Introduction
There are a some difficulties that students have at the very beginning of studying abstract math that are overwhelmingly important, not because they are difficult to explain but because too many teachers don’t even know the difficulties exist, or if they do, they think they are trivial and the students should know better without being told. These difficulties cause too many students to give up on abstract math and drop out of STEM courses altogether.
I spent my entire career in math at Case Western Reserve University. I taught many calculus sections, some courses taken by math majors, and discrete math courses taken mostly by computing science majors. I became aware that some students who may have been A students in calculus essentially fell off a cliff when they had to do the more abstract reasoning involved in discrete math, and in the initial courses in abstract algebra, linear algebra, advanced calculus and logic.
That experience led me to write the Handbook of Mathematical Discourse and to create the website abstractmath.org. Abstractmath.org in particular grew quite large. It does describe some of the major difficulties that caused good students to fall of the abstraction cliff, but also describes many many minor difficulties. The latter are mostly about the peculiarities of the languages of math.
I have observed people’s use of language since I was like four or five years old. Not because I consciously wanted to — I just did. When I was a teenager I would have wanted to be a linguist if I had known what linguistics is.
I will describe one of the major difficulties here (failure to rewrite according to the definition) with an example. I am planning future posts concerning other difficulties that occur specifically at the very beginning of studying abstract math.
Rewrite according to the definition
To prove that a statement
involving some concepts is true,
start by rewriting the statement
using the definitions of the concepts.
Example
Definition
A function $f:S\to T$ is surjective if for any $t\in T$ there is an $s\in S$ for which $f(s)=t$.
Definition
For a function $f:S\to T$, the image of $f$ is the set \[\{t\in T\,|\,\text{there is an }s\in S\text{ for which }f(s)=t\}\]
Theorem
Let $f:S\to T$ be a function between sets. Then $f$ is surjective if and only if the image of $f$ is $T$.
Proof
If $f$ is surjective, then the statement “there is an $s\in S$ for which $f(s)=t$” is true for any $t\in T$ by definition of surjectivity. Therefore, by definition of image, the image of $f$ is $T$.
If the image of $f$ is $T$, then the definition of image means that there is an $s\in S$ for which $f(s)=t$ for any $t\in T$. So by definition of surjective, $f$ is surjective.
“This proof is trivial”
The response of many mathematicians I know is that this proof is trivial and a student who can’t come up with it doesn’t belong in a university math course. I agree that the proof is trivial. I even agree that such a student is not a likely candidate for getting a Ph.D. in math. But:
- Most math students in an American university are not going to get a Ph.D. in math. They may be going on in some STEM field or to teach high school math.
- Some courses taken by students who are not math majors take courses in which simple proofs are required (particularly discrete math and linear algebra). Some of these students may simply be interested in math for its own sake!
A sizeable minority of students who are taking a math course requiring proofs need to be told the most elementary facts about how to do proofs. To refuse to explain these facts is a disfavor to the mathematics community and adds to the fear and dislike of math that too many people already have.
These remarks may not apply to students in many countries other than the USA. See When these problems occur.
“This proof does not describe how mathematicians think”
The proof I wrote out above does not describe how I would come up with a proof of the statement, which would go something like this: I do math largely in pictures. I envision the image of $f$ as a kind of highlighted area of the codomain of $f$. If $f$ is surjective, the highlighting covers the whole codomain. That’s what the theorem says. I wouldn’t dream of writing out the proof I gave about just to verify that it is true.
More examples
Abstractmath.org and Gyre&Gimble contain several spelled-out theorems that start by rewriting according to the definition. In these examples one then goes on to use algebraic manipulation or to quote known theorems to put the proof together.
- Rewrite according to the definitions in abmath
- Direct method in abmath
- Detailed proof in abmath
- A proof by diagram chasing, blog post
Comments
This post contains testable claims
Herein, I claim that some things are true of students just beginning abstract math. The claims are based largely on my teaching experience and some statements in the math ed literature. These claims are testable.
When these problems occur
In the United States, the problems I describe here occur in the student’s first or second year, in university courses aimed at math majors and other STEM majors. Students typically start university at age 18, and when they start university they may not choose their major until the second year.
In much of the rest of the world, students are more likely to have one more year in a secondary school (sixth form in England lasts two years) or go to a “college” for a year or two before entering a university, and then they get their bachelor’s degree in three years instead of four as in the USA. Not only that, when they do go to university they enter a particular program immediately — math, computing science, etc.
These differences may mean that the abstract math cliff occurs early in a student’s university career in the USA and before the student enters university elsewhere.
In my experience at CWRU, some math majors fall of the cliff, but the percentage of computing science students having trouble was considerably greater. On the other hand, more of them survived the discrete math course when I taught it because the discrete math course contain less abstraction and more computation than the math major courses (except linear algebra, which had a balance similar to the discrete math course — and was taken by a sizeable number of non-math majors).
References
- Abstractmath.org (“abmath”)
- Definitions in abmath
- Detailed proof in abmath
- Direct method in abmath
- Handbook of Mathematical Discourse
- Languages of math in abmath
- A proof by diagram chasing, blog post
- Rewrite according to the definitions in abmath
This work is licensed under a Creative Commons Attribution-ShareAlike 2.5 License.
Send to Kindle
is positive unless 
“. I have said that specific thing to my classes — calculus students tend not to remember that the parabola is below the line 
on that interval. (And that’s the way you should show them — draw a picture, don’t merely lecture. Indeed, make them draw a picture.)
“. I don’t think anyone would say that, but they might say “
“. I think that any who said that (among math people) would be told that they are wrong, because for example 
. That reaction amounts to saying that (3) is not a correct interpretation of “P unless Q”.
