Mathematics Education Research Seminar S (1)

  

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?

  5. 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 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? 2. 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 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”?

  3. 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? 12.

  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)?

  1. What teaching recommendations do the authors make? Do these follow from their findings?

  1. How does the author view definitions (p. 166 & p. 179)? 2.

  Ouvrier-Buffet, C. (2011). A mathematical experience involving defining processes: In- action definitions and zero-definitions. Educational Studies in Mathematics, 76, 165-182.

  What sort of study is this? Qualitative? Quantitative? Mixed methods? 7. What are some of the reasons the authors give for the students inconsistent responses? 8. How did the authors split the students into four groups: low, intermediate, high, math? 9. How did the authors select their final 307 respondents to analyze? Does this seem reasonable?

  5. Why do you suppose the authors only dealt with explanations to correct answers? 6.

  4. What were the six categories that the authors used to categorize participants’ definitions of function? Have you seen these in your own students?

  Do you think 20 minutes is enough time to give background information, to answer and explain your answers to this 7-item questionnaire?

  2. What was the methodology of the study? 3.

  Journal for Research in Mathematics Education, 20(4), 356-366.

  

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.),

Proceedings of the 14 th

  5. What were the four parts of the interview? What was their purpose? Vinner, S., & Dreyfus, T. (1989). Images and definitions for the concept of function.

  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

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

  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?

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

  

Annual Conference on Research in Undergraduate Mathematics Education and Powerpoint

presentation (both available on the web).

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

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

  According to this author, what is Lakatos’ view of the defining process? 3. What are naïve definitions, zero-definitions, and proof-generated definitions?

  4. According to this author, what is the end criteria of the defining process? 5.

  How does the author describe Larsen and Zandieh’s (2008) work on defining? 6. What does the author mean by the frameworks: Aristotelian, Popperian, and Lakatosian? 7. What were the research questions? 8. What is mean b the theory of didactical situations (Brousseau). What is meant by

  “mileau” and by “devolution”? 9. What is an “in-action definition” and an “in-action-proposition”? 10.

  What are the (theoretical) contributions of this paper? 11. What has Ouvrier-Buffet’s research program been? 12. What do you think of the sentence “An important part of learning mathematics is actually to become aware of the importance of defining just like the importance of proving” (p.

  179)?

Winicki-Landman, G., & Leikin, R. (2000). On equivalent and non-equivalent definitions.

  Part 1. For the Learning of Mathematics, 20, 17- 21.

  1. What do you think of “logical principle” #4 (p. 17) that the set of conditions for a definition should be minimal? Noting that the customary definition of a group given in most undergraduate abstract algebra textbooks is not minimal, what would you say?

  What do you think of the statement that for equivalent definitions “mathematically, there is no difference among them”?

  3. What do you suppose is meant by “elegance” and that it is desirable that definitions be elegant?

  4. What are “competing definitions”? Have you ever heard this term before? 5.

  Choosing a particular definition of tangent line to a curve has consequences (Table 1).

  Which definition is preferable for high school students, for pre-calculus students, for calculus students?

  6. What is meant by #1. on page 21? 7.

  Have you seen #3. on page 21 give students problems? 8. What is meant by the “spiral development” of a concept? 9. What are some didactic considerations when introducing the concept of tangent to students?

  10. Have you every introduced, or used, the concepts of “stronger” and “weaker” definitions as given by the authors?

  11. Have you ever had a discussion with students of the connection between properties of tangent lines to different curves (p. 21 and Table 1)?

  Leikin, R., & Winicki-Landman, (2000). On equivalent and non-equivalent definitions: Part 2. For the Learning of Mathematics, 20, 24-29.

  1. The authors have used A., B., C, D., and E. for five different definitions of absolute value on page 25, but later on page 26, refer to these as I., II, III, IV., and V. Also on page 26 they state, “Definition V and definition III were considered by some teachers as ‘appropriate only for real numbers’, while definition IV could be used for complex numbers, too.” Are any of the five definitions of absolute value applicable to complex numbers?

  2. Definition B involves two real numbers. How can this definition be considered equivalent to the other four definitions?

  3. What is the properties strategy that the teachers used to show the equivalence of two definitions?

  4. What didactical aspects of definitions did the teachers prefer? 5.

  The authors seem to use the term “arbitrariness” in several ways. Is it clear to you what the authors mean by each usage?

