Wednesday, April 29, 2015

My issue with hints

Yesterday, @mpershan asked for feedback on his Shadowcon talk regarding the usefulness of hints.

His contention is that the pedagogy of hints for 9 - 12 math teaching is not very developed and could be improved by thinking about the context, reasons, and specificity of the hints.

As much as I agree that hints could be improved by these things, I also have a lot of discomfort around hints in general. Too often, I find that they funnel student thinking in a predetermined direction... as in, the student is stuck, and the teacher is trying to direct them onto a path that they think is productive by using hints, but it's a predetermined path and therefore removes a lot of the exploration that one would presumably want a student doing in solving this problem. Michael argued that this was only true for bad hints, that good hints should not simplify the problem or do the heavy lifting for the student or close off avenues of thoughts and overspecify a direction. I'm still not sure if we're arguing semantics or if we genuinely have different views on whether hints are good or bad and thought it might be helpful to look at specific examples.

Random exhibit A from a recent assignment:

I gave my students a problem of the week from the IMP curriculum in which you are told that there are five bales of hay, but that instead of being weighed individually, they were weighed in all possible combos of two. We know all of these dual weights, but would like to know how much each individual bale weighs.

Lots of students were confused and stuck. Here were some things that I did not say (although I really, really wanted to) because I think of these hints as being too helpful and pushing kids in a certain direction in their problem solving.

• How many times does each bale of hay come up in all the weighings?
• What is the total weight of all of the combos? Why might this be helpful?
• How can you represent this using equations?
• Can you organize the combos in order of weight?
• Are there any combos the weights of which we can figure out? Any that we cannot?
• Can you make equations to represent what you know?
• Can you make a table to organize what you know?
• Can you make an easier version of this problem?
• I see that you have four equations, but five unknowns. How do you think that will play out in trying to solve this problem?

Here are some things that I did say:

• What have you tried?
• Have you talked to anyone else in the class?
• Where are you stuck? How do you know that what you're doing isn't working?
• What information would be helpful to get unstuck?
• What things do you think that you know? What don't you know?
• How are you organizing your thinking?
• How are you representing your understanding of this problem?
• Are you making any assumptions? Which ones? How will you know if your assumptions are correct?
• How will the person reading this understand what you did?
• What are strategies that might be helpful here that you haven't tried yet?
• You are making a lot of progress! Read through what you have already and see if you can restate it in a different way.
I make a distinction between teaching a specific procedure or specific content when you would want to channel students' thinking perhaps more narrowly - there may be multiple paths, but not an infinite number of them, and it is likely important that students understand which paths are more efficient under what circumstances and how they connect to each other - versus when you are asking students to work on a more open problem in which they are meant to develop problem-solving and sense-making. It seems like half the purpose of open problems are for students to come up with different approaches, persevere past sticking points, learn to think flexibly and independently, and make sense of unknown situations. And yes, that almost requires that they be stuck and frustrated for parts of it. If a problem can be solved by a student easily and without any false starts, then it's not much of an open problem. To me, hints like the ones I listed in the first section decrease this cognitive load significantly. I want students coming up with those ideas, not following mine.

I am trying not to get bogged down in the word "hint," but it just has this connotation of "I have the right answer in my head, but you can't figure out what it is so let me make it a bit easier for you to get it." If we redefine "hint" to also include questions or statements that push the student to think more deeply and develop their own internal resources rather than as a way to make the process smoother for them by external means, then I think that I can get behind good hints.

1. That was great, thanks!

2. I really appreciate that way of thinking about it. Thank you for sharing specifics!

1. I'm still mulling this over and may have more to say :) it's a tough topic

3. I really appreciate that way of thinking about it. Thank you for sharing specifics!

4. Great distinction between the two types of hints - thanks for sharing!

5. "I make a distinction between teaching a specific procedure or specific content...versus when you are asking students to work on a more open problem in which they are meant to develop problem-solving and sense-making."

This is a fine distinction, but 95% of my teaching takes place in between these two poles.

This year I'm teaching 9th grade geometry, and I often want them to solve geometry problems that they've never seen before. I want them to learn not just their general problem-solving tools, but I also want them to use their specific geometry problem-solving strategies. Something like, "If you're just starting off on a polygon problem, checking out the angles can help you get oriented." Or, even more broadly, "Have you tried playing with the diagram?"

This is not a specific procedure. There is a lot of student thinking and sense-making for students to do after this. And depending on the particulars of the student, the problem, the situation I might choose to not share this. But I think this lies in between the two poles you outlined and it's that vast middle ground that I'm interested in.

1. Hey Michael - that's a great point. I think that I would still like those specific strategies to be developed by students as much as possible, just like I would like content knowledge to be developed by them. So instead of telling them to play with the diagram, I would give problems where that would naturally fall out as a strategy for some students and have those students share their approach with the class so that over time, the class is developing a set of strategies together that they own. Now, if no one comes up with a strategy that I think is useful, I don't have an issue with providing it, but I would like for most of the strategies to be student-produced. Do you think that this is the root of our disagreement?

2. I think that I would still like those specific strategies to be developed by students as much as possible

This might be the root of our disagreement! I don't have much of a preference for students to invent the content or strategies of a math class.

6. We might begin with an easier example, say with 3 bales, A, B and C, A+B=5, B+C=9 and A+C=8. That's already challenging to someone who's never solved problems of this nature. It's possible for students to solve it without applying algebraic skills, then to develop a sense of the patterns involved without being given any hints, say by comparing B+C=9 and A+C=8 and realising that B>A.

Then transitioning to harder versions is only a small step up:
http://fivetriangles.blogspot.com/2013/03/50-mail-bags.html

Same concept, but pretty difficult, because they involve an ambiguity:
http://fivetriangles.blogspot.com/2015/03/225-integer-pairs.html
http://fivetriangles.blogspot.com/2013/05/69-weights_8.html

1. For older students (this was a task I gave to 10th graders), it may be reasonable to expect that some would create an easier example to play with on their own and then investigate its patterns and transition to harder versions rather than having those scaffolds provided by the teacher. For younger students, I think that this is a really helpful way to frame the problem, and thank you for sharing those versions!

2. Oh, and I think that I will offer the ambiguous versions as optional extensions for my students - fun!

7. A good hint should make the problem more interesting to the student. The hint is dependent on the problem and the student. This problem, I think, is likely more interesting if we avoid algebra & systems of equations. Perhaps give a hint that leads towards a less formal approach.

Would it be reasonable to ask something like "can you tell which of the bales is the heaviest"? This gets the thinking back to bales of hay and encourages reasoning and analysis rather than premature symbol manipulation. The student will immediately be able to start on this new question. Quickly narrow it down to one of two bales. And likely be motivated to figure out which of these two is the heaviest of the five. This of, course requires looking at a couple of the other pairs (you now have five triangles version of the problem without equations or symbols). Hopefully the student begins to think about "how much heavier", "what is the next heaviest", etc.

I guess I feel like the key to having good hints is having a good problem or collection or problem(s). If it is a good problem, you need to think about hints before you even ask the question. Should I include a hint with: part a) which is the heaviest bale. Or should I remove a hint by just stating that the weight of each pair is know, without providing those weights. Etc, etc.