Changing students' approach to math

Question

It's been quite a while since I was tutoring a high school student and even longer since not a gifted one.

However, this time, something was amiss. I have asked him to show me how he does some exercise, and then another and the only thing I wanted to do was to shout:

You are doing it wrong!

I hope I didn't let it show and tried to work it out in small steps. However, it was like he didn't wanted to learn, he just wanted to get the problem set done.

One mistake he would frequently do, is to mess the signs in trigonometric reductions (e.g. $\cos(90^\circ + x)$ would be $\sin x$ instead of $-\sin x$). In fact, he memorized all the formulas, but couldn't recall them properly. I've tried to teach him how to recover formulas from graphs of $\sin$ and $\cos$, but the response was along the lines of "I don't need to understand it" and "I would like to do it without thinking".

For example, there was a pair of tasks 1) prove that $X(\alpha) = Y(\alpha)$ and 2) calculate $X(30^\circ)$, where $Y$ was simple and $X$ was not. However, in the class they covered such calculations, so he evaluated the $X$ directly. Yet, pointing out the simpler way didn't changed his approach.

Another example could be an expression with non-round numbers, where I suggested to hide them behind $\alpha$ and its variations (esp. since such substitution made the necessary formulas and reductions easily visible). Nevertheless, he tirelessly rewrote them all the way through calculations and clung to their concreteness like to some kind of lifeline.

I am aware that an experienced tutor would do much better than I did, but that's not what I'm after. What shook me most strongly was this strange quality of his approach to mathematics that would produce a strong feeling of dissatisfaction (similar to the one you have when seeing something ugly). What I would like to ask about is:

How can I change such a student's approach to math?

It seems like a Catch-22. He sees math as horrendous, but won't learn otherwise, because he doesn't want to deal with it. On the other hand, when he does, it's so wrong that no wonder he sees math this way.

For a change I wanted to show him something engaging, e.g. how math allows us to build nice things using 3D animation software, but he would not make a connection. It was just a few cool pics useful only as long as they can be used to raise his status (e.g. his friends think them awesome).

I have no illusions on whether I could teach him to enjoy math. But, is it possible to make him not hurt mathematics? Right now, I admit defeat.

Answer 1

(This is a very, very long answer because I want to highlight both the likely mindset of such a pupil, and possible approaches to win them over. Also, please check Student Poisoned Experience with Math in case it may help you.)

Understanding the Mindset

Another example could be an expression with non-round numbers, where I suggested to hide them behind α and its variations (esp. since such substitution made the necessary formulas and reductions easily visible). Nevertheless, he tirelessly rewrote them all the way through calculations and clung to their concreteness like to some kind of lifeline.

You sound like you're teaching a younger me. I feel sorry for you.

This was exactly what I did. I would reduce the problem, not to its simplest abstract form from which a solution becomes more apparent, but to the form conceptually most recognisable to me, which would lead me around and often down a blind alley. And when you would say that there is an easier way to do it, and you'd even show it, I'd just ignore it because I didn't understand what you were saying, and I wanted to get there on my own steam, dammit.

In short, you have a pupil who is disenchanted, depressed, and close to belligerent.

I feel that, as tutors, we tend to make a mistake. We see someone take the long way around and we offer a short-cut. We approach the subject from the perspective of a master, with insights of simplicity and foresight. By imparting our knowledge, we're hoping something will stick, and that their performance will improve.

The perspective of our pupils, however, is that of a frog trying to cross a busy highway: at some point, they just make the plunge, race screaming and flailing for the other side, and hope not to get hit by a truck. They can't see the short-cut because they don't understand the short-cut, even when it's pointed out with neon signs.

And that's the really hardest part here, I think: debugging the misconceptions op your pupil. Stepping out of our frame of reference, and getting into theirs.

"I don't need to understand it" and "I would like to do it without thinking". ^(*)

How's that working out for you, partner?

Help him out with the rules by putting them on display. For trigonometry, show the unit circle, and put an example in each quadrant. It'll help him visualize and internalize the relations between the trigonometric functions. At some point, remove the unit circle--but let him draw it himself, if he so chooses.

