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Math Ed Seminar Specific Reading Questions, Paper-by-paper
Spring 2015
Erlwanger, S. H. (1973). Benny’s conception of rules and answers in IPI mathematics. Journal of Children’s Mathematical Behavior, Autumn 1973, 7-26. [Also, Chapter 5 in NCTM’s Classics in Mathematics Education Research, pp. 48-58.]
1. Discuss constructivism and its relation to the Benny paper. 2. What sort of paper is this?
3. How did Benny rationalize his inconsistent answers?
4. How does Erlwanger summarize Benny’s view towards mathematics? Is this so different from some of our students’ views of mathematics, say in Math 120?
5. Give an example where Benny adds two positive numbers and gets an answer smaller than either of them.
6. What is the danger of just giving examples (with no explanation) and then asking for lots of practice drill?
7. What is the problem with behavioral objectives?
8. How did the teacher in the Benny paper interpret her role in IPI instruction? Do you suppose this was necessary?
9. Erlwanger tried to work with Benny (p. 56). Was he successful? Why? Talk about eradicating bugs and misconceptions.
10.What are some of the weaknesses of the IPI approach? What do they stem from, according to Erlwanger?
11. Why do you suppose the Benny paper was selected as a “classic”?
McClain, K., & Cobb, P. (2001). An analysis of development of sociomathematical norms in one first-grade classroom. Journal for Research in Mathematics Education, 32(3), 236-266.
1. What is meant by “taken-as-shared”? (p.237). 2. What is meant by “experientially real”? (p. 238). 3. Why are sociomathematical norms important? 4. Discuss “constant comparative method”. (p. 241).
5. What is meant by “counting on”, “counting all”, and “counting back”? (p. 241).
6. How does a teacher make sure that she/he is not just valuing all students’ contributions equally?
7. What was the importance of the notational schemes introduced by Ms. Smith? (p. 251-252).
8. What struggle did Ms. Smith go through as she tried to change her teaching practice from a traditional one to a reform one? (p. 263).
9. What is the importance of notation in fostering the ideas of different, sophisticated, and efficient solutions ? (p. 264).
10. What is mean by “a sense of directionality”? (p. 253).
11. Was “thinking strategies” every defined? If so, how? (e.g., p. 260).
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Moore, R. C. (1994). Making the transition to formal proof. Educational Studies in Mathematics, 27, 249-266.
1. What is meant by “grounded theory”? What is meant by “constant comparative method?” 2. What is meant by “concept definition”? What is meant by “concept image”? (p. 252). 3. What is an “equivalence relation”? What is an “equivalence class”? (p. 253-254).
4. What does Moore say students’ concept images lack that are needed for writing a proof? (pp. 257-258).
5. Why, according to Moore, do students have trouble generating their own examples? (p. 260).
6. Note that what Moore calls the “skeleton of a proof” is similar to Selden & Selden’s idea of a “proof framework”.
7. What does Moore say about the two different definitions of 1-1 (for a function)? 8. What is meant by “cognitive load”? (p. 262).
9. What concept images of proof did the students in Moore’s study have? (p. 264).
Tall, D., & Vinner, S. (1981). Concept image and concept definition in mathematics with particular reference to limits and continuity. Educational Studies in Mathematics, 12, 151-169.
1. How do the authors use the term “concept image”? (p. 152).
2. How do the authors use the term “evoked concept image”? (p. 152). 3. What do the authors mean by the term “concept definition”? (p. 152). 4. What do the authors mean by a “personal concept definition”? (p. 152). 5. What do the authors see as a potential conflict factor? (p. 153).
6. According to the authors, how were the topics of limit and continuity taught in English schools (of the time)?
7. What do you think are the consequences of introducing sequences informally first (as on p. 156) and later reinterpreting sequences as functions?
8. Describe the potential conflicts that result from students considering 0.9999... < 1 and 0.9999… = 1? (pp. 158-159).
9. How does an intuitive, dynamic view of limit conflict with the formal definition of limit (given on p. 161)?
10.What sorts of everyday images of continuity do students bring to a calculus classroom? (pp. 164-165). Give examples. How do these differ from the formal definition?
11. Why is it important for a teacher to be aware of students’ incorrect or incomplete concept images of continuity (or some other concept)? (p. 168). What do the authors say a teacher who is aware of such incorrect or incomplete concept images might do? Do you think this would be effective?
12. This paper is considered a “classic”. Why do you think that is?
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1. What is the difference between “extracted” and “stipulated” definitions? Which of these terms refers to mathematical definitions and which refers to everyday dictionary
definitions? (p. 223).
2. What do the authors mean by saying “stipulated definitions create usage”? (p. 224). 3. According to the authors, what are some necessary features of mathematical definitions?
(p. 224). Do you agree with these?
4. According to the authors, what are some preferred features of mathematical definitions? (p. 224). Do you agree with these?
