Math is not linear

but as soon as your curiosity says it's time to go....

So why do we teach math

in hierarchical steps?

• Arithmetic
• Algebra
• Geometry
• More Algebra?
• Precalculus
• Calculus

It seems that to most people, my earlier statement...

A background of certain skills can be helpful before you set out exploring

...is a radical understatement.

Rather, the common opinion holds that mastery of one subject is a prerequisite for dabbling in another, and that, since mathematics is naturally ordered like a ladder, it is only correct for students to take one rung at a time.

Wrong!

Simpilico: ...You need to walk before you can run.

Salvati: No, you need something to run towards!

-From "Lockhart's Lament" by Paul Lockhart

Our students spend much time studying skills simply because they will need to use them in a later class. How motivating is it to say to a ninth grader that they need to learn about rational expressions because someone may ask her about them in two years? Has that ever been a good reason to learn anything?

How much more motivating would it be to give a student a problem that required rational expressions, wait for her to discover that she needed more technique, and then introduce the topic.

Students deserve a better reason to learn something than "this is the next chapter in the textbook".

Contrary to popular belief, high schoolers are generally capable of taking in a bit of topology, or set theory, or real analysis here and there.

And they like it.

Some students would benefit from a slower pace, or a different route through mathematics, but parents don't want their child to "fall behind".

Seriously, it's not the end all and be all of your mathematics education that you get to calculus by 12th grade.

I also worry about the prodigious young children who fly through the standard curriculum, without taking any side trips, and then get mighty smug about it.

Because our curriculum is so partitioned, students often don't see the connections between topics and miss out on many helpful metaphors.

It's easier to understand things when you can mentally "zoom out"

I believe that one of the things high school graduates should learn, in all the time they spend on mathematics, is what it means to be a mathematician. It's not particularly useful information, but I'm sick and tired of explaining to people I meet on the bus that I don't just sit around all day adding bigger and bigger numbers.

Most teachers do this anyway, to some extent. Embrace it! Don't be afraid to throw something out there that you think is way over their heads. You can be surprised.

One of my favorite classes was the day my geometry class dragged me into talking about infintely small differences and a student asked "Is there a number smaller than every other number, but bigger than zero?" The next twenty minutes were spent debating the properties of such a number and introducing hyper-real numbers.

To ninth graders.

Yet, I think this was a very natural question, and the student who asked it was no prodigy. The class didn't all understand it, but even those who were mystified felt they had just been let in on a secret: that there was more to math than they had thought. A few students walked out with formal logical symbolism written on their arms as though it were witchcraft.

It's quite common that you need students to do some problem that could be more easily accomplished by methods they haven't learned yet. Let them know! You don't have to teach them the more advanced method, but it will give them a sense of what their future classes will actually be about, and why someone would want to learn them (to save them from the drudgery of their current methods)!

For instance, try to have students find the area of curvy objects. In time, they'll probably try to fill the areas with polygons that are easier to deal with. At this moment, it's fun to mention that Calculus does exactly the same thing, only with a formula that makes it practically automatic.

Yeah, I've been known to taunt my classes with "Haha, too bad you guys don't know Calculus!"

Okay, wow, that sounds seriously sadistic, now that I've typed it out, but the point is, they got an idea of what Calculus is, and how it fits into what they already know, and they didn't have to wait until they were in Calculus class to get that.

The opposite of forshadowing. This is another necessary part of teaching; obviously we rely on our students' previously learned material (or what they can remember of it). You can help students see the relations within mathematics by pointing this out a bit.

Here's a cool question:

Find a rectangle (with integer sides) whose area and perimeter are the same.

Now find another one.

Now find another one.

What, you're stumped? Well, maybe there AREN'T any more. How could you prove that? This is one of my favorite problems for relating algebra and geometry, and I keep learning new ways to approach it!

As per the previous rational expressions example, try not to give more technique than students are asking for. Give them problems that require technique they don't have yet, and make the squirm a bit before helping out. The educational benefits of this go beyond a broader understanding of mathematics, but the reason I'm advocating this practice here is that it demonstrates to students that the structure of mathematics is problem-based and natural, rather than imposed from above.

To be clear, I am not advocating that students get to choose what they study any more than I would let five year olds choose what they eat. You still direct the class, but when possible, do it from behind the scenes by providing strategic problems. Math itself often makes it obvious what questions should be asked next. If you've just introduced exponents, what's the opposite of an exponent? Once you've got roots on your hands, what happens if you get a negative number inside of one? Students may make your job easier by asking these questions themselves.

If this is at all possible (and I know it rarely is) I think this is the best way to get students to take seriously the idea that there is not just one path through math, simply because it actually offers them an alternative. Here are some great math elective courses I wish were more common:

• Discrete mathematics
• Formal logic
• Set Theory
• Number theory
• Statistics and/or Probability
• History of Math
• Great proofs of mathematics
• How to make sense of the numbers you hear in the news
• Fermi problems

If you don't have time to teach an elective (who does?) you can ask your school to hire another math teacher who can. And if/when they say "no", here are some more alternatives:

• Start a math circle or club.
• Make a poster illustrating a cool mathematical question and put it up where everyone can see it.
• Set up a math library where students can read about these things on their own. (Okay, even I know this one is far-fetched.)

I know that there are much larger problems in math education than this.

And that most teachers don't have the time or the patience left to take on another crusade.

But if you enjoy math, there is a very simple thing you can do which would help.

because

Math is not linear

And whoever sits next to me on the bus is going to understand that.

Talk about math.

Talk about anything you like about math. Talk about it anywhere, with anyone.

Tell them why it's great. Tell them how you think. Tell them how it surprises you, how it charms you, and how it all fits together.

Linear Algebra

Topology

Fractal Geometry

Non-Euclidean Geometry

Group Theory

Geometry

Proof Theory

Algebra

Set Theory

Analytic Geometry

Synthetic Geometry

Mathematical

Logic

There are so many directions

you can choose!

(These are just a few)

Number Theory

Why, oh why, is Geometry sandwhiched between algebra classes??

Trigonometry gets covered somewhere...

Combinatorics

What the heck is "precalculus" anyway? It's not a natural subset of mathematics.

Calculus

Probability

Real Analysis

Differential Equations

Statistics

Complex Analysis

This is just so....

Sigh....

Go on tangents

It's not

motivating

It prevents

students from being exposed to topics

they might enjoy

It spreads

misunderstanding

about what

mathematics is

Teach an elective

Foreshadow

Yeah, I guess this is a selfish one.

But I bet you're sick of it, too.

Yeah, I agree, but

what can I do?

But why does

it matter that math is taught in this order?

It fosters anxiety

by turning mathematics

into a race

It hinders understanding by

obscuring the big picture

Be less helpful! Let students tell you what they need to know

Relate material back to previous classes

You know what I mean?