Reading Discussion Questions for Mathema

  

Math Ed Seminar Specific Reading Questions, Paper-by-paper

Fall 2014

Housman, D., & Porter, M., (2003). Proof schemes and learning strategies of above-average

mathematics students, Educational Studies in Mathematics, 53, 139-158.

  1. Does anyone have any comments/questions about how the study was conducted? 2.

  Did the paper say when the 2 interviews were conducted? One after the other? On different days?

  3. What kinds of data did the authors collect? 4.

  How was this data analyzed? 5. Why do you think the authors selected Anne, Carol, and Cathy to describe in detail? 6. What do you suppose is the importance of the study, i.e., why was it published? 7. What (technical) terms gave you trouble or would you like to discuss? 8. Why do you suppose the authors used some conjectures (pp. 143-4, Table 1) from the literature?

  9. What were the sources of data? Interview transcripts, written work. Was there enough triangulation?

  10. What else would you have liked to know about the students/study? E.g., the background of Chris and Anne and Cathy and Bonnie (Table IV)?

  11. How were the terms (Table IV) primary, significant, insignificant defined by the authors (p. 146). Is this enough description for you to get a sense of how they used these terms?

  N.B. Notice there are no right/wrong answers to interview questions. One is trying to get at how students think.

  For Alcock, L., & Simpson, A. (2002). Definitions: Dealing with categories mathematically. the Learning of Mathematics, 22, 28-34.

  N.B. This is a theoretical paper, some call it a “think piece”. For the Learning of Mathematics (FLM) is a different sort of journal than Educational Studies in Mathematics (ESM). Note that the authors do support their theoretical ideas with illustrative data.

  1. What ideas are being introduced in this theoretical paper? 2.

  Were the Figures (diagrams) helpful? 3. What did you learn from this paper? 4. Are there any (technical) terms you want explained? 5. What else might you have liked to have seen in this paper? 6. How do you suppose Greg (a student in the paper) got to where he uses definitions the right (i.e., mathematical) way?

  7. Why do you suppose it is that many mathematicians are unaware that students have such problems with (formal) mathematical definitions?

  8. Who read FLM? 9.

  Did the authors answer their two questions on p. 28? What were the answers they gave?

  

Vinner, S. (1991). The role of definitions in the teaching and learning of mathematics. In D.

Tall (Ed.), Advanced mathematical thinking (pp. 65-81). Dordrecht, The Netherlands: Kluwer Academic Publishers.

  1. Are the ideas of “concept image” and “concept definition” clear now that you have read this chapter?

  2. How are “concept image” and “concept definition used by Vinner? 3.

  What do you think of Vinner’s statement (p. 82), “ We do not believe in ‘mathematics for all’.”?

  4. Does Vinner suggest an “appropriate pedagogy” (p. 81)? What do you think would be an appropriate pedagogy? In what sorts of “technical contexts” do you think it is important to work with the concept definition vs. relying on one’s concept image?

  6. Is being able to use a concept correctly in lots of situation enough to say that the person has understood the concept?

  7. What do you think about the statement, “You can have grasped a concept and yet not know that you have done so.” [This is a questions about implicit learning.]

  8. What does it mean to “have a concept”? 9.

  “Definitions are arbitrary” ( p. 66, #5.) does not mean “anything goes”. What do you think it means?

  10. What do you think if the best definition of absolute value (of a real number) for pedagogical purposes, say in Math 120 (a very beginning algebra course, below college algebra)? Zazkis, R., & Leikin, R. (2008). Exemplifying definitions: A case of a square.

  Educational Studies in Mathematics, 69, 131-148.

  1. Are all the “odd ball” definitions (given by student participants” necessary for the authors to illustrate their points?

  2. What is the theoretical framework? Why do we have theoretical frameworks? 3.

  What do you think of the methodology of the study? Was it a good one for answering their research questions?

  4. What were the research questions? Were they stated? Where? 5.

  How does one’s personal example space differ, or be the same as, one’s concept image?