5. This is an example of a research-into-practice article, written for mathematicians. How does it differ from the research articles that we have read (to date)?
6. What is meant by “operable definition”, as discussed by Bills and Tall (1998)?
7. Describe briefly Edwards’ real analysis study. Also, describe briefly the Edwards and Ward abstract algebra study. (p. 225). What did these studies find out about students’ understanding of the role of definitions in mathematics? (p. 226).
8. How was the research in the two studies (real analysis and abstract algebra) conducted? (p. 226).
9. What were some of the results of the two studies (real analysis and abstract algebra)? (p. 227).
10.What objections did Freudenthal (1973) make to the “traditional pedagogical practice of providing for students extant definitions”? (p. 228). Who was Freudenthal?
11.According to the authors, what are some pedagogical objectives that could be fostered by using definition activities? (p. 229). What do the authors mean by a “definition activity”? Can you give some examples? (p. 229).
Dahlberg, R. P., and Housman, D. L. (1997). Facilitating learning events through example generation. Educational Studies in Mathematics, 33, 283-299.
1. What do the authors mean by a “learning event”? (p. 284).
2. Do you think the authors used the same students for both the pilot study and the study reported in this paper? (p. 284).
3. What do the authors mean by “base concepts”? (p. 286).
4. What does it mean for a function to be thought of as an object? As a process? (p. 287). 5. Describe the order and content of the (interview) pages given to the participants during
the interviews.
6. What were the four basic learning strategies that the students used “when present with the concept definition”? (p. 288).
7. What sorts of incomplete or incorrect concept images for function did the students in this study exhibit
8. What is a Dirichlet function? (p. 295).
9. What is a periodic function? Can fine functions be periodic? (p. 296). 10.According to the authors, when do learning events occur? (p. 297).
11.What are the advantages of having students generate their own examples? (pp. 297-298). 12.What do the authors see at the difference between textbook presentations (of a new
concept) and what they did in the interviews? (p. 298).
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Selden, A., & Selden, J. (2011). Mathematical and non-mathematical university students’ proving difficulties. In L. R. Wiest & T. D. Lamburg (Eds.), Proceedings of the 33rd Annual Conference of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 675-683). Reno, NV: University of Nevada, Reno.
1. Some people have observed that this is an observational paper, that no empirical claim is made. Is observation (such as in this study) a legitimate research technique in
mathematics education?
2. What is the theoretical background for this study?
3.Does the categorization of students’ proving difficulties amount to a (significant) contribution to the mathematics education research literature?
4. Have you observed any of these difficulties in yourself (over the years)? 5. Which difficulties resonate most with you personally?
6. Do you understand the difficulty (fourth page) described regarding mathematical
induction proofs (i.e., going from k to k + 1 versus going from k + 1 to k in applying the induction hypothesis)?
7.What is the difference between “minimum” and “minimal”? 8.How does one show “at most one” or “at least one”?
9. What is meant by the Axiom of Choice?
10.What do the authors mean by theorems of Types 1, 2, 3? 11. Discuss Anna Sfard’s idea of “commognition”.
Selden, A., & Selden, J. (2014). ). In M. N. Fried & T. Dreyfus (Eds.), Mathematics and Mathematics Education: Searching for Common Ground (pp. 248-251). New York: Springer.
1. Is this a mathematics education research paper? If not, what is it? 2. What are the contributions of this short paper?
3. Have you noticed these features/characteristics of (written) proofs?
4. Do you think mathematicians write (proofs) well? See title of Csiszar’s (2003) paper in the reference list.
5. Would letting transition-to-proof course students know these characteristics of (written ) proofs help them construct better proofs? Why or why not?
Resnick, L. B., Nesher, P., Leonard, F., Magone, M., Omanson, S., & Peled, I. (1989). Conceptual bases of arithmetic errors: The case of decimal fractions. Journal for Research in Mathematics Education, 20(1), 8-27.
1.What is meant by Brown and VanLehn’s “repair theory”? (p. 8). 2. What is meant by overgeneralization? (p. 9). Give some examples.
3. What are the three incorrect rules that previous researchers, Sackur-Grisvard and Leonard (1985) found? (p. 9). Explain their Rules 1, 2, and 3.
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elements of whole number knowledge? Give some examples of elements that are the same and elements that are different.
6. What does it mean to say that the 4th, 5th, and 6th grade students conflated the name of the columns to the right of the decimal point and to the left of the decimal point ? (pp. 11-12).
7. What is the difference in the organization of the curricula (with respect to fractions and decimals) in France versus in Israel and the US? (p. 13).
8. How did the research proceed (i.e., how was the data collected and analyzed)? (p. 13). 9.What is meant by “probing” in an interview?
