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Math Ed Seminar Specific Reading Questions, Paper-by-paper
Fall 2015
Gray, S. S., Loud, B. J., & Sokolowski, C. P. (2005). Undergraduates’ errors in using and interpreting variables: A comparative study. In Lloyd, G. M., Wilson, M., Wilkins, J. L. M., & Behm, S. L. (Eds.), Proceedings of the 27th annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education. Available online.
Because it is a proceedings paper whose size is limited to a certain number of pages, this is a very short paper, in contrast to an article in a print journal.
1. What is mean by variable (letter) as specific unknown, generalized number as functionally related quantities? Give examples of each. (Abstract, p. 1).
2. How do the authors define their Levels 1, 2, 3, and 4? Give examples of each. (p. 2). 3. What is mean by ignoring, evaluating or using letters as labels?(p. 2).
4. How was the study conducted? What was the methodology ? (p. 2) 5. What were some of the results on levels of variable use? (Table 1, p. e).
6. Why is having students at Levels, 1, 2, or 3 “far below expectations for entry into this course [calculus]”? (pp. 2-3).
7. Are you surprises at the percentages of incorrect responses to items 1a, 1b, and 2? Why? (Table 3).
8. Why do you think students in the three courses had as their most frequent error on Q1 and Q2 using variables as labels? (pp. 4-5).
9. The authors say that the errors they list for Q3a “seem t indicate reluctance to follow the instruction in the problem to let W represent total weekly wages.” What do you think of this interpretation? Could it have been the lack of parallel construction in the wording of the questions, and consequently, a matter of not reading carefully what was written? (p. 5). 10. What do you think of the authors’ two-sentence implications for teaching? (pp. 6-7).
Stavrou, S. G. (March 2014). Common errors and misconceptions in mathematical proving by education undergraduates. IUMPST: The Journal, Vol. 1 (Content Knowledge).
[www.k-12prep.math.ttu.edu] ISSN 2165-7874.
This online journal has an unusual way of numbering its contents. There are five volumes, the first of which is labeled “Content Knowledge”. When a paper has been accepted, it is placed in one of these volumes (i.e., content areas) and becomes the top entry in the papers listed
thereunder.
2 3. Where does the author get the 188 students? (p. 2).
4. Why do you suppose the author only told students in the second study about the previous students’ proving errors and did not tell them how to correct them?
5. Which areas of proof and proving have already been researched, according to the author? (pp. 1-2).
6. On page 2, the author states that Pfeiffer (2010) found students could and did identify proofs using examples as invalid. But the author says that he found proving with examples to be the most frequent among his participants. Why do you suppose this was?
7. What are some of the reasons, given by Edwards and Ward, as cited by the author, for why students misuse definitions (p. 4)?
8. What patterns did the author observe with the second group of 91 students who had been told about the proving errors of the first group of students?
Malisani, E., & Spagnolo, F. (2009). From arithmetical thought to algebraic thought: The role of variable . Educational Studies in Mathematics, 71, 19-41.
1. What is the difference between “index” and “a specific unknown”? (p. 21). 2. What are the three historical stages in the development of algebra? (p, 22). 3. What is mean by a “register”? (p. 22).
4. What were the research questions? (p. 25).
5. What is the “theory of didactic situations” (Brousseau)? (p. 25). 6. What is meant by the “didactic contract”? (p. 33).
7. What is the difference between a “concept” and a “conception”? (p. 24). 8. How do you suppose the authors did their a priori analysis to get Appendix 1? 9. How does one read/interpret the implicative graphs, such as Fig. 4? (p. 31).
10. What do you think of the authors’ four queries/problems given to the students? (p. 25).
Alcock, L., Hodds, M., Roy, S., & Inglis, M. (2015). Investigating and improving undergraduate proof comprehension. Notices of the AMS, 62(7), 742-752.
1. What kinds of research were e-proofs based on? (p. 743). 2. What were some of the results on e-proofs?
3. What sorts of data were collected on e-proofs? Students’ self-reports? On students’ comprehension?
4. What is meant by “surface features” of a proof? (p. 745).
3 6. Explain Fig. 9, which is the study design. ( p. 750).
7. What do the authors give as some implications of their research? (p. 751).
8. What can be the problem with using students’ self-reports of comprehension? (p. 752). 9. Since the research that e-proofs were based on came from Reference [2] on
ways to reduce cognitive load”, do you think it is always a good thing to try to reduce the cognitive load of students? (p. 743 & p.753).
