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Maths is art - not science,
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This review is from: A Mathematician's Lament: How School Cheats Us Out of Our Most Fascinating and Imaginative Art Form (Paperback)
Every so often, I read a book which I cannot put down. Paul Lockhart's book is one of them. I received it this morning and finished it this afternoon, including some time to work through one or two of his 'maths games.'
As reported in other reviews, Lockhart brings a wealth of experience as a university level maths teacher, who decided to take his talents to benefit K12 level students in school. Lockhart is exactly the kind of teacher everyone should have in their maths class. His approach is simple and intuitively sound; namely, that maths as it is currently taught in most school classrooms is not really maths per se; rather it is a training process that rewards those who are good at learning a multitude of facts in the shape of formulae and algorithms, but who are not necessarily inclined towards or even competent at thinking 'outside the box.' As the Forward to the book by Keith Devlin (a maths professor at Stanford University) points out, many successful high-school mathematics students come unstuck when arriving at university to study mathematics, since the approach and character of the subject is so very different. The analogy is that pre-university maths is similar to learning to paint by numbers and that only when one 'arrives' at university is true maths introduced into the curriculum and the student is allowed to pick up a blank canvas to construct a painting. Many cannot make the transition, largely because they lack the mind-set necessary for this unstructured approach.
Lockhart appeals to us to appreciate that this transition is not something which should simply occur for a minority of students arriving at university. Rather, real maths should be the starting point of a child's introduction to the subject, so that the beauty and creativity that is at the heart of mathematics can be truly appreciated and crafted by the student. Unfortunately, the existing educational system tends to wrongly assume that maths is really a branch of science and as such should be taught to prepare students to be competent in the use and manipulation of calculus to support their studies in the sciences. In order to reach this level at school, the student is therefore 'trained' from day one in basic maths, to be followed in sequence by more maths, algebra 1, geometry, algebra 2, pre-calculus and finally calculus. Of course, many students leave school or drop the subject at aged 16 and don't really even get exposure to calculus. Unfortunately, well before aged 16, many more students have simply been 'turned off' by maths, since its method of 'training' tends to reward the students who invest in the recipe of learning a mathematical operation, then practising the skill to a level where it is ingrained into the psyche. A good example being the algebraic formula for determining the factors of a quadratic equation where x=(-b+/-SQRT(b^2-4*a*c))/2*a. Whilst useful for solving the specific kind of problem, its relevance to the vast majority of students is such that once they have sat and passed or possibly failed their 16 year old maths exam, then like all the other formulae learnt for 'the exam' it will be willingly forgotten and never used again for the rest of their lives.
However, it would be wrong to paint Lockhart as being some free thinking spirit who denies the importance of learning certain facts, even formulae. His point is however, that frequently the student at school is introduced to such topics and concepts like the above formula as simply the next thing to learn and be mastered on the curriculum. Most of the time, the most important question of why does this formula work, or what is the history and reason behind its development is never mentioned. Lockhart's argument is that without expecting a student to be familiar with everything that has been developed in mathematical thinking during the past 3,000 years, it would at least make sense to introduce students to the various areas of the subject by way of exploration; by way of playing games and looking at maths as something to enjoy and experience without artificial exercises. One example of an artificial exercise that Lockhart uses which I enjoyed was his illustration of how in algebra one might be asked to solve the 'real life' problem of the age of your friend Maria, who is ". . . 2 years older than twice her age seven years ago." Lockhart's heartfelt retort to such attempts to make the subject interesting is that these kinds of unrealistic and ridiculous examples are not what algebra is all about. Instead, simply ask the question, "Suppose I am given the sum and difference of two numbers. How can I figure out what the numbers are themselves?" Lockhart states that "Algebra is not about daily life, it's about numbers and symmetry - and this is a valid pursuit in and of itself."
Furthermore, Lockhart is not out to attack school maths teachers. He fully recognises that most are 'trapped in the system,' but he appeals to maths teachers to rethink what they are trying to achieve in their classes.
Perhaps most importantly, Lockhart's observations go right to the heart of one of the problems of modern education in general, namely, that its objective is primarily to train people for the workforce. Setting aside any Orwellian undertones to such criticism, I wish that government ministers and policy makers would take note of Lockhart's messages. Maths is an art that should inspire and encourage thinking outside the box from the youngest ages. It should not be taught as a series of facts simply to be learnt for performing computations in a series of exams. Such an approach suffocates the intellectual development that real maths can so easily nurture.
One may not agree with everything said in this book, since it is first and foremost a lament, but also a call to arms as such it is naturally subjective in nature. However, like all good ideas, anyone reading it could not possibly fail to be stimulated into thinking about these important ideas.
Well worth the read.