  6. Why do you think the Noss (1998) quote included? What does it amplify? How does it help the article?

  7. Why is the Zaslavsky (1995) quote included? How does is help the article? 8.

  Is the article well-written? What were the confusing places (if any)? 9. Do you agree with the authors that “one of the main problems of mathematics education is learning to define”?

  10. What sort of article is this? They have no research questions. There is very little description of their methodology? What are their results (if any)? Why do you think it got published? 11. What difference do you think the authors are trying to make when they introduce both the two terms: “properties” and “indications” of a concept (p. 28)?

  12. What do you think the authors mean by “right-bisector” and “mid-perpendicular”? Why are they two names for the same set of defining conditions? Have you ever heard these terms used before? Weber, K. (2014). What is a proof? A linguistic answer to an educational question.

  

Proceedings of the 2014 Conference on Research in Undergraduate Mathematics Education.

  Available online.

  1. What is meant by the “sense of a concept” and by the “referent of a concept”? 2.

  What sorts of ways have mathematics education researchers attempted to define “proof”? 3. According to Weber, what is the problem with so many definitions of proof? 4. Who is Echeverria and why should we believe his view of mathematicians’ view of

  Goldbach’s Conjecture? 5. Who is Paseau and why should we believe him on proofs? 6. What sort of book is Littlewood’s A mathematician’s miscellany? Popular? Expository? 7. What epistemological error is Weber referring to on page 6? 8. Has Weber convince you that “mathematicians do not themselves agree on what constitutes a proof”?

  9. Is Weber saying something “deep” here? Or is he just saying that “proof” is not an “if and only if” (analytic) concept, but rather is learned by experience like other everyday concepts (defined in dictionaries)? 10. Why, according to Dawson (as reported by Weber() do mathematicians want to re-prove theorems?

  11. What cautions about (or implications for) research on proof does Weber make? 12.

  What does Weber advocate as a research question about proofs, instead of a binary choice (i.e., proof or not proof)?

  13. What suggestions/implications does Weber have for instruction? 14.

  What does Weber (following Aberdein) consider to be a “proof*”? How is a proof* related to a proof?

  Alock, L., & Simpson, A. (2002). Two components in learning to reason using definitions. nd

  In Proceedings of the 2 International Conference on the Teaching of Mathematics (at the

undergraduate level). John Wiley & Sons, Inc. ISBN 0-471-46332-9. Available on the web at

  1. What are the two components? 2.

  Valeria Aguirre Holguin conjectured four stages in learning a definition, within a context, in our joint Powerpoint presentation at the Vancouver PME & PME-NA Conference in July 2014): a.

  Understand there is a difference between dictionary/everyday definitions and mathematical definitions (as Edwards and Ward (2004) suggested).

  b.

  Understand when, and where, to use mathematical definitions.

  c.

  Recall, look for, and attempt to use/follow definitions, not necessarily with success.

  d.

  Use the definition successfully. How do Alcock and Simpson’s two components “fit in” with her four stages? Are they somehow orthogonal to Alcock and Simpson’s two components? Or, is there some overlap? 3. Is it true that proof problems “at beginning university level generally either require showing that a particular object is a member of a mathematical category … or showing that one category is a subset of another? That is, are U.S. students at the upper-level undergraduate level (mainly or exclusively) asked to show that a mathematical object satisfies a definition or that one mathematical definition implies another?\ 4. Do you think Cary’s diagram (Figure 1) “is drawn so as to illustrate some of the possible forms of non- monotonic sequences”, especially as they are eventually monotonic? 5. Do you think that the behavior of Students K and J, as exhibited in the interview excerpt, often appears “as an unexamined reaction to repeated experiences of being wrong”? What behavior are Alcock and Simpson referring to here?

  Education Development Center (2002). Understanding and Creating Definitions. Part of the Making Mathematics Project. Available online in PDF format at:

  1. What do you think of the six steps in understanding a definition (pp. 2-3). How do the six steps compare and contrast to those ways of understanding a definition in a.

  Dahlberg and Housman? b.

  In Valeria Aguirre Holguin’s handouts? 2. What do you think of the activity to provide “practice in reading, interpreting, and comparing definitions”? (pp. 3-4).

  3. What do you think of the possible alterative definitions (to test) on page 4?

  4. Consider the questions on page 5. Which is the best of definitions A, D, F, G (of convex)? Why don’t we use (in a classroom) one of these definitions of convex, instead of the one on page 1? Does the idea of convexity apply in three dimensions?

