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How lectures in advanced mathematics can fail to convey content Keith Weber Rutgers University

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Thanks To Michelle Cerillo, Bryan Crissinger, and the University of Delaware for giving me the opportunity to speak To my co-authors for helping me on the paper

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My co-authors Kristen Lew (collected all data and supplied leadership for the project) Tim Fukawa-Connelly (at Drexel University) Juan Pablo Mejia-Ramos

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The issue: The Feynman lectures In the early 1960s, Richard Feynman taught a two-year physics course that was recorded for posterity as The Feynman Lectures (Feynman, Leighton, & Sands, 2013). The Feynman Lectures became regarded as classics Feynman was “a great teacher, perhaps the best of his era” (Goodstein & Neugebauer, 1995, p. xix) His lectures were widely praised for their clarity and explanatory value (Davies, 1995).

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The issue: The Feynman lectures “Many of the students dreaded the course, and as the course wore on, attendance by the registered students started dropping alarmingly […] When he [Feynman] thought he was explaining things lucidly to freshman and sophomores, it was not really they who were able to benefit most from what he was doing. It was his peers—scientists, physicists, and professors—who would be the main beneficiaries of his magnificent achievement” (Goodstein & Neugebaur, 1995, p. xxii-xxiii).

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The issue: The Feynman lectures “Many of the students dreaded the course, and as the course wore on, attendance by the registered students started dropping alarmingly […] When he [Feynman] thought he was explaining things lucidly to freshman and sophomores, it was not really they who were able to benefit most from what he was doing. It was his peers—scientists, physicists, and professors—who would be the main beneficiaries of his magnificent achievement” (Goodstein & Neugebaur, 1995, p. xxii-xxiii).

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Lectures in advanced mathematics: Common perceptions “The teaching of abstract algebra is a disaster and this remains true almost independently of the quality of the lectures. This is especially true for some excellent instructors whose lectures are truly masterpieces” (Leron & Dubinsky, p. 227).

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Lectures in advanced mathematics: Common perceptions “The teaching of abstract algebra is a disaster and this remains true almost independently of the quality of the lectures. This is especially true for some excellent instructors whose lectures are truly masterpieces” (Leon & Dubinsky, p. 227).

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Lectures in advanced mathematics: Common perceptions “we go through the motions of saying what students ‘ought’ to learn while the students are trying to grapple with the more fundamental issues of learning our language and guessing at our mental models […] We assume that the problem is with the students rather than with communication: that the students either just don’t have what it takes, or else just don’t care. Outsiders are amazed at this phenomenon, but within the mathematical community, we dismiss it with shrugs”. (Thurston, 1994, p. 166)

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Lectures in advanced mathematics: Common perceptions “we go through the motions of saying what students ‘ought’ to learn while the students are trying to grapple with the more fundamental issues of learning our language and guessing at our mental models […] We assume that the problem is with the students rather than with communication: that the students either just don’t have what it takes, or else just don’t care. Outsiders are amazed at this phenomenon, but within the mathematical community, we dismiss it with shrugs”. (Thurston, 1994, p. 166)

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Lectures in advanced mathematics: Common perceptions “we go through the motions of saying what students ‘ought’ to learn while the students are trying to grapple with the more fundamental issues of learning our language and guessing at our mental models […] We assume that the problem is with the students rather than with communication: that the students either just don’t have what it takes, or else just don’t care. Outsiders are amazed at this phenomenon, but within the mathematical community, we dismiss it with shrugs”. (Thurston, 1994, p. 166)

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Lectures in advanced mathematics: Common perceptions Ample evidence that students do not learn as much from these courses as we desire. Limited concept understanding (e.g., Cottrill et al, 1996; Davis, 1996; Dubinsky et al, 2004; Limited ability to write proofs (e.g., Iannone & Inglis, 2010; Hart, 1994; Moore, 1994; Weber, 2001) Poor understanding of the proofs they read (e.g., Conradie & Firth, 2000; Cowen, 1990; Hodds, Alcock, & Inglis, 2014).