10. A number of studies are reported in this paper. The last one is on self-explanation training and three stages of self-explanation training are discussed. What are those kinds?
11. Compare Figs. 3 & 6. What differences do you see?
12. Why do you think the authors went to such a complicated research design for their study of the effects of self-explanation training on eye movements? (Fig. 9, p. 750).
Radford, L. (2015). The progressive development of early embodied algebraic thinking. Article accepted for publication in Mathematics Education Research Journal. Available on the web. Pages are numbered from 1 on.
1. What sort of paper is this?
2. What are the three conditions that Radford says “characterize algebraic thinking” (p. 4). 3. What were the research questions? (p. 6).
4. What is the meaning of “deictic”? (p. 7). 5. Give an example of arithmetic generalization.
6. What is meant by “seeing the general through the particular”? (p. 10). 7. Why is answering developmental questions tricky? (p. 11).
8. According to Radford, of what components does thinking consist? (p. 11). 9. What does Radford mean by a “semiotic contraction”? (p. 13).
10.The development of algebraic thinking from Grade 2 to Grade 4 is mainly illustrated using the example of Carlos. Do you think he was the only one who exhibited this progression?
11.What is meant by “sensuous cognition”? (p. 18).
12.Radford says his activities were “imbued with cultural significations and an intended teleological development direction”. (p. 18). Do you think this is similar to Marty Simons HLT (hypothetical learning trajectory)?
13.What does Radford mean by “objectification”? (p. 19). 14.What is NAEP? (p. 2).
15.Roger How, a mathematician interested in mathematics education, is “skeptical” that the study of patterns is part of algebra. (p. 2). Why/How is Radford’s work with Grades 2-4 different (i.e., not the study of patterns)?
16.What would you respond to your algebra students who ask “What is algebra”? 17.If you currently teach algebra, find an example of arithmetic thinking in your class.
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1. According to Yopp, what is the distinction between “earning how to prove theorems” and “understanding theorems”? (p. 122).
2. What is meant by by “vertical” and “horizontal” connections/understanding? (p. 123). 3. Is Yopp saying (p. 124) that his categories, WHY, UND, UND-B, LEA, classify the
distinction between “understanding math” and “explaining math”? (p. 121). 4. How is the author using “critical thinking”? (p. 124).
5. What is meant by “member checking”? (p. 124).
6. Which, if any, of the author’s categories are new to the literature?
7. What three distinct categories for variation in the roles of proof did the researcher find? (pp. 127-8).
8. What do you suppose is meant by a “hand waving proof”? (p. 127).
9. What are some roles of proof mentioned in the literature that were not mentioned by the professors in this study? (p.128).
10.What do you suppose is the difference between “hand waving proofs” mentioned by an applied mathematician and “informal proofs” considered central to secondary
mathematics by Knuth’s teachers (2002b)? 11.Who is Wu? (p. 128, etc.).
12.What is Fermat’s Last Theorem and why was its proof a “landmark” event? (p. 128). 13.How do you suppose secondary teachers form their “naïve notions” of proof? (p. 129).
Alcock, L, Hodds, M., Roy, S., & Inglis, M. (2015). Investigating and improving
undergraduate proof comprehension. Notices of the American Mathematical Society, 62(7), 742-752.
1. What kinds of research were e-Proofs based on? (p. 743). 2. What were some of the results on e-Proofs?
3. Why did the researchers do immediate and delayed post-tests? 4. What is meant by “surface features” of a proof? (p. 745). 5. Explain Fig. 9, which is the study design (p. 750).
6. What do the authors give as some implications of their research? (p. 751). 7. What can be the problem with self-reports of students? (p. 752).
8. Since the research that e-Proofs were based on came from Reference [ ] on ways to reduce cognitive load , do you think it is always a good thing to try to reduce the cognitive load of students? (p. 743 & p. 753).
9. A number of studies are reported in this paper, but they are three stages of a research program. What were those three stages of research about?
10.Compare Figs. 3 and 6. What differences do you see?
11.Why do you think the authors went to such a complicated research design for their study of the effects of self-explanation training on eye movements? (Fig. 9, p.750).
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1. Rule 2 has been observed and used by a large number of entering U.S. college students (Grossman, 1983). (p. 9). Have you seen this in your classes?