10. What do you think of the tasks (Table 4) used to see whether a child was using Rules 1, 2, or 3? The authors stated that these would allow them to “attribute a rule to a child on the basis of a series of responses, some of which are correct and some incorrect.” Do you agree this is possible? (This assumes the children are not answering randomly, but are using a rule.)
11. What is meant by a “conceptual framework”? (p. 16).
12. Give some of the additional interview tasks used to uncover the “children’s conceptual frameworks”? (p. 16)
13. What is a “malrule”? (Table 7, p. 18).
14.The zero insertion task (results in Table 11, p. 20) helped the researchers classify the children’s rules. What is it? (p. 19). In what way did it help?
15. What are the results of this study on the children’s use of Rules 1, 2, and 3? (pp. 16-20). 16. For analyzing the “cognitive sources of errorful rules”, what two sources of data did the
researchers use? (p. 20).
17.The authors state that they went beyond the findings of Sackur-Grisvard and Leonard in two respects”. What are these? (p. 24).
Schoenfeld, A. H. (2007). Method. In F. Lester (Ed.), Second Handbook of Research on Mathematics teaching and Learning (Chapter 3 of Part I). New York: MacMillan.
1.What is “positivism”?
2.What is meant by a study that offers an “existence proof”? Why are such studies valuable? 3. What does Schoenfeld see as the goals of Parts I and II of this chapter?
4.What does Schoenfeld mean when he writes, “In the 1960s and 1970s, an emphasis on experimental and statistical methods was stifling the field”?
5.What are Schoenfeld’s three aspects/dimensions of research (Section 2)? What do you think of them? What two of these three aspects/dimensions does Schoenfeld see the students-and-professors studies lacking?
6. What is triangulation? Why triangulate?
7.What “processes” does Schoenfeld consider as involved in “all empirical research”? 8.What is mean by the “cognitive revolution”?
9. How does Schoenfeld get the 88%, 91%, and 95% (at grades 3, 5, and 7) of students “rated proficient on the SAT-9” (Tables 1, 2, and 3)?How does Schoenfeld get the “roughly 1/3 of those declared proficient of the SAT-9”, but not on the MARS test (Tables 1, 2, and 3)?
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11. What are the four “steps [or phases] from test tube to new drug application review”? What are Schoenfeld’s equivalents for evidence-based educational R&D (Figure 6)? 12. What is an “acquisitionist theory of learning”?
13. What is meant by a “design experiment”? What are some common features of design experiments?
14.What sorts of questions could/should one investigate in Phase 2 of curriculum studies? 15.
DeBock, D., Verschaffel, L, & Janssens, D. (1998). The predominance of the linear model in secondary school students’ solutions of word problems involving length and areas of
similar plane figures. Educational Studies in Mathematics, 35, 65-83.
1. What is additive reasoning? What is multiplicative reasoning? Give some examples. (p. 66).
2. What three hypotheses di the authors make for Study 1? (pp. 69-70). What were the results (i.e., which of the three hypotheses, if any, were confirmed)?
3. How were the students selected for Groups I, II, and III in Study 2? (p. 75). 4. The same three hypotheses were tested in Study 2. What were the results? (p. 76). 5. The authors state that in getting the non-proportional items correct “the type of figure
made a difference”. (p. 76). Why do you suppose this is? 6. What is a Tukey test? (p. 76). What is it good for?
7. What three solution processes/strategies did the authors speculate that the students used on these proportional and non-proportional tasks? (p. 78). Which of these strategies did the researchers see in 90% of the cases?
8. What three explanations involving context did the authors propose for their findings? (p. 86). Do you find these reasonable/plausible?
9. What explanation (not based on context) did the authors propose for their results? (p. 81). 10.How do you see these results in relation to the work on the Students-and-Professors
problem (mentioned in Schoenfeld’s chapter on Method)?
11.The authors speculated that had the ready-made drawings been on graph paper, the results on the non-proportional problems might have been better. What do you think?
Johnson, H. L., Blume, G. W., Shimizu, J. K., Graysay, D., & Konnova, S. (2014). A teacher’s conception and definition and use of examples when doing and teaching mathematics. Mathematical Thinking and Learning, 16, 285-311.
1.What is meant by an “enactivist perspective”? (p. 285 & 290).
2. What is a metamathematical concept? (p. 287). How does one learn a metamathematical concept?
3.What is meant by the “form” of a definition? (p. 288).
4.What is meant by the “function” of a definition? (p. 288-289). 5.What is a “barely-not-minimal” definition? (p. 288).
6.How do the authors distinguish “engaging in defining” and “using definitions” (p. 290). 7.What is meant by “exemplification”? (p. 290).
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9.What sorts of data did the researchers collect on Lily? (p. 292).
10. The researchers distinguished between Lily’s use of examples as instances versus her use of examples as “exemplification”. What is the difference? Could you see the difference in the authors’ interview excerpts and descriptions of Lily’s actions?
