An ASCD Study Guide for Concept-Rich Mathematics Instruction: Building a Strong Foundation for Reasoning and Problem-Solving

This ASCD Study Guide is designed to enhance your understanding and application of the information contained in Concept-Rich Mathematics Instruction, an ASCD book written by Meir Ben-Hur and published in June 2006.

You can use the study guide before or after you have read the book, or as you finish each chapter. The study questions provided are not meant to cover all aspects of the book, but to address specific ideas that might warrant further reflection.

Most of the questions contained in this study guide are ones you can think about on your own, but you might consider pairing with a colleague or forming a study group with others who have read (or are reading) Concept-Rich Mathematics Instruction.

Chapter 1: Conceptual Understanding

This chapter emphasizes the importance of conceptual understanding in mathematics. As you read this chapter think back on your own mathematics learning experiences from elementary school through your present learning. Identify your strengths and weaknesses and try to conclude what has been the importance of conceptual understanding to your learning.

What do you think experts mean by the term “concept”? What are the key mathematical concepts you teach? What are the key mathematical procedures you teach?

Can students learn concepts and procedures at the same time? Do you teach concepts and procedures at the same time? How so?

What do you think teachers do appropriately to develop their students' conceptual understanding in mathematics? What is missing?

Chapter 2: Concept-Rich Instruction

This chapter presents the author's model of Concept-Rich Instruction. As you read this chapter reflect upon your instructional practices. Identify your strengths and weaknesses and try to conclude which of his components of instruction you want to adopt or improve.

The author argues that teachers often confuse the idea of “practice” with “exercise.” How important do you think it is to distinguish between the two?

Lev Vygotsky recognized that conceptualization is first “arrested” in what he called the Zone of Proximal Development. At this point students may say “the right thing” but fail to enact it or “do the right thing” but fail to explain it. If conceptualization does not continue beyond this level, the effects of practice are lost over time. What do you think this observation implies about instruction?

The author argues that teachers should offer sufficient amounts of practice for their students. What may be the most critical problems teachers must overcome if they apply this idea in their instruction? Is it possible for teachers to overcome these problems? How?

The author argues that practice with variable applications of new concepts provides students a framework for reflection and decontextualization. Do you agree? If so, what are the instructional implications?

The author argues that teachers contribute to student reflection by asking questions that would otherwise not occur to students. List the types of such questions.

How do you think teachers should identify and deal with student errors of misconception?

Identify a common misconception among your students. Why do you think Socratic questioning helps correct this misconception?

The author argues that teachers must guide students' reasoning toward the accepted mathematical view through careful “scaffolding” and guiding questions. The process involves encapsulating conceptual understanding in words. How is this view different from radical constructivist views?

Do you agree with the author that students should be encouraged to independently identify applications for new concepts? If so, what are the instructional implications?

There seems to be agreement among educators that parental involvement is necessary to ensure that students realize what they learn. What are some simple tactics teachers can use to enlist such support by parents?

Chapter 3: Misconceptions

Just as in the case of conceptual understanding, misconceptions represent the sense students make of their learning experiences. Think about a particular student misconception you have come across in your teaching. How do you the student developed it?

The author argues that misconceptions in mathematics are actually preconceptions. Do you think this distinction is important? Why? What are the instructional implications of this argument?

The author distinguishes preconceptions as either undergeneralizations or overgeneralizations of mathematical ideas. How helpful do you think this distinction is for mathematics instruction? Why?

Can you think about an example of a counterintuitive mathematical concept or conclusion? Can you use the author's ideas for intervention in this case?

What might be the negative consequences of the cognitive dissonance that arises when studentss misconceptions don't align with a new and different experience? Think about the author's examples of preconceptions in this chapter, and about the components of Concept-Rich Instruction in Chapter 2, and list instructional practices that can help prevent these negative consequences.

The author terms conceptual errors as the “seventh sense” of Concept-Rich Instruction. Do you agree or disagree? Why?

