There have been some great thoughts on hints recently. Michael’s talk at Shadow Con, Henri’s thoughts here and here, Annie here, and many more on blogs and especially on Twitter. This is my attempt to take a lot of really big ideas and pare them down into something I can put to work in my classroom.
One thing that has bugged me about the conversation about hints is that it seems to become idiosyncratic — every teacher has their own favorite hints, and their own justification for using them. Here are some approaches I’ve seen on Twitter:
- What do you know, and what are you trying to find out?
- What do you notice? What do you wonder?
- If you knew how to solve this problem, what would you do?
I think these are great, and there are plenty more out there, but I don’t want an endless list of all the possible hints. Instead, I want some general rules for how and when certain hints cause learning. Let’s start with the idea that the goal of a hint should be to promote mathematical thinking. How can we take these hints and create something concrete and useful for teachers?
- Locus of Attention – Feedback can draw the student’s attention towards something that matters for their learning.
- Motivation – You can ask a student to do something that they otherwise wouldn’t do.
This is interesting, and I think I buy it, but I think we can make this more specific and actionable. Here’s my proposal, which is really just breaking down the two ideas above:
Hints can promote learning in five ways:
- Redirect attention to features of the problem (What is it saying?)
- Redirect attention to student knowledge (What do I know?)
- Redirect attention to student cognition (How am I approaching it?)
- Promote students’ beliefs in their mathematical efficacy (I think I can solve it)
- Provide missing information (I know what I need to solve it)
Redirect attention to features of the problem (What is it saying?)
If a student has not attended to all of the relevant features of a problem, and will not on their own, they are in need of a hint. These come in all shapes and sizes, from “what do you notice?” to “what are the givens?” to “what does _____ mean?”. An important feature in the research on expertise and problem solving is that experts focus on structural features of problems, while novices focus on surface features. One of the roles of the teacher is to promote this type of thinking by redirecting student attention when appropriate.
Redirect attention to student knowledge (What do I know?)
This is the heart of learning — taking a problem, and relating the mathematics of the problem to the mathematics a student already knows. It’s also the most sensitive — I think if we are doing work here for the student, we are taking a great deal away from their learning.The goal is not just to solve the problem — “use this thing you’ve learned before” — the goal is for the problem to connect with students’ prior knowledge — “this is how and why the problem is related to that thing you’ve learned before”. If we make the connection for the student, they are less likely to make the connection. All that said, I think these hints have a place, and can be posed to promote thinking.
Redirect attention to student cognition (How am I approaching it?)
I really like these hints. “What have you tried so far?”, “What else could you try?”, “What are you doing?”, “How does it help you?” I think it’s important to note that this type of metacognitive hint is not always productive. It could interrupt students on the way to a solution, or cause them to second-guess useful ideas. I also think that an important features of these questions is that, in the end, they are often just refocusing student attention on features of the problem or their prior knowledge. Alan Schoenfeld has some great writing on metacognition, including classroom experience applying these hints.
Promote students’ beliefs in their mathematical efficacy (I think I can solve it)
Students need to be believe they are capable of solving problems in order to do so — go check out Carol Dweck and Jo Boaler’s work on the topic. This is usually wrapped up in another hint, and it’s essential features is that it does not do work for the student. Our hints send messages about the way math happens, and they must consistently send the message that students are have important mathematical ideas and are capable problem solvers. Questions like “What do you notice? What do you wonder?” promote this mindset if students can successfully solve the problem — we create a positive feedback loop of student success when we put them in situations to be successful problem solvers, whether that means solving the problem, or highlighting the great ideas that students produce.
Provide missing information (I know what I need to solve it)
This is the trickiest one, but I really believe it’s essential. Imagine a student needs to apply the Pythagorean Theorem to solve a problem. You ask them a few questions, and they are very clear — “I need to figure out this side length, and I know there’s a formula to find the third side of a right triangle, but I forget it.” They know what they piece of knowledge they are missing is, and they know where it fits, both into the problem and the rest of their mathematical knowledge. It seems like a waste of time to have them ask another student, or to search through their notes — I’m skeptical that doing so is fundamentally different from just telling a student. This situation is rarely cut-and-dry, but I think there is an important place and time for acknowledging that a student needs more information in order to solve, and just giving it to them.
This feels more concrete to me, in particular for categorizing what a student needs to solve a problem. I still think I need more specificity, and more examples of what these hints might look like in action, in order to have an impact on my teaching. I’m not sure how much farther I want to take this line of reasoning, but a next step might be to pick a task and plan what each of these hints might look like in that task, and then to see how effectively I can identify where a student is falling short in order to provide a useful hint.