11. How did the researchers “get at” Lily‘s metamathematical view of definitions? Carlson, J. P., & Bloom, I. (2005). The cyclic nature of problem solving: An emergent multidimensional problem-solving framework. Educational Studies in Mathematics, 58, 45-75.
1. According to Schoenfeld, as quoted in this article, what is the difference between an exercise and a problem? {p. 47).
2.How does the problem solving literature describe “resources”? (p. 48).
3.What does Vinner mean by “pseudo-conceptual” and “pseudo-analytical” thought processes? (p. 48).
4.What do the authors mean by “control”, “monitoring”, and “self-regulation”? (p. 48). 5.What is meant by “mathematical intimacy”? (p. 48) What is meant by “mathematical
integrity”? (pp. 48-49).
6.What behaviors did the participants in this study display during the “orienting phase”? (pp. 62 & 68).
7.What behaviors did the participants display during the “planning phase”? (pp. 62-63 & 68).
8.What behaviors did the participants display during the “executing phase”? (pp. 63& 68). 9. What behaviors did the participants display during the “checking phase”? (pp. 63 & 69). 10.Have you observed in yourself (when solving mathematical problems, rather than
exercises) any of the behaviors that the authors described mathematicians engaging in? 11. What sorts of metacognitive questions did the mathematicians in this study ask
themselves during problem solving? (p. 64).
12. What are the four phases of the authors’ multi-dimensional framework? What are the two embedded cycles (a.k.a., sub-cycles) of their framework? (Appendix II, p. 72). 13.How do you suppose one can promote problem-solving behaviors (such as those
described in this article?
Seaman, C. E., & Szydlik, J. E. (2007). Mathematical sophistication among preservice elementary teachers. Journal of Mathematics Teacher Education, 10, 167-182.
1. What is the distinction between content knowledge, knowledge of learning and the learner general pedagogical knowledge, and pedagogical content knowledge (PCK)? (p. 168). 2.What is meant by “social constructivism”? (p. 169).
3.What is meant by “mathematizing”? (p. 170).
4.What do the authors mean by the phrase “mathematical sophistication”? (p. 170). 5.What do you think of the nine points of the author’ framework (i.e., their list of what
mathematicians value)? (pp. 170-171).
6. What criteria did the authors use to select the participants for their study? What reasons do they give for this? (p. 172-173).
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8.What do the authors say is the “crucial role that mathematics content courses play in the preparation of teachers”? (p. 179).
9.These authors state that mathematicians are not aware of the “intuitive mathematical ideas held by children, with the sequencing of the school curriculum and with representations, examples and tasks that are most effective at bridging children’s mathematical models with more sophisticated ideas.” They also say that mathematics educators “lack mathematical sophistication” and that “neither mathematical sophistication nor the mathematical knowledge for teaching is easily attained.” What do you think might be done about this situation? If you are currently teaching a mathematics course for
preservice elementary teachers, do you feel that you have all of the above—knowledge of mathematics, knowledge of intuitive ideas held by children, knowledge of the sequencing of the school curriculum, and knowledge of representations and tasks that are effective at bridging children’s mathematical models with more sophisticated ideas?
Zazkis, R., & Hazzan, O. (1999). Interviewing in mathematics education research: Choosing the questions. Journal of Mathematical Behavior, 17(4), 429-439.
1. What three aims did Ginsburg (1981) see for a clinical interview? (p. 430).
2.What do the authors mean by assessing “strength of belief” and why is that important? (p. 430).
3. What were the research questions? (p. 430).
4. What is mean by a “phenomenological analysis of the content domain”? (p. 436).
5. Does reading this paper help one design clinical interview research questions? If yes, how? If no why not? What is missing?
6. Usually we (personally) have noticed something in teaching classes and wondered why students would do such a thing or answer the way they did. We have then often designed interview questions to find out why. What sorts of things have you noticed in teaching classes that you would like to explore further?
7. What kinds of mathematics task questions have researchers asked (according to these researchers)? That is, what were the categories of questions that they found?
8. Think of something that you have observed when teaching a class or that you have observed of fellow students that you are (or have been) curious about. Think about what sorts of specific research questions would “get at” information about the behavior that you have observed.
9. If researchers were to go beyond clinical interviews (and they have done so), what sorts of research studies could be (or have been) done?
10.How many mathematics education research papers did these authors consider for their study? Did they say? Would it make a difference if you knew this information? If yes, in what way?
Rasmussen, S. (2015). Common core mathematics tests are fatally flawed and should not be used: A critique of the smarter balanced tests for mathematics. Available at
www.mathedconsulting.com or from steve@mathedconsulting.com.
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However, it is topical, especially in the United States, where these tests are currently (i.e., in Spring 2015) being taken by K-12 students.
1. What differences do you see between this position paper and the mathematics education research papers that we have been reading? In what ways is it different? In what ways is it the same?
