Here we look at a series of structural arithmetic tasks that consist of simple expressions involving addition or subtraction.
Task 02A: This task can be solved by simply adding 10 to 213. Surprisingly perhaps, many children don't seem to spot this: they ignore the information about 165+48 and perform the calculation 175 + 48 from scratch. If this happens, encourage children to compare the expressions 165+48 and 175+48. How do they differ? How can we make use of this difference?
[It is 'natural' to see an expression like 165+48 as an instruction to perform an operation. However, it can be argued that to solve Task 02A structurally, one needs simultaneously to see the expression as a number in its own right, which changes as its components change. In an interesting paper by Gray and Tall, this 'amalgam of process and concept' is given the name procept. A similar phenomenon can be said to occur with algebraic expressions like 8 + g (see Task 02E, below); Kevin Collis refers to 'acceptance of lack of closure'; Anna Sfard to 'reification'.]
Gray, E. and Tall, D. (1994). Duality, ambiguity and flexibility: a proceptual view of simple arithmetic. Journal for Research in Mathematics Education, 26(2), 115-141.
In this task we have chosen fairly 'nasty' numbers to discourage children from performing the calculation 175 + 48. However, the very presence of such numbers might put some children off, in which case you might want to try a simpler version such as the one below. Of course, here it is very much easier to calculate the second expression from scratch, but if children do so you can still ask whether they could find its value a different way.
Here is another variant which is somewhat simpler than the original task, in that the change in the component 165 occurs in the ones rather than the tens column.
Task 02B: The first part of this task provides an opportunity to discuss the variety of ways one can get a 'derived fact' like '53 – 27 = 26'. For example, one could using 'adding on':
27+3+10+13 = 53, 3+10+13 = 26; or 27+30–4=53, 30–4 = 26; or 27+27=54, so 27+26 = 53.
This is more demanding than the original task, as it might lead some children to give the answer 162 rather than 1062:
And here's a variant where the underlying structure is rather different - and probably much simpler, once children spot it!
Task 02C: The structure in this task is different again. Some of the digits in the first expression have been switched from one term to the other, but keeping the same place value and so keeping the overall value of the expression. If one so wished (!), one could write both expressions as this:
800 + 900 + 10 + 40 + 5 + 7.
Finally, two of the digits have been switched within one of the terms (947) of the expression, so their place values have changed.
Task 02D: As with all this week's tasks, this task can be solved more easily if one pauses to take stock of the structure. If children attempt to solve this by simply working from left to right, the process is quite cumbersome and involves a negative number. On the other hand, if children look carefully they might notice that subtracting 53 (at some stage) and adding 63 is equivalent to adding 10, so the expression represents 19+10, or 29.
In this variant it is perhaps more tempting this time to simply work from left to right as it doesn't introduce a negative number. However, it is still more efficient to combine the later terms first.
Here the '53' terms cancel out, as long as children pause long enough to notice*; however, they also have to be comfortable about changing the order of the operations or accepting the relatively abstract idea that -53 will be cancelled by +53, even if the latter operation doesn't happen 'immediately'!
It might take a while to spot the contrivance that makes this variant easy to solve! It turns out that the sum of the first and third term is the same as that of the second and fourth term...
Task 02E: The items in this task don't quite match any of the other Week 2 items; their structure is slightly different and the expressions involve 'unknowns'. However, we have some interesting data about the items. Item a) is from the CSMS Algebra test that was given to large representative samples of secondary school children in Years 8, 9 and 10 in England in 1976 and again in 2008/9. Item b) was used in an informal study in 2019, involving an ad hoc sample of children mostly from Year 6.
Item a) turned out to be extremely easy. For example 97% of the 1976 Year 9 sample answered the item correctly, reduced slightly to 87% for the Year 9 sample in 2008/9. The younger children were equally successful on item b), with 87% giving the answer 40.
However, before we rush to the conclusion that these successful children can 'deal' with algebra, we should consider their performance on the items below, which also featured on the CSMS Algebra test and on the informal test with the younger children respectively. Here item a) was answered successfully by only 41% of the 1976 Year 9 sample, and by 37% of the corresponding sample in 2008/9. For many children, the correct answer, of 8+g, doesn't look like a proper answer - it is not closed. It looks like an instruction, not a number in its own right, just as for many children the expression 165+48 in Task 02A only looks like an instruction . As a consequence, some children gave the answer 8g instead of 8+g. Others came up with plausible numerical answers such as 9 (it's more than 8) or 12 (perhaps on the basis that e, f and g all equal 4) or 17 (perhaps on the basis that e, f and g equal 3, 5 and 7 respectively, or because g is the 7th letter of the alphabet). As for item b), only 1 child from the younger sample gave a correct open answer.
*Put another way, "Don't just do something, sit there!" Thanks to Tom Francome for alerting me to this phrase.
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