  6. What do the authors mean by “appropriate” and “inappropriate” definitions (as offered by the participants)?

  7. What do you think of Vasco’s (2006) classification (p. 138) of definitions of polygons?

  8. What do you think of their category of “richness” of a definition? Here the authors mean by “richness” the attributes (of a square) that refer to other than equal sides and right angles.

  9. What do you think of the ideas of minimal definitions vs. accessible definitions. Can a definition be both minimal and accessible?

  10. What did the authors find out, i.e., what are their results?

  11. What do you think of using the pedagogical tool (generating examples) as a research tool?

  Sandefur, J., Mason, J., Sylianides, G., & Watson, A. (2013). Generating and using examples in the proving process. Educational Studies in Mathematics, 83, 323-340.

  1. The authors state, “Students also need to recognize when exemplification is the most appropriate tool at the time.” (p. 339). However, from other math ed research papers, it is known that students have great difficulty just coming up with examples. How does one (as a teacher) get them to this point?

  For those who know the Carlson and Bloom problem-solving framework, it is interesting to note that these authors apparently did not consult it. How does these authors’ manipulating--getting-a-sense-of--articulating (MGA), syntactic/semantic distinction

  (S/S), conceptual insight (CI), and technical handle (TH) compare and contrast with the Carlson & Bloom problem-solvi ng framework? That is, is the authors’ MGA, S/S, CI, TH point of view compatible with the Carlson & Bloom framework? Sandefur, et al. only considered simple number theory theorems (proved by transition-to-proof course students). It might be that their MGA, S/S, CI, TH is a subset of the Carlson & Bloom framework. It may be that it adds something to the Carlson & Bloom framework. It might even be incompatible with the Carlson & Bloom framework. What do you think? [N.B. Like many math ed researchers, these authors consider proving as a kind of problem solving, so it is not “out of line” to compare their work with that of Carlson & Bloom.] 3. What is the difference, if any, between a “generic example” (p. 324) and a “generic proof”?

  4. What was the research question? What answer was given by the authors? 5.

  What were the 3 frameworks the authors state (at the beginning of the paper) that they used? What 2 frameworks do the authors say they are integrating (p. 339)? Is there a discrepancy here? How are the 3 frameworks (mentioned at the beginning of the paper) related to the 2 frameworks mentioned towards the end of the paper?

  6. What problems did the authors have using the syntactic/semantic distinction? 7.

  Note that, trio 1 (A1, B1, C1) do not consider both n=3L+1 and n+3L+2, just the former, so one wonders ether the authors jus t “cut short” the description of the proving process

  (on p. 330) or whether the students never realized (in 39 min) that they had another case to consider. Why do you suppose the authors didn’t mention this?

  8. In the analysis/commentary, the authors use “mod 3) language, but is that really the way the students are thinking when they consider m=3j, m=3j+1, m=3j+2 (p. 330)?\

  9. What claims do the authors make (relative to students’ example use) (pp. 333, 337-339)? 10.

  Why do you suppose these authors never put in the final proof that the students came up with?

  11. What is the organization (of the presentation of the data/results)? Is this a good one for helping readers understand the authors’ data/results? What could the authors have done to make their paper more readable? 12. Why do the authors analyze a second proof problem (about intervals)?

  13. Do the authors have an “axe to grind” relative to the previous study of Iannone, Inglis, Mejia-Ramos, Simpson, & Weber (2011)? That is, how was their study conceived in order to obtain d ifferent information on students’ example use?

  

Chesler, J. (2012). Pre-service secondary mathematics teachers making sense of definitions

of function. Mathematics Teacher Education and Development, 14(1), 27-40.

  1. What do you think of the literature review? Too long? Just right? Are there any references missing that you would have liked to have seen there?

  2. Is it true (in 2012) that there was “limited research on in- and pre-service mathematics teachers”? Do the phrases “Use this definition to justify” (Problem 1) and “Show that sequences satisfy the definition” mean different things to students? Would students behave differently in response to them, as they have been show to do with the words “prove” and “give a proof of”.

