I have recently read a little book: *A Mathematician’s Lament* by Paul Lockhart. Originally a 25-page essay, it was expanded into a book by popular demand and published in 2009. The author is a K-12 mathematics teacher who has a PhD in mathematics and used to teach at Brown University and U.C. Santa Cruz.

**Mathematics as an Art**

The central thesis of the book is that mathematics is an art, and should be taught like one. Mathematicians, like artists, are makers of patterns, except that they all follow the unifying aesthetic principle of “*simple is beautiful*“. Since the simplest possible things do not exist in the physical reality but in our imagination, and things that exist in our imagination are called ideas, it could be said that mathematicians make patterns with ideas, as the mathematician G. H. Hardy said in this quote:

A mathematician, like a painter or poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas. —G. H. Hardy

Just like art, there is no ulterior practical purpose to math. It is simply something that is fun to do, an activity that is done for the inherent pleasure derived from performing the activity. Mathematicians start by making up imaginary beings like triangles, then they ask questions about their intrinsic properties and behaviors. The art of mathematics lies in crafting satisfying and beautiful explanations (i.e. proofs) in response to these questions.

The major difference between ordinary art and mathematics, I feel, is that in ordinary art, you can control every aspect of your creation, whereas in mathematics, once an entity is born out of your imagination, what it then does is beyond your control; that would need to be *discovered*. And it is, I believe, through making such discoveries that mathematicians get to have a glimpse of divine beauty, which beyond any kind of beauty that human beings alone could create. This possibility to access divine beauty makes mathematics a higher form of art than any other.

To put it another way, although human beings are able to create imaginary beings in our mind, the patterns and structures in which our imaginary beings are embedded already exist out there in some “mathematical reality”. In this way, imagination is actually our personalized wormhole, our dokodemo door, to that fantastical world. Please note that this is my opinion, not the author’s. As expressed elsewhere on this blog, I am both a mathematical Platonist and a theist.

By the way, one can also look at this through a systems lens. I am currently reading *Thinking in Systems* by Donella Meadows, who defines a system as consisting of elements, interconnections, and a function or a purpose. If mathematics is a system, then its elements (numbers, spaces, etc.) would be those things that could be made up by the human imagination. But the interconnections among these elements (theorems) exist in the mathematical reality and would need to be discovered. As for the function/purpose of the system, well, that may be beyond our understanding at the moment!

**The Horror of Math Education in the U.S.**

Back to the book. Lockhart’s *lament* is about the status of math education in the United States. In the American classroom, the entire process of doing mathematics (wherein lies the “art”) is reduced to memorizing facts, following procedures, and mindless manipulating symbols. As Lockhart repeatedly writes: “The question has been asked and answered at the same time.” There is nothing left for the students to do.

Mathematics is the

art of explanation. If you deny students the opportunity to engage in this activity—to pose their own problems, to make their own conjectures and discoveries, to be wrong, to be creatively frustrated, to have an inspiration, and to cobble together their own explanations and proofs—you deny them mathematics itself. —p.29A Mathematician’s Lament

By removing real mathematics from the math curriculum and presenting a hollow shell of the real thing, many potentially gifted mathematicians end up hating the subject. Lockhart goes so far as to say “Better to not have math classes at all than to do what is currently being done.” This reminds me of the principle: “First, do no harm.”

I love this quote (p.33, my bold):

…

I don’t see how it’s doing society any good to have its members walking around with vague memories of algebraic formulas and geometric diagrams, and clear memories of hating them.It might do some good, though, to show them something beautiful and give them an opportunity to enjoy being creative, flexible, open-minded thinkers—the kind of thing arealmathematical education might provide.

I chuckled when I read that because I used to be that member of the society. As a typical victim of American math education, I had such a bad impression of the subject that I did not take a single course in mathematics when I went to college. Why would I? It was only until a few years after I graduated, when the bad memories had sufficiently faded and my interest in the subject had increased again thanks to my passion in logic and philosophy, that I had gathered enough courage and motivation to go back to college to study mathematics. Thankfully, I had wonderful professors the second time around (especially Professors Elwood Parker and Rudy Gordh, both now retired). They taught much much better than my high school teachers ever did. I am not blaming my high school math teachers. They were wonderful people. But unfortunately, they had to follow a particular curriculum that had little to do with real mathematics.

