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-0643734, DUE1245626) 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
INSTRUCTION
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A simplified model of research in
advanced mathematics teaching
INSTRUCTION
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A simplified model of research in
advanced mathematics teaching
VERY FORMAL
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A simplified model of research in
advanced mathematics teaching
ACTUAL
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A simplified model of research in
advanced mathematics teaching
INQUIRY-BASED
LEARNING
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A simplified model of research in
advanced mathematics teaching
LECTURE
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A simplified model of research in
advanced mathematics teaching
LECTURE
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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 {xn} with the property that
|xn – xn+1|<rn 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.
ResultsSummary
Content conveyed
By professor
To show sequence is convergent without a
limit candidate, show it is Cauchy
Group
#1
Pass 3
Group
#2
Pass 3
Group
#3
Never
Triangle inequality is important for proofs in
real analysis
Pass 2
Pass 3
Pass 3
Geometric series in one’s “toolbox” for working
with bounds and keeping quantities small
Never
Never
Never
How to set up a proof to show a sequence is
Cauchy
Pass 4
Pass 2
Pass 4
Cauchy sequences can be thought of as
“bunching up”
Pass 3
Pass 3
Pass 3
ResultsSummary
Content conveyed
By professor
To show sequence is convergent without a
limit candidate, show it is Cauchy
Group
#1
Pass 3
Group
#2
Pass 3
Group
#3
Never
Triangle inequality is important for proofs in
real analysis
Pass 2
Pass 3
Pass 3
Geometric series in one’s “toolbox” for working
with bounds and keeping quantities small
Never
Never
Never
How to set up proof to show a sequence is
Cauchy
Pass 4
Pass 2
Pass 4
Cauchy sequences can be thought of as
“bunching up”
Pass 3
Pass 3
Pass 3
ResultsSummary
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)
ResultsSummary
Content conveyed
By professor
To show sequence is convergent without a
limit candidate, show it is Cauchy
Group
#1
Pass 3
Group
#2
Pass 3
Group
#3
Never
Triangle inequality is important for proofs in
real analysis
Pass 2
Pass 3
Pass 3
Geometric series in one’s “toolbox” for working
with bounds and keeping quantities small
Never
Never
Never
How to set up proofs to show a sequence is
Cauchy
Pass 4
Pass 2
Pass 4
Cauchy sequences can be thought of as
“bunching up”
Pass 3
Pass 3
Pass 3
ResultsSummary
Content conveyed
By professor
To show sequence is convergent without a
limit candidate, show it is Cauchy
Group
#1
Pass 3
Group
#2
Pass 3
Group
#3
Never
Triangle inequality is important for proofs in
real analysis
Pass 2
Pass 3
Pass 3
Geometric series in one’s “toolbox” for working
with bounds and keeping quantities small
Never
Never
Never
How to set up proofs to show a sequence is
Cauchy
Pass 4
Pass 2
Pass 4
Cauchy sequences can be thought of as
“bunching up”
Pass 3
Pass 3
Pass 3
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 Nepsilon 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.
ResultsSummary
Content conveyed
By professor
To show sequence is convergent without a
limit candidate, show it is Cauchy
Group
#1
Pass 3
Group
#2
Pass 3
Group
#3
Never
Triangle inequality is important for proofs in
real analysis
Pass 2
Pass 3
Pass 3
Geometric series in one’s “toolbox” for working
with bounds and keeping quantities small
Never
Never
Never
How to set up proofs to show a sequence is
Cauchy
Pass 4
Pass 2
Pass 4
Cauchy sequences can be thought of as
“bunching up”
Pass 3
Pass 3
Pass 3
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
Conceptual explanation:
Written (N = 12, 27%)
Oral
(N = 33, 73%)
% in students’ notes
Inefficient note-taking strategies:
Does it generalize?
Type of content
Conceptual explanation:
Written (N = 12, 27%)
Oral
(N = 33, 73%)
% in students’ notes
65%
Inefficient note-taking strategies:
Does it generalize?
Type of content
Conceptual explanation:
Written (N = 12, 27%)
Oral
(N = 33, 73%)
% in students’ notes
65%
2% (9 out of 370)
Inefficient note-taking strategies:
Does it generalize?
Type of content
Conceptual explanation:
Written (N = 12, 27%)
Oral
(N = 33, 73%)
Heuristic methods:
Oral
(N = 10, 100%)
% in students’ notes
65%
2%
Inefficient note-taking strategies:
Does it generalize?
Type of content
Conceptual explanation:
Written (N = 12, 27%)
Oral
(N = 33, 73%)
Heuristic methods:
Oral
(N = 10, 100%)
% in students’ notes
65%
2%
3% (3 out of 116)
Inefficient note-taking strategies:
Does it generalize?
Type of content
Conceptual explanation:
Written (N = 12, 27%)
Oral
(N = 33, 73%)
Heuristic methods:
Oral
(N = 10, 100%)
Modeling mathematical behaviors:
Oral
(N = 14, 100%)
% in students’ notes
65%
2%
3%
Inefficient note-taking strategies:
Does it generalize?
Type of content
Conceptual explanation:
Written (N = 12, 27%)
Oral
(N = 33, 73%)
Heuristic methods:
Oral
(N = 10, 100%)
Modeling mathematical behaviors:
Oral
(N = 14, 100%)
% in students’ notes
65%
2%
3%
0% (0 out of 155)
Inefficient note-taking strategies:
Does it generalize?
Type of content
Conceptual explanation:
Written (N = 12, 27%)
Oral
(N = 33, 73%)
Heuristic methods:
Oral
(N = 10, 100%)
Modeling mathematical behaviors:
Oral
(N = 14, 100%)
Examples:
Written (N = 13, 93%)
Oral
(N = 1, 7%)
% in students’ notes
65%
2%
3%
0%
Inefficient note-taking strategies:
Does it generalize?
Type of content
Conceptual explanation:
Written (N = 12, 27%)
Oral
(N = 33, 73%)
Heuristic methods:
Oral
(N = 10, 100%)
Modeling mathematical behaviors:
Oral
(N = 14, 100%)
Examples:
Written (N = 13, 93%)
Oral
(N = 1, 7%)
% in students’ notes
65%
2%
3%
0%
75%
0% (0 out of 15)
Inefficient note-taking strategies:
Does it generalize?
Type of content
Conceptual explanation:
Written (N = 12, 27%)
Oral
(N = 33, 73%)
Heuristic methods:
Oral
(N = 10, 100%)
Modeling mathematical behaviors:
Oral
(N = 14, 100%)
Examples:
Written (N = 13, 93%)
Oral
(N = 1, 7%)
% in students’ notes
65%
2%
3%
0%
75%
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 largescale 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:
Mathematicians:
75%
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
keith.weber@gse.rutgers.edu
Some papers discussed in this talk can be found at:
pcrg.gse.rutgers.edu