2. What did the author actually do?
3. What are modal interfaces and what does the author say are a problem with them (for students taking these tests)?
4. Has anyone tried the practice tests for themselves (the way the author did)? Do you agree with the author’s criticisms of the particular practice test questions that he analyzed in detail? Did you find any other things about these practice test questions that you think would be confusing for students or difficult for them to interpret or answer?
5. Test development generally takes a long time. There is a video on the Educational Testing Service (ETS) website that explains their process of test development, see
http://www.ets.org/understanding_testing/test_development/. ETS develops tests such as the TOEFL, SAT, and GRE.
MacNell, L., Driscoll, A., & Hunt, A. N. (2014). What’s in a name: Exposing gender bias in student ratings of teaching. Innovations in Higher Education, published online December 5, 2014 by Springer. Available at http://link.springer.com/article/10.1007/s10755-014-9313-4.
N.B. This is not a mathematics education research paper, but it is an example of a quantitative research paper in education and has some general interest to those going into the academic teaching profession.
1. What differences do you see between this quantitative paper and the qualitative
mathematics education research papers we have been reading? What sorts of things does one find out that are different?
2. Do you think the results would have been different had the researchers not been
investigating an introductory-level course? A anthropology/sociology course? An online course?
3. What do you think of the researchers deciding to use an online course, and hiding the instructors’ real identities, to get information on the students’ biases about their instructor’s perceived gender identity?
Stylianides, G. J. (2009). Reasoning-and-proving in school mathematics textbooks. Mathematical Thinking and Learning, 11, 258-288.
1.What is meant by “scaffolding”? (p. 258).
2.How does the author use the term “reasoning-and-proving”? (pp. 258-259). 3. What is TIMSS? (p. 259).
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5.What are the two dimensions of the author’s analytic framework? (p. 262 and Table 1). 6.What is a “plausible” pattern for the author? (p. 263).
7. How does the author distinguish between proof and argumentation? (p. 265). If you know Duval’s work, how does the author’s distinction compare with Duval’s?
8. What is a generic example? (p. 265). Can you give one?
9. Can you give an example of 9i.e., do you understand) what is mean by proofs by counterexample, proofs by contradiction, proofs by reduction ad absurdum, proofs by mathematical induction, proofs by contraposition, and proofs by exhaustion? (p. 265). 10.What three reasons does the author give for the desirability of rationales as opposed to
empirical arguments? (p. 267).
11. How does the author distinguish between conjectures that are “proof precursors” and conjectures that are “proof non-precursors”? (p. 268). Can you given an example of each. If you know Harel’s work, compare the author’s distinction to Harel’s “result pattern generalization (RPG)” and “process pattern generalization (PPG)”. How does these concepts of Harel differ from those of the author?
12.According to the author, who is citing deVilliers, what purposes does proof serve? (pp. 268-9). Why does the author not include (in his analysis and framework)
“communication” as a purpose of proof? Also, why did he not include “systematization”? 13.What four stages did the author go through to produce his analytic framework?
(pp.270-271). Does this seem reasonable and thorough?
14.What were the content area comparisons for reasoning-and-proving that the author found for CMP? (p. 274). Were these adequate for a reform textbook series like CMP?
15. What were the author’s four levels of sophistication? (p. 280).
Rasmussen, C., Wawro, M., & Zandieh, M. (2015). Examining individual and collective level mathematical progress. Educational Studies in Mathematics, 88(2), 259-289. Published online: 30 November 2014.
1.Why are there no page numbers? (Because we worked from the online version and
numbered them ourselves, beginning with page 1. All page numbers are given in this way below.).
2.Which four parallel analyses did the authors make? (p. 2). 3.Explain Toulmin’s model? (p. 5).
4.What do the authors mean when they consider “the mathematical progress of the classroom community in terms of the disciplinary practices of mathematics? (p.18). 5.What is meant by “constructivism” and “social constructivism”? (p. 2).
6.What is mean by “the normative ways of reasoning” in a classroom community? (p. 3). How is this related to the “didactic contract”?
7.Note the use of the word “conceptions” when referring to the thinking of an individual. (pp. 3-4, Fig. 2).
8.What claim do the authors make for the contribution of their extension of the Cobb and Yackel interpretive framework? (p. 3).
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10.What do the authors take as evidence of an established norm in a classroom community? (p. 4). What two criteria do they use? (p. 5).
11.How do the authors describe the activity of algorithmatizing? (p. 6).
12.What do you think of how the authors decided to present their linear description of their data and analyses [i.e., episodes, then interpretations in the following order—classroom mathematical practices (p. 13), individual mathematical conceptions (p. 16), individual participation (p. 17), discipline of mathematics (p. 18)]? (pp. 8-20).
