Valuing Mental Computation Online Before you start…

Focus 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 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

 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

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

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 …,

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 down 35

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

The empty number line 300 – 158 =

Partitioning each number Can be recorded as: = = = =

Partitioning each number …or in diagrammatic form: =

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…

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  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.

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