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.
Focus for mental computation
With mental computation the focus is
twofold:

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.
Explaining mental methods

Recording student responses
Valuing Mental Computation Online

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.

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.

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 =

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
Problem: 52 – 17 =
524240
-10
-2
-5
35

First I took away the 2. Then I took away the 10.
Then I took away the other 5. My answer is 35.
Denzel
Problem: 52 – 17 =
52504035
-2
-10
-5

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
Problem: 52 – 17 =
17 20 50 52
33 …, 35
+2
+3
+30

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 =
50
10
40
2
7
5 down
35

… can also be used for larger numbers
300 – 158 =
The empty number line
300150142
150
8

300 – 158 =
The empty number line
300140
2
142
160

Can be recorded as:
20 + 30 = 50
3 + 8 = 11
50 + 11 = 61
Partitioning each number
23 + 38 =

…or in diagrammatic form:
Partitioning each number
23 + 38 =
20
30
3
8
50 11
61

Mental computation helps to
develop an understanding of
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.
Using models of 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…
Limits of models of place value
17
8
1
9
9
200
35
1
165
11

An over-reliance on the linguistic tags approach
leads to problems when the student breaks the
number into parts.
Limits of models of place value
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 models for 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 models for 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.

Mental computation

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Mental computation