# Some new thoughts on hints

A couple years ago I gave a talk (above, and here) about hints.

(What’s the deal with cardigans? Seems like they were totally in for a few years and now (for a fashion-forward fellow) they are practically unwearable. Maybe they were never in, I don’t know.)

Thanks to a conversation with Dave today, I started thinking again about hints, and I think I have something to add about hint-giving.

So, picture yourself in a classroom, a kid waves a hand and gets your attention. ‘Can I have some help? I’m totally stuck.’

You walk over: tell me your symptoms? when did it start? and then?

(My point being, you assess the situation…)

OK but once you have a sense of what’s going on, I think there are roughly three possibilities I experience and three corresponding suggestions I can make for how to react.

Possibility One: I need to teach the student something, so I sketch a quick example.

When an individual student needs to learn something new in the middle of the task, it’s never ideal.

I used to handle these moments by trying to nudge the kid along with questions about the task at hand. I’ve come to think that this is a mistake, and now I try to avoid it. Instead, I try to quickly write a related problem with a relevant, worked out solution.

Here’s a snapshot of what I mean. This student called me over because she was totally stuck on solving -1.7x = 4.3x + 3.6, and after some gentle questioning I saw what she meant. Knowing a bit about what this kid was already comfortable with and where she could go next, I quickly wrote a totally different problem [-2x = 5x + 7] and wrote a solution exemplifying what she might do for the problem at hand. (She still made mistakes, which I highlighted. Sorry for the overstuffed picture.)

I do this for two reasons.

The first is that, when one aims to nudge kids along at the task at hand, one runs the risk of ending up in a Polya-esque recursive suckhole of questions, digging the student deeper and deeper into their own brain, until the math is buried beneath piles of questions.

I call this ‘Polya-esque’ because when I think about the sort of interactions I try to avoid one-on-one, I often think of this passage from How to Solve It:

If the teacher, having watched sharply, cannot detect any sign of such intiative he has to resume carefully his dialogue with the students. He must be prepared to repeat with some modification the questions which the students do not answer. He must be prepared to meet often with the disconcerting silence of the students…

“Do you know a related problem?”

…..

“Look at the unknown! Do you know a problem having the same unknown?”

…..

“Well, what is the unknown?”

“The diagonal of a parallelpiped.”

“Do you know any problem with the same unknown?”

“No. We have not had any problem yet about the diagonal of a parallelpiped.”

“Do you know any problem with a similar unknown?”

…..

That’s issue #1 with focusing on the task

The second is that I worry that it’s very hard for a person to learn something from thinking about just one problem. I want to leave students with a chance to think about a whole problem, not just the little scraps that I didn’t solve for them via nudges. (These conversations just are “better-luck-next-time” interactions.)

Ideally, a kid has more than just one example and one practice problem to learn something new. Then again, ideally a kid isn’t missing crucial knowledge that’s keeping them from doing math in class. We’re well past ‘ideally.’

So, the first situation is that a kid really needs to learn something new. In the heat of the moment, I don’t try to feed them each step or to weave a series of vague questions into a meaningful discovery. Instead, when a kid is stuck because they need to learn something, I try to teach them that thing.

Possibility Two: I need to help the student make a connection, so I remind them of a similar problem.

This is the territory that is closest to what I was describing in my talk. If I’ve done a nice job with my teaching, the kids have some memorable examples, ideas, problems or techniques to refer back to when trying something on their own. That way, when a kid tells me that they’re stuck but I don’t think they’re missing something crucial, I can lead with…

• Remember the diagram we were studying at the start of class? This problem is actually really similar to that one.
• So this is a complex area problem, and there are always basically two options: add some lines to cut the shape up, or use negative space. Which do you want to try here?
• I see you solved this equation by adding two to both sides. Why not do something similar here?

Of course, this only works if the students have some prior knowledge. I often lead with this and see if I get a catch. If I don’t, maybe I’ll start thinking about Possibility One.

Possibility Three: I need to help a student realize that they can handle this on my own, so I redirect the student back to the problem.

Sometimes the only thing a student doesn’t know about some math is that they know it. (Which is something that they need to know.) In situations like these, my job is to either reassure the student that they’ve started down a good path, that they aren’t breaking math, or to deflect the question in a way that puts the work back on the kid.

I feel as if there isn’t much more to say about this possibility — it’s the one that math educators generally love to talk about, because it’s the most fun. And, come on, it is fun. How cool is it that the following interaction actually works, ever?

Student: Hey I’m totally stuck.

Teacher: OK what if you weren’t stuck?

This absolutely works, but only sometimes. To get roughly precise, it only works (roughly) a third of the time, because there are two other possibilities.

Other moves that are fun when we’re just trying to redirect attention back to the problem:

• What have you tried so far?
• What haven’t you tried?
• Why did you write this?

There’s not much magic here. When a student is in a situation like this it’s often just about getting them back into the problem. They probably got nervous about something and stopped early. I do that all the time when I’m doing math, it’s totally normal. Lending confidence is one of the many little things that a teacher can do.

***

Cardigans or not, I do need more sweaters, though at this point I could easily just wait until the fall.

Hints are a nice slice of teaching. Not too much, not too little. And it’s this interaction pattern that I have so many times a day. Question, response, question, response. It’s so easy to experiment and try out a slightly different pattern of response. If it works, I’ll likely try it again, and again, and then one year passes and then it’s another, and slowly a new pattern of interaction has replaced the old.

Some people that I talk to don’t like the connotations surrounding the word itself, ‘hint.’ I get that, and to that I offer three replacements that correspond to three possibilities. When a kid is stuck, in most cases I respond with either encouragement, a reminder, or a new example. If that’s simpler to talk about, then let’s stick to that.