decisions influence students’ opportunities to learn and they may become resistant to external influences such as new curriculum programs or professional development experiences. Consider Alicia, who seemed to adhere to a more conventional conception of mathematics teaching and learning during her planning. Because Alicia has quite extensive teaching and curricular experience in using more conventional mathematics curricula, she planned to focus students’ work on practicing computations and procedures and planned to modify lessons as wholeclass discussions rather than collaborative work groups. It appears that Alicia’s conceptions of teaching and learning have become somewhat cemented throughout her teaching career and seemed to have hindered her from planning for enacting CMP lessons in this unit in accordance with the curriculum’s underlying principles. The PMI Model also captures this common aspect of teachers’ practice – as this model represents an iterative process, allowing for teachers’ conceptions to become reinforced as teachers amass an increasing amount of knowledge and experience.

The PMI Model has the potential for even broader utility because the planning problems and teacher conceptions discussed here constitute only a handful of 19

the planning problems and conceptions that might influence teachers’ lesson planning. For example, teachers’ perceptions of limited time during a lesson may prove problematic for some teachers when deciding how much of the lesson to cover in the time allotted. Another planning problem may arise when teachers have to anticipate how to orchestrate the use of multiple solution strategies during a given lesson, thinking carefully about the order in which to present certain strategies and the mathematical affordances of discussing different strategies. In addition to accounting for a broad range of planning problems, the proposed model also can account for a broad range of teachers’ conceptions. For example, teachers’ conceptions of their role as teachers may influence how they engage with the curriculum materials during planning. This engagement, in turn, will give rise to new planning problems and ways of managing these problems. Regardless of the precise planning problems and conceptions that may influence teachers’ work, these key elements of the PMI Model help to explain teachers’ considerations and decisions made during the planning process.

Given the PMI Model and the notion of planning problems that provides its underlying structure, the question for future research becomes how reformers can work to improve teachers’ practice, and ultimately student learning. Viewed by many as a driving force of reform, mathematics curriculum materials have the potential to boost educational achievement while embodying new modes of instruction. However, teachers can hold diverse conceptions that stand in contrast to the conceptions of teaching and learning underlying the curriculum, which can hinder teachers from planning in accordance with the curriculum.

Teachers who have more experience with conventional curricula and exhibit a more conventional conception of teaching are desensitized to the modes of instruction entailed in implementing reform curricula. Researchers have found that for teachers with extensive experience teaching with more conventional methods and curricula questioned the value and relevance of reform curricula (Manouchehri & Goodman, 1998; Preston & Lambdin, 1995). Still, reformers can target teachers’ conceptions directly by designing professional development experiences that are aimed at helping teachers shift their views of what it means to know, learn, and teach mathematics. This is not to say that teachers should participate in the equivalent of a philosophy course, but rather reformers can situate teachers’ learning in the actual practice of teaching, wherein teachers can experience what learning and 20

teaching mathematics in reform oriented ways entails.

At the very least, the PMI Model underscores teachers’ conceptions as a target for reform efforts because they structure and provide a major resource for managing planning problems that arise in the course of teachers’ work.

In summary, the PMI Model highlights how teachers’ various conceptions frame and influence how teachers’ manage planning problems that arise when preparing for mathematics instruction. The model is structured around Lampert’s (2001) notion of teaching problems, which can illuminate processes teachers engage in during their daily planning, thus providing a useful lens to understand the nature of teachers’ planning routines and reasons underlying their decisions during this phase of teaching. The concept of teaching problems is useful for understanding teachers’ practice because it captures the interactions among teachers’ various conceptions, their engagement with actual curriculum materials, and their previous experiences. Although the discussion of the PMI Model is specific to teacher planning in a reform mathematics context, such a model of teacher planning is applicable to the planning that occurs in the context of conventional curricula as well, though the nature of teachers’ planning problems may be different. As the examples presented in this article illustrate, despite the principles of teaching and learning underlying a curriculum, teachers’ various conceptions heavily influence teachers’ engagement with the materials during planning, thereby influencing the ways in which teachers manage problems, and the types of planning problems teachers encounter. Applying the PMI Model to understand planning problems, how these problems change over time, and under what conditions they change highlights important elements in mathematics teachers’ planning processes.

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The Mathematics Educator 2008, Vol. 18, No. 2, 23–32

Communication Theory Offers Insight into Mathematics Teachers’ Talk Denise B. Forrest This article discusses how communication theory is used to understand the thoughts mathematics teachers employ when creating messages intended for students. According to communication theory, individuals have different premises about the act of communicating, and these thoughts, called message design logics, guide the process of reasoning from goals or intentions to actual messages (O’Keefe, 1988, 1990). Three distinct message design logics have been identified by communication theorists: expressive, conventional, and rhetorical. Depending upon which logic an individual employs, a very different message is said and heard. This theory was used to investigate the message design logics of 15 secondary mathematics teachers. It was found that teachers have varying logics in their message production and, depending upon the logic used, distinct characteristics correspond to different teacher premises for classroom communication.. The logic employed also results in different ways teachers encourage mathematical learning and evaluate classroom interactions.

In the last twenty years, a considerable literature base has been created by mathematics educators that describes effective verbal exchanges for classroom instruction and the critical role the teacher has in that process (Cobb, Wood, & Yackel, 1990, 1993; Cobb, Wood, Yackel, & McNeal, 1992; Cohen & Ball, 1990, 2000; Hiebert et al., 1997). The teacher uses verbal communication to articulate expectations, show care for students, and encourage discussion of specific content knowledge. During instruction, the teacher uses verbal communication to initiate questions and describe tasks in order to elicit, engage, and challenge student thinking. The teacher decides what topics to pursue in depth based on student feedback and content objectives, how to encourage every student to participate, and how to integrate further mathematical connections and representations of the topic.

Research on teaching and learning supports classroom discussions where the teacher focuses on students’ mathematical thinking and guides the discussion so the group can reach a consensus on an understanding of the particular mathematical content.

However, these interactions have not been typically found in mathematics classrooms (Goos, 1998; Jacobs, Hiebert, Givvin, Hollingsworth, Garnier, & Wearne, 2006; Weiss, Pasley, Smith, Banolower & Heck, 2003). Specifically in the United States, the Third Denise Forrest is an assistant professor of secondary/middle mathematics education at Coastal Carolina University. Her current research focuses on classroom verbal interactions for learning and specifically how teachers develop the skills and strategies for these interactions. For further contact, her e-mail is dforrest@coastal.edu..

Denise B. Forrest

International Mathematics and Science Study (TIMSS, Jacobs et al., 2006) reported that 78% of the topics covered during the eighth grade lessons were procedural, without ideas being explained or developed. Also in that report, 96% of eighth grade teachers stated that they had some awareness of current recommendations for mathematics education and 76% said that they kept up with these recommendations.

