Discussion Question 4
Marty
available in the binder (Thursday) and online at
http://www.msri.org/communications/books/Book36/files/tucker.pdf
Thomas Tucker gives six hypothetical reasons that proofs might be
considered important in the calculus classroom. Tucker also argues
against these reasons in some very short statements. It would
certainly be advisable to look at his article before participating in
this discussion.
His arguments are quite short -- there is room for much more
discussions. Argue for or against one of the following positions he
suggests.
1. Proofs help students understand concepts and believe results.
2. It is useful in later mathematics courses for students to see
proofs in
a calculus course.
3. Proofs are part of students' cultural heritage, which they should
appreciate the same way they appreciate the theory of relativity or
Huckleberry Finn|even if they don't understand it.
4. Proofs are what we mathematicians do, and students should see
what we do.
5. Proofs are beautiful.
6. Proofs build character.
When focusing on one of these positions, present your arguments
strongly and defend them against counterarguments. How is your
position based on your general beliefs about what the classroom is
supposed to be, and the goals of education? How is your position
practical? Can you find any existing research that supports your
position?
Feel free to respond to other posts and keep the debate lively!
Jen Mogel
Calculus classes I'm disappointed. I agree that we should include
proofs in calculus, but not for any of the reasons already given.
There is another reason not stated in Thomas Tucker's paper.
Universities make everybody, even those poor dear psychology majors,
take math classes. I am often asked why that is as many people won't
need calculus after they get out of school. There is another reason
why Universities require all students take a math class, as
illustrated in the UCSC General Catalog. The process of learning
math, doing homework, and working out problems teaches quantitative
reasoning skills, students learn to think logically through math. I
think we can all agree that learning how to prove things and
understanding proofs in general will have this desired side effect.
We should include more formal proofs in calculus class but practically
it's not so easy. Students are not as ready to understand some of the
proofs in calculus as we would like them to be. We have to prepare
them beforehand. We have to spend some time to teach students how to
solve problems properly, to do Proof II type proofs, as expounded on
in the second half of Thomas Tucker's paper, and actually call them
proofs. Take Tucker's example, on if f + g is concave up when f and g
are. By the time students are at the point where they could get a
problem like this, they should be able to write something more like a
formal proof of the answer, not an explanation. Most importantly,
this preparation should begin from the first day of algebra and
continue all the way through college. We need to get the students
used to math thinking instead of math as a formula early on. This
will produce more mature math students who will understand the formal
proofs with less difficulty.
I understand that this will be very hard to implement. It requires us
to go through all of our junior high and high school teachers and
probably tell them to change or fix they way they teach math. In my
opinion, we have to do this anyway and not just for math!
Some might also say that just thinking about math class in this way
cheapens the subject; math becomes a means to a not-so-very-math-
oriented end. I feel that the contrary is true. How many subjects,
science or not, have this side effect, especially at the freshman
level? In addition, this more formal approach to problem solving has
the potential to not only improve students' arguing skills but to
improve their math! We can relax with them a little after they've
decided to move on to upper division math classes.
UCSC General Catalog 2006-2008:
http://reg.ucsc.edu/catalog/html/programs_courses/06_08_Catalog/undergrad_acad.htm#graduation
Just a note: I have no actual evidence to back up my claims; I have
my gut. Stephen Colbert tells us that all we need is our gut to know
we're right; it helps us to work though impediments to our arguments,
impediments like facts. Have a nice day...
Marty
include proofs". To summarize, I think you are saying that: calculus
students should see proofs because they will learn to "think
logically" as a side effect.
Here are some questions, which I am interested. Perhaps another
person can find some research which approaches these questions when
responding to the post:
(1) Is there evidence of a causal relationship between taking proof-
based mathematics classes, and "thinking logically" outside of the
mathematical domain?
(2) Given that the vast majority of calculus students never attempt
to utter a statement about continuous functions on their own, can we
expect them to appreciate a proof of such a statement?
(3) What do calculus students think that a proof is?
(4) Is logical thinking more easily taught in a non-mathematical
course? For example, students are expected to present logical,
coherent arguments in their essays as well, and they seem likely to
understand why? (Why) do you think that mathematics is the best
medium from which logical thinking arises as a side-effect?
- MW
> UCSC General Catalog 2006-2008:http://reg.ucsc.edu/catalog/html/programs_courses/06_08_Catalog/under...
Jen Mogel
megan
students thinking in a logical way. This subject actually is very
important to me, and I have spent the last year trying to think of a
way to implement a completely new course in the high school system. I
actually think that there should be a completely separate course from
calculus (an alternative to calculus actually) offered at the high
school level on logic and problem solving. I think that this subject
is more beneficial to a larger range of students. As Thomas Tucker
mentioned in his article, only 10% of calculus students in high school
go on to be math majors. Now out of the remaining 90% I wonder how
many of them will move on to careers in which they regularly use
calculus. But I claim 100% of the students could benefit in logical
thinking. In fact when talking to my father, who works in the
business world, he told me that one of the big things that companies
look for in a potential employee is a problem solver. I have also
talked with lawyers, who have told me that a large majority of the
LSAT is based on logical reasoning. Philosophy students are required
in college to take their own logic course, which is similar to a basic
math logic course involving such things as truth tables. Clearly this
course would benefit those going on to study math as well, and the
other subjects that also use calculus regularly, such as Physics,
Computer Science and Engineering. I don't think there is a single
person that wouldn't benefit from this course, if even just to use in
their every day life.
