A Mathematician's Lament: How School Cheats Us Out of Our Most Fascinating and Imaginative Art Form

Once in a while we read books that we just know are especially important, and that we know we will be thinking and talking about long after reading them. This book is one of them for me. I am a returning adult student, and I am about to finish my training to become a math teacher. Having gone through my education program, my enthusiasm was just about completely drained, and I've been having trouble remembering why I ever wanted to become a math teacher in the first place. Why would anyone? Paul Lockhart knows, and his book has reawakened my desire to help students discover the joy of mathematics. His argument is concise, and he makes it forcefully. His book is a joy to read, mainly because his understanding of the subject and his passion for it are clear in every page. He reinforces ideas I already had about how school sucks the life out of math (and all subjects), but he also challenges some of my opinions. I think this will happen with most people who read it. Once he finishes making his argument about math education in about the first two-thirds of this short book, he devotes the remaining section to describing what he finds wonderful about mathematics itself. This section should make just about anyone want to become either a mathematician or a math teacher. I want people to read the book for the specifics of his arguments, but I want to discuss one important point that he makes. Many people in math education claim that in order to make math more understandable and interesting to students, we need to show how practical it is and how it is used in everyday life. I've always felt like this idea was wrong, or at least limited in its usefulness in that regard. Well, Lockhart demolishes the idea, essentially claiming that practical uses are simply by-products of math, and that the real excitement and beauty of mathematics is in the abstract, imaginary, and creative world of mathematical ideas that have no specific connection to the everyday. By-products and applications can make math seem boring and secondary to the uses it serves. I agree with him--and much more now after having read his argument. I honestly think just about everyone should read this book. Of course math teachers should, as should anybody involved in math education in any way. But I think people outside of math education should read it too. The specific mathematical ideas discussed in the book do not require a strong mathematical background, and I can't think of a better book that so concisely conveys the nature of the subject and the way it is viewed and misunderstood in society. I'm still not sure I agree with Lockhart's every point, but I love this book. (And I might agree with his every point after more thought and experience in the classroom.)

The author is lamenting the common (universal?) focus on failing to open students' eyes to the art of mathematics. The focus in schools is often authoritarian, based on transmitting algorithms and rote, mechanistic learning. On the other hand, there is a clever and fun side to the art of mathematics, a display of the elegance and beauty of mathematics, and a creative side to mathematics that is often missing in the teaching of mathematics. Prof. Lockhart laments this absence. Even very talented mathematicians, once in the school system, find it very difficult to introduce students to the fascinating, beautiful "why" questions and explorations of mathematics. This is common throughout the school system in the U.S. at every level, not just in grades K-12. Algorithms are rarely beautiful and elegant, rarely encourage our creativity, and rarely encourage the search for patterns and connections. The beauty and elegance of mathematics is the fascinating human side to mathematics. Of course, rote learning rarely engages students. The result is that, even the mathematically able and talented, can fail to connect with the raw human power of elegance and beauty in mathematics. Prof. Lockhart's book is very short, but it can be highly recommended as emphasizing a side to mathematics that is terribly important for us, and is lacking in the current education system. We live in an age when algorithms and the treatment of humans as machines dominates, while rote learning and drill is often viewed as necessary and expedient in education. Therefore, the author's lament seems to draw us to some ideal world that is very remote from today's world but worth thinking about and recognizing as important. In the real world, much of mathematics, at any level, and in any field, can seem overly hard, unmotivated, dry, and/or unappealing, even to mathematicians. There is some artistry about mathematics, as it is, and as we encounter it, but we do not have the privilege, none of us, of living in Prof. Lockhart's pristine and idealized version of the mathematical world.

I think the main contribution that this book provides is a basic narrative context of mathematical thought, which allows you to ask further mathematical questions. Unfortunately, this kind conversation about math is completely ignored in most educational systems around the world. Any criticism of modern education will need to be much larger and more expansive than this, but it does provide a good series of arguments that highlight the most important aspects of how our education can fail. Because of that, I think about this book often. So much so that I'm writing a review of it 7 years after I first bought it. It's a quick read and I highly recommend it.

... schön fände ich es, wenn möglichst viele es lesen - die, die Mathe unterrichten, an der Schule, auch an der Hochschule, in der Lehrerbildung und -fortbildung, in all den Kommissionen, die sich Gedanken darüber machen, was wann, wie und wozu unterrichtet und gelehrt werden sollte. Und wenn die, die es gelesen haben, sich darüber austauschen würden - im Lehrerzimmer, im Kollegenkreis, in all den Gremien und Kommissionen. Fände ich wirklich schön. Schön - wie die Mathematik.

Loved mathematics and hated art all my life. Had never thought about them being one and the same. Maybe not too late to question other perspectives. Great read

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.

J'ai toujours eu ce sentiment que le cursus traditionnel était nul. C'est en contact avec le monde des sciences informatiques que j'ai enfin compris la beauté des math appliquées.

Un regalo para mi hijo, que está estudiando matemáticas.