This inconsistency between research and practice needs more research; the National Research Council (Kilpatrick, Swafford, & Findell, 2001) called this area of research incomplete. Researchers should continue to make visible teachers’ decisions, and their consequences for students’ learning, as they manage classroom discourse.

Communication theory offers a different approach for mathematics educators to understand classroom interactions.

Communication researchers have developed a body of research describing how individuals create and understand verbal messages.

They view verbal communication as a strategic type of social interaction where “conversationalists create and modify their individual interpretations of their social world” (Stamp, Vangelists, & Knapp, 1994, p. 194).

According to communication theory, message design logics (MDLs) are systematic thoughts about a communication situation that an individual relies on when creating a verbal message (O’Keefe, 1988, 1990). Depending upon the logic used by an individual, a very different message is said and heard.

This paper examines how message design logic theory provides insight into secondary mathematics teachers’ verbal messages.

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Message Design Logic Theory Researchers have found evidence for three different message design logics used by leaders in specific communication situations (Hullman, 2004; Lambert & Gillespie, 1994; Lambert, Street, Cegala, Smith, Kurtz & Schofield, 1997; O’Keefe & McCornack, 1987; Peterson & Albrecht, 1996; Street, 1992). These logics are identified as expressive, conventional, and rhetorical and are developmentally ordered. Each has a constellation of related beliefs that describes the individual’s purpose for the message, choice of message context, management of the interaction, and evaluation of the interaction.

Expressive Message Design Logic Individuals employing expressive design logic operate under an assumption that verbal communication is a medium for expressing thoughts and feelings. When these individuals hear or see an event, they respond verbally with their immediate thoughts, conveying a clear and honest reaction.

Individuals using this logic believe listeners will understand the message provided that they speak openly, directly, and clearly. The conversation, being organized around immediate reactions, is quite literal, with little distinction between what is objectively and subjectively relevant in the situation (O’Keefe, 1988).

If another person in the exchange challenges the communication, the individual will again respond verbally, including some editing of previously stated messages. On average, 22% of participants in message design logic studies employ this type of logic in their verbal communication (Hullman, 2004; Lambert & Gillespie, 1994; Lambert et al., 1997; O’Keefe & McCornack, 1987; Peterson & Albrecht, 1996; Street, 1992). In the mathematics classroom when students ask clarifying questions, a teacher employing expressive design logic reacts by stating his or her immediate thoughts. These thoughts will likely focus on the teacher’s thinking, not the student’s thinking. As a result, this teacher will tend to simply repeat what was said earlier, attempting to be more clear and organized.

Conventional Message Design Logic An individual employing conventional message design logic believes communication is a cooperative “game” to be played using conventional rules and procedures. The individual organizes messages for the purpose of achieving a particular response, and expects everyone to play the game by listening to the communication context and inferring the individual’s 24

intentions. Communicators who employ conventional message design logic try to say things they believe are appropriate, coherent, and meaningful for the situation.

These messages are coherent and meaningful only when all parties involved agree on the same rules and norms. The individual hears and sees the response of others, assesses the response in the context of the situation, and continues the conversation using conventionally defined actions that they feel are appropriate. The individual judges the communication successful when he or she achieves the desired response, provided that everyone agrees on the communication rules and norms. This is the most common message design logic individuals employ in conversations, with studies reporting that 42% to 58% of individuals use the conventional message design logic (Lambert & Gillespie, 1994; O’Keefe & McCornack, 1987; Peterson & Albrecht, 1996).

The mathematics teacher employing conventional design logic will focus on using conventional norms and practices for communication in the mathematics classroom. Upon hearing and evaluating students’ responses, the teacher says what is needed to move them in the direction he or she thinks is appropriate. As teachers develop their professional expertise they learn responses they should employ in various situations; this newly developed expertise guides their communication. Unlike the expressive design logic, where responses are immediate, this communication is more purposeful and guided by conventional rules for communicating, though it may not necessarily address the students’ needs or questions.

Rhetorical Message Design Logic Rhetorical message design logic is based on the belief that “communication is the creation and negotiation of social selves and situations” (O’Keefe, 1988, p. 87). The individual employing this message design logic realizes that the intended meanings of his or her messages are not fixed, but are part of the social reality being created with others. Rather than merely being immediate reactions or conventional responses to situations, messages are explicitly designed toward the achievement of goals. Words shared in the exchange are not treated as givens, but as resources that can be called on in transforming the situation towards attaining the desired goal. These communicators use language to transform the situation to be more motivational and to give explicit re-descriptions of the context so that goals are achieved.

Communicators using rhetorical message design logic will also modify their language style to define a Communication Theory

symbolic reality so that listeners can make an acceptable interpretation and be motivated to give an acceptable response. Successful communication is viewed as a smooth and coherent negotiation among all participants towards a desired goal. Although this logic is used by 22% to 32% of adults, researchers have found that individuals typically preferred messages consistent with a rhetorical message design logic (Lambert & Gillespie, 1994; O’Keefe & McCornack, 1987; Peterson & Albrecht, 1996).

Of the three message design logics, rhetorical message design logic seems to best resemble the current literature describing preferred classroom communication (Franke, Kazemi, & Battery, 2007).

The rhetorical message design logic emphasizes a dynamic negotiation in communication. Mathematics education literature describes classroom communication where the teacher, as facilitator, focuses on student thinking and encourages dialogue so that students negotiate mathematical understanding.

Teachers who use this logic realize that communication is a dynamic negotiation process and that the students’ thoughts, the current situation, and the teacher’s goals must all be taken into account. They do not respond with prescribed statements, but are more reflective in their interactions with students.

In summary, expressive message design logic is a system of talk that simply reacts to circumstances, whereas conventional message design logic is a system that responds to exigencies with some appropriate preconceived remedy. In conventional message design logic, responses are limited by historically evolved structures. Rhetorical message design logic, on the other hand, draws on a wider range of structures, while containing within it the knowledge of conventional social forms and relations. Further, rather than seeing

people and situations as givens in a conventional system of rules or seeing meaning as fixed in messages by their form and context, “meaning is instead treated as a matter of dramaturgical enactment and social negotiation” (O’Keefe, 1988, p. 87). The relation of message and context is reversed in the conventional and the rhetorical view. In the conventional view, context is given and the relevant features of the context anchor meaning, but in the rhetorical view, context is created by the message or the process of communication. Table 1 summarizes the three message design logics.

Message Design Logics of Secondary Mathematics Teachers Message design logic theory provides a framework for studying classroom interactions. Consider the following description of classroom communication summarized from the National Council of Teachers of Mathematics’ Principles and Standards for School Mathematics (NCTM, 2000): Students should engage in conversations in which mathematical ideas are explored from multiple perspectives. They should participate in discussions where they are expected to justify solutions—especially in the face of disagreement.