It will also help him realise that these 'rules' are not axiomatic, independent laws, but properties and relations; tools rather than obstacles.

^(*) _{Objects follow the path of least physical resistance. People follow the path of least intellectual resistance.}

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Changing the Approach

How can I change such a student's approach to math?

Sneak the math onto him. Really. Seriously.

My brother had the same problem tutoring me in physics. I was horrible. I didn't get anything right. I hated it. I wanted to burn my course book after smothering it in the blood of a sacrificial goat.

We had been covering potential and kinetic energy, and then he decided we needed a better exercise. So we were going to calculate the horsepower of throwing a horse off the Niagara falls. Because the exercise was so far removed from what I personally experienced as the boring, horrible physics, I did not have the instant mindblock I would otherwise have ^(*). Suddenly, physics was a means to an end, and I had the (admittedly morbid) drive to find the answer.

A logic and philosophy professor at our university had this approach to teaching where he would start with anecdotes and slowly, sneakily slide into his courses. By the time students 'caught on' that he was back to teaching, he brought up another anecdote to reel them in.

Our probability and statistics professor would take topics from recent news for exercises and show either why the news report was highly misleading, or what the implications of something would be: error margins on polling, probabilities on game shows, casino games, poker, stuff like that.

^(*) _{We didn't actually throw a horse off a cliff. My brother wasn't too big on empirical physics.}

For a change I wanted to show him something engaging, e.g. how math allows us to build nice things using 3D animation software, but he would not make a connection. It was just a few cool pics useful only as long as they can be used to raise his status (e.g. his friends think them awesome).

3D animations can be cool and flashy, but they don't immediately, intuitively link back to math for a layperson.

Stick closer to home. Does he like video games? Maybe he played Call of Duty, and heard of the Coriolis effect. Or maybe he played Realm of the Mad God; that uses Voronoi diagrams for world generation. Maybe he's addicted to his cell- or smartphone; introduce satellite orbits or talk about GPS systems. Math itself is difficult to connect directly, so you may have to go through physics.

You're right: you can't make him go from loathe to love in an instant. Even now, I don't love mathematics, but I am no longer terrified of it, and I regularly use it as a powerful tool of abstraction.

Answer 2

1. JvR's answer is very good.
2. I'd like to point out a missing aspect:

Emotion

Your student has developped highly negative emotions concerning math. He has lost motivation and is in despair concerning math. He feels, he'll never understand math also he knows, that it's important. He does not understand, what he's doing at the moment, but he also doesn't want to do something new because of two reasons:

1. Learning something new has the risk of forgetting or loosing the little he thinks he knows or understand. (That's wrong, but can't see it in the right way.)
2. Trying to learn something new has the risk of failure. That'd increase the negative feelings towards math.
3. The new stuff is gonna be harder than the current stuff. As he doesn't even understand the current stuff well, he won't understand the next stuff at all.

You cannot win against these emotions. You have to reduce them.

Bring in good emotions

Forget about the current subject, if you hamper with it, give him the recipes he wants, but don't try to change him.

Take the next subject or an alternate subject to teach him something new, that really works for him. He needs success with math. The way of solving, you choose to teach him, need not be the "right" way. He need not understand it fully. It only must work and bring him from a given task to its solution.

Let him stay with that successful way for a while

He must gain confidence in math. He must gain confidence in you. He must gain confidence in himself being able to do math. He only has this one working way. Don't take that away from him. His school teacher is already doing that regularly.

When he's so confident, that he thinks himself, something else -- some development might be due, then you can show him better ways. He'll trust you then.

Answer 3

What does he want? The question says more about what he doesn't want.

Maybe he wants to pass an exam to avoid math for a few years, and wants to improve his chance of passing. Maybe he wants to run a business, and would be interested in business algebra.

Starting from his goals can make the relevant activities more appealing for him. That may get him in a frame of mind to appreciate other approaches to math.