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Lectures in advanced mathematics: Limited research base Speer, Smith, and Horvarth (2010) reported a literature review on studies of collegiate teaching –Research on the teaching of advanced mathematics was limited –Only one study (Weber, 2004) met standards of rigor, including observing instruction and seeking the professor’s perspective on the instruction –No studies examined teachers’ and students’ perceptions of this lesson

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Blame the mathematicians “A typical lecture in advanced mathematics… consists entirely of definition, theorem, proof, definition, theorem, proof, in solemn and unrelieved concatenation” (Davis & Hersh, p. 151). If examples are presented at all, it is “parenthetical and in brief” (p. 151). Teaching “almost exclusively the one very convenient and important aspect which has been described above, namely the polished formalism, which so often follows the sequence theorem- proof-application” (Dreyfus, 1991, p. 27).

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Blame the mathematicians This dry limited form of instruction is attributed to poor behavior of mathematicians. Mathematicians are rewarded for research rather than teaching (Kline, 1977) Mathematicians are more interested in establishing truth than in providing explanation (Hersh, 1993) Mathematicians use effortless deduction to appear brilliant or to hide what they do not know (Davis & Hersh, 1991) Mathematicians are indifferent since they assume most students are not capable of learning the material (Leron & Dubinsky, 1995)

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Blame the mathematicians Hersh (1991) distinguishes between The front of mathematics that the public sees. Formal precise definitions, theorems, and proofs The back of mathematics that is hidden from the public, including sketches of diagrams, guesses, visual arguments, idioms, etc. In short, mathematicians only show the front of mathematics, not the back, in their lectures, giving students in an inaccurate and harmful view of the discipline.

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Blame the mathematicians These views have no empirical support. In fact, the few studies of mathematicians’ teaching advanced mathematics have found that: Mathematicians regularly use diagrams and examples in their lectures (Fukawa-Connelly & Newton, 2014; Mills, 2014) Mathematicians say they are trying to provide methods and insights in their proofs. Conviction is secondary (Weber, 2012; Yopp, 2011) Mathematicians attempt to explain their practices to show how proofs can be written (Fukawa-Connelly, 2012) Mathematicians’ instruction is based on coherent beliefs and a good deal of thought (Weber, 2004)

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Blame the students “We assume that the problem is with the students rather than with communication: that the students either just don’t have what it takes, or else just don’t care” (Thurston, 1994, p. 166). “To understand syllogism is not something that you can learn; you are either born with the ability or you are not” (Halmos, 1970, p. 124).

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Blame the students Several interview studies with mathematicians find they feel that some students are mathematically “tone deaf” and will never understand advanced mathematics (Harel & Sowder, 2009; Weber, 2012). M: Basically the class consists of two groups. There are groups that understand it, and probably hardly need it, and then there are those who really need it, and are not learning it. M: Everyone thinks that those things are somehow so obvious that anybody who is not a complete moron and has some ability to do mathematics should be able to see that. Therefore it is a waste of time to teach that because the people who need to be taught are the people who are hopeless anyhow. [... ] And I would have been vehemently opposed to that idea a few years ago. And now I’m less sure because I’m certainly trying to teach that and I can so maybe these people have a point (Alcock, 2010).

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Blame the format Some have blamed conventionally presented proofs. The proofs are non-optimal for conveying information. –The rigor and symbols are intimidating (Kline, 1977; Thurston, 1994) –Mask non-formal reasoning processes (Dreyfus, 1991; Thurston, 1994) –Hides decision-making processes (Anderson, Boyle, & Yost, 1986; Leron, 1983) Some have proposed alternative formats: –Structured proofs (Leron, 1983), that present proofs in an “outline” format –Generic proofs (e.g., Tall, 1979; Rowland, 2001) which demonstrates the theorem with a particular example that generalizes to any case –E-proofs (Alcock, 2010), an interactive computer demonstration bringing up things students should be thinking about

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Blame the format Only trouble is, changing the format does not improve comprehension: E-proofs did no better than text and (not statistically significantly) worse than a lecture (Roy, Alcock, & Inglis, 2010) Similar results for structured proofs (Fuller et al, 2014) and generic proofs (Lew et al, 2012) “Changing the presentation in these ways requires substantial instructor effort, and given the underwhelming empirical results, this may not be effort well spent” (Hodds, Alcock, & Inglis, 2014).

