Math is coming under attack recently at education conferences, in the literature, and in conversations about school reform. Fully half of the subject-specific workshops at a recent California Association of Independent Schools conference were about math. Relevance, engagement, projects – all of the presentations in one way or another sought to justify the teaching of math at the high school level. In question is whether high school students really need all the math they get in the typical high school curriculum. Is trigonometry useful for the average high school student? Maybe. What about calculus? That one is a little harder, testing even dyed-in-the-wool math teachers, most of whom concede eventually that calculus in high school is more about the challenge than its usefulness to life.

How much math is enough math and what is its value beyond a facility with numbers and quantities? If you had the resources to offer either a calculus class or a statistics class but not both, how would you choose? And just under the surface is a deeper question, is a subject’s connection to the real world, what we are calling *real-world application,* the highest value in school curriculum?

It seems it is. Or is becoming so. There are strong calls lately for a new three R’s – Rigor, Relevance, and Relationships. There are high school programs emerging all over that incorporate job skills, internships, and real-world learning. Linked Learning, which has its origins in what used to be known as Vocational Training (i.e.: shop class), is a wonderful and long overdue evolution of Career Technical Education. Linked Learning is a comprehensive, pathways-based approach to learning in the world that creates a meaningful and mutually educative partnership between school and career by involving students in internships and working with schools to create programming that supports their experiences. One of their mottos is, “Life comes to school.”

So relevance is playing a larger role in high school education. And that makes math teachers nervous.

For many educators, many of whom are now in positions to affect high school curricula, math in high school was an interminable series of puzzles that neither had nor claimed to have relevance in the world outside of math class. At best it was amusing. At worst, dehumanizing. But regardless of whether one liked it or not, it was unapologetically divested from the world outside.

“Solve for x,” we were told.

“X what? What is x?” we all thought and some of us asked.

“No, no,” we were told. “X isn’t something. It isn’t anything. Its a quantity of whatever.”

And I need to solve for it? Even though it isn’t anything, doesn’t intend to be anything, has no basis in life or reality? We aren’t talking about apples or oranges, not slices of pie or distances between stars. We are not talking about anything identifiable or recognizable. It is a pure abstraction. Do I have that right?

For many that is a hard truth. Add to that the fact that for most students the advanced topics in math will never be used and you can see the argument for math’s irrelevance in high school.

What do math teachers say? I recently sat with a great group of inspired and inspiring colleagues, most of them math teachers, in a workshop that touched on these topics. What is mathematics? What is mathematical thinking? What is the argument for math in high school? The workshop put us into the shoes of students and walked us through a lesson designed to help us identify and recognize mathematical thinking in ourselves and our students. The math challenge – or puzzle – was to figure out how many squares can be made with twelve straight lines. And from there we went on to wonder and explore what we could understand about lines and squares. Was there a relationship between the number of lines and the number of squares? Could we generate an equation?

It was an engaging lesson, no question. We enjoyed the challenge and it felt fun to do the math, work creatively in a small group of interested folks, and feel like we were figuring something out. And when the question came up, “So, beyond the fun, what is the application for this in the world outside of math class?” things got a little dicey.

“Why isn’t fun enough of a justification?” was one response.

“Not everything in math has a real-world application,” came another.

“They are learning how to think.”

There was some genuine defensiveness in the room. Also a great deal of compassion. And I did feel some pity for the poor, recovering, misunderstanding English teacher for whom the deeper relevance of finding squares with lines remained elusive. But we had been encouraged at this conference to “go hard on the content, easy on the people” so I persisted in spite of my better judgement and strong internal messaging that to back away slowly.

In the end, the workshop ended and I had the familiar feeling of being pitied for my ignorance. As I began to scratch some feedback for the presenter on the back of a handout, I felt one of my table mates lean in. I stopped writing and scanned my peripheral vision. What was about to happen? My eyes wandered up and met hers. She was grinning at me.

What ensued was a wonderful and very helpful 45-minute conversation that went from real world utility of math to conferences to education to what we believe most about students and learning – and finally to three of the biggest concepts out there: truth, beauty, and goodness – three things that need no further justification.

The study of math, she proposed, has a grander relevance than simply utility. We do math, she said, like we play piano. Because it is beautiful. Not because it is useful. A simple, elegant solution to a problem is beautiful. A relationship between numbers that proves true in all cases is beautiful. Math, she said, is much more like art than tooth brushing.

And truly, for the first time, I got it. Beauty, truth, and goodness – these are things I get and I look for them in all worthy endeavors.

So, what I understand now is that math does have a real-world relevance – it is beauty (and truth, maybe not goodness). Beauty is relevant, in and of itself. Like art. And that, I think, is that much stronger argument in favor of math than the somethings-just-don’t-connect line. Math teachers should stop saying that fun is enough of a justification for math. It isn’t. There is no other subject in high school or education at any level that is comfortable relying solely on fun as its reason for being. Fun is wonderful – anyone who knows me knows that I go for fun at the expense of productivity too often. But it is not enough to justify a high school curriculum. And math teachers should stop saying that not all math needs a real world application. It does. To spend time doing it, it has to have relevance. No question.

