What is prompting me to return to this topic today is that I just read an interesting post about it by Annie Fetter of the Math Forum. In it, she argues that the standard hint "what is the problem asking?" and variations on it are constricting, and prevent students from trusting their own thinking. So far so good, but Annie seems to believe that the only valid hints are universal hints about problem-solving, never hints with any math content. Her favorite is "What do you notice? What do you wonder?" I agree that in some situations, that is a great hint, for example with students who are used to the "this is how you do it, now practice" style of teaching.
But what you notice and what you wonder is limited by what you know and understand. I discussed this some years ago in an article about the use of electronic graphing on the early graphing calculators.
Teacher: "Graph y=x, y=2x, y=3x, y=4x. What do you notice?"
Student: "The line is getting jagged".
Teacher: "Graph y=4x+1, y=4x+2, y=4x+3, y=4x+4. What do you notice?"True, and possibly the start of an interesting conversation about screen resolution, or the x-intercept, but almost certainly not what most of us would hope for. More effective is to ask students to create a design using y=mx+b (a starburst, or parallel lines.) That forces the noticing of the relationship between m and the line's slope, or between b and the y-intercept. And then, asking students to fill in the gaps in their starburst leads to negative slopes, slopes with absolute value less than 1, and so on. Any time we can ask kids to make something instead of consuming something, we should jump on it. (See for example the designs kids made on Desmos at one school.)
Student: "The line is moving to the left".
But I digress. My point here is that there are many situations where generic hints are useless. Take the problem I worked on a few months ago, and reported on in this blog. I literally spent hours noticing, wondering, exploring, and experimenting. I tried it with small numbers. I decided drawing a picture would not help. I kept organized records of my investigation. I slept on it. Still, I could not crack it. Then I tried to write a computer program to complement my manual explorations, and again did not get very far.
As I reported in the subsequent post, I asked for help. I had the good fortune to get help from my son, who spent maybe 20 minutes, maybe more, showing me how to sketch out a strategy for the computer program. That was sufficient hintage for me to write the program, and it was very satisfying. Continuing to struggle without his hints would have been pointless. Then I tried to prove the conjecture that was suggested by the computer experiment, and again I hit a brick wall. I asked a mathematician friend to help, and a few hours later, he out-and-out told me the answer. Far from feeling cheated, I was thrilled, and applied what I learned to a similar problem.
These problems were too hard for me to solve on my own. Just like adding 1/4 and 2/3 was too hard for "Melissa". In those situations, it is healthy to ask for help, and it is kind to offer it. If my son or my friend had withheld their help, it would have seemed mean-spirited to me, even if they had the best intentions. The no-mathematical-hints-ever strategy will not work when a problem is too hard. And really, you should once in a while put your students in situations where they have to face problems that are too hard. If you never do, your class is probably not challenging enough.
As I once tweeted: trust your intuition, avoid dogma, be flexible, be kind.
I return to this topic here.