# Teaching High-Level Math to Young Students

There’s been a lot of talk lately about fast-track systems, separating classes based on ability, and promoting earlier and earlier start times for higher concept learning. But some research has also shown that underestimating students means they may not be challenged.

We’ve all underestimated students at one point or another: It can be hard to believe in some students when they struggle to grasp many concepts. What I’ve found through experience, however, is that students can achieve a very high bar, if the teacher is brave enough to set it for them.

I taught at a small, individualized private school that provided some escape from the frenzy of fast tracking, with a math culture that allowed freedom in pacing and curriculum. No one had to specify their expectations for students—instead I figured this out myself and had the power to challenge them as I saw fit. But I’m aware that many readers find themselves in a more constrained teaching situation.

### Giving Middle Schoolers High School Math

One class I created for a junior high program was an elementary algebra course. The class included eighth graders along with two seventh graders and even a sixth grader. It covered much of what a typical student sees in high school: solving rational expressions; heavy emphasis on quadratics; graphing linear functions, complex functions, and polynomials; combinatorics; geometric proofs; and multistep equations, to name a few points on the curriculum.

I didn’t tell the students the material was high school level. What I discovered was that they naturally bounced ideas off of each other and connected what they had previously learned within the course to build a knowledge base that was comfortable to each of them personally—this engagement is crucial.

A lot of times I introduced an idea and gave extremely difficult problems but didn’t instruct deeply. Instead, I anticipated that the examples and problems the students worked through by themselves or with a partner would invite them to connect ideas into what made sense in their own mind.

For example, one middle school lesson about the properties of exponents involved providing one property, X^{A}X^{B}=X^{A+B}, and having students use that to methodically derive each following exponent property using only the ones they had previously proved. In one such case, a sixth grader noticed that performing X^{A}/X^{B} was the same as multiplying X^{A}X^{-B}, by definition of inverse (a sixth grader!). She then asked if we could use the first property of exponents to add a positive A power and a negative B power.

At first, I was shocked by the quickness to relate the properties to each other in such a short time (this was all a single one-hour class). Then I realized that I had never phrased the material as outside the realm of what the students were expected to achieve, so to them these concepts were just what typical middle school students should see—they weren’t aware it was high school material, and they were less prone to give up easily. I had been wondering if I was teaching this material a bit prematurely, but the students rose to the challenge.

Another example: Much of my students’ work with rational expressions came down to how much the students could simplify them, to show an increasing familiarity with the process so that down the road this was second nature and more advanced material could be taught without delaying on the “little things”—in fact, an entire week was dedicated to practicing this idea.

Take the basic example of (X-1)/X. The question I posed was whether this could still be simplified further. At first, a common (and incorrect) response was that the variables could cancel, leaving -1. My response was simply to ask for examples of this hypothesis succeeding (without accepting or denying their previous result). The students already knew that a few examples were not enough to prove a conjecture, and that just one situation where the example doesn’t work should be enough to disprove (we were close to discussing the concept of a counterexample).

Suddenly, a seventh grader realized that subtraction and division were not opposite operations, so they couldn’t cancel like they do when solving equations. This was a fantastic revelation, and everyone agreed.

### Encouraging Perseverance

These experiences were less about the above-grade level solutions, and more about exploring the concept of reasoning, something that John Holt often highlights in his works, particularly his book How Children Fail, which I cannot recommend enough for math teachers of all levels of experience.

I’m reminded of a story a fellow educator once told me. A teacher performed an experiment with two classes learning the same material. In one, she praised students for their work ethic leading to success. In the other, she placed value on being smart. At the end of the unit, the kids who were told to work hard to be successful were less likely to give up, and they got more out of the class.

My students were not aware that this material was considered above their capacity. Again, I’m aware that we can’t all teach in this way, but the material that does appear in the curriculum can be framed in encouraging ways. It all depends on how we anticipate students’ abilities and on not underestimating them.

Change it up, persevere, explore. Our kids will rise to the challenge.