This will allow them to gain better mathematical understanding and develop the ability to acquire and recognize conventional mathematical styles of dialogue and argument. Through the grades, their arguments should become more complete and should draw directly on the shared knowledge in the classroom. The role of the teacher is to support classroom discourse by building a community where students feel free to express their ideas. (pp.

60 – 61)

.

Table 1 Characteristics of Message Design Logics Fundamental Premise

Key Message Function Message/Context Relationship Method of Managing the Interactions with Other(s) Evaluation of Communication

Denise B. Forrest

Expressive Verbal communication is a medium for expressing thoughts and feelings.

Self-expression Little attention to context Editing

Conventional Verbal communication is a game played cooperatively by social rules.

Secure desired response Action and meaning determined by context Politeness forms

Rhetorical Verbal communication is for the creation and negotiation of social selves and situations.

Negotiate social consensus Communication process creates context Context redefinition

Expressive clarity, openness and honesty, unimpeded signaling

Appropriateness, control of resources, cooperativeness

Flexibility, symbolic sophistication, depth of interpretation.

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The teacher’s role in this communication is to create the opportunity for students to talk and share their ideas. From the message design logic perspective, the emphasis is not on the teacher being clear and organized in presenting the mathematics, nor on securing a desired response from students. Instead the focus is to allow a dynamic conversation to take place where negotiations and consensus by all parties is the desired outcome; this idea is consistent with rhetorical message design logic.

The present study was designed to investigate message design logics of secondary mathematics teachers. Because message design logic theory informs us that individuals hear and say different messages depending upon which message design logic they use, this could be an informative perspective for mathematics educators who are trying to better understand the verbal communication practices in mathematics classrooms. In particular, this study used the message design logic framework to identify a) the fundamental purpose for secondary mathematics teachers’ verbal messages to students, b) the key reasons teachers gave for their verbal messages, c) the primary ways secondary mathematics teachers account for students and content in their messages, and d) the perceived success of teachers’ verbal messages.

Methods Fifteen secondary mathematics teachers participated in this study. They were purposefully chosen (Patton, 1990) to reflect a range in experience, school setting (urban and suburban), and education. In interviews, teachers were asked to 1) provide a sample verbal message to address two hypothetical classroom vignettes, 2) recollect two classroom situations where they felt their verbal communication with students was successful and two situations where they felt it was not and 3) provide general information about their experiences with classroom verbal communication and 4) discuss their development of verbal communication skills.

The first part of the interview, responding to the written hypothetical vignettes, was consistent with other message design logic studies. The vignettes in this study were designed with the guidance of mathematics educators, previous message design logic studies (Lambert & Gillespie, 1994; O’Keefe & McCornack, 1987; Peterson & Albrecht, 1996), and a communication professor who has formally studied message design logics (Kline, 1984, 1988, 1991; Kline, Hennan-Floyd & Farnell, 1990). The vignettes needed 26

to contain three key features in order to elicit a variety of responses and determine the message design logic being used by the teacher. First, there is a lack of conformity in the expected response to the situation, allowing salient beliefs from the past that may not particularly be relevant for dealing with the present situation to be accented. Second, actions or processes are included that could be subject to renegotiation but that are relevant to the current situation. Third, the subject is assigned an authoritative role in the group.

This last criterion was easily met in this study, because teachers are assumed to be the leaders of classroom instruction. Factors relative to the other two criteria were incorporated into the vignettes, by embedding two to four problem situations that require teachers make decisions about curriculum and instruction.

This paper focuses on the first vignette, where an algebra class is working on the following open-ended problem on the board: ‘If the value of -7abc2 is negative, what do you know about the signs (positive or negative) of a, b, and c?’ While the students begin working, the teacher walks around monitoring their work and checking homework. The teacher notices that students are struggling with the problem as a number of them had not completed the homework assignment and some were socializing. (See Appendix A for vignette.)

Teachers were asked to state whether they thought the given vignette was realistic, and to provide a sample response message for the vignette. Though in previous message design logic studies the participants were asked to give their response in writing, in this study the response was audio-taped in order to use a cued-recall procedure (Waldron & Applegate, 1994; Waldron & Cegala, 1992). This procedure entails playing back the response and stopping intermittently to get the participant to share thoughts that are relevant to the specific statements. (See Appendix A for interview protocol.) A member check was conducted following each interview. Teachers were supplied with their sample messages and reasons for each message, their positive and negative classroom communication experience, along with general information provided in the interview. After all member-check documents were validated, each message was coded separately by two researchers as reflecting either an expressive, conventional, or rhetorical message design logic; there was 100% agreement between the two coders.

Findings All three message design logics were found to exist amongst the secondary mathematics teachers. Twenty Communication Theory

percent of the messages were coded as employing an expressive design logic, 53% conventional, and 26% rhetorical. Sample messages given in response to the first vignette representing each message design logic follow.

Mathematics Teachers Using an Expressive Design Logic Twenty percent of the teacher responses provided for the above vignette were coded as expressive. These messages were characteristically a set of statements in reaction to the situation that often included observations by the teacher that were irrelevant points to solve the immediate mathematical tasks. Here is an example of one such response.

Folks, we need to get on task here, I need everyone working on this problem. That’s important because math is not a spectator sport, you just can’t listen to me talk and expect to understand it. Now, get to work so we can get going with this lesson. (If students continue to be off task then I’ll tell them I’m going to grade this problem.)

This message was a reaction to the students’ offtask behavior. The teacher said the first idea that comes to mind, with little attempt to reorganize or address the students’ understanding of the mathematics. There was some irrelevant information in the message about mathematics not being a spectator sport and there were consequences for students who continued not to work on the problem.

The teachers who used this message design logic seemed to have a genuine desire to get students to learn. They designed their messages to guarantee a certain responses, but these messages were predominantly past-oriented, incoherent, and might have failed to engage the immediate mathematical problem at hand. In summary, these teachers used their messages to express their immediate reaction to the current situation.

Mathematics Teachers Using a Conventional Design Logic Conventional message design logic was employed in 53% of the messages. These messages focused on an appropriate action in the current situation in order to get students engaged with the mathematics. The teacher’s main purpose was to secure a desired response from the students, manage the situation, and encourage student cooperation. The context of the message was centered on the action, meaning, and justification of the students’ response. A sample of a conventional message is: Denise B. Forrest

OK class, there seems to be some confusion with the problem. Let’s work it out together and we’ll talk about the thinking I am asking you to do and why that might be valuable. (After working on the problem together) The problem involved using some critical thinking which is an important part of mathematics. Looks to me like we need to think about and work out more problems like this. (Make up several other problems that are similar.)

