College Math Teaching

May 16, 2011

Why is 0! = 1?

Filed under: advanced mathematics, calculus, integrals, mathematics education, media, uniqueness of solution — collegemathteaching @ 2:39 pm

This is in response to the following article which says, in part:

I do want to bring up one interesting case study I came across that points in favor of the “math is invented” side of the debate. My friends over at the popular blog Ask a Mathematician, Ask a Physicist did a great post a while ago addressing one of their readers’ questions: What is 0^0?

The reason this question is a head-scratcher is that our rules about how exponents work seem to yield two contradictory answers. On the one hand, we have a rule that zero raised to any power equals zero. But on the other hand, we have a rule that anything raised to the power of zero equals one. So which is it? Does 0^0 = 0 or does 0^0 = 1? […]

Indeed, the Mathematician at AAMAAP confirms, mathematicians in practice act as if 0^0 = 1. But why? Because it’s more convenient, basically. If we let 0^0=0, there are certain important theorems, like the Binomial Theorem, that would need to be rewritten in more complicated and clunky ways.

I provided an answer to the 0^0 question in another post. But the binomial theorem (which indeed starts with (a + b)^0 uses the binomial coefficients; and for each n we eventually have to do with n choose k notation {n \choose k} which involves a 0! in the denominator when k = n or when k = 0 . And yes, the formulas don’t work unless 0! = 1 .

But is that the only reason that 0! = 1 ?

Hardly. As before, we’ll use the properties of analysis to see why this should be true.
Note: I can recommend this internet article for those who are unfamiliar with some of the topics that we are going to discuss; in fact, that article is better than what I am about to write. 🙂

So, can the factorial function n! = n(n-1)(n-2)....(3)(2)(1) be extended so as to be continuous on at least the non-negative part of the real line? The answer is “yes”; the usual extension is the so called gamma function \Gamma (x) = \int^{\infty}_{0} t^{x-1} e^{-t} dt

It is an easy exercise in improper integration, integration by parts and mathematical induction to see that \Gamma (n + 1) = n! for all n \in {1, 2, 3, ....} and that \Gamma (1) = 1 ; hence it makes sense to set 0! =1 .

It is a more advanced exercise to see that the gamma has continuous derivatives of all orders defined for all positive reals (hint: differentiate under the integral sign) and that, in fact, the gamma function is real analytic (has a Taylor series defined on some open interval at every point ).

Strictly speaking, the gamma function is NOT the only possible way to extend the factorial function to an analytic function (example: \Gamma (x) + sin(\pi x) = \Gamma (x) on the non-negative integers ). But if we add the following restrictions:
1. f(1) = 1
2. f(x+1) = xf(x) (a property the factorial function has) and
3. f is logarithmically convex (e. g., ln(f(x) is convex in x then
the Bohr-Mollerup theorem shows that the gamma function is the only extension.

Now of course, the key here is the recursive property (2); this forces the 0! =1 definition.

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Teaching Mathematics in this era: The Helicopter Parent

Filed under: calculus, mathematical ability, mathematics education, student learning — collegemathteaching @ 12:45 pm

Mathematics Education
I was somewhat taken aback at this Daily Kos diary:

I was asked how my daughter is doing. She has 16 credits to go to her bachelor’s degree but it might as well be 2 more years. She’s up in Michigan licking her math wounds. She’s trying to gear up for what she has to do to complete her degree. She had her math ass kicked this last term – again. My daughter, the high school mathlete can’t hack collegiate Calculus II, Differential Equations or Multivariate Analysis (aka Calculus III on steroids) well enough to get a 73.5% or higher which is a requirement for her Physics B.A. degree. She can get 70% and 71%, but not the 73.5% deemed necessary. I mentioned that our current strategy is to find an instructor who’s actually teaching these classes in another school in town (we have 6 major colleges and universities in town to choose from) and transfer in the credits so she can get her degree.

Well, that statement set the mouse amongst the pooties.

How could I say that! Was I saying the instructors weren’t teaching! (That would be sacrilegious in this group.) What’s strange is that I wasn’t criticizing Chibi’s elementary, middle or high school math experience. Her high school math classes were the last classes where actual math instruction occurred. We complain about education and math education in particular, but Chibi got decent math education through to her Senior year in high school. She did well there. It’s university math outside the physics department that’s giving her trouble. She aces all her Physics & Calculus classes that require her to calculate vortexes, how moisture flows through tail pipes, predicting the sizes of hail stones or whatever; these aren’t difficult for her. It’s the math for math’s sake classes, where there’s a bare equation with no real world application or context for solving it that kicks her butt. The lack of practical application simply stymies her mathlete abilities. […]

One thing is for certain, as long as we continue to teach collegiate math using 100 year old methods, the U.S. math competencies will remain where they are.

So do you get this: his daughter did well in high school mathematics but didn’t do well in college mathematics, so it must be the fault of the mathematics professors….ALL of them. Then he blames our position with respect to the world in mathematics education on the university mathematics faculty.

Let’s examine this for a bit:

The disappointing performance of U.S. teenagers in math and science on an international exam, in scores released yesterday, has sparked calls for improvement in public schools to help the country keep pace in the global economy.

The scores from the 2006 Program for International Student Assessment showed that U.S. 15-year-olds trailed their peers from many industrialized countries. The average science score of U.S. students lagged behind those in 16 of 30 countries in the Organization for Economic Cooperation and Development, a Paris-based group that represents the world’s richest countries. The U.S. students were further behind in math, trailing counterparts in 23 countries.

The gap is already there by the time the kids are 15 years old….and exactly how is this the fault of university mathematics faculty?
Remember, his complaint was that his daughter got good grades in high school mathematics but didn’t get them in college mathematics…in several classes. What was the constant there? It was his daughter, of course.

In short, this person makes a sweeping claim based on the lack of success of HIS KID. THIS is part of the “helicopter parent” era.

I’ll add a few thoughts from my experience: in high school, I made A’s in foreign language but had to work to earn a C in my junior level class in college. The professor was excellent; the material was just difficult for me. Even in mathematics: I did well in analysis, algebra, topology and ok in complex variables. I struggled badly in numerical analysis. Yes, my numerical analysis professor…was quite good and I said so on the student evaluations. I just didn’t do well IN THAT CLASS. It just took a long time for that material to make sense to me.

I am not saying that there aren’t some horrible mathematics instructors: there are. That is unfortunate, but there are there. But that is not an excuse for repeated poor performance in classes.

Upshot: there are some parents who will never admit that their student really isn’t that good; the fault will always lie elsewhere.

Note: I am NOT saying that not doing well in mathematics makes anyone a failure. For example: my step son took the multi-year path to get through the required calculus sequence required by his computer science program. He got his degree and is now earning (at least) triple my salary in the database industry.

The founder of Jetblue Airlines didn’t do well in mathematics.

May 10, 2011

Non-portability of mathematical skill

On my calculus final exam, I gave two questions about a metal plate of uniform density. The plate was easy to describe: it’s boundary was the x axis and the parabola y = 1-x^2 . In the first question, I asked for M_x (the moment about the x axis and in the second question, I asked for the center of mass (they could use symmetry to deduce \overline x = 0 ). So to find \overline y , they needed to find the area (mass) and M_x .

What astonished me is that a number of students missed the question “find M_x  ” completely but then went on to solve for \overline y correctly!

This says something about the intellectually immature mind, but I am not sure what is says.

A Point about Probability Made in a Cartoon

Filed under: media, probability — collegemathteaching @ 1:09 am

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