13.Note that it took the linear algebra class three days to get to the concepts of span, linear dependence, and linear independence. Compare this to the amount of time spent in an ordinary lecture class to develop these notions—the class time taken by such
participatory methods is often greater than the time taken in an ordinary lecture class. What is gained by doing this? What is lost, if anything, by doing this? In answering this question, consider the “coverage issue”, which mathematics education researchers do not often mention, but is of interest to the mathematics lecturers teaching the course.
14.What do you think of using Krummheuer’s terms (i.e., author, relayer, ghostee,
spokesman and conversation partner, co-hearer, over-hearer, eavesdropper) for analysis of individual participation)? What does this add to their analysis? Does this have any implications for teaching?
15. What do you suppose the authors mean by “theoremizing continued through a process of increased rigor as students pushed to understand why a theorem is true or to be fully convinced of its veracity”? (p. 19). What do you suppose the mean by “increased rigor”? What view(s) of proof, if any, do you think the students in this linear algebra class gained? Compare this to Harel and Sowder’s (1998) ideas, where they speak of proving in terms of “ascertaining” and “persuading”. If this linear algebra course (insofar as you can tell from the article) were used as a transition-to-proof course in a U.S. university, would the students, who had successfully taken this course, be prepared for traditional lecture courses in undergraduate abstract algebra or real analysis, that is, would they be able to write proofs that their professors would accept (within a short time of entering those courses)?
Maher, C. A., & Martino, A. M. (1996). The development of the idea of mathematical proof: A 5-year case study. Journal for Research in Mathematics Education, 27(2), 194-214.
1. What sorts of data were collected? (p. 199).
2. Why do you suppose the researchers selected combinatorics as the topic to introduce these children to?
3. What themes were looked for in the analyses of the transcripts? (p.201).
4.What do the authors mean by “local organization” in Stephanie’s solution? (p.204). 5. Was there an actual research question?
6. Do the critical events selected for Table 1 provide convincing evidence for the growth (over time) of Stephenie’s argumentation ability?
7.What are Balacheff’s three forms of argument? (p. 198 & p. 212). 8. Why do you think this particular paper was published in JRME?
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10. According to the authors, what conditions “encourage student conversation in classrooms”? (p. 196). What do you think of these?
11.Why did the authors select Stephanie for this case study?
Shepherd, M. D., Selden, A., & Selden , J. (2012). University Students’ Reading of Their First-Year Mathematics Textbooks. Mathematical Thinking and Learning, 14, 226-256.
1.What were the authors’ research questions? (p. 232).
2. Was the methodology (reading aloud to the interviewer, who was also their teacher) appropriate for answering the research questions?
3.Why do you suppose the authors defined “reading effectiveness” the way they did? (p. 257).
4. Why do you suppose so much of the textbooks read by the students was included in the appendices?
5. The authors describe U.S. students first-year university mathematics textbooks as
containing exposition, definitions, theorems, less formal mathematical assertions, as well as graphs, figures, tables, examples, and end-of-section exercises. (p. 230). Does this description agree with your experience?
6. Do you agree that, in the case of mathematics textbooks, “everyone who reads the definition of a concept with comprehension will have basically the same basic understanding of the definition”? (p. 230).
7.What do you think of the debriefing questions (Appendix C, p. 256)? Are there other questions that you would like to have had asked?
8. Have you observed the three kinds of difficulties (p. 238) discussed by the authors in yourself or in others (e.g., your students)?
9. What do you think of the kind of advice given to students in the precalculus textbook used by the students in this study? (pp. 226-227). Is it likely to help? Will the students know how to carry any of it out? Will they actually try to carry it out?
10. What do you think of the differences in style of writing between mathematics textbooks vs. other books? (#1. - #9., pp. 228 & 230)? Do these resonate with your experience. Have you now, or in the past, had trouble with any of these differences?
Noss, R., & Hoyles, C. (1996). The visiblity of meanings: Modelling the mathematics of banking. International Journal of computers for Mathematical Learning, 1, 3-31.
1. What sort of journal is/was this? It is now called Technology, Knowledge, and Learning. Why do you suppose the name was changed?
2.What do you think the authors mean by “a simplistic notion of transfer”? (p. 3). 3. What two positions on modelling do the authors distinguish? (p. 4).
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6.What is meant by providing the bank employees with “ a new ontology”? (p. 10). 7.What is meant by “constructionist”? (p. 11).
8.What were the two themes that underpinned the authors’ courses for the bank employees? (p. 12).
9. How did the authors view graphs vs. how the bank employees saw graphs? (pp. 15-16). 10. What did the authors see as their goal in teaching the bank employees? (p. 17).
11. What do Noss and Hoyles mean by “situated abstraction”? This term is not used in this particular paper, but one can see the germ of this idea here. You will probably need to look this term up.
Eisenhart, M. A. (1988) The ethnographic research tradition and mathematics education research. Journal for Research in Mathematics Education, 19(2), 99-114.