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Blame the format Maybe it’s the lecture itself. No lectures can be effective. “Telling students about mathematical processes, objects and relations is not sufficient to induce mathematical learning. Hence the sorry state of affairs with the best of lecturers” (Leron & Dubinsky, 1995, p. 241) Many have proposed student-centered inquire instruction: In group theory (Leron & Dubinsky, 1995; Larsen, 2013) Real analysis (Swinyard & Larsen, 2012; Roh, 2010).

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Blame the format Wu (1999) states that lectures are required to cover all the required content. Reform-oriented usually instruction covers less content: The cited abstract algebra instruction (Larsen, 2013; Leron & Dubinsky, 1995) does not reach the first isomorphism theorem Larsen and Swinyard (2012) had students spend ten hours “re- inventing” the definition of limit Most assessments deal with concept understanding, not the ability to write proofs This requires instructors to change their pedagogical beliefs and goals, which is difficult and some are not willing to do this (Johnson et al, 2013)

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An alternative explanation: Playing different games Skemp (1976, 1978) wrote a classic article about instrumental (procedural) and relational (conceptual) understanding. “Let us imagine that school A sends a team to play school B at a game called ‘football’, but neither knows that there are two kinds (called ‘association’ [soccer] and ‘rugby’). School A plays soccer and has never heard of rugger, and vice versa. Each team will rapidly decide the others are crazy, or a lot of foul players” “The problem here is one of mismatch… and does not depend on whether A or B’s meaning is the right one” (Skemp, 1978, p )

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An alternative explanation: Playing different games The idea that the instructor and students might perceive different goals (conceptual understanding vs. procedural fluency) and different expectations of students has had a profound influence on mathematics education. –Good inquiry-based instruction activities will not be implemented faithfully or meaningfully engage students if students have different beliefs about math instruction (e.g., Herbst, 2003; Herbst & Brach, 2006). Is there a difference between how mathematics professors and students perceive instruction in advanced mathematics? Can this account for students’ failures to learn?

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Different expectations: Some examples 28 math majors were interviewed. One interview question was what makes a good proof? (Weber, 2010, 2012). 16 of 28 students said a good proof contained all logical details and the reader shouldn’t have to infer how new statements were deduced. –It’s got to be really detailed. You have to tell every detail. Every step, it is very clear. I like doing things step by step. –It has to cover all the bases so that it is in fact a complete rigorous proof. For me, as a student, what else I would like to see are all the intermediate sorts of steps, things to help along, graphs, and visual things. Things that recalled facts that perhaps I should know but you know, maybe not immediately at the tip of my tongue. That’s to me what makes a good mathematical argument.

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Different expectations: Some examples 175 math majors and 83 math professors were asked whether they agreed with the following statement on a survey using a five-point Likert scale. In a good proof, every step is spelled out for the reader. The reader [A mathematics major] should not be left wondering where a new step in the proof comes from.

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Different expectations: Some examples 175 math majors and 83 math professors were asked whether they agreed with the following statement on a survey using a five-point Likert scale. In a good proof, every step is spelled out for the reader. The reader [A mathematics major] should not be left wondering where a new step in the proof comes from. Math majors:75% Mathematicians:27% * *- A Mann-Whitney test indicated the difference in math majors and mathematicians’ responses differed significantly.