The challenge then is to recast math as relevant in grander and more subtle ways. It is relevant not because you use it all the time. Most people don’t. Math is relevant because it is elegant truth. Math is beautiful.

tcuknappIntriguing thoughts on how math is and should be viewed. I completely agree that an argument that math does not always need a real world application is very weak. Truthfully, every math equation or problem does have a real world application, but many of them only apply in very technical and specialized career fields, of which most students will not be involved. In such cases the individual relevance for a particular student decreases. Most students in high school do not have a full grasp on the exact career they will have or even want to have anyways. In some cases it is about exposing them to something they may not have chosen on their own. What they learn in a particular class could excite them to pursue a particular career or learning path they otherwise never would have considered.

However, there must be a balance between immediately relevant curriculum and intentional exposure with the goal of inspiration. If we only teach the basics of everything in an effort to teach just what students will need to know for everyday life, we risk losing the creative thinking and inspiration that drives students to explore a new field. But if we delve too far into exposure classes just for the sake of challenge and attempting to inspire, we may not be providing the majority of students with a solid foundation in practical, everyday knowledge.

I can agree with the sentiment that a simple and elegant truth discovered through math has an element of beauty. I can even understand the comparison of Math to art. However, from a curriculum standpoint, art is typically an elective course. So are we suggesting that math is beautiful and because of it’s level of beauty and intrinsic value, it needs to be a core curriculum subject everyone must learn? It seems this line of logic would end up offering math as an optional elective class in high school.

Again, I don’t disagree with the ideas presented, but there must be a balance and I’m afraid too many people are trying to pull the scales one way or the other instead of recognizing the necessity of both.

always strive higher

Peter PoutiatineYes. I am with you on the balance idea. And you nailed the big issue with likening math to art. Art tends to be the first thing cut from tight budgets. Given the decisions educational bureaucracies make beauty as a justification for math is cold comfort. It is kind of the best we can do, though, right? I mean, what other reason is there for most of out math curriculum?

And the really dangerous question that I find many educators simply unwilling to entertain: Can we we teach the beauty of math and the mathematical thinking modes through real life projects and authentic activities that do have real relevance? Would that not be better for students?

Thanks for your thoughts tcuknapp.

BeauYou will not be surprised to know that I love your entry, Peter.

I wonder if the topic of creating and re-enforcing neural pathways came up in your discussions? I know I have experienced a mental second wind when thinking about math topics for an extended time. I felt like I was making my physical brain stronger, just like my own grade school math teachers hinted might happen if I persevered.

I have the great opportunity now of working with A student who suffered injury to his brain. As he heals, we hope he will find new ways to process information through his work with algebra. He relearned to walk and talk. Can he relearn how to complete multistep mental procedures even when he has limited short term memory? Can he rewire his circuitry through math? And here I mean, math that is not beyond his ability. If we keep him at a comfortable level, say 70% of his ability, I believe that this will be 70% of a larger and larger level of ability. Next month’s 70% may be today’s 90%. So far we are seeing progress.

Also, did you discuss valuing experiences where students get so involved that they lose track of time and surroundings? I have long thought that such experiences inform them about how they might want to spend their future working hours. It is a beautiful thing to get so enmeshed in a topic that the whole world drops away. By offering high school students a variety of topics where they can identify their areas of interest, passion, and talent we set them up to go on to study topics they care about and love. We need to be sure they have the opportunity to find the topics that grab them. Perhaps this results from reading literature, writing fiction and nonfiction, figuring our how to design an vehicle in a physics project, learning about the Russian revolution, or even thinking about how to find the volume of an oddly shaped figure in a calculus class.

I remember coming home from my own high school calculus class one day when I perceived a drinking glass in a whole new way. I had learned to see it as a vessel resulting from the revolution of a two dimensional region about its axis is symmetry. That concept fascinated me. Did I use it to figure out the volume of the drinking glass? Not at all. I could have. Did it add a little magic and pleasure to my life? Yes, it did.

I will stop here. Thank you for writing your blog entry.

BrianPeter,

As usual, I love your thinking–and the sass of your question. I’m an English major and teacher who loves math–chose lit because it was more challenging than math–so the beauty argument works on me as well. I’ve been engaged in these questions for a long time and my answer starts with thinking of math the way I think of other languages.

I’m sure you’ve had fresh young 8th graders ask why they should take English class when they already speak it “perfectly good”? My answer is that 8th grade fluency is fine for USA Today, but it’s well short of the skill level needed to grasp and wrestle with the kind of issues adults really care about. Isn’t it my responsibility to keep them moving toward the kind of fluency that someone like you demonstrates–the skill to deal with big questions without language being a barrier in and of itself?

Well, in the world of mathematics, calculus is considered basic fluency–a good foundation, but not the skill you’d need for an important conversation. Our whole culture is essentially math illiterate (and I heard that in Washington half of Ed majors require remedial math in college, so it’s our whole school culture too). The “when will I ever use this” question feels to me like learning the difference between “good” and “well”; just because we don’t see it done right very often doesn’t make it irrelevant. In fact, it makes it more relevant.

For me, the bottom line is that the goal of ( public) education is to create the kind of literate citizens required to own and operate a democracy. Math is one form of literacy we could do a lot better at.

Art too, despite what our budgets say, but that’s a different post.