It is interesting to note that Lockhart included a whole chapter on high school geometry. To be honest, I liked that class a lot, especially the proofs part (despite the rigidity of their format). Math had never been my strong subject in school, but that year I was actually picked by my teacher to enter into a geometry math competition. As the author says on p.81: “It’s hard to completely ruin something so beautiful; even this faint shadow of mathematics can still be engaging and satisfying.” Unfortunately, my interest in math subsided again after Precalculus and AP Calculus. I would not see math again until seven years later.

**Math is Useless and That’s OK!**

I also love the author’s emphasis on the inherent uselessness of mathematics. Sure, mathematics has lots of practical applications, but these are just a trivial by-product of mathematics, and not what mathematics is about.

Algebra is not about daily life, it’s about numbers and symmetry—and this is a valid pursuit in and of itself. —p.38

Isn’t it wonderful to see that written out?

Now hold on a minute, Paul. Are you telling me that mathematics is nothing more than an exercise in mental masturbation? Making up imaginary patterns and structures for the hell of it and then investigating them and trying to devise pretty explanations for their behavior all for the sake of some sort of rarefied intellectual aesthetic?Yep. That’s what I ‘m saying.

In particular, pure mathematics (by which I mean the fine art of mathematical proof) has absolutely no practical or economic value whatsoever. 🙂—p.120 (my bold and my smiley)

**The Ideal Math Teacher**

So what does Lockhart think should be done instead? How should mathematics be taught? Since I have been job-hunting for some time now (unsuccessfully, I might add… if anyone knows of an accounting position in NC or MO, please contact me on LinkedIn) and read more job postings than anything else, I thought it would be interesting to write a job description of the K-12 Math Teacher according to Lockhart’s ideals (I made up the numbers for years of experience and salary, lol).

**K-12 Math Teacher – Entry Level**

American public schools are looking for talented and passionate teachers to conduct daily lessons in Math.

**Responsibilities (c.f. p.43):**

- Choose engaging and natural problems suitable to the students’ tastes, personalities, and levels of experience
- Give students time to make discoveries and formulate conjectures
- Help students to refine their arguments and creating an atmosphere of healthy and vibrant mathematical criticism.
- Be flexible and open to sudden changes in direction to which their curiosity may lead.
- Have an honest intellectual relationship with students and with mathematics.

**Qualifications:**

- Open and honest
- Have the ability to share excitement
- Have a love of learning
- Have enough of a feeling for mathematics to be able to talk about it in your own voice, in a natural and spontaneous way

**Experience:**

- Proofs (done by self): 1+ year
- Field experience in Mathematical Reality: 1+ year

**Job Type**: full-time

**Salary**: $35,000 – $40,000 per year

Lockhart believes that “There should be no standards, and no curriculum. Just individuals doing what they think best for their students” (p.82). In other words, teachers should have the freedom to do what they think is best. Generally, professors are granted that freedom, and I think that is why mathematics at the university level is so much more fun.

By the way, here is the proper way to teach techniques (p.42):

Give your students a good problem, let them struggle and get frustrated. See what they come up with. Wait until they are dying for an idea,

thengive them some technique. But not too much.

This would seem counter-intuitive to many people, since most of us are inflicted with the Do Something Syndrome. But it is important to do *nothing* sometimes. Teaching is not transmitting information; it is giving students just enough guidance so they can learn by themselves. Here, I am reminded of the importance of white space in Chinese landscape paintings. If the entire paper is covered by a mountain, you wouldn’t be able to appreciate the majesty and beauty of the mountain. In the same way, if a question is asked and answered at the same time, students would not be able to appreciate a mathematical theorem for what it is. Struggling in ignorance provides the white space that is necessary to appreciate the mountain that is mathematics.

After all, isn’t the universe born out nothingness? The void is the beginning of many things. OK, someone needs to stop me from philosophizing.

**My Question for the Author: Role of Definitions?**

Overall, I mostly agree with the author. However, he has a viewpoint that I am still not sure I understand, and that is “…you don’t start with definitions, you start with problems” (p.79). Here is his explanation (p.79-80, my bold):

In an effort to create an illusion of clarity before embarking on the typical cascade of propositions and theorems, a set of definitions is provided so that statements and their proofs can be made as succinct as possible. On the surface this seems fairly innocuous; why not make some abbreviations so that things can be said more economically? The problem is that definitions matter. They come from aesthetic decisions about what distinctions you as an artist consider important. And they are

problem generated.To make a definition is to highlight and call attention to a feature or structural property.Historically this comes out of working on a problem, not as a prelude to it.The point is you don’t start with definitions, you start with problems. Nobody ever had an idea of a number being “irrational” until Pythagoras attempted to measure the diagonal of a square and discovered that it could not be represented as a fraction.

