Saturday, July 4, 2009

Arthur Benjamin about Math Education

I found this video on my usual bookmarking sites. It got Front Page on Digg and lots of Stumbles also. The video is very interesting from my point of view as a mathematics teacher. Arthur Benjamin, a mathematics professor and also the mathemagician offers a bold proposal on how to make math education relevant in the digital age.

I think he is a really good performer but he cuts some corners and speaks like a politician convincing people of something they know very little about.

Before I utter my criticism, I have to remind you that my perspective is Finnish and our mathematics education. I probably don't know enough of the conditions in USA. I have learned that there are schools where the students don't advance very well, but on the other hand the most of the best universities are located there. Anyway calculus and statistics are universal.

The biggest problem in my opinion is that statistics is not a science we can put under the main category mathematics. It belongs to social sciences (in the University of Turku for one) and the interpretations of statistic data and their relevance has very little to do with math. Probability is a part of applied mathematics and also the calculations regarding statistics use math, but something as vague as predicting the future can not be a part of a science which gives only absolute solutions or shows that there are not any! I have a master's degree and my major is mathematics, I have even had some predoctoral studies, but I never had a single course in real statistics.

In Finland we have two different levels of mathematics the students can choose from, the short and the long course. Both levels have one course containing probability and basic statistics. The courses are very similar on both stages and I have noticed a certain trend with these courses: there are always students who get fascinated by the common-sense-math and perform much better than on the other courses BUT also some of the top students are finding this course very hard.

Calculus (we actually use the word Analysis) is taught about 2 courses on the shorter course (no integrals) and 5 on the longer course but also extra courses for those who want to choose them.

I have heard the statement: "I don't remember anything about derivatives" or: "I have not used derivatives after I left school" numerous times. I also know that most people don't, not even those who choose their career in engineering. However the basic calculus is needed in order to understand the further studies in engineering, physics, chemistry, mathematics and even statistics! A tool for the continuous probability distributions is integrals.

Teaching basic mathematical statistics is not as simple as it may sound. When I teach mean values and standard deviations the examples are either too simple to show anything about real statistics or too complicated to be handled without the calculator's built-in statistical functions. Mental mathematics skills are not very good even now, they would be worse if we take more calculators or computers in use. Regardless of the methods we use the examples take time. Typing all the input data is not so simple.

Arthur Benjamin mentions the digital age. I have a better suggestion, let's add Number Theory. It is simple, it is fun and it is a very relevant field of mathematics regarding computers, coding, cryptography and sharing secrets which will be needed in internet voting. These fields of math are hot at the moment, but the basics have been there for ages. I teach one extra course in Number theory (+ Math Logic) but there are many students who don't choose it.

It was actually Gauss (more known from the Normal Distribution) who said: "Mathematics is the queen of the sciences and number theory is the queen of mathematics."

Image credits: Wikipedia and zazzle


Anonymous said...

"I have a master's degree and my major is mathematics, I have even had some predoctoral studies, but I never had a single course in real statistics."

I'm afraid that I think that your view of Statistics is probably based on not having studied it in any depth. Results like the Central Limit Theorem are very broadly applicable, entirely rigorous, and statistical. Similarly for properties of maximum likelihood estimation, etc.,

(Whether the assumptions hold exactly in a given application may be open to debate, but the same may be said of much Newtonian mechanics. I never encountered a frictionless slope or "point mass" in my life - presumably because the existence of the latter would lead to a black hole!)

Statistics and probability underlie things that we use every day, e.g. google search, efficient micro-processors, insurance.

As science becomes more and more data-intensive almost every field now relies heavily on statistics. Examples are: DNA sequencing, mass spectroscopy, phylogenetic reconstruction (i.e. working out what are different species and how they are related), proteomics, climate prediction, evaluating new medical treatments, etc., etc.,

Differential calculus alone is quite insufficient for any of the above modern scientific fields.

The real problem is that the theory of mathematical statistics is actually somewhat harder than the calculus - understanding the calculus is really a pre-requisite to understanding mathematical Statistics. Consequently at a high-school level the temptation is to teach statistics in "cookbook - recipe" form without giving any explanation of the underlying theory.

"something as vague as predicting the future"

This statement really betrays a simple lack of knowledge.

To take an example, it may be very hard to predict when a specific individual is going to die, but we may be able to make very accurate predictions about what proportion of people in a certain population are going to die.
This is the entire basis for insurance, pension funds, etc., etc.,

Similarly, it is hard if not impossible predict whether a couple will conceive a boy or a girl, but
for several hundred years we have been able to predict with great accuracy what proportion of births lead to males vs. females.

(I suggest you read up on the weak law of large numbers, central limit theorem, etc.,)

A look back through history also reveals that many of the minds we associate with classical mathematics have been involved in the development of Probability and Statistics: Newton, Bernoulli Gauss, Laplace, Pascal, Kolmogorov, etc., etc.,

I think your post actually serves rather well to illustrate what Arthur Benjamin is saying.


Incidentally, you maybe interested to know that Sudoku puzzles may be solved easily via probabilistic Markov chain Monte Carlo algorithms, and further these puzzles are closely related to the theory of experimental design in statistics.

lekahe said...

Actually your comment shows I was not very clear in my post.

I think what you need to teach the students in school is the basic tools and leave the applications to the higher education when the students understand the limitations.

Mathematics has actually two different schools of scientists, the pure math and the applied math. I feel like a mathematician lost on track to become a teacher but my field is the pure math, combinatorics, number theory, coding theory etc. I prefer the applications which have exact answers which also has applications like RSA coding.

If the other scientists don't have enough knowledge about the basics and limitations, the results can be disastrous. In Finland in the University of Turku Statistics is part of the social studies and the math they are using is not expressed the way I would need it to be. That is the reason I never studied it, expect as a part of the probability theory which covers as much as I need it, enough to show how much people using statistics really ignore the initial conditions and draw wrong conclusions.

As for Sudokus, I am not interested in the fastest way to solve them. I am much more interested in questions like how many valid sudokus you can create, what is the minimum amount of clues needed and so on. These problems are hard. Since the amount of 9x9 sudokus is finite, these problems are a bit easier but the theory of nonograms and their solvability is in NP.