How to use questions to inspire mathematical thinking

Take a look at this puzzle. What do you think it might be about?

Check out this puzzle. What might it be about?  The outside numbers give clues about what to put inside. Does that change what might be going on?

Only the numbers 1 to 4 can be used. Now, look at the puzzle again and consider: does that change what might be going on?

We'll come back to this puzzle – and what it's all about – in a bit.

The difference between instructions and questions

In the puzzle you've been looking, what you weren't given, was a list of step-by-step instructions on how to solve it:

Step 1: ...
Step 2: ...
Step 3: ...

Instead, you were given a series of prompts designed to get you focusing on the information in front of you, and thinking about what might be going on:

The word 'instructions' crossed out.
What might this puzzle be about?
(Then, a hint)
Does that change what might be going on?

If you step into the shoes of a learner, consider what these questions offerered you. For example:

  • Time to take in complex information
  • An opportunity to form your own ideas, before ‘getting started' with the maths
  • A space for being creative
  • A chance to make observations without needing to be right or wrong.

But, how do you know what questions to ask? What kinds of questions can encourage thoughtful, reflective mathematical thinking rather than mindless rule following?

Helpfully, there's a categorisation that can make this easier for you.

Introducing funnelling and focusing questions

Questions are the bread and butter of a teacher's day. They're used for:

  • teaching new skills and concepts,
  • help students getting unstuck
  • extending and challenging thinking,
  • affirming and confirming ideas,
  • redirecting attention.

While questions are obviously present in classroom conversation, what's less obvious are the types of questions that get asked.

Enter: funnelling and focusing questions.

I first came across Funnelling and Focusing questions from maths educator, Mark Chubb. It was one of those lightbulb moments that suddenly helped me to look at the purpose of questioning and its value to student thinking in an entirely new way.

So, what are they?

Funneling questions guide students through a procedure or to a pre-determined endpoint. For example:

  • What should you do first?
  • What's the next step?
  • So, what's the answer?

The message that's being communicated to students through funnelling questions is to correctly follow your steps.

In contrast, focusing questions guide students based on their ideas and responses. For example:

  • What's this question asking you?
  • What do you notice about this example?
  • Can you explain your approach to me?
  • Why do you think this symbol/word is used?
  • Look at these solutions. How are they similar/different?

Focusing questions are a way of you understanding a student's thinking. These questions put you in a position to then provided meaningful, targeted support.

There's a subtle value to focusing questions, which I believe is worth surfacing here. These questions:

  • help students to think for themselves,
  • to become reflective and analytic, and
  • they can inspire confidence and curiosity about what’s going on.

All of these things are absolutely worthwhile in the maths classroom.

Why should we care about focusing vs funnelling?

I believe that both focusing and funnelling questions can have a role in maths class, depending on a combination of factors, such as:

  • the purpose of a lesson,
  • where students are at in their understanding and confidence,
  • student motivation,
  • the culture and values that you want to promote around doing maths.

We all get into patterns of speaking, instructing and communicating with students – and many of these patterns are, unknowingly, built up over time. Reflecting on the extent to which you're using funnelling and focusing questions, is a way of making these patterns visible – and then putting yourself in a position to adjust your practice.

So…. what was that puzzle about?

The puzzle at the very start of this article is called a Skyscraper puzzle. It's one of my favourites!

The numbers inside the grid represent the heights of skyscraper buildings. Each row and column contains one building of each size. The clues outside the grid tell you how many skyscrapers are visible (in a row or column) as seen from that vantage point.

Sometimes it helps to look at the answer to make sense of what you're first given.

See if you can understand what's meant by the number of visible buildings from each vantage point that's outside the grid:

If you want to play around with Skyscraper puzzles, here are 3 great sites:

  1. Krazy Dad has loads of printable puzzles, starting easy and getting really tough
  2. The Art of Puzzles has standard versions, plus many fantastic variations
  3. Mark Chubb shows how you can make the puzzle concrete using linking cubes.

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