  1. Do you think this paper was well-written? Doe it follow the “classic” format of an empirical research article?

  9. What does the author recommend/suggest in the way of teaching implications?

  Why does the author speak of these student images (of convergence of a sequence) as epistemological obstacles?

  7. Did the author appropriately situate this study in the literature? 8.

  6. What were the three images of limit the author found? Have you observed students with these images?

  Why do you suppose the interviews were once a week for 5 weeks? Could the participants have changed their images of sequence convergence over the 5 weeks? Does the author say anything about this? After all, concept images evolve, grow, and change. What do you think of the author’s statement on page 224 that the images reported were “stable and consistent”? 4. Why did the author say the two -strip definitions came from students? 5. What does the author claim is new about this study, that is, what does this study add to the literature?

  2. What do you think of the authors’ criteria/checklist for selecting participants? 3.

  13. What sorts of competencies would a high school teacher need to try out such activities with a class?

  5. What do you think of the activities having to do with writing definitions (page 5 onward)?

  What are some of the teaching suggestions in this paper? 12. Where in the present curriculum would such activities (as those described in this paper) fit?

  9. Consider the definition of semicenter on page 13 and answer the questions on the bottom of pp. 13-14. How does the semicenter idea relate to the idea of convex? Does a circle have every point as a semicenter? 10. This paper has advice/suggestions to teachers. Do you think it is specific enough for a teacher to conduct a class on definitions and defining? If not, what else would you (as a teacher) want?

  8. Answer Carl’s question in the box on page 9.

  7. What do you think of discussing (with a high school class or your won class) the degenerate case of a definition? (p. 8).

  Unambiguously define the thing of interest c. Only use words whose meanings are clear.

  Avoid unnecessary words b.

  6. How do the criteria for definitions on page 7 compare to those we have read in other papers (e.g., minimal, use only previously defined concepts, etc.)? a.

Roh, K. H. (2008) . Students’ images and their understanding of definitions of the limit of a sequence. Educational Studies in Mathematics, 69, 217-233

  10. What do you think of the way the quantifiers are written in the Apostol definition of sequence convergence (p. 218)? Do you think it would be preferable to have all the quantifiers at the beginning of the definition? If so, why or why not?

  Marriotti, M. A., & Fischbein, E. (1997). Defining in classroom activities. Educational Studies in Mathematics, 34, 219-248.

  1. What point, or points, is this paper trying to get across? 2.

  What is meant by a “figural concept”/ 3. What do you think of the authors’ statement that “the pertinence of a property for classification may remain hidden to pupils”? Do you know of any examples from your own teaching? 4. What do you think of the authors’ statement that “it seems useful to find a problematic context within which the significance of a definition arises”? What might the authors mean by the “significance of a definition”? Can you think of such a problematic context for a definition that comes up in one of your courses?

  5. How did the teacher (described in this article) “mediate that part of the defining process necessary in order to transform an indefinite description into a ‘definition’”? What are the authors referring to wh en they write about an “indefinite description”? Can you give an

  6. Fischbein developed a theory of figural concepts that is used in this paper. [See Fischbein, E. (1993). The theory of figural concepts. Educational Studies in Mathematics,

  24(2), 139-162.] What is that theory? How is it used in this paper? 7.

  What was the role of the teacher in this paper? Why do the authors say that “the intervention of the teacher is determinant” (p. 243)?

  1/3 Tirosh, D., & Even, R. (1997). To define or not to define: The case of (-8) . Educational Studies in Mathematics, 33, 321-330.

  1. What is the issue that the authors raise in this paper?

  1/n 2.

  What is your response to the authors’ question, “Why is it so important to ensure that x has exactly one value?”

  3. If one considers the set of complex numbers, instead of the set of real numbers, there are exactly three (different) roots of -8. What are those complex roots? Are any of them equal to 2? Would going to the complex numbers (with your students) be a useful thing to do?

  1/3

  Or, would it be a confusing thing to do? Would it help resolve the issue that (-8) = -2,

  2/6

  whereas (-8) = 2? 4. What do you think of the idea that a definition should give an unambiguous determination of the concept being defined? Is this the same criterion as that one should be able to use the definition to determine what is an example of the concept and what is a nonexample of the concept?