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Different expectations: Some examples Mathematicians’ rationale: –The goal of lectures is to provide higher-level ideas. The proof in the textbook can be used for reference (Lai & Weber, 2014). –Going over every detail is not feasible. It’s time confusing and students would lose the big idea amid the logical details. –Students could benefit from filling in the gaps themselves (Lai, Mejia- Ramos, & Weber, 2012). Students’ rationale: –Students are expected to justify everything, even trivially things, in geometry and transition-to-proof courses (cf., Herbst, 2002). –Students do not appreciate that the proofs they hand in have different epistemological purposes from the ones professors present.

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Different expectations: Some examples The students were also asked what they thought it meant to understand a proof. Many indicated that understanding consisted of being able to provide justifications for each step in the proof. –To understand a mathematical argument? That you read it, that you understood it step by step, and certainly any facts that it appeals to, that you either know them at the top of your head or you’ve gone and looked them up and come to a point where you could, with very little prompting, reproduce it or certainly explained it to somebody else. –To understand every logical step, and be able to know the reason why each step was done, and understand that it was a valid step, logically.

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Different expectations: Some examples The students were also asked what they thought it meant to understand a proof. Many indicated that understanding consisted of being able to provide justifications for each step in the proof. –To understand a mathematical argument? That you read it, that you understood it step by step, and certainly any facts that it appeals to, that you either know them at the top of your head or you’ve gone and looked them up and come to a point where you could, with very little prompting, reproduce it or certainly explained it to somebody else. –To understand every logical step, and be able to know the reason why each step was done, and understand that it was a valid step, logically.

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Different expectations: Some examples 175 math majors and 83 math professors were asked whether they agreed with the following statement on a survey using a five-point Likert scale. If an individual [a mathematics major] can say how each statement in a proof follows logically from previous statements, then that student understands this proof completely. Math majors:75% Mathematicians:23% * *- A Mann-Whitney test indicated the difference in math majors and mathematicians’ responses differed significantly.

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Different expectations: Some examples Mathematicians’ rationale: –In interviews, mathematicians stress that the purpose of presenting a proof is not to provide conviction, but to provide insights into why things are true or how proofs can be written (Weber, 2012; Yopp, 2011). Students’ rationale: –Students are rarely assessed on their understanding of a proof. When they are assessed, it is often recalling the proof by rote (Weber, 2012). –Students are told to avoid giving their problem-solving process when writing a proof, only to say what the logic is.

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Different expectations: Some examples Mathematicians’ rationale: –In interviews, mathematicians stress that the purpose of presenting a proof is not to provide conviction, but to provide insights into why things are true or how proofs can be written (Weber, 2012; Yopp, 2011). Students’ rationale: –Students are rarely assessed on their understanding of a proof. When they are assessed, it is often recalling the proof by rote (Weber, 2012). –Students are told to avoid giving their problem-solving process when writing a proof, only to say what the logic is.

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Goal of the current study Case study– One professor (Dr. A) with 30 years experience and an excellent reputation as a real analysis instructor Three student pairs– rated collectively as above average by the professor One 11-minute proof that a sequence {x n } with the property that |x n – x n+1 |

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Goal of the current study The purpose is not to present sample-to-population generalization. –I do not wish to claim that all lecturers behave like Dr. A. –Indeed, Dr. A is chosen in part because he is atypical. He’s an excellent teacher. The purpose is an analytic generalization –The goal is to develop theory about why high quality lectures might not convey understanding –This includes the creating of relevant constructs and distinctions –A fine-grained analysis of how understanding failed to occur –Hypotheses that could be tested in a larger study

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Theoretical frames for the study Purposes of proving: Conviction– Students should have increased (or absolute) certainty that the claim is true. Explanation– Students can understand alternative ways of thinking about mathematical concepts that illustrate why the claim is true. Discovery- The students will perceive new methods that they can use to prove other theorems. Communicative– the norms for how a proof should be written and should appear. (adapted from deVilliers, 1990)

