1. Substance versus Presentation of Mathematics Ideas
George W. Bright
Professor Emeritus
University of North Carolina at Greensboro
TDG Seminar, February Substance and Presentation of Ideas

2. Acknowledgements
• Based on work completed as part of a project funded by a grant from the National Science Foundation (# ); George W. Bright and Jeane M. Joyner, co-Principal Investigators.
• All conclusions and opinions expressed are those of the authors and do not necessarily reflect the position of the Foundation or any other government agency.
• Professional development materials published by ETA/Cuisenaire as Dynamic Classroom Assessment.
TDG Seminar, February Substance and Presentation of Ideas

3. • Distinguish between substance and presentation of a mathematics idea
Goals for the Session
• Distinguish between substance and presentation of a mathematics idea
• Apply this distinction to thinking about the kinds of feedback that might be helpful to students
TDG Seminar, February Substance and Presentation of Ideas

4. Classroom Assessment
• Classroom assessment is the process by which teachers gather information about what students know and can do and then use that information to make more effective instructional decisions.
• Classroom assessment involves planning effective mathematics instruction based on understanding how students think about key ideas.
TDG Seminar, February Substance and Presentation of Ideas

5. Model for Classroom Assessment
Learning
Targets
Purposes
Instructional
Assessment
Communication
Decisions
Methods
Documentation
Inferences
About
Thinking
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6. Direct Effects
Back and Wiliam (1998) conclude from an examination of 250 research studies on classroom assessment that “formative assessment does improve learning” -- and that the achievement gains are “among the largest ever reported for educational interventions.” The effect size [is] 0.7, on average.
(Wilson & Kenney, 2003, p. 55)
Wilson, L. D., & Kenney, P. A. (2003). Classroom and large-scale assessment. In J. Kilpatrick, W. G. Martin, & D. Schifter (Eds.), A research companion to Principles and Standards for School Mathematics (pp ). Reston, VA: National Council of Teachers of Mathematics.
Black, P., & Wiliam, D. (1998). Assessment and classroom learning. Assessment in Education, 5, 7-74.
TDG Seminar, February Substance and Presentation of Ideas

7. Outcomes for Students
If mathematics teachers were to focus their efforts on classroom assessment that is primarily formative in nature, students’ learning gains would be impressive. These efforts would include gathering data through classroom questioning and discourse, using a variety of assessment tasks, and attending primarily to what students know and understand.
(Wilson & Kenney, 2003, p. 55)
TDG Seminar, February Substance and Presentation of Ideas

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27. Reflecting on Feedback
• Students’ errors may reflect any of many different misconceptions or confusions. Sometimes the difficulties are conceptual; sometimes the difficulties are procedural.
• It is important to explore causes of errors before choosing the feedback to give.
In short, ……
TDG Seminar, February Substance and Presentation of Ideas

28. Talk Less.
Listen More!
TDG Seminar, February Substance and Presentation of Ideas

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33. Instructional Decisions
Decisions are based on the inferences made from assessment data.
• Choose problems that reveal different kinds of thinking
• Thinking about ways of representing ideas
• Provide feedback to students about their work
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34. Feedback
• Understanding how students think about key mathematics ideas provides a context for feedback to students.
• Feedback is most effective when it responds to the perceived strengths and weaknesses of students’ thinking.
TDG Seminar, February Substance and Presentation of Ideas

35. Residual Effects
[S]tudents learn more when they receive feedback about particular qualities of their work, along with advice on what they can do to improve. They also benefit from training in self-assessment, which helps them understand the main goals of the instruction and determine what they need to do to achieve.
(Pellegrino, Chudowsky, & Glaser, 2001, p. 38)
Pellegrino, J. W., Chudowsky, N., & Glaser, R. (Eds.). (2001). Knowing what students know: The science and design of educational assessment. Washington, DC: National Academy Press.
TDG Seminar, February Substance and Presentation of Ideas

36. Goals for Feedback
• Feedback should help students reorganize or rethink their understanding.
• Trying to substitute the teacher’s thinking for the student’s thinking probably does not help students rethink what they know and what they don’t know.
• The goal for feedback is to help students understand more or understand more deeply, not just to say the correct answer.
TDG Seminar, February Substance and Presentation of Ideas

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43. Learning is the Bottom Line
• We all care deeply about how much and how well students learn mathematics.
• Understanding students’ thinking is an important first step in planning instruction that helps students learn.
• Using a variety of tasks can help generate more accurate information about what students know so that instructional decisions can be better aligned with the needs of students.
TDG Seminar, February Substance and Presentation of Ideas

44. gbright45@comcast.net George W. Bright Dynamic Classroom Assessment
Activities and examples from:
Dynamic Classroom Assessment
published by ETA/Cuisenaire
TDG Seminar, February Substance and Presentation of Ideas