  5. What do you think of the requirement that “a definition should not depend on the representation of the numbers (or objects) involved in the operation”? (p. 327). Think of the concept of a linear transformation given in linear algebra. Is it unambiguously

  n

  defined? There are many matrix representations of a linear transformation in R (depending on the basis one selects). Does this affect the definition of linear transformation? What’s the difference between this situation and the one in the paper? An answer might be: A linear transformation is defined by a property that is independent of any (matrix) representation. Can one do this sort of thing when defining rational roots? 6. What do you suppose the authors mean by the question, “Should definitions be proved?” (p. 329).

  7. What do you think of the authors’ recommendation that “such problematic cases could be used in teacher education programs as springboards to facilitate teachers ’ mathematical and pedagogical knowledge?” (p. 329).

  Weber, K., Brophy, A., & Lin, K. (2008). Learning advanced mathematical concepts by reading text. Proceedings of the 11 th

   Annual Conference on Research in Mathematics Education. Available online.

  1. What three conjectured reasons (as discussed in this paper) does Chi give as to why generating self-explanations facilitates learning when reading scientific text? What do you think of these. Do they apply to reading mathematical text? 2. What do you think of how the SMM and LSMM participants were selected? Is this a good way to do it?

  3. Have you noticed with any of your students a tendency to write template-based proofs? There are typos in this paper. Are any of them confusing? 5. There are a couple of unfortunate wordings in this paper. For example, on page 18, the authors state that the participants “were unable to”, but they did not know that. It would have been more appropriate to have written “did not”. There is a similar sentence that goes beyond the data on page 19. Can you find it?

  6. Is the research question on page 1 too broad? 7.

  How was the research conducted? Does this seem a reasonable was to proceed? 8. What is inductive coding? Fernandez, E. (2004). The students’ Take on the Epsilon-Delta Definition of a Limit.

  1. What sort of paper is this? What sort of journal is PRIMUS? 2.

  What do you think of the author’s intervention—eliciting misconceptions, etc.? 3. There are no mathematics education research references, despite there being a lot of research on limit prior to 2004 (when the paper was published). What are some of the references that this author could have consulted.

  4. Note the references are not in APA Style. Each journal has its own requirements regarding the formatting of references.

  5. What were the results of the intervention by the end of the semester? 6.

  What do you think of the observations that a- < x < a+ is easier for students to understand than |a-x| < ? Why do you think this might be so? Have you noticed this with students in any of your classes?

  7. What kinds of things about the - definition bothered the author’s calculus students?

  Mejia-Ramos, J. P., Fuller, E., Weber, K., Rhoads, K., & Samkoff, A. (2012). An assessment model for proof comprehension in undergraduate mathematics. Educational Studies in Mathematics,79, 3-18.

  1. What kind of paper is this? Empirical? Theoretical? Think piece? Research Commentary? Something else? 2. How common is it that professors have students reproduce proofs? In the U.S.? In U.K.?

  In S. Africa (where Conradie and Frith are)? Elsewhere? 3. What are Yang & Lin’s four levels of proof comprehension (for geometry)? What do you think of them?

  4. What are the seven types of proof comprehension questions posed by the researchers in this paper? What do you think of them? What two groups are the seven types of proof comprehension questions separated into by the researchers in this paper? 5. What steps did the researchers take to generate their holistic understanding questions? 6. What criterion of inclusion did the researchers use for including what the mathematicians said was important for proof comprehension? What four facets did the reserchers find?

  What do you think of the criteria of inclusion and of the facets themselves? 7. What is the difference between comprehension and apprehension?

  Men of Inquiry-Based learning in College Mathematics: A Multi-Institution Study. Brief Report. Journal for Research in Mathematics Education, 45(4), 406-418.

  1. What is a “brief report”? Why would you (as a researcher) want to publish one? 2.

  What is IBL instruction? How did the researchers define it? 3. How much do you suppose interviews with 110 students cost? How much time would it take?

  4. Why do you suppose the researchers (on p. 409) did not write 60% vs. 13%, rather than 60% active learning (for IBL) vs. 87% listening to instructors (for non-IBL)? This is not parallel construction.

  5. In Table 1, what is an episode? Did the researchers explain that anywhere? 6.

  What is propensity analysis (p. 412)? 7. What did the researchers mean by “course type” (p. 413)? Where is course type in Table

  2? 8. According to the researchers, what are the “twin pillars that support student learning in

  IBL classes ” (p. 413)? 9.

  What time would it take to do all the classroom observations that were gathered from 100 course sections at four different universities in the U.S.? What might the entire large scale study have cost?