When these teachers were asked why they chose to say this message, responses were consistent with O’Keefe’s (1988) interpretation: “Either the speaker said this because he or she wants X and saying this is a normal way to obtain X in this situation; or the speaker is responding to prior message M, and the relevant response to M” (p. 87). In summary, the speaker said what was believed to be appropriate to accomplish the intended purposes. When one of the teachers was asked to clarify why he gave a message that was coded as conventional design logic, he stated, I want them to think about the logic, the problem solving; it’s going through a situation where something is given to you, here’s a problem, now what do you do, and they have to realistically think through it, think about what are the things I need in order to solve this problem, do I need to converse with someone else about it, do I need to get input, do I not, do I have the material in front of me, what are my resources, you know, there are three different variables here and so I go through that with them, I say ‘OK those are the kind of things you need to be thinking about in this’. And then after that, then they start to understand.

The teachers who used this message design logic also expressed a genuine desire to encourage student learning. They talked about saying what needed to be said in order to accomplish specific learning goals. The teachers, not the students, defined the direction of classroom discussion and activity. These teachers assumed that they knew what the students needed to hear to move students closer to the desired outcome.

Mathematics Teachers Using a Rhetorical Message Design Logic Rhetorical message design logic was employed in 26% of the messages. These messages allowed for student input, setting the stage for negotiation. The aim of the verbal communication was to build a social consensus. The teacher tried to manage the situation and move the communication strategically towards a desired context. An example of a message employing rhetorical message design logic is:

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(Moves to front of class, and asks for everyone’s attention.) “I’m noticing that there is something about this problem that is causing confusion for some of you. Take a minute and write down at least one thing that confuses you, or the rest of you write down at least one key thought that helped you get started on it. (Listen to responses and depending upon what was said would determine what I do next.)

The teachers who employed this design logic in their message were cognizant of the social negotiation.

For example, one teacher described the reasoning behind her message as follows: I have some ideas about what’s going on in this situation, but it’s always good to get the students’ input first, you know, it could be something I haven’t thought of at the time. I don’t just want to assume I have all the facts.

Rhetorical message producers placed importance on harmony and consensus. They tended to ignore power and resource control as a means in conflict resolution. They persistently underestimated the force of social convention and routine, and overestimated individuality and creativity. (O’Keefe, 1988). This was also evident when these teachers clarified the reasoning for their messages. One teacher said, Students should be given a voice in the classroom, it’s so easy to answer and speak for them and move on, when in fact they have a lot to say and contribute, and if we just listen, we learn a lot from them.

These messages were neither a reaction to some prior condition nor a taken-for-granted feature of the classroom. Rather than being a conventional response to some prior state of affairs, they were forwardlooking and goal-connected.

Discussion The main finding of this study is that mathematics teachers have varying knowledge and beliefs about verbal communication and these seem to influence what teachers hear and say when they talk to students.

This is a notion to consider as mathematics educators try to understand classroom discourse better. When mathematics teachers have the common goal of engaging students in learning mathematical content, message design logics provide an explanation for the different paths a teacher’s verbal message can take towards achieving this goal. In particular, these logics help explain the possible thoughts teachers use as they communicate with students. Because this study found that all three message design logics could be identified in secondary mathematics teachers’ verbal messages, it is natural to consider how these message design logics might influence classroom interactions.

Message Design Logics and Classroom Interactions A teacher employing expressive design logic generally creates messages in response to what is heard and seen in the current situation. The teacher responds with the thoughts that come to mind based on what is happening at that moment. A figure representing this situation is shown below. (See Figure 1.) Even though the classroom interaction includes student talk, the diagram is focused on the teacher’s verbal message and the space the teacher provides for students to interact in the discussion. Students’ mathematical learning may be the teacher’s desired outcome, but the verbal path towards that learning is viewed as more random. The random arrows represent the messages that are expressions of the teacher’s initial thoughts. The path from teacher message to desired learning outcome is implicit, as indicated by the dashed line. The space available for students to interact in the discussion is also indicated by the dashed rectangle.

Figure 1. An illustration of expressive design logic.

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Figure 2. An illustration of conventional design logic.

The teacher who employs conventional design logic, where the verbal exchange is more controlled and fixed, uses thoughts that move students one step closer to the goal of mathematical learning. The figure below represents this classroom interaction. (See Figure 2.) The teacher focuses his or her message on a piece of information, determining the appropriateness of responses to the next piece of information. Each message is an effort to move the students closer to the desired learning goal. The space for student interaction can be narrow or broad depending upon the teacher’s intention for that piece of information. The teacher’s message encompasses elements that relate to the desired outcome for the current piece of information.

For a teacher who employs rhetorical message design logic during classroom instruction, the goal is to create verbal messages that allow students to have space to discuss their thinking, and allow the teacher to redirect the conversation when needed in order to achieve mathematical learning goals. (See Figure 3.)

With these goals in mind, the teacher creates his or her verbal messages. The teacher begins the interaction with a message to open the space for negotiation.

Focusing on the learning goal, the teacher creates messages to collect everyone’s thoughts and directs the student interactions as needed.

In summary, the theory of message design logics provides mathematics teacher educators a way to explain teachers’ knowledge and beliefs about communication: what they believe is important to say and why it is important. These different beliefs influence the message design logic used, thereby impacting how the message is stated and heard by the teacher. This perspective can inform mathematics

teacher educators’ thinking about teachers’ classroom communication.

Further Research Using Message Design Logic Theory Because this study established that different message design logics do exist in secondary mathematics teachers’ communication, this theory is being used to study interactions in the classroom setting. For a study in progress, teachers have agreed to have their classroom interactions audio recorded and follow up with an interview similar to the protocol, shown in the Appendix. This investigation aims to identify how the teacher verbally addresses the challenges that arise in the classroom. Other issues being considered are the consistency of a teacher’s message design logic across conversations, and the identification of the influence of contextual factors.

Preliminary findings indicate that there is a consistent message design logic that a teacher uses during classroom conversations.

A second study in progress is investigating how preservice teachers develop their logical reasoning for classroom verbal interactions while participating in their teacher preparation program. Data has been collected throughout the preservice teachers’ university experiences during related coursework, field experiences, and student teaching. This data include an initial survey, written responses to classroom episodes, a self-evaluation of classroom discussion, student teaching evaluations, and interviews at the conclusion of their program. In these two studies described above, message design logic theory continues to provide an effective lens for studying teacher communication.

Figure 3. An illustration of rhetorical message design logic.

Denise B. Forrest

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References Cobb, P., Wood, T., & Yackel, E. (1990). Classrooms as learning environments for teachers and researchers. In R. Davis, C. Maher, & N. Noddings (Eds.), Constructivist views on the teaching and learning of mathematics. Journal for Research in Mathematics Education Monograph No. 4 (pp. 125–146).

