College Maths vs Arithmetic

 Proof-based mathematics is normal mathematics, and has been since the ancient Greeks. Unfortunately, many school curricula focus almost entirely on being able to perform computations, with nary a thought about why any of this works, or what it means. As a simple example, I am quite certain that virtually no one who has not taken some intermediate level math courses in college would be able to provide a definition of the real numbers that I would not be able to tear to shreds. Considering that I have taught college students who were able to show exactly how you multiplied fractions, but were not able to properly explain why that was the right thing to write down, my confidence in this assertion is extremely high.

However, if you only have mechanical understanding of procedures, then you cannot write proofs, because that requires conceptual understanding. If you have no experience in explaining your reasoning (and most people are quite terrible at this), then you cannot write proofs. If you don’t have a good feeling for how logic works, then you cannot write proofs. Most people don’t really have a good understanding of logic—to wit, I accidentally tripped up a lot of my students on an exam by giving them the following question: they had to decide whether the statement:

Suppose that f′(x)=x3−2x+1$f^{\prime }(x)=x^3-2x+1$. Then f(x)=14x4−x2+x$f(x)=\frac{1}{4}{x}^4-x^2+x$.

was true or false. Virtually all of them answered that this statement was true, because they checked that the given function f(x)$f(x)$ really does have the desired derivative… and so you must be done, right? Well, no, because there are all sorts of different choices for f(x)$f(x)$ that would also give that same derivative, hence the statement is actually false. I am fairly certain that this was a problem of logic rather than a misunderstanding about anti-derivatives, because I did a good job of drilling into them that if you want to find an anti-derivative, you find a particular anti-derivative plus a constant. In other words, if I had asked them to write down the general anti-derivative of x3−2x+1$x^3-2x+1$, I wager that most of them would give a correct answer of 14x4−x2+x+C$\frac{1}{4}{x}^4-x^2+x+C$.

Writing proofs is not hard once you have developed the skills required to do so. The trouble is that most students are dumped into this all at once, without training any of the myriad skills that you need to be successful in this endeavor.

   

    -Senia Sheydvasser,PhD in Mathematics,BA-Physics