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Valuing Mental Computation
Online
Before you start…
Focus for mental computation
With mental computation the focus is
twofold:
 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.
Explaining 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 Online
Recording student responses
Recording 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 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 =




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
-10
-5
-2
35
40
42
52
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
-10
-5
-2
35
40
50
52
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
33 …, 35
+30
+3
+2
17
20
50
52
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
50
10
40
2
7
5 down
35
The empty number line
… can also be used for larger numbers
300 – 158 =
150
8
142
150
300
The empty number line
300 – 158 =
160
140
142
2
300
Partitioning each number
23 + 38 =
Can be recorded as:
20 + 30 = 50
3 + 8 = 11
50 + 11 = 61
Partitioning each number
23 + 38 =
…or in diagrammatic form:
20
3
30
8
11
50
61
Mental computation
helps to develop an
understanding of
place value.
Using models 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.
Limits of 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…
1
17
8
9
1 19 1
200
35
165
Limits of models of place value
 An
over-reliance on the linguistic tags
approach leads to problems when the
student breaks the number into parts.
23 + 18
2
 Instead
3 1
8
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.
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