Thoughts On “A Mathematician’s Lament”

In Defense of Math, but Not Those Who Teach It

What is Paul Lockhart Lamenting?

Dr. Lockhart is lamenting two issues: 1. the essence of math is widely misrepresented and misunderstood and 2. the education system is terrible at teaching math. Much of the book addresses the first issue by explaining what math is really about. He does this very well by exploring “Mathematical Reality” with the reader and sharing some of his favorite problems, which highlight how math is something to play with and not something to be passively learned.

As for the second issue of how poorly the education system teaches math, Dr. Lockhart is not so helpful. He denigrates everyone involved with teaching math without seeming to understand the crux of the problem. He refers to math teachers as “little devils” who are just doing the best that they can. As one of those “little devils” myself, his sweeping generalizations about math education are hard to stomach because – like most generalities about society – they are not always right.

The open enrollment high school I teach math at uses the Interactive Mathematics Program (IMP), which is complete with mathematical play just as Lockhart describes. Furthermore, it defies Lockhart’s “ladder myth”. IMP was first published 10 years before “A Mathematician’s Lament”. In the end, “A Mathematician’s Lament” comes off as more of a therapy session for Paul than an earnest effort to make a difference in math education.

I Get Why, But Not How

In the book, Dr. Lockhart mentions that the problem with math education is that it focuses more on the what of math and not the why. Math education does not need as much help with the why as he thinks. Take a look at the current national standards, which list several mathematical practices mentioned in the book. These include problem solving, persevering, looking for and expressing regularity in repeated reasoning, and even constructing arguments and critiquing them. These were launched in 2009, which was the same year “A Mathematician’s Lament” was published. Lockhart and those who set the curriculum standards he despises are in agreement with regards to why math is important. What is missing from the standards, the book, and everywhere else is how to do it in the classroom.

In his book, Dr. Lockhart presents the problem of the shortest path between two points that touches a line and then writes, “Were you my student (and assuming the problem interested you) I would simply say, ‘Have fun. Keep me posted.'” The math teacher, he recommends, should do nothing. I will assume by nothing he means do not try and dictate the path the student chooses as they work the problem. There is clearly something that math teachers do or what’s the difference between a math teacher and a piece of paper that has the problem written on it? If he knows the answer to this he chooses not to discuss it.

He also mentions in one sentence that his class spent several days to get a result after a long sequence of failures. I cannot help but wonder how a class is led through that. Learning to accept the failures, reflect, and then try something different is why math is so important to spend time with and it is what students struggle with the most. Students tend to give up and avoid, complain that the teacher isn’t teaching, or they end up attacking themselves saying they just don’t get it and never will. I wonder how this is overcome in his classroom. This is exactly what math teachers need to improve on and if he has some advice, he leaves it out completely.

As a final example of Lockhart glossing over the crux of the problem for math educators, he writes, “So, let’s imagine you’ve been playing with this problem for a while…” Are we to assume that student’s just naturally know how to play with a problem? If they do not, how do we teach it? How do we entice them to try? What do we do when they fail? How do we get them to reflect on what they’ve done and most importantly, get them to try again without feeling too much frustration or shame? This is what math teachers need to be working on and he does not even acknowledge that this is a challenge, let alone provide any advice.

Maybe I’m just missing the whole point. Maybe I’m part of the problem and I just I can’t really wrap my head around math class being more like art class. Maybe if society saw the beauty in the works of famous mathematicians like they see it in the works of famous painters and writers and musicians, students would come to class already knowing how to play with math and would find the curious things in “Mathematical Reality” more appealing. Then, students might bring to math class the same expectations that they bring to music and art class – let’s play! Maybe! But, it is very far from the world we live in today.

The Crux of the Problem

In the world we live in today, math students feel the need to get right answers and when they do not, they feel wrong. From the beginning of a problem – before they even lay eyes on it – students need to feel like they are making progress along the way towards the correct answer. Anything else leads to frustration, shame, and negativity that is difficult to overcome. Overcoming this negativity has very little to do with the teacher’s math ability and how many proofs they do in their spare time. The reason math teachers do not teach more mathematical play is because they want to have a high level of order in their classroom. Co-workers and principals expect that classrooms are well managed. The crux of the problem is how to maintain order while teaching mathematical play.

Order in the math classroom can easily be achieved by completely perverting the way math is taught, hence Lockhart’s and many other mathematicians’ laments about math in our public schools. Worksheets, tests, notes, and lectures lead to a quiet and orderly classroom. Students are trained to know what to do in this environment and principals and co-workers who really have no idea what math is, walk by and are impressed with the “classroom management”. 

If math teachers had confidence that they could maintain an orderly classroom while each student was playing with a math problem, then teachers would devote more time to it. Also, if teachers were supported by co-workers and principals that a bit more dis-order in the math classroom was OK, the teachers might be more willing to try mathematical play. Let’s not be mistaken that teachers are afraid to try new things. They are more than happy to as long as order is maintained. Order in the classroom is the highest priority.

Why Is An Orderly Classroom So Important?

Because it is very stressful having 25 teenage kids in a room without a clearly defined task! Even with clearly defined tasks, there are breakdowns all over the place for a multitude of reasons. Depending on the school, a class might have zero students who know how to play mathematically without explicit instructions on what that means. Those who do not are immediately in some version of “I don’t know what to do” and are on the verge of giving up if they have not already. This leads directly to sidebar conversations, off-task loudness, negativity, and discipline referrals. Repeated discipline referrals from the same teacher leads that teacher to gain a reputation for not having control of their classroom, which is a negative thing to be known for.

As I mentioned, I teach mathematical play at an open enrollment high school. It is an incredible amount of work to lead this and keep everyone engaged. Some days are complete failures and some days are amazing successes. What was an amazing success one day doesn’t work the next day or with the class I have the following hour. I’m young and handle the stress. I run around my classroom until I’m sweating trying to keep everyone engaged. I’m lucky that my principals do not expect math class to be quiet and they certainly don’t want to see me giving students too much help. They actually want the students coming to their offices complaining that Mr. Meyer doesn’t teach anything!

I’m lucky for this but it’s still incredibly stressful. I’ve had to deal with fights, shouting, cursing, crying, sleepers, avoiders, and so on when students lose their attention with the math (which is at least one student every day). It is so tempting to give the students long problem sets of stuff they already know how to do. It’s tempting not because it’s what I think is best for my students. It’s tempting because I think it’s what’s best for my sanity!

Making A Difference

If Lockhart wants to make a difference, he needs to acknowledge that the stress of disorderly conduct in the classroom is why mathematical play is not more widely used. Unlike him, we all can’t teach at a private high school that costs more than many colleges. Creating an environment in the math classroom where students are OK with being wrong and will not give up after failures is what math teachers need to do. It takes more than an appealing problem and “Have fun. Keep me posted.” What’s more, this is something that all the little devils – er, math teachers – can immediately start working towards with explicit examples of tools, dialogs, and methods, for leading a class full of students through mathematical play who have never done it before.

I’m not looking for examples of when it goes right, which are the only examples I am ever able to find. What do you say or do with a student who has given up? What do you say or do with students who are angry because they feel nothing is being taught or because they feel no progress is being made? How do you head off these inevitable issues on the first day of class so you are ready for them when they come up? The answers to these questions will truly make a difference.