2. What was the authors’ working hypothesis going into the study? p. 9-10).
3. Do you agree with Table 1 on which aspects of whole number knowledge support decimal knowledge? (p. 10).
4. What is the whole number rule? (p. 11). 5. What is the fraction rule? (p. 13).
6. What is the zero rule? (p. 13).
7. What did the authors expect (going into the study) about use of these three rules by children in the U.S., in France, and in Israel? (p. 13).
8. How many students from each country participated in the study? (pp. 13-14). 9. What is meant by probing ? p. .
10.What sorts of questions and responses di the authors use to determine whether a child has a fraction rule? (p. 15).
11.What two additional tasks were used to confirm the rule classification of the children? (pp. 15-16).
12.What do you think of the authors’ tasks and methodology? pp. -16). 13.What was the hidden number comparison task? (pp. 15 and 19).
14.What data did the authors use for determining cognitive sources of errorful rules? (p. 20).
15. Which of the authors’ initial hypotheses were confirmed? p. .
16.What three categories of incorrect answers did the authors find for converting fractions to decimals? (p. 23).
17.In what two ways do the authors claim their study goes beyond that of the earlier Sackur-Grisvard and Leonard (1985) studies? (p. 24).
18.In their discussion, what do the authors say about the conceptual understanding of whole number rule children? (pp. 24-5).About fraction rule children? (p. 25).
19.Why do the authors speculate the French curriculum (decimals first, then fractions) might be better? (p. 25).
20.What do the authors see as possibly wrong with teachers telling students to first ad zeros and then compare the decimals as if they were whole numbers? (pp. 25-6). 21.Since the creation of errorful rules by learners cannot be avoided, what do the
authors suggest teachers should do? (p. 26). What do you think of this suggestion?
Tabach, M., & Nachlieli, T. (to appear). Classroom engagement toward using
definitions for developing mathematical objects: The case of function. Educational Studies in Mathematics. A copy was obtained from the author and the pages were numbered from 1 on.
1. What is the difference between object-level learning and meta-level learning ? 2. What is meant by the genus and the differentiae of a defined term? p. .
6 4. What was the research questions? (p. 8). 5. Who were the participants? (p. 8).
6. Do you think that mapping an easier word for students to grasp than function ? (p. 9).
7. In the commognitive framework, how is learning viewed? (p. 9).
8. What did the authors use to analyze the small group discussions (according to the commognitive framework)? (p. 9).
9. What three actions do the authors state contributed to the group’s changed resolution in Vignette 3? (p. 18).
10.What do the tables (e.g., Tables , , … add to the paper that was not already explained in the text?
11.According to the authors, what three basic aspects need to be present in the communication process for meta-learning to happen? (p. 22).
Kantorovich, I., & Zazkis, R. (2015). Development of researcher knowledge in
mathematics education: Towards a confluence framework. Roceedings of the 13th
International Conference on Higher Education. London.
1. The authors claim that their framework provides vocabulary to capture, at least in part, the researcher’s knowledge development and explain her decision making . p. 5). What is some of that vocabulary?
2. What answers would you give to the questions at the top of page 4 (left-hand side)? 3. (ave the empirical papers we’ve [you’ve] been reading conform to the standard
empirical research paper p. ., left-hand side), as described by the authors? 4. . What do you think of the authors’ analysis of the excerpts from Artigue [p. , lef
-hand side and p. 4 right--hand side]?
5. On page1, the authors state The insights that emerge from this exploration [of the authors] will be useful for the refinement of the current approaches used in
educating prospective ME researchers and designing new approaches . Did the authors do this (i.e., did any new approaches emerge)?
6. What is modified analytic induction ? pp. - . What are the steps in modified analytic induction ?
7. [In a previous semester] We have read an empirical paper by Zazkis and Hazzan in which they asked math ed researchers how they developed their research questions. Why do you suppose it is not cited/referenced here? That paper is: Zazkis, R., & Hazzan, O. (1999). Interviewing in mathematics education research: Choosing the questions. Journal of Mathematical Behavior, 17(4), 429-439.
8. What is meant by the practice of formulating a research question is much more prescriptive than considering mathematics p. , right-hand side).
9. The researchers/authors suggest math ed researchers have much tacit/implicit knowledge (p. 1) Is this any different for mathematicians physicists, historians, etc.? 10. What is propositional knowledge? What is procedural knowledge? (p. 2, left-hand
side).
7 knowledge (PCK)? (p. 2, right-hand side, top).
12.Where is Leikin’s notion of dimenions of knowledge used as claimed on page , right-hand side, top)?
