Why mathematics? What purpose does it serve? Why should we teach it to students? There tend to be roughly three broad categories of answers to this question:

- It is useful. Math is needed both in daily life and in work, and without a strong math education you are closing doors to excellent careers. In this vision, the math curriculum should be designed to teach specific knowledge and skills.
- It trains you to think. Math poses deep and abstract problems. Facility in solving these problems strengthens your critical-thinking skills across the board. In this vision, the math curriculum should be designed to build abstraction and pose challenging problems. (Warning: there may be some research that says that critical thinking in mathematics does not transfer to other disciplines.)
- It is pretty. Math is gorgeous; it should be explored and enjoyed. It is one of the great achievements of humankind, and every student should have the opportunity to appreciate and understand it.

I believe deeply in all three. It is dangerous to ignore any of them. If you design a class that explores the beauty of pure and applied mathematics, builds abstract thinking, and poses deep problems, you can easily meet all three criteria together. But when you ignore one of them, mathematics suffers.

It’s for this reason that I am not nearly as enthralled by Lockhart’s Lament as many of my fellow mathematicians. Lockhart’s Lament is a powerful and eloquent description of why mathematics is *beautiful*. I couldn’t agree more with all of the points Lockhart makes for the beauty of mathematics. However, when he proposes to scale all teaching towards the exclusive goal of beauty, I believe the result is both dangerous and unrealistic. (Lockhart later wrote a clarification that he did not mean to be making a proposal nor to imply a dichotomy between pure and applied mathematics; although he walks back from what the Lament says, it fairly clearly speaks for itself and it is what is being passed around between mathematicians.)

Teaching only for beauty is dangerous because without encountering specific content, students would not learn mathematics that is important for their lives. Those who will not pursue math-oriented careers would not have the tools to manage their lives or interpret the world around them, while those who will pursue math-oriented careers would find themselves without the necessary tools. Besides, not everyone finds mathematics beautiful for the same reasons. Some find it beautiful for its applications and relation to the physical world. Some might not find mathematics beautiful at all, and I do not consider that a failing. Just as I do not particularly enjoy opera and do not want its appreciation forced upon me, I would not want to force appreciating mathematics on others.

Teaching only for beauty is unrealistic because we simply do not have the teaching force that is capable of teaching mathematics purely for its beauty (as Lockhart notes). Moreover, even if teachers *were* in a position to do this, if each teacher designed their own curriculum with no guidance then classes would be filled with students with very different background knowledge and designing coherent lessons would be impossible.

Paul Lockhart teaches at Saint Ann’s School, a private school in New York City whose tuition ranges to over $30,000/year. It has a free-form curriculum focused on the arts (and with many children of artists attending). It is a wonderful school, ideal for Lockhart’s self-described teaching style, and I have known several teachers from there who are quite remarkable people. However, to take that experience and generalize it to all of K-12 education is neither wise nor truthful.

In the end, I believe that it is possible to construct a school mathematics curriculum that combines all of points (1)-(3) above. We should not focus on one at the expense of another; we should instead see the wonderful ways that they can all work together. Lockhart’s essay is a wonderful demonstration of the power of mathematical beauty in teaching. It is now time for the discussion among mathematicians to move beyond this very satisfying lament and on to incorporating *all* aspects of mathematics.

In his follow-up/reply to critics, Lockhart writes, “My point is that at present we have neither Romance nor Practicality – nothing but a jumbled, distorted mishmash of pseudo-mathematical vocabulary, symbols, and mindless procedures.” With this I completely agree. With his proposed solutions and measures of success—well, I’m not there yet.

Great post. One thought is that our ancestors began teaching philosophy (“love of knowledge”) and then this single subject began a process of subdivision as our knowledge became broader. At the time of Newton, “natural philosophy” was a separate discipline but it still allowed him to dabble in optics, physics and mathematics. Since then we have sliced and diced the pursuit of knowledge into narrower fields.

So today we have “mathematics” which is treated as a single sequence separated from other problem solving disciplines like computer science, physics or economics…but also trying to serve the distinct purposes that you outline well.

To what degree are we just victims of path dependency? Could we trace back to philosophy / natural philosophy and think about a breakdown of disciplines that are more effective and appropriate to the 21st Century? Instead of a traditional curriculum of 1 math class, 1 english class, 1 history class, 1 art class etc., could we not reorganize around:

* Practical life skills – mathematical and other skills needed to be successful in our society

* Abstract problem solving – mathematical and other approaches to solving problems (IMO starting with a grounding in epistemology)

* Aesthetics – training our children to appreciate and create beautiful things which will require a mathematical understanding

The ratio of these three “classes” might change based on the inclinations and talents of a particular student, but some minimum threshold of performance would be expected.

FWIW practical life is one of the strands if the Montessori curriculum, at least for younger kids.