  4. Do you think students may not have read part (b) of Problem 1 carefully and that might be why they did not use the definition they supplied in part (a)?

  5. Consider the last sentence on page 37. What is being conjectured here? What is meant by “accommodate this action-object connection”? For example, one of the PSMT’s said “if a function is an association, then a sequence ‘associates’.”? [N.B. Research on the reading of science textbooks has indicated that nominalizations are hard for students (especially middle school students), whereas action verbs are more easily understood. Examples of verb-noun nominalizations: combine-combination, apply-application, fail-failure, move- movement react-reaction. Nominalizations turn actions into things. They also increase the noun-to-verb ratio. Readers expect the action to be conveyed by the verb in a sentence.] 6. The author uses “their habits using definitions” (p. 33). But what do you suppose were these PSMT’s habits in regard to definitions? How many times before in their mathematics course had they had to use definitions, other than perhaps in proofs? 7. What do you think of the use of the word “special” in Problem 2, Definition (ii)? Would most university mathematics students know what it meant?

  8. Why are there no percentages in Table 1? 9.

  What is an Action level of understanding, a Process level of understanding, an Object level of understanding as in APOS theory? This was referred to on page 37 in connection with the Dubinsky and McDonald citation.

Sanchez, V., & Garcia, M. (2014). Sociomathematical and mathematical norms related to definition in pre- service primary teachers’ discourse. Educational Studies in Mathematics, 85, 305-320

  1. What is the difference between a social norm, a sociomathematical norm, and a mathematical norm? One can think back to how Yackel & Cobb first defined these terms.

  2. What do the authors say is the purpose of a mathematical definition in the Spanish school curriculum?

  3. What are endorsed narratives? What are routines? [N.B. These terms were introduced by Anna Sfard.] 4. What is a “commognitive conflict?” How does it differ from a cognitive conflict?

  5. How do commognitive conflicts 1 and 2 differ from each other? 6.

  What do you think of the statement (quoted by the authors) that “any new concept must be describes as a special case of a mor e general concept.” 7. Can one relate some of the excerpts in the paper to the van Hiele levels of the pre-service teachers who uttered them?

  8. What is meant by a socio-cultural perspective? 9.

  What do you think primary teachers (of math) need to know about mathematical definitions?

  10. Are some of the authors’ claimed sociomathematical norms really just social norms? In answering this, consider Yackel & Cobb’s original definitions.

  11. What is the emergent perspective? How are commognitive conflicts related to misconceptions? 13.

  What do the authors mean by “school sociomathematical norms”? 14. The authors seem to have identified both general conflicts (e.g., CG1, CG2, etc.) and commognitive conflicts. What’s the difference? [N.B. It seems that CG1, CG2 were not defined in the article, which makes answering this question somewhat difficult.]

  Furina, G. (1994). Personal reconstruction of concept definitions: Limits. A MERGA Conference paper.

  1. What is meant by “encapsulation”? 2.

  What is meant by “procept” and “proceptual”? 3. What is the difference between “interiorization” and “internalization”? 4. What is meant by “re-present”? 5. What is meant by “the algebra of limits”? 6. What is meant by “constructivist theory” and “constructivist teaching experiments”? 7. Where is the “personal reconstruction” of the title found in this paper? 8. What might the author mean by the integration of schemas and by dissociated schemas?

  Van Dormolen, J., & Zaslavsky, O. (2003). The many facets of a definition: The case of periodicity. Journal of Mathematical Behavior, 22, 91-106.

  1. What kind of article is this? Empirical? Theoretical? Think piece? Opinion piece? 2.

  Why do you think this paper got published in JMB? 3. What issues (about definitions) does this paper bring up? 4. What did Lakatos mean by monster-barring? 5. What do the authors mean by “pedagogy before logic” and “logic before pedagogy”? 6. What do you think of introducing a paper with a vignette? Is this appropriate for JMB? 7. What do you think of the sentence (p. 96) “One can only talk about concepts if they exists, but one cannot prove they exist”?

