Discussion Questions
9/5/06:
Before
It’s Too Late
1. What was the purpose of this report?
2. How would the commission describe the problem with mathematics education?
3. Why do they feel the problem is important to solve?
Highlights
from TIMSS 2003
1. What is TIMSS?
2. How can it be used?
3. How was the study conducted?
4. How would you summarize the 2003 results?
The
Nation’s Report Card: Mathematics 2005
1. What is The Nation’s Report Card?
2. How is it used?
3. How was the study conducted?
4. How would you summarize the 2005 results?
Skemp Chapter 1
1. Consider the quote on page 3 as well as the first line of the last paragraph. The author claims we fail to teach mathematics to children in school and in fact train them to dislike the subject. Do these observations fit with your own experience?
2. On page 5, in paragraph 3, Skemp explains that because problems of learning and teaching are psychological, one might expect to find a solution to the problem of learning mathematics in psychology. At the time he began looking, learning theory was dominated by behaviorism. What is behaviorism, and what implications would it have for teaching mathematics? Why do you think the author was skeptical about this approach to teaching?
3. Why do we study learning theory?
4. According to Chapter 1, what is this book about? What will we gain as a result of studying it?
Skemp Chapter 8
1. In this chapter Skemp argues that human learning includes habit learning as described by behaviorists, but is not limited to it. How does the author argue that there is more to learning than operant conditioning?
2. The author describes intelligence as the ability to engage in intelligent learning. What is intelligent learning, and how did it evolve?
3. How would Skemp describe the process of learning?
4. Throughout the book the author repeated uses the word “schema”. What does this term describe?
Skemp Chapter 12
1. What is the difference between instrumental understanding and relational understanding? Which do you think most students prefer, and why? Do you believe the instrumental approach to teaching mathematics is really as widespread as Skemp claims it is?
2. Do the advantages of the instrumental approach to teaching mathematics listed on page 158 justify its use? Do you agree with the justifications for the instrumental approach listed on page 160?
3. What do you think is the real reason the instrumental approach to teaching is so widespread?
4. Were you taught mathematics relationally or instrumentally? Can you remember specific instances of each? How did you react in each case?
9/12/06:
Teaching Mathematics
in Seven Countries
1. What was the purpose of the TIMSS 1999 Video Study?
2. Why is it important to study teaching?
3. Why is it useful to study teaching in different countries?
4. Why does video provide a meaningful way to study teaching?
5. Which three aspects of teaching seem to both have an effect on student learning and tend to differ from country to country?
Lessons in Perspective
1. Which observations made by the author do you find to be the most interesting, surprising, or important? Does he make points that you disagree with?
2. What tips for teaching will you take away from this article?
Skemp Chapter 2
1. List the major points made by Skemp in Chapter 2. What implications does each have for teaching mathematics?
Skemp Chapter 3
1. What is a schema? What is the function of a schema? Give an example of the functioning of a schema which is not in the book.
2. What are the advantages of schematic learning? Why is it especially useful in learning mathematics?
3. What are the disadvantages of schematic learning? How serious are the threats to schematic learning posed by its disadvantages?
4. What outward behavior in a student might you recognize as requiring reconstruction of a schema?
5. How does Skemp define “understanding”? He claims his definition “explains the subjective nature of understanding”. What does he mean by this?
6. How do you react to the experiment described on pages 30-32?
7. What implications for teaching mathematics does our understanding of schematic learning have?
9/19/06:
Skemp Chapters 4 and 5
1. If you were to lead a discussion of these chapters, what questions would you ask?
(To prepare discussion questions, you should first outline the chapter. This means, while reading, you should pause every 2-3 paragraphs and either compose a sentence summarizing them on a separate piece of paper, or highlight parts of the text that contain the main points of those paragraphs. When you finish your outline, read it and compose questions which draw out the main points. Every important idea contained in the chapter should be addressed by one of your questions. You may have to revise your questions several times to accomplish this in a natural way.)