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Theoretical frames for the study Learning from lectures: Listening Processing –Encoding –Connect to one’s own knowledge and understanding Note taking –Professor says far more words than students can record –If students do not note a point, they recall this point only 5% of the time. –This places importance on what students prioritize Reviewing (from Williams & Eggert, 2002)

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Theoretical frames for the study Learning from lectures: Listening Processing –Encoding –Connect to one’s own knowledge and understanding Note taking –Professor says far more words than students can record –If students do not note a point, they recall this point only 5% of the time. –This places importance on what students prioritize Reviewing (from Williams & Eggert, 2002)

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Methods- Lecture analysis Video-recorded a lecture, chose the most interesting proof, and transcribed that proof. –If a sequence {x n } has the property that |x n – x n-1 |

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Methods- Lecture analysis by professor Instructor was audiotaped in an interview on lecture. –First asked to describe why he presented this proof to students –Then asked to stop the video recording at every point he thought he was trying to convey mathematical content –We coded each content as explanation, method, or communication.

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Methods- Lecture analysis by students Three student pairs were interviewed where we made four passes through the data. Pass 1: Students were asked to refer to their notes and state what they thought were the main ideas of the proof. Pass 2: Students watched the lecture again in its entirety, taking notes, and were asked the same question. Pass 3: Students were shown individual clips of the video and asked what they thought the professor was trying to convey. Pass 4: Students were told one thing that you might get from some proofs of this theorem was the content that Dr. A highlighted and asked if they got that from this proof.

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Results- Summary Content conveyedGroupGroupGroup By professor #1 #2 #3 To show sequence is convergent without aPass 3Pass 3Never limit candidate, show it is Cauchy Triangle inequality is important for proofs inPass 2Pass 3Pass 3 real analysis Geometric series in one’s “toolbox” for working NeverNeverNever with bounds and keeping quantities small How to set up proofs to show a sequence isPass 4Pass 2Pass 4 Cauchy Cauchy sequences can be thought of asPass 2Pass 3Pass 3 “bunching up”

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Results- Summary Content conveyedGroupGroupGroup By professor #1 #2 #3 To show sequence is convergent without aPass 3Pass 3Never limit candidate, show it is Cauchy Triangle inequality is important for proofs inPass 2Pass 3Pass 3 real analysis Geometric series in one’s “toolbox” for working NeverNeverNever with bounds and keeping quantities small How to set up proofs to show a sequence isPass 4Pass 2Pass 4 Cauchy Cauchy sequences can be thought of asPass 2Pass 3Pass 3 “bunching up”

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Results- Summary Content conveyedGroupGroupGroup By professor #1 #2 #3 To show sequence is convergent without aPass 3Pass 3Never limit candidate, show it is Cauchy Triangle inequality is important for proofs inPass 2Pass 3Pass 3 real analysis Geometric series in one’s “toolbox” for working NeverNeverNever with bounds and keeping quantities small How to set up proofs to show a sequence isPass 4Pass 2Pass 4 Cauchy Cauchy sequences can be thought of asPass 2Pass 3Pass 3 “bunching up”

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Results: Cauchy to establish convergence At three points in the proof, Dr. A emphasized if you want to show a sequence is convergent when you do not have a limit candidate, you can show it is Cauchy. Dr. A: How can we proceed to show that this is a convergent sequence? Anybody have a guess? Student: [Incomprehensible utterance] Dr. A: Well that’s not quite the right term. What kind of sequences do we know converge even if we don’t know what their limits are? [pause] It begins in ‘c’. Student: Cauchy. Dr. A: Cauchy! We’ll show it’s a Cauchy sequence.

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Results: Cauchy to establish convergence At three points in the proof, Dr. A emphasized if you want to show a sequence is convergent when you do not have a limit candidate, you can show it is Cauchy. We will show that this sequence converges by showing that it is a Cauchy sequence [writes this sentence on the board as he says it aloud, then turns around to face class]. A Cauchy sequence is defined without any mention of limit.