  8. What is the difference between a “global” and a “point-wise” definition? Is there any reason to prefer one over the other?

  9. How, and when, should/could one explain/discuss with students such issues about definitions? In particular, choosing a definition based on its consequences.

  10. Why is it dangerous to mix-up doing something for logical reasons, conventional reasons, and pedagogical reasons? Give an example of each.

  11. What do you think of having so many (16) footnotes, when APA Style discourages footnotes as distracting?

  Antonini, S., & Mariotti, M. A. (2008). Indirect proof: What is specific to this way of proving? ZDM Mathematics Education, 40, 401-412.

  1. What uis meant by “minima; logic’ and by “intuitionist logic”? 2.

  What do you think of the authors’ model of indirect proof and its levels (p. 405)? 3. Why do you suppose Marie sees as “absurd” word in the one proof by contradiction and not the other (i.e., 2 is irrational) (pp. 406-407)?

  4. Why do you suppose considering possibilities/cases is easier for Paolo and Riccardo and for Valerio and Cristina?

  5. What is a reference theory? 6.

  According to the authors, what cognitive processes can be hidden in the production of an indirect proof (p. 411)?

  7. According to the authors, why is a conjecturing phase helpful before trying to make an indirect proof?

  8. Did the authors answer the questions in the title: What is specific to this way of proving? (from their point of view)? 9. What is Fischbein’s answer to why proof by contradiction and proof by contrapositive are hard?

  10. According to Duval, what is the difference between argumentation and proof? 11.

  What is meant by Cognitive Unity in proving? 12. Who were the inutitionists? 13. Where do the authors say the students see ;’gaps’ between the secondary statement (S*) and the principal statement (S)?

  14. Implicit in this paper is the idea that examining alternative possibilities, rather than negating the conclusion, is easier for students. Do you agree with this implication of what is at the end of Section 6?

  

Weber, I., Inglis, M., & Mejia-Ramos, J. P. (2014). How mathematicians obtain conviction:

Educational implications for mathematics instruction and research on epistemic cognition. Psychologist, 49(1), 36-58.

  1. What is meant by the “received view”? 2.

  What is meant by “epistemic cognition”? 3. What are some of the “nuanced instructional recommendations” of the authors? (p. 37).

  4. What are the following: authoritarian proof scheme, empirical inductive proof scheme, and deductive proof scheme?

  5. What are the two theoretical assumptions upon which the article is based ? (p. 42).

  6. What four epistemic aims do these authors focus on? (p. 42).

  7. What sorts of articles does Educational Psychologist accept? Who do you suppose reads this journal? Why do you suppose the authors published this paper there?

  8. What do you think of the assertion that mathematicians must present arguments that are impersonal and transparent? (p. 43). Do you agree?

  9. How is this paper structured and what do you think of that organization (for a scholarly essay)?

  

Kinsel, M. T., Cavey, L. O., Walen, S. B., & Rohrig, K. L. (2011). How do mathematicians

make sense of definitions? In S. Brown, S. Larsen, K. Marrongelle, & M. Oehrtman (Eds.),

th

  Proceedings of the 14 Annual Conference on Research in Undergraduate Mathematics Education, Vol. 1 (pp. 163-175). Portland, Oregon. (available on the web).

  Also, Cavey, L., Kinzel, M., Kinzel, T., Rohrig, K., & Walen, S. (2011). How do th mathematicians make sense of definitions? (Preliminary Report). Proceedings of the 14

Annual Conference on Research in Undergraduate Mathematics Education and Powerpoint

presentation (both available on the web).

  1. What are the main findings of this short paper (and Powerpoint)? 2.

  How was the research conducted? Does this short paper tell you enough about how the research was conducted to get a sense of the methodology used?

  3. How were the data analyzed? Does the article/Powerpoint say? 4.

  What do you think of the authors’ claim that mathematician so one thing (i.e., understand the words and notation of a definition first) and say they do another (i.e., test the definition with examples and nonexamples)? 5. What were the four parts of the interview? What was their purpose?