One of the challenges that I see with students learning mathematics is their confusion with what qualifies as the content of mathematics and the language of mathematics.

In a famous and enduring article, “Relational Understanding and Instrumental Understanding”, Richard Skemp pointed out that teaching concepts instead of procedures will be difficult if students think that mathematics is about procedures. A key point was that, in order to teach something, there needs to be an agreement about what that something even is.

I fear the rift between the mathematical education theorist and the high school student might be even broader than this. One of the current key concepts offered in mathematics teacher preparation programs is the five strands of mathematical proficiency. These are:

- Conceptual understanding
- Procedural fluency
- Strategic competence
- Adaptive reasoning
- Productive disposition

Which of these are involved in “Solve \(x^2 + 3x – 5 = 0\)”?

It would seem that this involves procedural fluency, and the article mentions that the first two of these strands are “often seen as competing for attention in school mathematics. But pitting skill against understanding creates a false dichotomy” (122). Even so, four of the five standards focus on understanding what a mathematics problem means, and only one seems to require applying the language of mathematics exclusively.

I say “seems to”, though, because that doesn’t really require it, either. Let’s look at the Pythagorean Theorem (PT). A conceptual understanding of PT might involve understanding that we can make three squares, aligned with each side of a triangle. If the area of the lesser two squares equals that of the third, the triangle is a right triangle, and vice versa. A procedural understanding of PT might involve understanding that, given three lengths, \(l_1\), \(l_2\), and \(h\), where \(l_1\le h\) and \(l_2\le h\): \(l_1^2 + l_2^2 = h^2 \Leftrightarrow \{l_1, l_2, h\}\) form a right triangle.

A student who understands the latter fact can be fairly said to have a procedural understanding of PT, especially if they can apply this understanding to find, for instance, that \(l_1 = \sqrt{h^2 – l_2^2}\). Such a student clearly has procedural fluency. Such a student has yet to have written \(a^2 + b^2 = c^2\).

I find that most of my students come into my geometry class thinking that the Pythagorean Theorem is \(a^2 + b^2 = c^2\). It’s not that they lack the conceptual understanding, although they generally do. It’s that they also lack the procedural understanding that variable symbols are ultimately arbitrary. They appear to thnk that there’s something immutable about the symbols being used. This gets in the way of even procedural fluency: If PT is \(a^2 + b^2 = c^2\), a fact that simply must inscribed in the stone beneath the bust of Pythagoras somewhere in Athens, then it cannot be \(a = \sqrt{c^2 – b^2}\), let alone \(l_1 = \sqrt{h^2 – l_2^2}\), \((x – h)^2 + (y – k)^2 = r^2\), or \(d = \sqrt{(x_2 – x_1)^2 + (y_2 – y_1)^2}\).

Jo Boaler continues to lead the charge to move away from a focus on specific notation towards conceptual understanding. And while I’m supportive of that view, notation is important. Concepts are how mathematicians attack problems; notation is how mathematicians record their thoughts. Recording our thoughts in a tangible form is important for two reasons. First, it puts less stress on our memory. We have limited mental bandwidth, but if we write things down, we can release that memory for other purposes.

Secondly, and I think more importantly, physical notation allows us to communicate our thoughts to others. It’s how mathematicians share ideas.

In “Great Moments in Mathematics Before 1650″, Howard Eves promotes teaching geometry by starting with common sense reflection (“subconscious”), proceeding to experimentation and conjecture (“scientific”), and then finally formal proof (“demonstrative”) (21). Only the last of these requires formal notation, so it makes sense that we at least minimize notation until after we’ve explored the concepts. At the same time, though, adept mathematicians need to be able to “speak math”.

Teaching notation before concept is putting the cart before the horse. Hand-waving notation to a corner is getting rid of the cart entirely.

I think the overall effect has been to confuse students about what it is they’re supposed to learn in mathematics class. And when there’s confusion, students will employ Occam’s Razor and focus on the simple stuff. In this case, that’s memorizing formulas in fixed notation as if they’re some sort of impenetrable Harry Potter incantations.

Here’s a fairly typical mathematical explanation. It’s the article the inspired me to write this entry, but it’s not egregious at all. It’s the way that mathematics writers tend to write new concepts, thinking that this will be accessible to the student. On the one hand, it’s useful for serious students to see how mathematicians process their thoughts. On the other hand, though, mathematicians writing as mathematicians think for non-mathematicians isn’t likely to be successful overall.

Many famous musicians can’t read sheet music. And that’s okay. Sheet music is one way that melodies get communicated between musicians, but if you’re only playing songs that you’ve either made up or heard directly, you may well not need it. Prior to the invention of the gramophone, musicians would had to have either memorized all their melodies or recorded them in some fashion, although even then, an individualized method would have done the job. The symbols on sheet music are fairly arbitrary: Other symbols would do the job.

Likewise, oral story-tellers don’t need to be literate. Written texts are for recording information in a set form and for communicating it to people who can’t be present physically. We as a society have bestowed a level of formal significance to written texts that they don’t deserve: The text is not the tale, it is a codification of the tale.

Writing things down means we don’t have to remember them. It also allows us to reflect visually on things we’ve written, and it allows us to communicate more easily with each other. Mathematics can be done without writing anything down, but it gets progressively more challenging as puzzles get more complex. There are people who can play chess, and play chess well, without having access to a board, but it’s very challenging.

However, the ability to write and read sheet music doesn’t make you an adept musician. It might be considered a key for most people learning how to perform, but it’s not needed. If Paul McCartney, Eric Clapton, and others don’t need to know how to read sheet music, it’s clearly possible to succeed as a musician without being able to do so.

To summarize: Mathematics is not its notation. The Pythagorean Theorem is not \(a^2 + b^2 = c^2\): That is the standard notation for the Pythagorean Theorem. When we teach \(a^2 + b^2 = c^2\), we’re teaching something important: How mathematicians keep notes, and how they communicate with each other. A challenge of effective mathematics education is to teach the notation while also teaching that mathematics is not its notation.