The Learning Gap Chapters 1 – 3
1. Why did the authors write this book?
2. How do our stereotypes of Japanese and American cultures get in the way of improving our educational system?
3. According to the authors, who is to blame for the “crisis” in American education? Why? Do you agree?
4. Do the results presented in this book about elementary education have relevance to high school teachers?
5. Chapter 3 explores several differences in the school and home lives of Asian and American children. These differences may partly explain the gap in achievement between the two groups. Which characteristics of Asian life likely contribute to their relative success? These characteristics suggest possible directions for change in American homes and schools. Which do you think are feasible, and which would probably not work here?
How
We Measure Up
1.
The authors of this article claim that due to a “deluge of confounding
factors”, the “inference that the
The
Case for Quantitative Literacy
1. Which points of the case for quantitative literacy do you find the most compelling?
2. Are there points you feel the author neglected to mention?
9/26/06:
The Learning Gap Chapters 4 – 6
1. Chapter 4 debunks several stereotypes about Asians that Americans call upon to explain their poor relative academic performance. Which stereotypes are they, and what do the authors claim is really going on?
2. Why do you think Americans are so reluctant to use models for academic achievement?
3. Chapter 4 also discusses stereotypes that Americans hold about themselves that get in the way of academic achievement. What are these views, and how do they hinder American children’s progress in school?
4. Some American students fail to reach their academic potential in part because their families don’t value educational achievement. Why do other students fail despite having families that encourage success?
5. What are the causes and
effects of the effort model of learning found in Asia and the ability model
found in the
6. Do you tend to agree more with the goals of intellectualism or anti-intellectualism as described on page 107?
7. Why do you think Americans are reluctant to use groups of students with mixed abilities in the classroom in place of tracking?
8. Why do American parents accept such low levels of academic achievement from their children?
9. Why are low standards a bad thing?
10/3/06:
The Learning Gap Chapters 7 – 10
1.
How would you respond to the idea that the problem with education in the
2.
What are the differences in goals for education in the
3. How would you respond to the claim that Asian students outperform their American counterparts because they spend more time in school each day, and more days in school each year?
4. Describe the pros and cons of the American insistence on attention to the needs of individual students in the classroom.
5. How does Asian teacher training compare with American teacher training? What features of the Asian educational system make such training possible?
6. How do Japanese teachers avoid the burnout American teachers complain about?
7. How do Asians and Americans define the ideal teacher?
How does the American ideal hamper
progress in improving mathematics education in the
8.
Do you think Asian methods of maintaining discipline could work in the
9. How would Americans describe the role of the teacher in a mathematics classroom? How do Japanese view this role?
10.
What is meant by ``coherence" in a mathematics lesson? What are some of the threats to coherence
found in American mathematics classrooms?
11. How do American and Asian classrooms
differ in their use of language and manipulatives?
12. How do American and Asian teachers differ in their approach to student errors? In their use of questions? How are these differences reflected in American textbooks?
13.
Why do we need national standards for mathematics achievement in the
14. What else do we need?
15. Do you agree with the recommendations concerning teaching to the whole class on page 211?
16. How do you feel about increasing class size?
Skemp Chapters 14, 16, 17, and 18
1. On page 179, Skemp writes, ``What we can devise for the surface structure of our symbol system is inevitably much more limited than the enormous number and variety of relations between the mathematical concepts, which we are trying to represent by the symbol system." What does he mean by this statement, and what are the implications of it?
2. What does Skemp mean when he writes, ``mixed emotions are to be expected in a learning situation" on page 195? What does this mean for teachers?
3. In what ways can teachers create unnecessary frustration and anxiety in students?
10/10/06:
A
Coherent Curriculum
1. How would you characterize the intended American mathematics curriculum relative to others in the world?
2. To what extent do teachers deserve the blame for an unfocused, repetitive, undemanding, and incoherent curriculum?
3. What does it mean for a curriculum to be “coherent”?
4. How would you describe the “A+ Composite”? Which countries are considered to be “A+”? How did the authors decide whether or not a country should be considered “A+”?
5. What are the “math wars”?
6. What are the benefits of a coherent national curriculum?
10/24/06:
Cognitive
Science and Mathematics Education
1. Explain how each of the following theories view learning, and describe the implications each has for pedagogy: associationism, gestaltism, behaviorism.