1.What does the author claim are the “main goals of the research” for mathematics education researchers vs. educational anthropologists? (p. 100). Is this still true (if it ever was) of mathematics education researchers today?
2. What sorts of researchers does the author label as “educational researchers”? (p. 101). As “educational anthropologists”? (p. 102).
3. What, according to the author, is the central idea of interpretivism? (p. 102). 4. According to the author, what are the four methods of data collection used by
ethnographers? (pp. 105-106). Do mathematics education researchers today use these methods?
5.What does the author mean by “ethnographic interviewing”? (p. 105).
6. What other methods of data collection (other than the four discussed in detail by the author) are used in ethnographic research? (p. 106).
7.According to the author, what is the ultimate goal of ethnographic research? (p. 107). 8.On page 107, the author suggests that (in ethnographic research) the “sorting procedure”
can be done by the researcher alone or by the researcher and the participants together. Do you think that a mathematics education researcher should pass his/her research results past the participants for their comments/reactions?
Palha, S., Dekker, R., & Gravemeijer, K. (published online June 2014). The effect of shift-problem lessons in the mathematics classroom. International Journal of Science and Mathematics Education. [N.B. Pages are number consecutively below.]
1.What do the authors mean by a “quasi-experimental” study? (p. 1).
2.What do the authors mean by “theoretical knowledge”? ( p. 1). Is this ever defined/ 3. What distinction do the authors make between “enrichment” process and “construction”
process? (p. 3 ).
4.Why do the authors say “simply devising good problems” is not enough to “foster
construction processes in classes familiar with more traditional forms of education”? (p. 4).
5. What three main activities did the authors develop for learning about geometric proof? (p. 6).
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7.What is meant by an “operational or action conception of function”? (p. 7). What is meant by a “structural or more object-oriented conception of function”? (p. 7).
8.What is meant by “covariational reasoning”? (p. 7). 9. What sort of journal is this?
10. What do you suppose the authors mean by “a course”? (p. 3 & p. 18 & p. 23). 11. What do the authors mean by a “process model for group discussions”? (p. 4). 12. What do the authors mean by “passes through” in the “Exploring and Conjecturing”
learning activity described in Table 1? (p. 7).
13.Do the authors provide enough description on how the groups of students were formed? (pp. 9-10). What else might you like to know?
14. What do you think of the authors “three hierarchical categories”--TR, ER, RR--given in Table 3? (p. 11).
15.Would you have liked to have seen a specific task (or tasks) analyzed according to these various criteria, e.g., TR, ER, RR and correct, partially correct, incorrect? (p. 12). 16.Why do you suppose integer values (with a mean score of 3) were used for the geometric
proof test? (p. 14). Why do you suppose decimal values (with means scores of 0.50 and 0.31 for the SPLA and RLA groups) used? (p. 14).
17.The information on the top of page 14 is very difficult to follow. Would there be a better way to convey this information?
18. The authors examined “student engagement during the shift-problem lessons by
analyzing the solutions constructed collaboratively in these lessons.” This is a proxy for direct observation, say by using video. What do you think of this method? (p. 18). 19. What do the authors mean by examining “the extent to which the students enriched their
knowledge”? (p. 19).
20. The authors say that they judged “complexity” by “the number of theorems, definitions and interrelationships “ and “the unfamiliarity of the concepts that had to be recalled and applied in innovative ways”. How do you suppose they did that counting? (pp. 19-20). 21.Why do you suppose the authors write “We do not know why this happened. [i.e., finding
a much higher percentage of answers in the empirical-reasoning or no-answer categories in the SPLA(G) group than in the RL(G) group for task 3]”? Do you suppose they did not ask the teachers their opinions about this? (p. 20).
22. Which do you suppose are the proof validation tasks in the tasks given in Appendix 1 and Appendix 2? (pp. 25-26 & p. 31).
Dawkins, P. C. (2015). Explication as a lens for the formalization of mathematical theory through guided reinvention. Journal of Mathematical Behavior, 37, 63-82.
1. What is neutral axiomatic geometry?
2.What are the author’s three criteria for assessing whether a proposed formalization could sucessfully explicate an informal one”? (pp. 63 & 65).
3.As explained by the author, what are Gravemeijer’s four stages of mathematical activity when implementing guided reinvention? (pp. 63 & 67).
4.What is mean by “model of” and “model for”? (p. 67).
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6. What is meant by having similarity, exactness, and fruitfulness? (p. 65).
7. According to the author, what are some of the ways secondary and undergraduate curricula have explicated the concept of “line”? (p. 65).
8.How does the author define “f-similarity” [formal similarity] and “i-similarity” [informal similarity]? (p. 66).
9. Who is this article written for? What does a reader need to know in order to understand this article?
10. What do the elements of formal theory explicate or formalize? (Section 3.4). 11.How does the author use the organization tool (Fig. 2)?
12.What does the author mean by “recognizing incompatibilities”? (p. 69). Give an example of some incompatibilities from this paper.