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Results: Cauchy to establish convergence At three points in the proof, Dr. A emphasized if you want to show a sequence is convergent when you do not have a limit candidate, you can show it is Cauchy. And now we’ll state what it is we have to show. We will show that there is an N- epsilon for which x_n minus x_m would be less than epsilon when m and n are greater than this number N-epsilon. [Dr. A writes this sentence on the board as he says it aloud] This is how we prove it is a Cauchy sequence. [Turns around and faces class]. See there is no mention of how the terms of the sequence are defined. There is no way in which we would be able to propose a limit L. So we have no way of proceeding except for showing that it is a Cauchy sequence or a contractive sequence. So let’s look and see how we proceed.

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Results: Cauchy to establish convergence Our research team highlighted this as the main point of presenting this proof. The other instructor described this as “the main objective”. Dr. A highlighted these three points where he was trying to convey important content No student mentioned this in Pass 1 or Pass 2, but two groups noted this in Pass 3 (when shown particular clips) –S1: We should recognize it, figure out it's a Cauchy, we should know that it's converging, but it's limit is not necessarily given. So that we recognize it instantly S2: Because we have no way of figuring out what the limit is. All we have is them in relation to each other. Cauchy makes sense.

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Results: Cauchy to establish convergence The written work on the board consisted entirely of the proof. The comments about the necessity of Cauchy sequences (and all the content we coded for) was only said orally. In his interview, Dr. A mentioned that it is important for students to pay attention and not just transcribe what’s on the board. Dr. A: If the lecture is going to be of use to people, that during the lecture at times their minds are picking up something useful. Otherwise they're just copying off the board, which is what we always do sometimes too. But, it makes it a little more exciting for me to be able to ask questions and talk to the class rather than stand up there and write stuff on the board.

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Results: Cauchy to establish convergence The written work on the board consisted entirely of the proof. The comments about the necessity of Cauchy sequences (and all the content we coded for) was only said orally. In his interview, Dr. A mentioned that it is important for students to pay attention and not just transcribe what’s on the board. Dr. A: If the lecture is going to be of use to people, that during the lecture at times their minds are picking up something useful. Otherwise they're just copying off the board, which is what we always do sometimes too. But, it makes it a little more exciting for me to be able to ask questions and talk to the class rather than stand up there and write stuff on the board.

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Results: Cauchy to establish convergence Students did not record any of the professor’s oral comments. –One student recorded nearly everything that the professor said. –Four students only wrote down what was on the blackboard. –One student did not take any notes. Unclear why students focused on what they did. –But one student noted that he didn’t have time to write anything other than the student wrote at the board. Previous research shows students remember less than 5% of a lecture that they do not record. –Indeed, two groups of students were capable of processing the information, but did not note it when asked what the professor was trying to convey.

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Results: Cauchy to establish convergence A plausible mathematician’s rationale: Mathematicians are charged with two goals: (i) showing student what a proof is, and (ii) showing students how to write a proof In particular, students need to learn the process of writing a proof is not always contained in the final product. One plausible compromise is to distinguish between the product and the process is by writing the product on the board and saying the process aloud. A plausible student’s rationale: Not everything can be noted and the stuff on the board is most important. Students are responsible for producing proofs like the type on the board.

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Results: Cauchy to establish convergence A plausible mathematician’s rationale: Mathematicians are charged with two goals: (i) showing student what a proof is, and (ii) showing students how to write a proof In particular, students need to learn the process of writing a proof is not always contained in the final product. One plausible compromise is to distinguish between the product and the process is by writing the product on the board and saying the process aloud. A plausible student’s rationale: Not everything can be noted and the stuff on the board is most important. Students are responsible for producing proofs like the type on the board.

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Results: Expanding one’s toolbox to work with inequalities Dr. A mentioned real analysis proofs as beginning with quantities that are assumed to be small and showing other quantities are small Dr. A: Now once again we ask the question. If we were to show this is small, we must represent it in terms of what we know is small. Well what do you know is small? For n large enough [gestures toward the statement of the theorem], the difference between two consecutive terms is small. [Turns and faces the blackboard]. So what we must do is represent that as a sum of consecutive terms.

