Barriers to students’ comprehension of proofs in mathematics lectures Keith Weber Rutgers University

Thanks To Tara Holm and the AMS COE for inviting me to speak To the National Science Foundation (DRL , DUE ) for funding much of this research To collaborators: –Tim Fukawa-Connelly (Temple) –Kristen Lew (Rutgers) –Pablo Mejia-Ramos (Rutgers) –Eyob Demeke (University of New Hampshire)

A simplified model of research in advanced mathematics teaching S i S f LEARNING INSTRUCTION

A simplified model of research in advanced mathematics teaching S i S f LEARNING INSTRUCTION

A simplified model of research in advanced mathematics teaching S i S f LEARNING VERY FORMAL LECTURES

A simplified model of research in advanced mathematics teaching S i S f LEARNING ACTUAL LECTURES

A simplified model of research in advanced mathematics teaching S i S f LEARNING INQUIRY-BASED LEARNING

A simplified model of research in advanced mathematics teaching S i S f LEARNING LECTURE

A simplified model of research in advanced mathematics teaching S i S f LEARNING LECTURE

Overview: Four studies Qualitative research –A case study of the meaning that one professor and six of his students saw with one proof presentation –Goal is to generate hypotheses that can account for the phenomenon that students do not learn from a clear lecture Quantitative research (test the hypotheses with large scale studies) –Inefficient note-taking strategies –Unproductive beliefs about proof –Failure to understand colloquial mathematics

Naturalistic case study of a real analysis lecture Case study– One professor (Dr. A) with 30 years experience and an excellent reputation as a real analysis instructor One 11-minute proof that a sequence {x n } with the property that |x n – x n+1 |<r n for some 0<r<1 is convergent Our aim: –Find out what Dr. A was trying to convey –See the extent to which six of Dr. A’s students recognized that the content that Dr. A was trying to convey in his presentation

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

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.

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.

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.

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 a proof to show a sequence isPass 4Pass 2Pass 4 Cauchy Cauchy sequences can be thought of asPass 3Pass 3Pass 3 “bunching up”

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 proof to show a sequence isPass 4Pass 2Pass 4 Cauchy Cauchy sequences can be thought of asPass 3Pass 3Pass 3 “bunching up”

Results- Summary What did the students say they learned in Pass 2? Cauchy sequences will be on the mid-term (Pair 1) The triangle inequality is important in real analysis (Pair 1) There is a consistent format to follow when writing a proof that a sequence is convergent (Pair 2) Dr. A expanded the toolbox for simplifying expressions (Pair 2) You can use prior knowledge from courses like calculus when writing proofs in real analysis (Pairs 1, 2, and 3)

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 3Pass 3Pass 3 “bunching up”

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 3Pass 3Pass 3 “bunching up”

Results: Cauchy heuristic At three points in the proof, Dr. A emphasized the Cauchy heuristic: 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.

Results: Cauchy heuristic At three points in the proof, Dr. A emphasized the Cauchy heuristic: 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: 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.

Results: Cauchy heuristic At three points in the proof, Dr. A emphasized the Cauchy heuristic: 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.

Results: Cauchy heuristic Our research team highlighted the Cauchy heuristic as the main point of presenting this proof. The other real analysis instructor described this as “the main objective”. Dr. A highlighted these three excerpts where he was trying to convey important content No student mentioned this in Pass 1 or Pass 2.

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 3Pass 3Pass 3 “bunching up”

Account #1: Inefficient note-taking strategies Note-taking is an important part of learning from a lecture. –Students learn through both the process of recording the notes and so they can study them at a later time (Kiewra, 1985) Notes that are not recorded during a lecture are recalled by students as little as 5% of the time (Einstein et al., 1985; Kiewra, 2002) Note-taking is challenging for students –A typical lecturer speaks 100 to 125 words per minute (Wong, 1985) –A typical student writes at a rate of 23 words per minute (Kiewra, 1987). Effective note-taking requires students to prioritize.

Inefficient note-taking strategies 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: By asking questions, and asking people by names, they will have their minds alert, saying ‘he might ask me, I'd better think about what's going on.’ Now we all fall asleep in classes at times, so it's not clear you're always going to be alert. But hopefully, that 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.