Reston, VA: National Council for Teachers of Mathematics.

Cobb, P., Wood, T., & Yackel, E. (1993). Discourse, mathematical thinking, and classroom practice. In E. A. Forman, N. Minick, & C. A. Stone (Eds.), Contexts for learning (pp. 91–119).

New York: Oxford University Press.

Cobb, P., Wood, T., Yackel, E., & McNeal, B. (1992).

Characteristics of classroom mathematics traditions: An interactional analysis. American Educational Research Journal 29: 573–604.

Cohen, D., & Ball, D. L. (1990). Relations between policy and practice: A commentary. Educational Evaluation & Policy Analysis, 12 (3), 331–338.

Cohen, D. K., & Ball, D. L. (2000, April). Instructional innovation: Reconsidering the story. Paper presented at the meeting of the American Educational Research Association, New Orleans, LA.

Lambert, B., Street, R., Cegala, D., Smith, D., Kurtz, S., & Schofield, T. (1997). Provider-patient communication, patientcentered care, and the mangle of practice. Health Communication, 9(1), 27–43.

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

O’Keefe, B. J. (1988). The logic of message design.

Communication Monographs, 55, 80–103.

O’Keefe, B. J. (1990). The logic of regulative communication: Understanding the rationality of message designs. In J. Dilllard (Ed.), Seeking compliance: The production of interpersonal influences messages (pp. 87–106). Scottsdale, AZ: Gorsuch-Scarisbrick.

O’Keefe, B. J. & McCornack, S. A. (1987). Message design logic and message goal structure: Effects on perceptions of message quality in regulative communication situations. Human Communication Research, 14, 68–92.

Patton, M. Q. (1990). Qualitative evaluation and research methods (2nd ed.). Newbury Park, CA: Sage.

Peterson, L. & Albrecht, T. L. (1996). Message design logic, social support, and mixed status relationships. Western Journal of Communication, 60(4), 290–309.

Franke, M. L., Kazemi, E., & Battey, D. (2007) Mathematics teaching and classroom practice. In F. K. Lester, Jr.(Ed.), Second Handbook of Research on Mathematics Teaching and Learning (pp. 225–256). Charlotte, NC: Information Age Publishing.

Stamp, G.H., Vangelists A.L., Knapp, M.L. (1994) Criteria for developing and assessing theories of interpersonal communication. In F. L. Casmir (Ed.) Building communication theories: A socio/cultural approach (pp. 167– 208). Hillsdale, NJ: Lawrence Erlbaum Associates.

Goos, M. (1998). We’re supposed to be discussing stuff: Processes of resistance and inclusion in secondary mathematics classrooms. Australian Senior Mathematics Journal 12(2), 20–31.

Street, R. L. (1992). Analyzing communication in medical consultations: Do behavioral measures correspond to patients’ perceptions? Medical Care, 30, 159–173.

Heibert, J., Carpenter, T. P., Fennema, E., Fuson, K. C., Wearne, D., Murray, H., et al. (1997). Making sense: Teaching and learning mathematics with understanding. Portsmouth, NH: Heinemann.

Waldron, V. R., & Applegate, J. L. (1994). Interpersonal construct differentiation and conversational planning: An examination of two cognitive accounts for the production of competent verbal disagreement tactics. Human Communication Research, 21, 3–35.

Hullman, G. A. (2004). Interpersonal communication motives and message design logic: Exploring their interaction on perceptions of competence. Communication Monographs, 71(3), 208–225.

Waldron, V. R., & Cegala, D. J. (1992). Assessing conversational cognition: Levels of cognitive theory and associated methodological requirements. Human Communication Research, 18, 599–622.

Jacobs, J., Hiebert, J., Givvins, K., Hollingsworth, H., Garnier, H., & Wearne, D. (2006). Does eighth-grade mathematics teaching in the United States align with the NCTM Standards?

Results from the TIMSS 1995 and 1999 video studies. Journal for Research in Mathematics Education, 37(1), 5–32.

Weiss, I. R., Pasley, J. D., Smith, P. S., Banilower, E. R., & Heck, D. J. (2003). Looking inside the classroom: A study of K–12 mathematics and science education in the United States.

Retrieved February 11, 2007 from University of North Carolina at Chapel Hill, Horizon Research, Inc. Web site: http://www.horizon-research.com/reports/2003/ insidetheclassroom/highlights.php.

Kilpatrick, J., Swafford, J., & Findell, B. (Eds.) (2001). Adding it up: Helping children learn mathematics. Washington, DC: National Research Council.

Lambert, B. L., & Gillespie, J. L. (1994). Patient perceptions of pharmacy students’ hypertension compliance-gaining messages: Effects of message design logic and content themes.

Health Communication 6(4). 311–325.

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Communication Theory

Appendix: Interview Documentation A classroom vignette and a set of semi-structured interview questions were among the documentation taken to each interview. An abridged interview outline focusing on the cued recall questions and general questions is shown below. It should be noted the interview outline here is the one used during the study discussed in the article, but changes have been made in subsequent studies to reduce the amount of time needed to study each individual and allow for a greater number of participants. For example, the evaluation of another teacher’s classroom transcript has been omitted because this data focused less on the verbal interaction and more on the environmental and situational factors.

I. Introduction, Expectations, Info Sheet II. Vignette 1/Vignette 2: • Directions Here is a hypothetical classroom situation. Please take a minute to read this and think about how you would respond. When you are ready, tell me exactly what you would say in this situation to the student and/or students.

•

How realistic is this situation likely to happen in a mathematics classroom?

• Cued-Recall Task Now we are going to play back pieces of your response OR Now I am going to review some of the things you said.

Try to remember what you were thinking at that time. You are going to be asked to answer three questions the best you can about your thoughts during this period of the response. If you cannot remember or are not sure, just indicate so, do not try to guess.

1) What were your reasons for saying that?

2) Were you thinking about other things that you might do or say in the near future or later in the conversation?

3) Was there something you thought about saying but didn’t? Why? • 1) 2) 3) 4)

After Cued-Recall What would you like your students to think about/do/say after hearing your message?

What do you believe students thought were your reasons for saying that?

In summary, what do you believe are the most important ideas needing responded to in this situation?

What do you think will happen next?

• Critique Participant was given a transcript of an actual lesson where the teacher employs an expressive design logic. That is, the teacher in the lesson just reacts to the questions being asked, focusing on one student at a time. Participants were asked to evaluate the transcript, providing examples from their classroom experiences.

• General communication questions We have been talking about particular messages you would create in the classroom, based on specific situations.

Now I’d like to step back and ask some general questions about this.

1) On a scale of 1 – 5, one being lowest and five highest, how would you rank your classroom communication and tell me why.