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Results: Expanding one’s toolbox to work with inequalities Dr. A mentioned the importance of having things in your toolbox, which we interpreted as consisting of tools to keep something small. On the board: ≤r n (1+r+r 2 +…+r m-n ) Dr. A: Now we know this is small [circling r n ]. Now what can we say about this expression right here [circling (1+r+r 2 +…+r m-n ]? [pause] Anybody have a vague idea? I’ll give you a hint. Calculus two... Student: Geometric series? Dr. A: … thirty or forty years ago? [gestures to student who spoke] Student: Geometric series. Dr. A: Geometric series! You have to always keep geometric series in your toolbox.

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Results: Expanding one’s toolbox to work with inequalities In his interview, Dr. A spoke of the importance of working with bounds. Dr. A: Once you get into the area where you're doing approximations, you can't do equal, equal, equal. You have to have bounds, bounds, bounds […] The objective is to show how bounds, using the triangle inequality, can be used to show that something is small using information that they're given is small. And this instance turns out that the information which is small is given in a form that allows us to use the geometric series as a bound.

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Results: Expanding one’s toolbox to work with inequalities Students never mentioned the idea of having techniques to work with bounds or keep quantities small. –In Pass 1 (what students recalled from the lecture), one group noted the use of geometric series, with a student saying a main idea is you can use things from calculus 2 in real analysis. –In Pass 2 (after seeing the lecture again), all groups mentioned the importance of using ideas from calculus in real analysis. One student mentioned geometric series as a tool to “simplify expressions”

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Results: Expanding one’s toolbox to work with inequalities Students never mentioned the idea of having techniques to work with bounds or keep quantities small. –In Pass 1 (what students recalled from the lecture), one group noted the use of geometric series, with a student saying a main idea is you can use things from calculus 2 in real analysis. –In Pass 2 (after seeing the lecture again), all groups mentioned the importance of using ideas from calculus in real analysis. One student mentioned geometric series as a tool to “simplify expressions”

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Results: Expanding one’s toolbox to work with inequalities In Pass 3, students were shown this clip. Dr. A: Now once again we ask the question. If we were to show this is small, we must represent it in terms of what we know is small. Well what do you know is small? For n large enough [gestures toward the statement of the theorem], the difference between two consecutive terms is small. [Turns and faces the blackboard]. So what we must do is represent that as a sum of consecutive terms. No student mentioned the word “small” or a synonym when describing what Dr. A was trying to convey here. –“basically manipulating the information that we're given so that we can show that a sequence fits the definition” –“Given on the problem to see like what we could, how we can manipulate the problem statement. Just how we can start the proof in general”

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Results: Expanding one’s toolbox to work with inequalities In Pass 4, each group was explicitly asked: “One last thing you might get from this proof is that mathematics students need to have a toolbox of ideas that help them to prove things are small. Is this something that you got from this presentation?” All six students answered yes, five students did not mention the word “small” or a synonym in their responses, referring to to other techniques in their toolbox. –“I think if he structures the way that he does, and you keep seeing it, it stays in your toolbox memory area […] not just in this specific proof itself, but it carries over to any other areas of math when you want to start to prove something”

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Results: Expanding one’s toolbox to work with inequalities The student who did mention the word small gave a revealing response. S5: We can use Mathematica, or like a tool to convert to make something small. I: So right so mathematics students need to have a toolbox of ideas to help them prove things are small. S5: Things are small. Oh you mean that they're not so complicated. When you say that things are small? I: No I mean like in terms of convergent sequences. Is that something that you think you got from this presentation? S5: I mean, in terms of simplifying them and deriving for approximating the answer, I think it's on the path, it's like it's working.