Inefficient note-taking strategies 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: By asking questions, and asking people by names, they will have their minds alert, saying ‘he might ask me, I'd better think about what's going on.’ Now we all fall asleep in classes at times, so it's not clear you're always going to be alert. But hopefully, that 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.

Inefficient note-taking strategies Only one student recorded any of the oral comments that Dr. A made. –The rest of the students’ notes consisted only of what Dr. A wrote on the blackboard. As a result, for five of the students, all of the material that Dr. A highlighted as important was not recorded in their notes. –which, as previous research shows, makes it unlikely that they will recall it at a later time.

Inefficient note-taking strategies: Does it generalize? (1) Do professors try to convey conceptual content and heuristic proof methods in their lectures? (2) Do professors tend to state these ideas orally and not write them down? (3) Are students’ notes comprised of board work but not oral comments?

Inefficient note-taking strategies: Does it generalize? We recorded 8 lectures in proof-based advanced mathematics courses. If a student agreed, we photographed their notes. We analyzed the lectures, flagging for every time we felt the professor was: –Giving a conceptual explanation –Providing a heuristic or method for how to write a proof –Modeling mathematical behaviors –Giving an example of a concept We looked to see if these ideas were present in any form in students’ notes

Inefficient note-taking strategies: Does it generalize? Type of content% in students’ notes Conceptual explanation: Written(N = 12, 27%) Oral(N = 33, 73%)

Inefficient note-taking strategies: Does it generalize? Type of content% in students’ notes Conceptual explanation: Written(N = 12, 27%)65% Oral(N = 33, 73%)

Inefficient note-taking strategies: Does it generalize? Type of content% in students’ notes Conceptual explanation: Written(N = 12, 27%)65% Oral(N = 33, 73%)2% (9 out of 370)

Inefficient note-taking strategies: Does it generalize? Type of content% in students’ notes Conceptual explanation: Written(N = 12, 27%)65% Oral(N = 33, 73%)2% Heuristic methods: Oral (N = 10, 100%)

Inefficient note-taking strategies: Does it generalize? Type of content% in students’ notes Conceptual explanation: Written(N = 12, 27%)65% Oral(N = 33, 73%)2% Heuristic methods: Oral (N = 10, 100%)3% (3 out of 116)

Inefficient note-taking strategies: Does it generalize? Type of content% in students’ notes Conceptual explanation: Written(N = 12, 27%)65% Oral(N = 33, 73%)2% Heuristic methods: Oral (N = 10, 100%)3% Modeling mathematical behaviors: Oral (N = 14, 100%)

Inefficient note-taking strategies: Does it generalize? Type of content% in students’ notes Conceptual explanation: Written(N = 12, 27%)65% Oral(N = 33, 73%)2% Heuristic methods: Oral (N = 10, 100%)3% Modeling mathematical behaviors: Oral (N = 14, 100%)0% (0 out of 155)

Inefficient note-taking strategies: Does it generalize? Type of content% in students’ notes Conceptual explanation: Written(N = 12, 27%)65% Oral(N = 33, 73%)2% Heuristic methods: Oral (N = 10, 100%)3% Modeling mathematical behaviors: Oral (N = 14, 100%)0% Examples: Written(N = 13, 93%) Oral (N = 1, 7%)

Inefficient note-taking strategies: Does it generalize? Type of content% in students’ notes Conceptual explanation: Written(N = 12, 27%)65% Oral(N = 33, 73%)2% Heuristic methods: Oral (N = 10, 100%)3% Modeling mathematical behaviors: Oral (N = 14, 100%)0% Examples: Written(N = 13, 93%)75% Oral (N = 1, 7%)0% (0 out of 15)

Inefficient note-taking strategies: Does it generalize? Type of content% in students’ notes Conceptual explanation: Written(N = 12, 27%)65% Oral(N = 33, 73%)2% Heuristic methods: Oral (N = 10, 100%)3% Modeling mathematical behaviors: Oral (N = 14, 100%)0% Examples: Written(N = 13, 93%)75% Oral (N = 1, 7%)0%

Account #2: Different beliefs about learning from proofs The way that students attend to a proof is dependent on: –What students think the role and function of proof is –What their responsibilities are as they read a proof I have identified discrepancies between the way that math majors and mathematicians view proof. –I first interviewed math majors and mathematicians to pose hypotheses and qualitatively illustrate these differences –I then tested the extent that they generalized by the use of a large- scale survey

Different beliefs about learning from proofs M8: There are different levels of understanding. One level of understanding is knowing the logic, knowing why the proof is true. A different level of understanding is seeing the big idea in the proof. When I read a proof, I sometimes think, how is the author really trying to go about this, what specific things is he trying to do, and how does he go about doing them. Understanding that, I think, is different than understanding how each sort of logical piece fits together.