Definitions make sense when a point is reached in your argument which makes the distinction necessary. To make definitions without motivation is more likely tocauseconfusion.

“Starting with problems” is probably how most professional mathematicians work, and seems to be the natural way for mathematics to develop. However, as a student I find definitions to be really important, and I rely a lot on them. Most of my professors seemed to think so too, and made us memorize definitions verbatim. Almost every math textbook I had in college follow this format:

- Definitions
- Theorem
- Proof
- [Repeat]

So even though I understand Lockhart’s argument conceptually, I cannot say that I truly understand the practical implication of his suggestion. Maybe he only means to attack definitions that are not necessary (as in the following quote found on p.58)? If so, then I would understand.

No mathematician in the world would bother making these senseless distinctions: 2 1/2 is a “mixed number,” while 5/2 is an “improper fraction.” They are

equal, for crying out loud. They are the exact same numbers, and have the exact same properties. Who uses such words outside of fourth grade?

That last sentence made me lol.

Or perhaps he is only complaining about the “timing” for introducing definitions? Maybe he is just saying that definitions should not be introduced at the very beginning, but only after students have struggled with problems themselves and have an initial understanding of the problem’s structure (much like how techniques should be introduced to students)?

**Ending Remarks**

Finally, I just want to say that I love all the little proofs in the book. They are a perfect demonstration of the exquisite elegance and the miraculous nature of mathematics.

I agree with Keith Devlin in his forward to the book (p.11-12):

In my view, this book … should be obligatory reading for anyone going into mathematics education, for every parent of a school-aged child, and for any school or government official with responsibilities toward mathematics teaching.

I don’t belong to any of these groups of people, but I still enjoyed the book. So I think if you are reading this, you will too!

You mentioned confusion on whether to start with problems or with definitions. I think a good example of people starting with problems are educators who look subscribe to the Realistic Mathematics Education movement. Sean Larsen is one such education researcher. Here’s the citation for an example of how such an approach looks in education:

Larsen, S. P. (2013). A local instructional theory for the guided reinvention of the group and isomorphism concepts. The Journal of Mathematical Behavior, 32(4), 712–725. https://doi.org/10.1016/j.jmathb.2013.04.006

Thank you so much for sharing! I had never heard of the RME movement before. Their approach sounds pretty revolutionary. The problem with “starting with the definitions”, as I understand now, is that it is a method that provides students with “ready-made” mathematics, and does not require as much effort from the students. As a result, students don’t learn as much as if they had started with the problems instead (= “guided reinvention” in RME terminology, as I understand).

I found this explanation from the paper you cited to be helpful in clarifying the distinction: “The concept is considered a model-of when an expert observer can describe the students’ activity in terms of the concept. The concept is considered to be a model-for when students can use the concept to support their reasoning in a new situation. In this way, the transition from model-of to model-for can be seen as a transition to more general mathematical activity.”

Thank you again for sharing!

You also mentioned systems thinking. For me, that calls to mind Piaget’s book called structuralism and how his ideas have been used in mathematics education. Your prior post on analogy in particular also calls to mind the notion of ‘reflective abstraction’ and how Piagetians (in math ed.) have focused on building models for mathematical understanding (such as L. Steffe). Just some sources if you are interested

Very nice connections and I am interested in exploring Piaget’s idea more in depth. Thank you for sharing! Btw I found this chapter of a book by Ed Dubinsky when googling “reflective abstraction” that seems to be relevant here: http://www.math.wisc.edu/~wilson/Courses/Math903/ReflectiveAbstraction.pdf

Thanks again! So happy to see comments 😀

I would mention though that the RME movement and (separately) those who build off Piaget don’t really ascribe to a Platonist conception of mathematics as you do. If there is a real mathematics ‘out there’ in the real world, they aren’t concerned with it. Their primary focus is on student’s reconstruction of mathematical concepts as a human activity/endeavor.

Yeah, I noticed that a little bit too. Thanks for pointing that out. But I think specifying the ontological place of mathematics is not really a prerequisite for finding out the best way to teach mathematics, so maybe it is not contradictory to be a Platonist and a supporter of the RME movement at the same time 🙂