Team up with a group of your colleagues and brainstorm questions related to the applications of the author's six instructional principles for conceptual remediation. What do you believe are the three most important questions to ask about these principles? How would you answer them?

Think about the author's discussion on alternative mental representations as means for altering misconceptions. Do you agree that teachers should recognize this strategy as a principle for conceptual remediation? Why?

Among his six instructional principles for conceptual remediation, the author distinguishes “reciprocity” (between the teacher and students) from “constructive interaction among learners”? Do you think this distinction is helpful or not? Why?

Chapter 4: Solving Problems Mathematically

The author promotes the idea that problem solving is an effective means of mathematics instruction—not just its ends. Do you agree or disagree with this position? What are the instructional implications?

Some argue that problem solving is not content knowledge but an ability. Do you agree, or disagree? Why?

Think about misconceptions students hold about mathematics. Do you agree with the author that problem-based mathematics instruction can correct these misconceptions? Why?

In his discussion of the self-regulation challenge, the author insists that teachers must help students develop the ability to decide autonomously what needs reading or rereading, and what else to do in order to understand the quantitative information that is given and the relationships among the various concepts. What do you think is the import of this idea, if any?

Word problems challenge students to reconstruct language into mathematical problems that can be solved. Identify one or two word problems in the textbook you use with your students. What features of reconstruction can you identify as necessary in the solution of these problems?

The author shows that word problems may appear to be similar (by context), yet be substantially different (in structure and complexity). Do you agree with his conclusion that teaching how to solve word problems should focus on structure rather than on appearance (form and key words)? Do you find his analysis of problem structure helpful? How?

The author presents research that shows students have the most difficulties solving word problems in which the “problem” is vaguely or not at all defined. Do you find this to be true in your classroom? Why? Reflect upon the instructional implications.

Reflect upon your own instruction on problem solving. How important is it for you to encourage students to develop different representations for word problems? What measures do you take to provide opportunities for your students to view alternative representations? Do you find the author's suggestions helpful for improving this aspect of your instruction? How?

Comparing solutions for different problems helps students develop their metacognitive awareness and ultimately their problem-solving skills. Think about your own learning and your own teaching in mathematical problem solving. Were such comparisons common practice? Why or why not? How important is this practice?

Chapter 5: Assessment

The author argues that formal achievement tests cannot help teachers much to “identify developmentally appropriate content, recognize student misconceptions, evaluate the meaning students make of what they learn, see whether instruction is effective in altering misconceptions and distinguish diverse learning needs”. Do you agree? Why?

The author argues that Concept-Rich Instruction relies on dynamic, authentic, process-oriented methods of assessment. Do you agree? Why?

The author identifies classroom communication and observations as one of the elements of dynamic assessment. Do you agree? If so, what are the instructional implications of this method of assessment? How would you organize it?

Reflective examination and comparisons over time of students' homework and students' weekly entries in problem solving notebooks is time-consuming. Does the kind and quality of information teachers can obtain with the help of this method justify the investment it requires? Why?

In your opinion, how practical is the use of individual interviews around problem-solving activities as an assessment tool in your classroom? What do you see as the major challenges? What types of information will you be able to collect through this method that are not otherwise available to you?

Should student journals provide you with types of assessment information that the author identifies? Which of his ideas for implementation do you find helpful? Why? What would you do differently with students' journal writing? How would you organize the information you collect? How would you use this information?

Do you agree with the author's argument that student self-assessment should be included among the assessment tools of Concept-Rich Instruction? Why? What is unique about the data it provides about student learning? How do you think such data can affect instruction?

Which of the author's suggestions about portfolio assessment do you find particularly helpful? Why? If you already include student portfolios among your assessment tools, what would you change based on these suggestions?

What do you think are the limitations of performance-based assessment? What are the benefits of this assessment method? How can you use it in your classroom?

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