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Mental Mathematics

Almost all of mathematics could be described as ‘mental’ in the sense that engaging in a mathematical task involves thinking. Thus every mathematical problem a pupil tackles must involve several stages of mental mathematics. Pupils actively involved in mental mathematics might be engaged in any combination of:

interpreting, visualising, analysing, synthesising, explaining, hypothesising, inferring, deducing, judging ,making decisions & justifying

These ideas are prevalent throughout mathematical endeavour and underpin the work of using and applying mathematics.

These 6 booklets cover a few key areas likely to influence pupils’ progress beyond level 5.

Attainment Target Mensuration and Measures
Number Algebra Shape Number Algebra Shape

Number Talks

A Number Talk is a short, ongoing daily routine that provides students with meaningful ongoing practice with computation. A Number Talk is a powerful tool for helping students develop computational fluency because the expectation is that they will use number relationships and the structures of numbers to add, subtract, multiply and divide

PowerPoint Explanation               Number Talks 1                Number Talks 2

Level 4 Problems                           Level 5 Problems            Level 6 Problems                 Level 7 Problems                 Level 8 Problems

Maths Talks

"I love this so much that I think I could do this all day, not just for 5 to 10 minutes at the start of class. I can't get enough of what kids share about how they think mathematically. And boy, they have stuff to share."

This new blog is at

I want to share this with you because...
  1. it's important to allow students to verbalize their thinking.
  2. it's generous for kids to share and give feedback. This act hones each kid's own reasoning.
  3. it's powerful to be flexible in one's thinking, how we see patterns and structures allows us to dig deeper mathematically.
  4. number sense is just so good to have.
  5. you can do math talks in any class. To say that students are too young or too sophisticated for math talks is like saying that the love of mathematics has a defined age range.
  6. we want to give voice to the kids who don't always get high marks on tests. 
There's a how-to page in case you could use it. 

Mental Imagery

There is compelling evidence that imagery plays a significant role in mathematical reasoning. For example, a young child may add 7 + 5 by mentally "moving" 1 from the 7 to the 5 to form 6 + 6, a known double. Or a child might determine how many one-inch cubes there are in a rectangular solid 3" by 3" by 4" by visualizing the solid as composed of three layers. Whether working in a numerical or geometric context, when students are engaged in meaningful mathematics rather than rote computation, it is quite likely they will be using some form of imagery.  Click here for the whole document.  

Mental Imagery Activities

A Decimal Line:

Close eyes – see a line segment (or a line where you can see both ends) – place a zero at one end and a number of your own choice at the other end – make a mark on your line -  put the number you think should be on the mark.

Open eyes – draw your line and mark the three numbers.

Select examples and discuss with the class (group).

To download the complete document in word format click here

Quick Draw

Quick Draw is an engaging mathematical activity that helps students develop their mental imagery. A figure such as the one shown below is presented briefly to students. They are asked to "Draw what you saw." When students have drawn their figure, give them a second look. Finally, uncover the figure and ask students to describe what they saw. Encourage a wide range of interpretations.

To download the complete document in word format click here and for a pdf download of activity 1 click here and of activity 2 click here

Exploring Mental Imagery

I use the term mental imagery to refer to anything and everything that happens inside you when you are thinking, planning, considering, and reflecting. For some it may be predominantly visual, for others predominantly verbal, and for others predominantly visceral; for most there will be elements of all of these.

The power to imagine is perhaps the most important and fundamental of all the many powers which children possess when they come to school, indeed when they emerge from the womb. In particular, it forms a world which lies between the outer world of material objects, and the outer world of abstract symbols. The issue here is how we can exploit that power to the full. My aim is to illustrate different ways in which imagery and imagination can be used effectively in the mathematics classroom and in preparation for lessons.

To download the complete document in pdf format click here

Try these three sets of activities
Visualising Squares
Visualising Triangles
Visualising Loops