Making Maths Meaningful How can we use practical

Making Maths Meaningful

How can we use practical resources to support mathematical thinking?

What do we mean by practical resources/manipulatives and why use them? • • • Any practical apparatus we use in the classroom e. g. Dienes, counters, cubes, Cuisenaire rods, hundred squares, digit cards … Practical resources are models which ‘stand in’ for something else Dutch research (1999): models of models for tools for

A good model … • Gives insight into mathematical structure • Extends different aspects of mathematics

The power of arrays 3 x 4 = 12 4 x 3 = 12 4 + 4 = 12 3 + 3 + 3 = 12

Exploration Current feature on NRICH - Counters NRICH task: Making Shapes

13 x 4 = (10 x 4) + (3 x 4)

15 x 14

34 x 9

22 lots of 8 is the same as 176 ÷ 8 = 22

• Mental image to draw upon as children begin to use and memorise number facts • Leads into multiplication of large numbers using blank arrays • In turn, leads to understanding division as inverse of multiplication • Also provides an image for multiplication of fractions

Ways in which counters can support mathematical thinking • To embed a mental image of a concept, which can then be used as a tool for thinking • To keep track of moves or mark a place • To ‘stand in’ for something nonmathematical

What do you notice?

Square Corners What is the greatest number of counters you can place on the grid below without four of them lying at the corners of a square?

Key messages • • Manipulatives can be powerful tools to support sense making, mathematical thinking and reasoning All children of all ages should have access to as wide a range of manipulatives as possible Learners need lots of opportunities to play feely with manipulatives before mathematical properties are drawn out Manipulatives should not be seen as ‘crutches’ to be used without understanding to follow a learned procedure

Liz Woodham Email: emp 1001@cam. ac. uk Twitter: @emw 1001 http: //nrich. maths. org/borehamwood