I have talked with quite a few teachers that teach or have taught high
school math, and some people in administration. And while all of them
feel like this would be something worthwhile it is very hard to get
such a program recognized in the system, due to what one person called
the "Math Wars". This stems from the "No child left behind" act, in
which they have to prepare all students to pass government issued
standardized testing. The math section of these tests is basic
computational problems, and don't have much in the way of logic and
problem solving. Of all the people I have talked to about this idea,
they believe this will be the biggest hurdle for me. I also think that
this may be one of the reasons that they do not enforce proofs in
calculus. Our country (I mean government) at the moment is more
concerned with our children passing these standardized tests than
getting a better education, and I intend to change that.
Jacob West
"understanding" proofs is at least as wide as that between "seeing"
calculations in calculus and "understanding" them. Nearly every
student has experienced the feeling of confidence in understanding a
calculation or proof when seeing it carried out in lecture, while only
later discovering that they are completely unable to reproduce even
the simplest parts of it on their own. Bridging the gap between
"seeing" and "understanding" necessarily requires "doing" on the part
of any student pursuing any subject. In short, while I would argue
that introducing proofs as early as is admissible in elementary
education is beneficial for the development of logical thinking, I
would also argue that "thinking logically" is not a side effect of any
endeavor, even mathematics. Granted, logical argument is a
prerequisite to the proper presentation of any mathematical proof, but
despite this fact producing "good" proofs remains difficult even for
veteran mathematicians. Rather, I would argue that "logical thinking"
must be actively pursued in its own right, and it is only highly
correlated with mathematicians because as a group they are among the
most motivated to pursue it, as a means to understand the work of
others and improve the quality of their own. On the other hand, one
can study mathematical proofs as a means to better understand the
structure of logical argument, but I would argue that the two pursuits
are complementary rather than causally related.
I should admit now that underlying the thesis presented in the
previous sentences is in part a distinction between being logically
sound and logically complete. The vast majority of mathematical
proofs are logically sound without being logically complete, if for no
other reason than completeness can often be "easily seen" and would
unnecessarily clutter the key ideas in a proof. This is perfectly
reasonable, and in fact usually desirable, when the completeness is
indeed "evident" to the reader, but usually poses an enormous
challenge for the student still struggling with such distinctions.
This difficulty is only amplified when mathematical definitions or
theorems themselves are stated in an incomplete form, since the
student does not have the necessary experience to fill in the missing
details.
In her paper entitled "The Role of Logic in Teaching Proof" [Epp03],
Epp reports on studies of whether abstract reasoning developed in one
domain can transfer to other domains. In particular, a cited study
that tested the understanding of the logic of conditional statements
"found no difference in performance between university students who
had taken an introductory logic course and a control group of students
who had not." Another cited study indicated that an explicit course
in logic had no effect on high school students' ability to prove
theorems in geometry. On the other hand, another study "suggested
that instruction in logic could, in fact, lead to greater success in
geometry if the logic units were interwoven with the geometry and if
cues were given to help students realize the relevance of the logic to
the specific geometry tasks." That is to say, the role of logic in
the process of formal reasoning about mathematical statements is only
evident to the inexperienced student when those connections are made
explicit, and their importance emphasized. The intimacy of these
connections is only appreciated after a great deal of experience has
been amassed diligently struggling to understand their interplay.
Also, see [Bako] for more discussion on teaching logic in mathematics.
The evidence produced by the studies cited above agree with my
personal experience, both as a student of mathematics and as a
teacher. Tucker's comment that "the only character built by seeing
proofs is the ability to sit while some else is talking to you" has
been my experience exactly when initially learning or attempting to
teach the machinery of formal reasoning and proof construction.
"Seeing" proofs in this way is only beneficial once a solid foundation
in logical reasoning has been developed together with a sufficient
scaffolding of mathematical language and structure within which one
might incorporate something new.
There is clearly much more to say on this subject.
References:
[Bako02] "Why we need to teach logic and how can we teach it?", Maria
Bako. International Journal for Mathematics Teaching and Learning,
October 17, 2002. Web address: http://www.cimt.plymouth.ac.uk/journal/bakom.pdf
[Epp03] "The Role of Logic in Teaching Proof", Susan S. Epp. The
Mathematical Association of America, Monthly 110, pgs. 886-889,
December 2003. Web address: http://condor.depaul.edu/~sepp/monthly886-899.pdf