Scenario: Sharon teaching subtraction
Grade (student age): Year 4 (age 8-9)
Contributed by: Ray Huntley, Brunel University, UK
Context – national, curricular, professional, other
Subtraction is taught to primary children as being either ‘take-away’ or ‘difference’, with various models and images used to help children understand the processes involved. In terms of finding a difference, the notions of ‘counting back’ or ‘counting on’ are often introduced, and the hundred square can be a useful image to assist with this process. Sharon, a final year undergraduate trainee was reviewing these ideas with her class, starting with subtracting single digits from 10, before going on to discuss other examples which extended the method into subtracting single-digit and two-digit numbers from multiples of 10.
Sharon was teaching a Year 4 class and in her lesson on subtraction, the key vocabulary was the phrase ‘take away’, suggesting a model for subtraction which involved physically removing one number from a larger number and counting what remains. This was supported by the use of ‘minus’ which also represents the take away method of subtraction.
In the first part of the lesson, since number bonds to 10 are being considered, 10 is the selected larger number in each subtraction problem. The first example listed is
10 – 1 = □
which might be considered a good starting example since it involves the smallest amount of taking away from the selected number. However, the choice of numbers in this example is not supported by an appropriate choice of method, as the children are asked to draw a block of ten squares and colour one square to represent the ‘one’ to be taken away. From this coloured diagram, the children are to complete the number sentence as 10 – 1 = 9 by counting the uncoloured squares on their diagram.
This example provides a straightforward calculation but does not model the ‘take away’ method faithfully, since nothing is actually removed from the representation of 10. The children then worked on some other similar problems, each time drawing a block of ten squares and colouring in a specified number, an exercise in partitioning, rather than ‘take away’.
These problems were listed on the interactive whiteboard (IWB) but there is no record of the actual sequence of the problems. By examining the transcript of Sharon’s interview, it is possible to gain insight into her approach when choosing examples (Code CUE) for children to work on independently. When asked about the factors involved when choosing examples, Sharon responded with:
I normally start off quite easy because it’s better to start off with a really easy one that they can do, because they already get the confidence up that they can do it…if they can’t do what you think is easy, ‘cause you can sometimes pitch it too high, if you think that’s easy they can still struggle with it, so then you can lower it or higher it as you need to.
This approach was also evident from Sharon’s description of how she might choose examples specifically for a worksheet or a set of problems for the IWB:
It often took quite a while to explain it, we would do a lot on whiteboards and then they’d have a sheet and that would be about it, but in my own planning I would always write out, so instead of using a worksheet I wrote the questions on the board sometimes, so then I could go to the children who’d finished and write questions in their book, and I would get more…progressively more difficult.
It might be surmised that Sharon’s examples for the children continued in some sort of sequence, 10 – 1, 10 – 2, 10 – 3, and so on. Such a choice is certainly apparent in the extension work on the lesson plan, where the activity is carried out using number bonds to 20.
It is not evident whether the children are required to draw a line of 20 squares for each example and colour in the amount to be ‘taken away’, but the number sentences are required to be recorded in sequence:
20 – 1 = 19
20 – 2 = 18
20 – 3 = 17, and so on.
Again there is no element of physically removing the smaller numbers, and so in this activity the choice of examples appears to lead to pattern formation rather than an understanding of the subtraction process, although this is not supported by direct evidence. Continuation of this sequence of subtractions will lead the children eventually to the case 20 – 20 = 0 and it is likely that they will stop at that point.
In the main teaching part of the lesson, Sharon demonstrates how to subtract on the number square by either moving upwards to subtract tens or moving backwards along the rows to subtract units. The hundred square in this case acts as a calculation aid and whilst it produces the correct answer to the subtraction problem, it is by a method which fails to demonstrate the physical connection between taking 20 items from a group of 99 and the number sentence 99 – 20 = 79.
Finally for this lesson, examples from the number bonds to 10 are used in the plenary to demonstrate the inverse connection between subtraction and addition. Children are asked to consider 10 – 1 = 9, and ‘walk backwards through the sum’ to arrive at 9 + 1 = 10. This serves to reinforce the idea of the inverse. Notwithstanding the use of ‘sum’ to describe a subtraction, there is a problem with these examples not enabling learners to relate the physical action of taking away and the inverse operation of combining two sets to produce an additive total.
Knowledge Quartet Coding Commentary
Contributed by: Ray Huntley, Brunel University, UK
Knowledge Quartet Dimension: Transformation
Knowledge Quartet Code: Choice and Use of Representations (CUR)
Scenario: Sharon teaching number bonds and subtraction
There is mention on the lesson plan that Sharon will point out to the children that ‘subtraction is the opposite/inverse of addition’ which, whilst being correct, may lead to a mixed model being demonstrated if the examples selected are worked using a complementary addition approach, where the difference between the larger and smaller numbers in the calculation is determined by adding on from the smaller amount. This calculation is characterized by use of a minuend and a subtrahend, the difference between these representing the answer to the calculation. As Rowland (2008) points out, each of these elements when choosing examples of this type of calculation can be a dimension of variation, although the choice of numbers for any two of these elements will naturally determine the value of the third.
Choosing different numbers for each of the three elements helps learners to discern the purpose of each of the minuend, subtrahend and difference within the calculation, although in a lesson analysed by Rowland, ‘Naomi’ used as a first example of subtraction 4 – 2 = 2. In this case the subtrahend and difference took the same value, thereby obscuring the role of the variables. With regard to Sharon’s examples of taking a single digit from 10, this situation would occur with 10 – 5 = 5.
The first method used by Sharon involved identifying the difference, for example, between 10 and 1, by drawing 10 squares and colouring one, then counting what remained. By moving onto 2-digit subtraction, the efficiency of such a method is questionable since it is unlikely that children will be asked to draw a block of 99 squares and colour 20 before counting the 79 that remain. By using the hundred square, subtraction is perhaps best carried out by making use of the place value arrangement of the numbers. The chosen example, 99 – 20 can then be easily seen by starting at 99 and moving upwards 2 rows to reach 79.
In summary, this lesson uses a range of numbers with which to carry out subtraction problems, but it is the choice of representation rather than the choice of numerical examples which could be misleading to learners.
Rowland, T. (2008) ‘The purpose, design and use of examples in the teaching of elementary mathematics’, Educational Studies in Mathematics, (69) pp.149 – 163.
Watson, A. and Mason, J. (2005) Mathematics as a Constructive Activity: Learners generating examples. Mahwah, NJ, Erlbaum.