2) Can you remember a time when it was higher than this number and describe that situation to me?

3) What made that situation “better”?

4) Can you remember a time when it was lower than this number and describe that situation to me?

5) What made that situation “worse”?

6) What is your role in classroom communication?

7) What is the students’ role?

8) What factors do you think effect your verbal communication in the classroom?

If extended message stated, use cued-recall Denise B. Forrest

31

Now we are going to play back pieces of your response OR Now I am going to review some of the things you said.

Try to remember what you were thinking at that time. You are going to be asked to answer three questions the best you can about your thoughts during this period of the response. If you cannot remember or are not sure, just indicate so, do not try to guess.

1) What were your reasons for saying that?

2) What would you like your students to think about/do/say after hearing your message?

3) What do you believe students thought were your reasons for saying that?

Vignette #1: The bell has rung, you asked students to get out last night’s homework and while you go around and check to see if they have it done, students are to work on a problem you’ve written on the board to start the day’s lesson.

If the value of –7abc2 is negative, what do you know about the signs (positive or negative) of a, b, and c?

As you walk around, you notice there are many of them who had not completed the homework assignment, and even more are taking this time to socialize instead of work on the problem. You remind them to work on the board problem. Some students begin working on the problem, others just sit there, and Max, a student on the other side of the room says “Why do we have to do this?” Another student, sitting right next to you adds, “This problem is stupid.”

Describe exactly what you would say to the student(s).

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Communication Theory

The Mathematics Educator 2008, Vol. 18, No. 2, 33–39

Rethinking Mathematics Assessment: Some Reflections on Solution Dynamics as a Way to Enhance Quality Indicators Elliott Ostler Neal Grandgenett Carol Mitchell This paper is intended to offer some reflections on the difficulties associated with the appropriate use of rubric assessment in mathematics at the secondary level, and to provide an overview of an assessment technique, hereafter referred to as solution dynamics, as a way to enhance popular rubric assessment techniques. Two primary aspects of solution dynamics are presented in this manuscript.

The first aspect considers how the tasks assigned in mathematics classrooms might be better organized and developed to demonstrate an evolving student understanding of the subject. The second aspect illustrates how revised scoring parameters reduce the potential for scoring inconsistencies stemming from the non-descript language commonly used in rubrics.

Introduction Professional teacher organizations have established the importance of assessment as the vanguard of instructional decision making. Specifically, in mathematics, the Principles and Standards for School Mathematics (National Council of Teachers of Mathematics [NCTM], 2000), emphasize assessment as a cornerstone to effective instruction and illustrate the need for teachers to have a solid grasp of what it means to effectively assess their students’ abilities. Of course, how specific assessments are carried out in different environments will always vary according to individual needs; nevertheless, the authors still see a great need for innovation in assessment, both in interpretation and in technique.

The U.S. educational industry makes a staggering number of decisions, fiscal and otherwise, based on the “snapshot” results of standardized tests. These tests Dr. Elliott Ostler is a professor of mathematics education at the University of Nebraska at Omaha. He has interests in assessment and technology as they relate to mathematics education and has consulted with institutions such as the College Board to develop teams for implementing curriculum for vertical articulation.

Dr. Neal Grandgenett is a professor of mathematics education at the University of Nebraska at Omaha. He has extensive expertise in technology based learning in mathematics and has authored more than 100 articles in mathematics education. He is currently assisting in building a curriculum for robotics in mathematics.

Dr. Carol Mitchell is a professor of science education at the University of Nebraska at Omaha. She has been involved public education for the past 37 years and is an alumna of the 2002 Oxford Roundtable, Oxford, England. She has numerous publications and provided leadership for several National Science Foundation Grants.

Denise B. Forrest

cause part of the assessment dilemma, forcing a teacher to decide whether to use the results of standardized measures or focus on assessment methods that are more contemporary and meaningful. The standardized assessments most appropriate for large-scale policy decisions are not necessarily those most suitable for instructional decision making. Ostensibly, the most appropriate small-scale assessments would be those allowing teachers to make decisions about their instruction (NCTM, 2000). Yet standardized test results continue to capture the lion’s share of attention even for teachers gauging their own success. In fact, despite the research-supported utility of rubric-based assessments that allow teachers to examine quality indicators (Arter & McTighe, 2001; Goodrich, 2000; Stiggins, 2001; Wiggins, 1998), there still appears to be great resistance to transferring the scope of pedagogical decisions made from standardized tests to those more appropriate for evaluating the quality of students’ mathematics work.

The purpose of this manuscript is to offer some reflections on item selection and scoring difficulties associated with appropriate use of rubric assessment in secondary mathematics and to introduce an interpretive assessment strategy, hereafter referred to as solution dynamics, as a way to enhance popular rubric assessment techniques. Two primary aspects of solution dynamics are presented in this article: first, how mathematical tasks might be better organized and developed to allow students to demonstrate evolving understanding as they progress through the subjects, and, second, how revised scoring parameters reduce the potential for scoring inconsistencies stemming from the non-descript language commonly used in rubrics.

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Figure 1.

Solution Dynamics Defined Solution dynamics can be thought of as a way to analyze, organize, and rank student solutions based on the inherent level of sophistication represented in the tasks. This is akin to how a performance rubric might be used, but instead of measuring student performance with vague descriptors, we will make statements concerning the complexity of the tasks. Specifically, solution dynamics considers what that complexity implies for student understandings needed for completing the tasks. In some sense, the analysis of mathematical tasks for solution dynamics assessment will also determine which tasks are most effective for instructional purposes.

The solution dynamics process uses the same general techniques for ranking the complexity of problems that are used to rank the difficulty of problems in standardized tests, but the nature of the tasks require that student solutions be more openended. For example, if a student correctly completes a math problem of moderate difficulty on a standardized test, we may come to the conclusion that the student understands the nature of mathematics related to solving such problems. However, given the opportunity to investigate further, we may find that the student took a long time to solve the problem by using a low level trial-and-error technique, or that he or she may even have simply guessed. A rubric assessment of the same type of problem could possibly determine that the correct solution illustrates some understanding of how to complete the task, but this type of scoring would not necessarily be able to provide specific references to quality because of the nature of the way the task was presented. For example, the student’s solution may receive a score of “progressing,” or a 1 34

on a 0-3 scale, which is actually no more effective for instructional decision making than a multiple choice answer to such a question.

On the other hand, in the solution dynamics model a group of teachers would first look specifically at the task and provide an organizational structure of possible solution techniques, each of which would be ranked by the complexity of the mathematics needed. Student solutions would then be mapped to the ranked structure template (See Figure 1) for a score. At first glance, this may appear to simply be a subtle new twist on an existing rubric technique, and to some extent, it is; however, by creating a ranked structure of possible solutions for a given task, teachers have not only been forced to analyze the importance and validity of the task, but also to review a template which provides the vertically articulated concepts immediately above and below what the student’s solution illustrates.