13.What are the three key components of the authors’ framework of research
knowledge development in conducting a study in mathematics education? (p. 3). 14.(ow well do you think the authors’ framework fits the six criteria of a theory?
Steinle, V., Gvozdenko, E., Price, G., Stacey, K., & Pierce, R. (2009). Investigating
students’ numerical misconceptions in algebra. In R. Hunter, B. Bicknell, & T. Burgess
(Eds.), Crossing divides: Proceedings of the 32nd annual conference of the Mathematics
Education research Group of Australasia (Vol. 2, pp. 491-498). Palmerston North, NZ: MERGA.
1. What new does this paper add to the literature?
2. What is meant by Pattern Recognition script? (p. 494).
3. What is meant by EB, DLDN, New 1, New 2, New 3, New 4? (pp. 493-496).
4. In Fig. 3, why do you suppose the percentage of DLDN_total = DNDL + New 2 is 31% for Yr. 8 and only 19% for Yr.7? (p. 496).
5. What is meant by readiness to move to expertise ? pp. 9 -497). 6. What is SMART? (p. 491 & 497).
7. What is mean by response pattern analysis? (p. 497). What advantages does it have? 8. What is item analysis? (p. 497).
9. What is meant by pedagogical content knowledge (PCK)? (p. 491).
10.What is meant by non-numerical ways of thinking? (p. 491). Given an example. 11.What algebra is studied in Year 7 in Australia?
12.What algebra is studied in Year 8 in Australia?
Iannone, P., Inglis, M., Mejia-Ramos, J. P., Simpson, A., & Weber, K. ( 2011). Does generating examples aid proof production? Educational Studies in Mathematics, 77, 1-14.
1. The authors state Abstract that many math ed researchers have suggested that asking learners to generate examples of math concepts is an effective way of
learning about novel concepts. (owever, the authors tested this by having students produce proofs. Is that a good way to test learning about novel concepts?
2. Are there any papers on the teaching of example generation? This seems to be a non-trivial task.
3. What limitations do the authors see to previous studies (e.g., Dahlberg & Housman)? (pp. 2-3).
4. What do you suppose the authors mean by a rich understanding of the concept ? (p. 3, top).
8 in it? (p. 3).
6. What strategies di the authors use (from Watson & Mason, 2005) for constructing their example generation tasks? (p. 3).
7. I believe there is a hierarchy of example generation in Watson & Mason (2005). The authors indicate they randomized the order of the task for each participant (p. 4). Doesn’t that negate some of the effect of using a hierarchy of tasks?
8. What is meant by ceiling or floor effects ? p. .
9. What three strategies of example generation did Antonini (2006) find with his participants? (p. 6). Explain each.
10.What is meant by a real-valued, periodic, non-constant function with no minimum period ? p. .
11.In the second interview study, the participants had 30 minutes to complete 20 example generation tasks (p. 6). Is this enough time?
12.There were 9 interviews and 20 e3ample generation tasks, making 9x20=180 tasks altogether, but the participants only attempted 62, meaning 118 were not even attempted. Do you think it would be interesting to know which these were? Why? 13.Which property does f(x)=1/x not satisfy for a function f from R to R that is
preserved on R-(0)? (p. 7).
14.What do the authors speculate about the examples generated by trial and error? (p. 8).
15.The authors state (p. 9) that the students had extensive proving experience. How do they know this? Did they check it?
Dimakos, G., Nikoloudakis, E., Ferentinos, S., & Choustoulakis, E. (2010). The rile if examples in cognitive apprenticeship. Quaderni di Ricera in Didattica (Mathematica), 20, 161-173. Department of Matheamatics, University of Palermo, Italy.
1. What is cognitive apprenticeship ?
2. Explain why the infinite series 1-1+1-1+1- + … diverges. (p. 165).
3. Hat function do the examples on pp. 164-168 serve in this paper? What do they illustrate?
4. The authors state (p. 165) suing the paradox of Zenon of Elea the teacher can stimulate students’ interest. What do you think of thi9s claim?
5. What purpose do the examples 1. -5. (pp. 166-167) serve in this paper? In particular, how specifically would one use Example 5 to teach parabola ? 6. What do you think about how the authors sated their research goal? (p. 168). 7. How would one interpret and solve pre-test problem 1? (p. 172).
8. The paper does not say which examples were presented in the experimental group and how they were taught. It also does not say what is meant by traditional