2. What event inspired the “new math” curricular reforms of the 1960’s? Why did it have this effect?
3. In the 1970’s the “new math” was replaced by a “back to basics” movement. Why?
4. Why did Americans in the 1980’s become disillusioned with the “back to basics” reforms of the 1970’s? What was the result?
5. What explanation does Schoenfeld give for the repeated failures of reform movements?
6. How do cognitive scientists differ from classic educational researchers?
7. In pages 17 – 20 Scoenfeld discusses Polya’s influential work on problem solving using the methods of cognitive science. How does Schoenfeld claim to have improved on Polya’s work?
8. In the last part of this chapter, Schoenfeld introduces the constructivist point of view on learning. What is this point of view? What implications does it have for the way we teach?
Polya
1. What tips for teaching do you take away from this chapter?
10/31/06:
New
Knowledge About Errors
1. What is a bug? Why are bugs important to study?
2. Where do bugs come from? What do they say about the way people learn?
3. On pages 171 – 172, the author explains why some students choose to approach mathematics instrumentally, perhaps even despite having a teacher who encourages relational learning. How does he account for this phenomenon?
4. On page 173, the author writes, “The bug explanation of errors does call into question our universal grading procedure: the more problems wrong, the lower the grade.” Do you agree? How should we evaluate student work?
5. How can we reduce the occurrence of bugs in our classrooms?
Some Classical Errors
1. What did you learn from this chapter about some of the errors students make?
2. How can you tell whether poor student performance is due to carelessness, conscious misunderstanding of a concept, or a bug?
3. What does this chapter contribute to the debate over learning with understanding versus learning by rote? About teaching by discovery versus teaching by telling?
Why Johnny Can’t Add
1. Many of Kline’s criticisms of the traditional curriculum are familiar to us at this point in the course. One criticism we haven’t yet discussed appears on page 8 when he notes that deductive proof is required in geometry but not algebra. Why do you think this might be the case?
2. On page 10, the author writes, “Clearly one cannot defend algebra, geometry and trigonometry on the ground that they will be of use later in life.” He continues on page 11, “Much of mathematics taught is often defended as ‘training of the mind.’ There many very well be some training, but the same effect can be achieved withy subject matter that is far more understandable and agreeable.” Do you agree with either of these statements?
3. How does Kline propose we defend the study of mathematics?
4. What is Sputnik and what effect did it have on mathematics education?
5. How did members of the Commission on Mathematics justify the need for curriculum reform? What was the problem with this justification?
6. On page 26, Kline claims that in order to examine the changes offered by the “new math” curricula and determine “why these changes are desirable, and what reasoning or evidence can be proffered to support the desirability of this changes, one is faced with a problem of considerable magnitude.” Why?
7. On page 26, Kline claims that in order to examine the changes offered by the “new math” curricula and determine “why these changes are desirable, and what reasoning or evidence can be proffered to support the desirability of this changes, one is faced with a problem of considerable magnitude.” Why?
The Original New Math
1. Are the authors for or against the “new math”?
2. What positive consequences of this curricular reform movement do they cite?
11/7/06:
The
Math Wars
1. What are the math wars? Who is at war with whom over what?
2. How does each side support its case?
3.
Why does mathematics education in
4.
How does reform mathematics in
5. Why is it so difficult to measure whether or not students learning reform mathematics obtain a deeper understanding of the subject than those learning in traditional ways?
6. What do Richard Askey and Phil Daro believe is missing from the debate? Would James Stigler agree or disagree?
7. How would Dick Stanley describe the pros and cons of the reform movement?
What’s All the Fuss About Metacognition?
1. What is metacognition?
2. What kind of metacognitive behavior is typical of students working on mathematics problems? Of mathematicians?
3. How can students’ beliefs about mathematics affect their problem solving performance?
4. Schoenfeld suggests four classroom techniques to improve students’ metacognitive skills. What are they, and how do they work? Would you consider using any of these techniques in your classroom? Why or why not?
5. In the last section of this chapter, the author suggests an explanation for why the techniques considered above work for him in his classroom. What is this explanation?