Different beliefs about learning from proofs KM: [In a lecture] the point of the instructor is that the instructor can give a big picture view of things that is harder to get from a textbook, than details.

Different beliefs about learning from proofs 28 students were 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.

Different beliefs about learning from proofs 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 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.

Different beliefs about learning from proofs Other beliefs held by the majority of mathematics professors: Well-written proofs in math classrooms can contain gaps. Math majors are expected to work to supply some of the justifications in the proofs that they read. Math majors may require more than 30 minutes to understand some of the proofs that they encounter in their courses. Math majors should think about how they would prove a theorem before reading its proof. After reading a proof, a math major should compare the method in the proof to the one that he or she would take. This can help expand their arsenal of proving techniques.

Different beliefs about learning from proofs But the majority of mathematics majors saw things differently: Math majors should not be expected to work to supply some of the justifications in the proofs that they read. A well-written proof will make these things clear for the student. One should not spend more than 30 minutes studying a proof. Survey participants said that they would not prove a theorem before reading its proof. Survey participants said that after reading a proof, they would not compare the method in the proof to the one that he or she would take.

Different beliefs about learning from proofs 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.

Different beliefs about learning from proofs 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. Students saw this as: –“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”

Account #3: Colloquial mathematics 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.

Account #3: Colloquial mathematics 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. The use of the word “small” was an instance of colloquial mathematics, expressing a technical mathematical idea using informal English so students can use their intuitive understanding of “small” to help comprehend the technical mathematics.

Colloquial mathematics We note that the colloquial meaning of “small” in mathematics is contextual. Small could mean: –Negative quantity with a large magnitude –Small quantity in comparison to another magnitude (when dealing with millions of dollars, $10 is a small amount) –Quantity with a small magnitude in an absolute sense (one billionth) However, in real analysis settings, small means “arbitrarily small” or “sufficiently small” –Show that a quantity can be made sufficiently small if the terms that it is composed of can be made arbitrarily small

Colloquial mathematics In Pass 4 of our naturalistic study, each pair of students 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”

Colloquial mathematics 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.

Colloquial mathematics: Does it generalize? Oehrtman (2009) analyzed 120 introductory calculus students’ reasoning as they worked on tasks about limits. –He found that students continually used the language of arbitrarily small and sufficiently small, although this did not seem to significantly affect their reasoning. –By arbitrarily small and sufficiently small, students appeared to interpret this as meaning “very, very small” in an absolute sense. Jacob: [Arbitrarily small] would be so small for practical purposes that it doesn’t really matter, you know? Int: Another phrase that [the professor] used in class was sufficiently small. How would you interpret that phrase? Jacob: I guess larger than arbitrary. So it doesn’t have to be so microscopically small, it would just be like a decimal number, you know? Which is sufficient.

Summary We studied an excellent instructor who was clear (to us) in his communication and emphasized important math content. –On the one hand, students did learn useful information, such as that they can use techniques from earlier calculus courses for writing real analysis proofs. –However, students generally did not recognize the points that Dr. A highlighted as essential to convey. –Note that this was a privileged environment. Students were watching the proof for a second time. They were given all the notes that he wrote on the board. They knew they would be asked about the proof so they would be more likely to pay attention Students were rated as collectively above average Dr. A had a reputation for being a good instructor.

Summary Written vs. oral comments –Dr. A presented the most important content of the proof orally. –Students only recorded written work in their notes. Beliefs about proof –Dr. A attempted to convey the overarching method of the proof and an important heuristic for writing proofs. –Most math majors think understanding a proof consists of knowing how new assertions were derived from previous assertions, but do not concentrate on the methods used in the proof. Colloquial mathematics –Students did not attend to Dr. A’s use of the word “small” and did not appear to know what he meant by this.

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