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Results: Expanding one’s toolbox to work with inequalities A plausible mathematician’s perspective: Speaking strictly in terms of formal symbols is cumbersome and difficult. It’s common to use mathematical idioms like toolbox and small to facilitate communication. “Small” is a loose term, but it does have a (semi-)precise meaning (e.g., it refers to magnitude, not sign, and it doesn’t mean less than a particular value like x<10 -6 ). A plausible student’s perspective: The work in these proofs involves simplification so what is needed is terms to simplify. As students are focused on the connections between local steps, overarching goals like keeping things small, are ignored. Words like “small” are insignificant hedges (Oehrtman, 2009)

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Results: Metaphors In his interview, Dr. A stressed the importance of metaphors (although no metaphor appeared in the proof that we studied). Dr. A: And the whole objective is to get them to have in their mind a certain way of approaching problems. That's learning how to do mathematics, is learning structures to carrying out certain types of proofs. Having pictures and a structure for how they develop their understanding of the pictures. Because the proofs relate to the pictures. Let epsilon be greater than zero be given tells us that there's a neighborhood. The epsilon is the error which is the neighborhood in which the approximation will be. It can't be more than epsilon away from where the answer lies. So those are the things they've been seeing over and over again. Repetition, epsilon means error. And if you can make the error smaller than any epsilon, then you know that you have a sequence that's approximating something.

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Results: Metaphors In pass 4, students’ were asked: ““another thing that you might get from this proof is that the epsilon used is the error in approximations. Is that something that you got from this presentation?” All participants said yes, but none described the meaning of error or mentioned the word approximation. S3: Yeah. I think the first day or the second day, he explained what epsilon was so I think every time I see epsilon, I think about the error. S4: Yeah. I: So in this, so this proof is included in every time you see the epsilon? S3: Yeah. S4: [Nodding] Yeah, every time, every time I begin a proof I see let epsilon greater than zero be given. And we've had the same opening for that proof for, ever since we started dealing with like lower bounds and upper bounds.

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Results: Metaphors In most cases students were able to recognize and repeat the words of the metaphors that Dr. A used in class. However, these words seemed to be mainly used as “labels” by students, not as signifying the conceptual structures that were inherent in the metaphors. The repetition that Dr. A employed was useful in giving students repeated exposure to these idea. This gave them multiple opportunities to engage with the material and implicitly emphasized importance. However, each instance did not provide students with the opportunity to build the conceptual structures inherent in these metaphors.

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Results: Metaphors In most cases students were able to recognize and repeat the words of the metaphors that Dr. A used in class. However, these words seemed to be mainly used as “labels” by students, not as signifying the conceptual structures that were inherent in the metaphors. The repetition that Dr. A employed was useful in giving students repeated exposure to these idea. This gave them multiple opportunities to engage with the material and implicitly emphasized importance. However, each instance did not provide students with the opportunity to build the conceptual structures inherent in these metaphors.

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Summary We studied an excellent instructor who was clear (to us) in his communication and emphasized important math content. His students learned little of this content, even being asked directly after viewing the proof. We give several accounts for why this happened: –Students focused on the written work and seemed to ignore what was stated orally. –Students viewed the proof as doing algebraic manipulations rather than working to keep quantities small. Indeed, students did not know what the idiom “keeping quantities small” meant. –Students could repeat the metaphors that Dr. A used, but could not map them to a coherent conceptual structure where they could do real work.

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Recommendations Students need to be aware that what is said is as important as what is written in a lecture. Giving an informal argument is only useful if students have an accurate working definition of the semi-formal terms in the argument. –In many cases, ideas like “small” are quite sophisticated. –Students would need activities or instruction to build an understanding of this for these terms to aid comprehension. Students’ comprehension will be limited if they view real analysis proofs in terms of manipulations. –Giving proof comprehension tests that highlight the main ideas of the proof is a start, both in emphasizing the importance of global understanding and conceptualizing what it means to understand a proof.

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Thank you Some papers discussed in this talk can be found at: pcrg.gse.rutgers.edu

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