6. What is Schoenfeld’s explanation for why mathematics curriculum reforms are destined to fail?
11/14/06:
Principles and Standards
1. What is a principle? What is a standard? What is the purpose of this document?
2. How is the equity principle achieved in the classroom?
3. What does it mean for a mathematics curriculum to be coherent? Why is a coherent curriculum important?
4. On page 18 we read that “effective teaching conveys a belief that each student can and is expected to understand mathematics”. How can teachers do this?
5. On page 19 we read that “worthwhile tasks should be intriguing, with a level of challenge that invites speculation and hard work”. Where do we find such tasks?
6. On page 22 the document claims “the tasks used in an assessment can convey a message to students about what kinds of mathematical knowledge and performance are valued”. What kind of message does a multiple-choice exam convey?
7. What is a formal assessment? An informal assessment? Why does the document advocate the use of both formal and informal assessments?
8. What is the difference between process and content standards?
9. Do the standards proposed in Chapter 3 seem reasonable? Achievable?
10. Do you agree with the statement that systematic reasoning “is found in all content areas and, with different requirements of rigor, at all grade levels” (page 57)? Can first graders reasonably be expected to prove things?
11. Do you think that “high school students should be able to present mathematical arguments in written forms that would be acceptable to professional mathematicians (page 58)?
12. If we allow students to make informal mathematical arguments in the early grades, are we allowing the development of bad habits such as “proof by example”? How do we make sure this doesn’t happen?
13. How do we make students “explicitly aware” of the mathematical connections between the topics we teach (page 64)?
14. Why does this document discourage “tracking” of students? What does it offer as an alternative?
15. The document acknowledges that “disengagement” is a serious obstacle to reaching the vision it describes. What is disengagement and how does society encourage it?
Curriculum
Focal Points
1. What are the focal points?
2. Why were the focal points developed?
3. How do the focal points relate to NCTM’s principles and standards?
11/28/06:
Standards-Based
Curricula, Selection Criteria, and Selection Instruments
1. In what ways do classrooms implement standards-based curricula appear different from those implementing traditional curricula?
2. What are selection criteria in the context of mathematics curricula? What does it mean to design selection criteria? How should one go about doing so? Why are selection criteria important?
3. Which of the sample selection instruments do you feel is most useful? Least useful? Why?
Cognitive
Science and Algebra Learning
1. What is the objective of the research presented in this chapter?
2. What assumptions were made prior to conducting the research? Do these assumptions seem reasonable?
3. What are the instructional consequences of these assumptions?
4. Why do examples play such an important role in mathematics instruction?
5. What kinds of “side effects” do we see as a result of examples and sets of practice problems in textbooks?
6. What explanation does this chapter suggest for why some students perform very well on routine homework exercises, yet cannot solve similar problems outside the homework context?
7. What explanation does the author offer for why students dive into computations without first planning to reach a solution?
8. What method does this chapter suggest for generating practice problems?
The
Roots of Belief
1. What does it mean to say that many students take an empirical approach to problem solving? Where does this tendency originate? Can it be overcome?
2. Why do students view deductive argument as irrelevant to discovery?
3. How doing students come to believe that “being mathematical” means “expressing oneself via the prescribed forms”?
4. Why do students give up attempting to solve a problem after only a few minutes?
5. Why do students lack problem-solving and control strategies?
12/05/06:
Beliefs
That Block Equity
1. Why do gaps in mathematics achievement between rich and poor, minority and white, persist despite decades of reform?
Through Ebony Eyes
1. In Chapter 1, Dr. Thompson briefly describes ten theories that may explain the gap in school achievement between black and white students. Do any of these theories seem more or less plausible that the others? Do any of them match you own experiences? Do you have a theory of your own?
2. Which attitudes and beliefs tend to be shared by effective teachers of black students?
3. Of the instructional practices considered effective for teachers working with black students, which can you imagine implementing in your classroom? How might you do so? Which can you not imagine implementing? Why?
4. What does it mean to make education culturally relevant for students? How can teachers make their math classes culturally relevant?
Performance
without Anxiety
1. What is stereotype threat? How can it affect students of mathematics?
2. What message should teachers take away from this article?