Stimulating mathematical thinking

Sometimes we use simplistic recall and knowledge questions, for example:

  • Is 29 a multiple of 3?
  • What is ¾ of 48?
  • What is the probability of rolling a 6 on a fair dice?
  • What shape is this image 1 ?

We tend to use them because we want to move the lesson on at a pace that we are comfortable with, responding to the time pressures involved in our teaching.  These sorts of questions feature heavily in many assessments and sometimes they do have their place, but how much do students get out of these sorts of questions?  Sometimes there is very little thinking actually involved.  In the case of the first question some students do no thinking at all, as they have a 50% chance of guessing the right answer.

So what can we do to open up our questions, thereby encouraging our students to achieve more stimulating mathematical thinking?  Here are some ideas.

Good questions… Examples
provoke discussion and encourage speculation ·      Would it be better if…?

·      What happens when we…?

·      Can you predict the outcome of…?

·      What do you think?

·      Why….?

are challenging and encourage diverse responses ·      How would you justify…?

·      How many different … can be found?

·      Which is the odd one out? Why?

·      Explain how you would…

stimulate mathematical thinking and promote reasoning and problem-solving ·      What is the same/different about these?

·      Can you group these…?

·      Do you have any suggestions…?

·      What do you know about….?

·      Is this sometimes, always or never true?

explore possible misconceptions and encourage reflection. ·      What would happen if?

·      Have we found all the possibilities?

·      What have you discovered?

·      Why did you do it that way?

·      Who has a different answer/solution/method?

·      Give me an example where this doesn’t work.

It can be straightforward to convert a simple question into something more stimulating.

Here are a few examples.

Simple question More stimulating mathematical questions
Is 29 a multiple of 3? Why is 29 not a multiple of 3?

What do you know about multiples of 3?

What is ¾ of 48? Explain how you find ¾ of a number.

Is ¾ of a number always even?

When is ¾ of a number a whole number?

What is the probability of rolling a 6 on a fair dice?


Is the following statement always, sometimes, or never true?

‘It is harder to roll a 6 than a 1 on a dice.’

What shape is this?

image 1

What is the same and what is different about these shapes?

image 2

What is 44% as a decimal? Which is the odd one out?

image 3

What is the area of a rectangle with length 9 cm and width 4 cm? How many rectangles can you find with an area of 36 cm2?
Write down the next number in the sequence 23, 30, 37, 44, 51, … Explain how you can work out if 87 is in this sequence.
What are all the factors of 20? Which number under 25 has the most factors?
Write down the fraction of this shape that is shaded.

image 4

How many different ways can you find to shade 3/8 of this shape in 2 minutes?

image 5

What are the first four prime numbers? Explain why 7 is a prime number and 1 is not.

What is the best method for finding all the prime numbers under 100?

What is the lowest common multiple of 90 and 72? These answers were all given as the lowest common multiple of 90 and 72 by students.

18      6     360      6480       1

Which is correct?  Write a question involving 90 and 72 and factors or multiples that give the other answers.

What are the first four square numbers? Why are 1, 4, 9 and 16 called square numbers?

The sum of two square numbers is 20, what are they?

Which square number is closest to 30?


image 6

Sort these into groups.

image 7


Kind regards,

Debbie Barton

DebbiDebbie Bartone Barton is a teacher, examiner and maths consultant with over 20 years’ experience. She’s written a number of books including Complete Mathematics for Cambridge Secondary 1. She also worked as a Gifted and Talented trainer and is passionate about ensuring able students are challenged with exciting mathematical stimulus.