The Learning Framework in Number

Julie ReesK-6 Quality Teaching Consultant

New England Regional OfficeNoel Park House

A thought to begin

Teacher: ‘Who can tell me what 7 times 6 is?’

Student: ‘It’s 42!’ Teacher: ‘Good, and who can

tell me what 6 times 7 is?’ Student: ‘That’s easy. It’s 24!’

What research found

Students arithmetical learning develops through two complimentary processes: counting and grouping.

The transition from count-by-one strategies to collection-based strategies underpins the structure of the framework.

Underlying the Underlying the framework is a belief framework is a belief that it is important to that it is important to observe and take observe and take account of children’s account of children’s knowledge and knowledge and strategiesstrategies..

3 components

Assessment – what can they do?

Framework - ‘where are they now’

Teaching tasks – ‘where to from here’

The details of the learning framework:

Early Arithmetic Strategies

Where on the LFIN?

Some breakthrough moments

in this part.

Emergent – can’t count or match name to count. Perceptual – can count but needs to see, feel, hear

or touch. Figurative – can count concealed collections but

needs to count from one. Counting on and back – can add on to or take from

a group count. Facile – confidently uses a variety of strategies to

count, add or subtract.

Addition and subtraction

What key concepts are you trying to assess? To solve the task does the student:

count perceptual items using 1:1 correctly? rely on perceptual items? count from one when visualising the groups? count on from the larger number? use groups to solve tasks? use knowledge of combinations of numbers to 10 or 20?

CMIT

Number/name confusion

Sounding out the letters …b…a…t Sound images for 1 to 10 Imagine a young child who learns how to

read 13, 15, 16, 17, etc correctly (i.e. sound image starts from the right). How might he or she read 43?

Confusion between the names of the numerals 19 and 90

What is the next number after 19?

Order these!

13 30

3220

12

28

33

Numerals

The clothesline.

Children can all be given the same challenge but this can be differentiated to draw out more sophisticated strategies.

The Clothesline

Emphasis is on the order of numbers rather than the exact position

Can start at any point. Not always zero or one Models the concept of an empty number line

50 100?

Just for funHow many times can you subtract 7 from 83, and what is left afterwards?

As many times as you want, and it leaves 76 every time.

Tape DiagramsTape Diagrams

Tape diagrams are developed in the Japanese curriculum from Grade 2 or Grade 3.

7 ?

13

The empty number line

Represents the linear model of numbers Is used to record student thinking Can be used flexibly by varying the

parameters

Using a diagram

John had 29 buttons. His brother gave him some more buttons. Altogether John had 73 buttons. How many buttons did John get from his brother?

Beyond addition and subtraction

What is 14 x 10 = ?

Can you use your answer to work out

14 x 12?

What is 16 x 25 = ?

What is 16 x 24 = ?

How many groups of ten can be formed from 611?

Review

Multiplication and Division relies on the structure of a composite group.

A composite group is a collection of individual items that can be viewed as one thing.

For example, the student must be able to see a group of 5 items as one group of 5 and for it to be one unit repeated.

Building multiplication and division through equal grouping and counting

Equal groupsPerceptual multiplesFigurative unitsRepeated abstract composite units

Constructs composites and coordinates the countUses multiplication and division as binary operators

Multiplication & division as operations

The effect of question type on strategy

NUMBERS POSSIBLE STRATEGIES

40 + 20 "40, 50, 60" or "4 and 2 is 6, so

60"

47 + 20 "47, 57, 67" or "40 and 20 is 60. 7

more makes 67."

47 + 5 "45 and 5 is 50. 2 more is 52" or

"47 + 3 is 50 and 2 more is 52."

47 + 22 "40 and 20 is 60. 7 and 2 is 9. So

69."

47 + 25 "40 and 20 is 60. 7 more is 67

and 5 is 72." Or "40 and 20 is 60.

7 and 5 is 12, so 72."

Elicit as many strategies as possible

Multiplication & division

What key concepts are you trying to assess? To solve the task :

Does the student form equal groups? Does the student need concrete material? Does the student use materials to represent

each item or each group? Does the student keep track of the groups while

counting? What counting strategies does the student use

to find the total?

CMIT

The why chart…

What does a maths lesson focus on?

What will the deep knowledge and deep understandings

be?What will a maths lesson sound like?

This will be the metalanguage and substantive communication.

What will a maths lesson look like?

This will be H.O.T.S. background knowledge, problematic knowledge.

Learning Plans