Mathematician’s Lament : Summary

In US, it is widely believed that math education at school level needs a BIG reform. The comparison is with respect to Asian kids who supposedly do well in math.  My strong belief is that math education is broken in some of the Asian countries too, including India. One might mistake the immense problem solving exercises that most of the Indian kids go through to get to engineering colleges as a strong foundation for mathematical thinking. Nope. Having gone through the system myself, I think that most of the Indian students are good at pattern recognition in problems that they are posed, that too a specific kind of problems that figure in the entrance examinations.

The other day I was reading about students putting in15 hrs of practice for engineering exams. The kids might be brilliant, but I would guess that their enormous practice at solving1000’s of problems relating to algebra, trigonometry, coordinate geometry , calculus, etc make them Excellent Pattern Recognition Engines who work brilliantly till they crack an engineering exam like JEE or State level engineering entrance and then they STOP FOREVER, at least for most of the kids. Most of them , as they reach adult stage, fondly recall math as their favourite subject in their childhood, but struggle to formulate problems using math, think mathematically about the problems. 

Yes , this enormous focused Pattern recognition exercise does sometimes help some of the students sail through their entire lives with out really exercising their brains as far as real math is concerned.  If you crack open a recently graduated engineering student, I bet that the ability to pose problems and solve them creatively would be very less and in the name of math, there would be tons of methods, techniques, formulae, notations, symbols floating around in his mind.  This cannot be called math education. 

The same is the case , sadly , with statistical education at a PG level course in India. Methods, techniques gain upper hand over the narrative . Many students would rattle the formulae for ordinary least squares estimate  but spend very little time to get to the story behind least squares estimate, the epic battle between Legendre and Gauss to stamp their supremacy on OLS. Stories always gives an opportunity to be creative. But sadly, statistics which is a result of enormous trial and error is presented in the curriculum in an extremely dry , placid and boring manner. Ask any PG student who has taken statistics in his academic course, to talk about statistics with out equations and techniques for just 5 minutes, and you will know how badly the education system sucks in India. Anyways coming back to this book,

This book is an essay by Paul Lockhart which goes over the lack of MATHEMATICS in school. It is a book which offers no solutions per se, but points to the specific problems of math education at the school level.

Mathematics is Art, says the author.  Math education has become a set of techniques , often irregularly spaced and disorganized. No effort is put in to provide the context. Math, like painting , is a result of hard creative process and often the result which is an outcome of the process is far more joyous and wonderful for a student rather than the other way around. I can relate to this aspect somewhat, as I have spent 1.5 years teaching Pre Calculus and Calculus to undergrad students at CUNY. One incident I particularly remember is that of a student asking me the relevance of knowing Difference Quotient. I told him that it would be used in Limits and conveniently ignored so as to save class time. But the question stuck in my mind since then. Why should I teach the definition of Difference Quotient , quiz the kids on the definition and related exercises and leave it for them to come back in some other class to appreciate its usage in limits. Almost all the sections I had taught, I just could not deviate from the syllabus which comprised, giving definitions, framing questions around those definitions, asking students to work on some exercises etc. Useful for cracking the exams but utterly useless in the long run.

Author, calls, Mathematics curriculum as  a confused heap of destructive disinformation . He shatters something called the ladder myth, where curriculum is designed as a set of ladders where in the initial classes , students are taught definitions, notations, proofs and statements thus supposedly preparing them for higher class math. This is similar to teaching painting to kids with a tremendous amount of focus on paint theory, theory about colour, than allowing them to take a brush and start off painting and discover the techniques in the process.

With a few insightful examples from the book, the author makes a point that techniques are useless with out thinking about the context and the process of mathematical education. BTW, this process is not to be confused with the process term used in the industrial world context. Here process is highly customized , individual specific where the education is loosely coupled with an underlying theme, Problems and the context matters and not specific techniques or a Russian doll kind of mathematical education where you proceed from definitions to notations to functions to dy/dx , integration with no real context.

Yes , school has become a training ground for children to perform so that they can be sorted. Math is not a collection of facts but is about reason and understanding. We want to know WHY and not for any practical purpose.  In that sense, this book is a wake up call to all the teachers and the curriculum designers of math education to relook at the entire Math education

This book also serves as a reminder that “Its not notations, but notions that help in our progress”. Unfortunately , it is former that the school is focusing on , leaving no time for the kids to indulge in the latter activity.

  Takeaway :

Math education does not need reform as it equates to rearranging chairs on a sinking Titanic.

We need to build a new ship .