13. What does the author mean by a “design experiment”? (p. 69).
14.In Table 3, the author mentions the data collected via pre/post-meta-mathematical questionnaires (p. 70). Why do you suppose he collected this data? Was it used in this paper?
15. What is meant by “retrospective analysis”? (p. 70).
16. What does the author mean when he writes “… the primary unit of analysis … [is] the ecology of student meanings: their history, their evolution, and their implications for students’ sense making and proving”? (p. 70).
17. What does the author mean by “semantic restraint”? (p. 80).
18. Do you think a course based on Blau’s geometry book, which was used in this course, is suitable for preservice secondary teachers? Will taking an axiomatic geometry course, based on Blau’s textbook, help preservice teachers teach geometry to high school students?
Ely, R., & Adams, A. E. (2012). Unknown, placeholder, or variable: What is x? mathematics education Research Journal, 24, 19-38.
1. How do the authors define “unknown”? (p. 21).
2.How do the authors define “variable”? (p. 21). What do the authors see at the two distinguishing properties of variable?
3.How do the authors define “placeholder”? (pp. 21-22).
4.How do the authors define “generalized number”? (pp. 22-23). 5.What is meant by “ theory of epistemological obstacles”? (p. 23). 6.What is meant by “genetic epistemology”? (p. 23).
7.What two important practices distinguish unknowns from variables? (p. 23).
8. What two important shifts in point of view did mathematicians need to make in order for the above two practices to became available to them? (p. 23).
9. What historic idea(s) prevent the historical development of our generic idea of “number” and of generic methods for manipulating quantities? (p. 24).
10.What is the difference between (quantitative) extensive and (qualitative) intensive quantities/? (p. 27).
11.What do the authors claim is a crucial step for middle grades students? (p. 30). 12. What sort of article is this?
16 numbers, and variables? (p. 22).
14. What is meant by the phrase “ontogeny recapitulates phylogeny”? What might this phrase have to do with this paper?
15.Who is credited with the use of z, y, x to represent unknowns?
Selden, J., & Selden, A. ( 2015). A perspective for university students’ proof construction. Submitted of the Proceedings of the 18th Annual Conference on Research in Undergraduate Mathematics Education. Available currently on www.academic.edu or www.Researchgate. net. [N.B. The pages are numbered consecutively below.]
1. Would it have been helpful for you (as a reader) to have described the students in more detail in this theoretical paper?
2. What is meant by a “proof framework”? (p. 3). Illustrate the idea of a proof framework using the Theorem: The sum of two odd integers is even.
3. What is an “operable interpretation” of a definition? (p. 4). Illustrate the idea of an operable interpretation of a definition using the definition AB = {x| xA and xB}. 4. Why do journal articles, etc., have lists of key words?
5. Would the article benefit a reader like yourself from the inclusion of some sample student scratch work?
6. What ideas from psychology would you like to discuss further?
7. How is the idea of a mental action, the same as and different from, the idea of a physical action?
8. What do the authors mean by an “unreflective guess”? (p. 8). 9. What is a “design experiment”? (p. 1).
10. What is meant by S1 cognition and S2 cognition? (p. 2.).
11. What do the authors mean by the “formal-rhetorical” part and the “problem-centered” part of a proof? (p. 3). How do these ideas relate to the idea of a “proof framework”? 12.What do the authors mean by a “behavioral schema”? (pp. 5-7).
13.What do the authors mean by “cognitive feelings”? (p. 9).
14.What do the authors see as the importance of automating certain proving actions such as the construction of a proof framework?
Math Ed Research Seminar Final Take-Home Examination
Spring 2015
I. Briefly explain the following concepts/terms/ideas:
1. Concept image, concept definition, evoked concept image 2. Social norms, sociomathematical norms
3. Cognitive conflict
17 6. Experientially real
7. Grounded theory 8. Cognitive load
9. Stipulated definition vs. extracted definition 10.Conceptual framework, theoretical framework 11.Semi-structured interview
12.Triangulation
13.Example, nonexample, counterexample 14.Exemplification
15.Behaviorism 16.Metacognition
17.Problem (as used by Schoenfeld) 18.Mixed methods study
19.Conceptual understanding 20.Procedural fluency 21.Cognitive revolution
22.Acquisition (container) vs. participation metaphors (as explained by Sfard) 23.Design experiment
24.Repair theory (as applied to bugs, a.k.a., misconceptions) 25.Positivism
26.Constant comparative method
27.Generality (as applied to math ed research) 28.Trustworthiness (as applied to math ed research) 29.Falsification (as applied to math ed research)
30.Explanatory power (as applied to math ed research) 31.Design experiment
32.Social constructivism (as used in mathematics education) 33.Situated abstraction vs. situated cognition
34.Cognitive demand (of a task) 35.Qualitative vs. quantitative research
