1. Valuing Mental Computation Online Before you start…

2. 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:

3. 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.

4. Valuing Mental Computation Online Recording student responses

5.  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.

6. 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.

7. 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

8. 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

9. 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 525040 35 -2 -10 -5

10. 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 17205052 33 …, 35 +2 +3 +30

11. 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

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

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

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

15. Partitioning each number …or in diagrammatic form: 23 + 38 = 20 30 3838 50 11 61

16. Mental computation helps to develop an understanding of place value.

17. 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.

18. 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… 17 8 1 9 9 20 0 35 1 165 1 1

19. 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. 2318 + 2318  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.

20. 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.

21. 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.

22. PC users close the new browser window to return to the website. Mac users - quit PowerPoint to return.