Example Solution Dynamics Task: Optimizing the Volume of a Box Problem: Suppose a rectangular (threedimensional) box is to be created by using a 20-inch by 20-inch square sheet of plastic (See Figure 2). Square corners will be cut from the original sheet of plastic and the rectangular tabs on each side will be folded up to create the sides of the box as illustrated below. What size corner pieces need to be removed so that the box will have the greatest possible volume? This is considered a good solution dynamics task because there is great potential for a number of possible unique solutions, starting at an arithmetic level and ending at a calculus level. The same problem can be used in each of a number of successive courses but the solutions will change (become dynamic) as the material in the courses becomes more sophisticated.

Rethinking Mathematics Assessment

Figure 2.

Level 1 (arithmetic-based) solution. The student creates a chart (See Table 1) that records the volumes of all possible boxes with whole number increments being removed from the corners. Such a chart might look something like Table 1.

Table 1 Example of arithmetic-based solution Corner Removed 1 x 1 inch 2 x 2 inch 3 x 3 inch 4 x 4 inch 5 x 5 inch

Resulting Bases 18 x 18 inch 16 x 16 inch 14 x 14 inch 12 x 12 inch 10 x 10 inch

Height

Volume

1 inch 2 inch 3 inch 4 inch 5 inch

342 inch3 512 inch3 588 inch3 576 inch3 500 inch3

By the time the student has reached the fifth entry in the chart, they will probably be able to recognize that the volume is decreasing and that the optimal corner piece to remove is a 3-inch by 3-inch section. This kind of solution indicates the student recognizes that the corner piece removed has the same dimension as the height of the box and that the base of the box decreases steadily as larger and larger corner pieces are removed.

They are likely to make a number of other observations as well; however, at this level they may not yet have the ability to efficiently test fractional increments, making their solution incomplete.

Level 2 (algebraic) solution. Students will use the same basic diagram to provide context, but this representation of the solution indicates that they recognize the volume of the box is a function of the corner piece removed. When examining the pattern Elliott Ostler, Neal Grandgenett, Carol Mitchell

that emerges from the chart in the first level, students may derive the following formula: V = (x)(20 – 2x)2.

Using this formula, students can test both whole number and fractional increments of corner piece dimensions much more efficiently than was possible with a chart. Yet this solution is still limited in that it does not allow for an efficient determination of an exact solution.

Level 3 (advanced algebra/calculus-based) solutions. Once again, students will use the same diagram to provide context for the problem. An advanced understanding of this problem will illustrate that students not only recognize the functional relationship between the volume of the box and the dimension of the corners removed, but that they understand that the volume can be graphed as a function of the dimension of that corner. They may also recognize that a maximum volume can be determined by closely examining the resulting graph or that by calculating the derivative of the function, they can determine an exact maximum point, which would represent a maximum volume of the box.

By analyzing the solution to a problem in terms of levels of sophistication, not only can we place a student on a scale, we can surmise with some accuracy what they know, and what they need to know in order to achieve the next level of complexity. The general tree diagram in Figure 1 adapted from Craig (2002) can help determine the complexity of mathematical tasks based on a continuum, which progresses from simple to complex. Galbraith and Haines (2000) conducted research that clearly indicated that mechanical 35

processes, here referred to as algorithmic processes, were easier than interpretive problems, which, in turn, were easier than constructive problems. Algorithmic processes consisted of mechanical solutions where students needed only to follow a sequenced set of steps to solve a problem. Interpretive problems were those problems presented in more abstract forms (i.e. word problems) from which the correct processes had to be interpreted. Constructive problems were those that required a combination of the two lower categories.

Certainly the use of the model in Figure 1 does not allow for the ranking of mathematical tasks to be an exact science, but it does guide teachers to focus on the hierarchy of difficulty innate to a task.

The following example illustrates how solutions on another simple mathematical task might be ranked on a solution dynamics rubric as the mathematics used to solve the problem becomes more sophisticated. Note that the same problem is used year after year so that growth in the understanding of the processes related to this specific problem can be tracked. The differences in the complexity of the mathematics at each scoring level have been greatly exaggerated in this example in order to help differentiate between the elements in Figure 1. With an actual solution dynamics task, the differences would be more subtle and require the attention of a team of mathematics teachers to study the nuances of expected students’ solutions.

The levels of the task shown in Figure 3 are certainly subject to interpretation, but illustrate how solutions become dynamic by focusing on the sophistication of the mathematics and the process of derivation rather than on the actual formula for the area of the circle as an answer. This particular task is one of the most basic examples of solution dynamics and one that has been used successfully by the authors in calculus courses. Allowing students to observe the evolution of complexity in a mathematical task provides context to the processes of integration.

The mathematical tasks assigned would be used to help reinforce concepts being taught at each course level. Certainly a teacher would not expect a student to use a complex mathematical technique to solve a very simple problem, but often a simple problem can provide a very powerful context for illustrating how complex mathematical ideas can be applied to various situations. In the example above we saw that a simple task can be used to demonstrate how both simple and complex mathematics can be applied to a situation. The derivation of the formula for the area of a circle is simply a convenient task that can be repeated through multiple levels of instruction to allow students to 36

demonstrate an understanding of increasingly sophisticated thinking within a familiar context.

Because the task remains the same, teachers can get a sense of what students know about evolving levels of mathematics based on how they might approach the solution.

Why Not Rubrics Alone Rubrics are popular tools for assessment and can no doubt provide insight to student understanding in a variety of subjects and contexts if they are carefully constructed. Rubrics by themselves, however, have some inherent flaws that inhibit consistent scoring and decision making (Popham, 1997). The three most problematic flaws are as follows: rubrics are actually secondary scoring instruments but are often misunderstood to be the primary instrument; the language used in the quality descriptors, although consistent, is too vague to make meaningful decisions; and quantity indicators are often mistaken for quality descriptors. We elaborate on each of these three flaws below.

Rubrics are secondary scoring instruments Students do not perform on a rubric. Students perform a task that is then scored by a rubric. This simple misunderstanding creates confusion about the nature of rubrics and how they should be used. It is not unusual to hear people talk about how students performed on the rubric, when in fact they mean how students scored on a preset task as interpreted by the rubric. This being the case, it should be at least as important to consider the innate value of the mathematical task as it is to consider the performance level descriptors used to rank the students’ understanding of the task. Unfortunately, task considerations tend to be passed over in lieu of more careful consideration of the rubric scale.

The language used in the quality descriptors of rubrics is too vague Tierney & Simon (2004) argue for the need to state the performance criteria and the attributes clearly.

