A few days ago, I saw a raging debate on Twitter about hint-giving in math class. It was triggered by a short talk by Michael Pershan, a teacher in NYC. Michael argues that high school teachers need to share good hints with each other, and he proposes some guidelines as to what makes a good hint. There was a strong reaction to his stance from educators who worry that providing hints, or the wrong kind of hints, robs students of agency and of learning opportunities.
This is a great topic, and it gets at some big ideas, so I will jump into the conversation.
– If understanding some foundational mathematical content is at stake, it may be irresponsible to leave a couple of students to their own devices, while others are proceeding merrily along. I’ll take the example Michael gives: a 4th grader not named Melissa is stuck when asked to add 1/4 and 2/3. His hint was “draw a picture”, which did not help. I don’t know what the students in that class have seen before, but if they were supposed to already know how to do this, I would guess that Melissa needed some serious hintage.
I would not hesitate to give her a piece of graph paper, and ask her to draw two 3 by 4 rectangles. Then, I might say “Let’s say that each of these rectangles represents a whole. See if that helps you add the fractions.” I don’t know Melissa, and I don’t know what that 4th grade had been doing, so my hint may be too much, or not enough. But it’s better than saying “draw a picture” or “don’t you remember what we learned about common denominators?” To use Michael’s language, the first hint is too vague, and the second would kill thinking. We need to aim somewhere between those extremes.
– Michael suggests that we need to give reasons for our hints. I’m afraid that in the middle of a high-stress situation, when a student just wants to get through the damn problem, they are not in a good place to listen to reasons. They just want the hint. However later, I might ask Melissa “Why did I suggest 3 by 4 rectangles, and not 2 by 5, or 15 by 17?” That would equip her to deal with this in future problems of this type without requiring my assistance. So yes, we can give reasons, but we should be attuned to whether a student is able to listen to those reasons at a particular time.
– Finally, Michael claims we improvise too much. I cannot agree with that. Improvisation is absolutely required in every facet of teaching, whether lecturing, handling disruptions, supporting group work, or providing hints. Of course, a first-year teacher improvising because of having no clue about what the student needs will not do it well. But an experienced teacher is likely to do a lot better by improvising during a frequently-taught lesson than by looking at a pre-made list of hints.
Instead of the negative “we improvise too much”, I would put it positively: let’s share our best hints, so that they will be available next time we are improvising. The reason that a repository of hints may be useful is that in the end, nothing is new under the sun. Student misconceptions do not vary all that much from one school to another, or even from one country to another. A list of hints that range from slight nudge to sledgehammer would be a nice thing to consult and discuss.
– In her response to Michael, Anna Blinstein’s blog post provides two lists of hints for a particular problem: hints she approves of, and hints she objects to. Her main point is that you should not direct students to a teacher-chosen solution path. They should find their own way. In my view, that position is too harsh. Should the teacher also prevent them from getting ideas from their classmates? Are hints from classmates better than hints from the teacher? Most likely they are worse.
Or take the case of a student who has been so damaged in their previous schooling as to feel very fragile and have zero self-confidence. Such a student may be unable to proceed without some support, and it may take a long time before we get them to a place where we can make our hints vague or withhold them altogether.
In teaching, context is everything. All the hints Anna lists may be useful, depending on the situation.
As Deborah Ball says, requesting that students solve problems is not the same as equipping them to do it. A student-centered, problem-rich, investigative classroom requires a teacher who engages with the students, and supports them. Sometimes that support is manifested by just standing there, saying nothing. (Mysteriously, that is often enough.) Sometimes that support is manifested by saying something like “trust yourself”. Sometimes that support is manifested by providing a vague hint like “draw a picture” or “try it with smaller numbers.” Sometimes that support is manifested by out-and-out guidance, with the hope and intention of decreasing that over time. No one approach to hints will work in every situation.
PS: In these two posts (I’ve Got a Problem, Getting Help) I describe my own struggle with a problem that was too hard for me to solve on my own. I am so happy that I got help of the very specific kind. Super-vague advice would almost certainly have been useless. (I had already tried it with smaller numbers, thank you very much.) I spell this out in a follow-up post about hints. I return to the topic of hints again here.