Valuing Mental ComputationValuing Mental Computation

OnlineOnline

Before you start…Before you start…

Focus for mental computationFocus for mental computation

students explaining their own mental strategies

students listening to and evaluating, in their own minds, the methods other students are using.

Your questioning needs to facilitate this.

With mental computation the focus is twofold:

Explaining mental methodsExplaining mental methods

When explanation and justification are central components of mental computation, students learn far more than arithmetic.

They learn what constitutes a mathematical argument and they learn to think and reason mathematically.

Valuing Mental Computation OnlineValuing Mental Computation Online

Recording student responsesRecording student responses

Recording student responsesRecording student responses

It is important to record student responses so that all students can see the thinking.

It is important not to judge the methods students offer.

Students will be able to see the variety of methods and may choose to try a different one next time.

Recording student responsesRecording student responses

The way you record the student responses so that all students can visualise the thinking will depend on the method.

The empty number line is a useful tool when the student begins with one of the numbers and deals with the second number in parts.

Recording the steps is better when the student partitions both numbers and then recombines.

Problem: 52 – 17 =Problem: 52 – 17 =

I took 10 from the 52 to give me 42. Then I took away 2 more gives me 40. I have 5 more to take away gives 35. Lawrence

First I took away the 2. Then I took away the 10. Then I took away the other 5. My answer is 35. Denzel

I started at 17 and added 3 to make 20 and then 30 more makes 50 and I need 2 more to get to 52. My answer is 33 …, 35.Kate

First I take 10 from 50 to get 40. Then I take 7 from 2 to get 5 down. My answer is 35. Dominique

Problem: 52 – 17 =Problem: 52 – 17 =

I took 10 from the 52 to give me 42. Then I took away 2 more gives me 40. I have 5 more to take away gives 35. Lawrence

524240

-10

-2

-5

35

Problem: 52 – 17 =Problem: 52 – 17 =

First I took away the 2. Then I took away the 10. Then I took away the other 5. My answer is 35. Denzel

52504035

-2

-10

-5

Problem: 52 – 17 =Problem: 52 – 17 =

I started at 17 and added 3 to make 20 and then 30 more makes 50 and I need 2 more to get to 52. My answer is 33 …, 35. Kate

17 20 50 52

33 …, 35

+2+3

+30

Problem: 52 – 17 =Problem: 52 – 17 =

First I take 10 from 50 to get 40. Then I take 7 from 2 to get 5 down. My answer is 35. Dominique

501040

27

5 down

35

The empty number lineThe empty number line

… can also be used for larger numbers 300 – 158 =

300150142

150

8

The empty number lineThe empty number line

300 – 158 =

3001402

142

160

Partitioning each numberPartitioning each number

Can be recorded as:

20 + 30 = 50

3 + 8 = 11

50 + 11 = 61

23 + 38 =

Partitioning each numberPartitioning each number

…or in diagrammatic form:

23 + 38 =

20

30

3

8

50 11

61

Mental computation helps to develop an

understanding of place value.

Using Using models ofmodels of place value place value

The current approach to developing written algorithms is through forming a place value rationale of “trading”.

This begins with models of place value:– bundling– multi attribute blocks (MAB, Dienes)– place value charts.

Limits of Limits of models ofmodels of place value place value

The sense of numbers students need is more than reading the positional tag of a numeral.

Activities using trading with models of place value do not always translate into understanding of place value and we see…

178

1

9

9

20 035

1

165

11

Limits of Limits of models ofmodels of place value place value

An over-reliance on the linguistic tags approach leads to problems when the student breaks the number into parts.

23 18+

2 3 1 8 Instead of 2 in the tens column and 3 in

the units column, 23 needs to be seen as a composite — 20 and 3 or 10 and 13.

Developing Developing models formodels for place value place value

Mental computation practices often preserve the relative value of the parts of the numbers that are being operated on.

That is, hundreds are treated as hundreds and tens are treated as tens.

Developing Developing models formodels for place value place value

In written algorithms, the relative values are set aside and digits are manipulated as though they were units.

Mental computation is more likely to be meaning-based than written algorithms which are rule-based.

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