They also argue for the need to describe the qualitative degrees of performance more consistently between the performance levels of the rubric. They indicate that these modifications make the task, criteria, and attributes clearer to students and allow a broader use of the rubric. These are noble concepts, and the claim they make about clarity may be true, but the terminology they suggest is part and parcel of the problem with broad-use assessments: non-descript language. In one example, the terms they suggest using Rethinking Mathematics Assessment

Figure 3. Levels of solutions for the task of deriving the area of a circle. to provide consistency and clarity are few, some, most, and all. These are not bad terms, but they are only indications of clarity or quality when antecedent to some very specific requirements provided in the initial task. For example suppose a timed, 100-item, singledigit multiplication test were being used as an assessment. A student answering 45 items correctly would probably fall into the “few” or “some” category of the rubric. We might surmise from that score that the student has difficulty with multiplication. However, suppose the student only answered 45 questions and was correct on all completed items. It is possible that the student simply writes slowly but knows the information very well. The terms few, some, most, and Elliott Ostler, Neal Grandgenett, Carol Mitchell

all generally do nothing more than a checklist would, particularly when they are applied in the manner indicated above. If however, the assessment instrument included items that gradually became more difficult, the terms few, some, most, and all would be more appropriate because they would be antecedent to levels of difficulty within the test rather than just looking at quantity of similar items completed. This idea leads into the next point.

Quantity indicators are often mistaken for quality descriptors As far as student performance on a given task is concerned, the demonstration of basic knowledge does not necessarily require a rubric. Once again, the nature 37

of the task needs to be a primary consideration. For instance, if a teacher wants students to know basic facts like multiplication tables, a rubric is probably not necessary. If a teacher were to create a rubric where the scale indicators showed increased student performance by the number of problems they correctly answered (i.e. “Beginning” = 20 problems, “Progressing” = 30 problems, “Advanced” = 40 problems, etc.) the categories would not be indications of conceptual quality nor are the descriptors assigned to the scales necessarily set by any externally valid criteria. It is therefore unnecessary to provide a rubric scale that counts or quantifies the number of correct answers. For a task such as this, a checklist would be more appropriate. Quality indicators are more appropriate to tasks that require some higher-level thinking and rubric levels that clearly indicate the quality of thinking, or the lack thereof.

Conclusion Ultimately, there are two primary factors that make a solution dynamics approach a potentially effective way to clarify and increase the accuracy of rubricbased assessment. First, a solution dynamics model considers the evolution of a mathematical solution over time. Second, this approach specifically considers the quality of the student performance and the difficulty of the task within the same instrument. Both of these factors, though somewhat obvious, emphasize ideas that are generally absent in the explanation of rubric assessment. Problems such as the derivation of the area formula for a circle, as illustrated earlier, have been used with great success in a solution dynamics format by the authors to show not only the evolution of students’ simple mathematical models to complex ones, but also to illustrate natural connections and applications between scientific and mathematical content. This has been particularly true in calculus courses where students tend to lack the conceptual understanding behind processes like integration.

Though it is probably not realistic to expect large gains in mathematical understanding to come in a single academic year for every student, the selection of the right kinds of dynamic mathematical problems can better illustrate the dynamic nature of the mathematics the students are learning and therefore help facilitate the conceptual evolution of mathematical knowledge that represents a transition from algorithmic to abstract thinking. It is important and appropriate to engage in assessment techniques that measure students’ progress over a successive period of years. Attention to evolving representations of student solutions allow for this to 38

happen. A focused effort on vertical articulation, and in particular, efforts to build dynamic solution exercises (specific mathematical tasks that lend themselves well to solution dynamics assessment) will provide a more comprehensive view about students’ understanding of mathematics and its various components, concepts, and skills.

Romberg (2000) argues that, with appropriate guidance from teachers, students can build a coherent understanding of mathematics and that their understanding about the symbolic processes of mathematics can evolve into increasingly abstract and scientific reasoning. This, of course, happens through opportunities to participate in appropriate kinds of mathematical tasks. As mentioned previously, a coherent understanding of anything does not happen with most students over the course of a single academic year. The evolution in a student’s thinking that allows them to demonstrate a transition from algorithmic to abstract semiotics presumably happens over a period of years. It follows then that developing the kinds of appropriate mathematical assessments, the dynamic kinds that allow for this transition to be measured over time, can most appropriately be done by a team of mathematics educators. Each considers the nuances of what the others do, and then documents their part in the process through thoughtful solution dynamics assessments.

References Arter, J., & McTighe, J. (2001). Scoring rubrics in the classroom: Using performance criteria for assessing a improving student performance. Thousand Oaks, CA: Corwin Press/Sage Publications.

Craig, T. (2002). Factors affecting students’ perceptions of difficulty in calculus word problems. 2 nd International Conference on the Teaching of Mathematics. Crete, Greece, July 1-6, 2002. Retrieved November 28, 2008 from http://www.math.uoc.gr/~ictm2/Proceedings/pap411.pdf.

Galbraith, P., & Haines, C. (mis(understandings) of beginning undergraduates. International Journal of Mathematical Education in Science and Technology. 31(5). 651–678.

Goodrich, A. (2000). Using rubrics to promote thinking and learning. Educational Leadership, 57, 13–18.

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

Popham, W. J. (1997). What’s wrong—and what’s right—with rubrics. Educational Leadership, 55, 72–75.

Romberg, T. (2000). Changing the teaching and learning of mathematics. Australian Mathematics Teacher, 56(4), 6–9.

Stiggins, R.J. (2001). Student-involved classroom assessment (3rd ed.). Upper Saddle River, NJ: Prentice-Hall.

Rethinking Mathematics Assessment

Tierney, R. & Simon, M. (2004). What's still wrong with rubrics: focusing on the consistency of performance criteria across scale levels. Practical Assessment Research & Evaluation, 9(2). Retrieved November 28, 2008 from http://PAREonline.net/getvn.asp?v=9&n=2 .

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The Mathematics Educator (ISSN 1062-9017) is a semiannual 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 the mathematics education community locally, nationally, and internationally. The Mathematics Educator presents a variety of viewpoints on 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).

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In this Issue, Guest Editorial… Motivating Growth of Mathematics Knowledge for Teaching: A Case for Secondary Mathematics Teacher Education AZITA MANOUCHEHRI Planning for Mathematics Instruction: A Model of Experienced Teachers’ Planning Processes in the Context of a Reform Mathematics Curriculum ALISON CASTRO SUPERFINE Communication Theory: Another Perspective to Think About for Mathematics Teachers’ Talk DENISE B. FORREST Rethinking mathematics and science assessment: Some reflections on Solution Dynamics as a way to enhance quality indicators ELLIOTT OSTLER, NEAL GRANDGENETT, CAROL MITCHELL