A Taxonomy of Hints: First Draft

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?

Michael cites Kluger & DeNisi (1998), and argues that hints can cause learning in two ways:

  • 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.

In sum:
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.

9 thoughts on “A Taxonomy of Hints: First Draft

  1. Michael Pershan

    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

    Helpful, great post, and I think this would be an interesting next step.

  2. Pingback: Dylan: “Hints can promote learning in five ways” | Problem Problems: All About Problem Solving

  3. revuluri

    Thanks for this post! A couple thoughts:

    To your first category, I might add or expand to features of the solution. Deborah Ball, in a couple sessions at NCSM 2014, called these the “conditions of the problem”. Here’s some bits from my notes:

    One strategy is to identify and use the problem “conditions” – the constraints that shape what would count as a solution to the problem… by making conditions explicit (visible) and naming them, can ask students to use the conditions to check proposed solution. Makes more accessible to more children the thinking that’s required in the problem – what the problem is asking for and what a solution would look like. More is explicit about what’s involved in doing the problem.

    The role of conditions:
    • Making sense of what the problem is asking and what a solution would look like
    • Conditions are a useful tool to help persevere when difficult
    • Conditions help construct a mathematical argument
    • Referring to conditions can help critique a mathematical argument

    To your fifth category, I think this is a nice way to “create a need” (as I think Dan Meyer puts it) for a new concept or procedure. I like how you’ve articulated it — in particular, this seems to show how this kind of approach is not only motivational, but also highlights connections and thus is likely to promote the creation of a “knowledge package” rather than ideas that are disconnected and therefore fragile.

    1. dkane47 Post author

      That’s really interesting — it’s both metacognitive (asking students to make a prediction and then check their answer with that prediction) and pushing students to pay attention to critical features of the problem. I like considering the relationship with argument — it’s something I’m ignoring here, but it’s a huge part of what needs to happen with these tasks in the classroom.

  4. mrdardy


    Late to the conversation here – apologies.

    This post reminds me of one of my most treasured interactions with a student. In my early aching years I worked at a small school where I was one of three full time math teachers for grades 6 – 12. I had the unique opportunity of moving through the curriculum with a small group of students who I taught Alg II, Precalculus, AP Calc AB, and then AP Calc BC. About a week before their BC exam I was talking to a girl named Ashley and asked her how she felt about the upcoming exam. She said she was confident and that she knew that if she struggled with a problem she would hear my voice in her head and hear some of the questions that I always asked her. This made me feel SO good about my habit of guiding questions. Now I know that I had the benefit of four years in a row with this group of students, but I have always held on to the idea that I can transfer some good questioning habits by my insistence on asking questions in response to student questions. I suppose that my questions would fall into your description of hints.

    1. dkane47 Post author

      Do you think those fall into the category of metacognitive hints? Is that a useful way to categorize the hints? I’ve been thinking more and more that there are two purposes for hints — one is to learn content, and one is to shape beliefs/dispositions/mindset. I think what you’re talking about is the second part? But I’m not sure whether these are totally separate purposes, or how we can work to meet both of them.

      1. mrdardy

        Many – if not most – of my reflexive questions fall into one of two patterns
        1) Remember when we talked about ______? What were our strategies in this situation?
        2) What does this problem remind you of? Why? What were our strategies in this situation?

        I point out to my students that they do as much telling (if not more) than I do during class conversations. I also admit that I am asking helpful questions, ones that they should try to train themselves to ask themselves.

        Does this make sense in the context you are examining?

  5. dkane47 Post author

    Makes sense. I feel like those two fit into the category of “Redirect attention to student knowledge” — make the student aware of the connections between what they know and what they need to figure out. And you’re talking about the habit of mind where students do this naturally? I think that’s a really hard thing to do, and I like the idea of a “voice in your head” to help students do it.

  6. Pingback: Problem Solving: Classroom Application | Five Twelve Thirteen

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