Scenario: Suzy teaching multiplication methods
Country: UK
Grade (student age): Year 5 (age 9-10)
Contributed by: Ray Huntley, Brunel University, UK
Context – national, curricular, professional, other
Suzy was reviewing ideas about multiplication of tens and hundreds with her class, before moving on to discuss other examples which extended the method into thousands. Suzy was a final year undergraduate pre-service teacher, and the lesson took place in a school-based placement towards the end of her three-year teacher preparation.
Scenario
Suzy, an undergraduate student-teacher, was reviewing multiplication methods with a Year 5 (pupil age 9-10) class. She was teaching the children to multiply using the ‘grid method’, and it is the third lesson of a sequence of lessons on multiplication that is described here. It opens with a worked example, 34 x 26, using a method outlined in lesson 2, and then continues with a series of six word problems which require children to extract the appropriate multiplication problem and calculate the answer using the same method. Suzy teaches the grid method by working through the six examples with the children, demonstrating the method using a set of examples which she had prepared in advance.
The first question is as follows: There are 40 crayons in a pot. How many crayons in 15 pots? This question provides a relatively simple calculation in multiplying a multiple of ten by a 2-digit number containing the digits 1 and 5 which are among the easier multiplication tables to calculate with. The grid and column method for this calculation then becomes:
40 x 15 Th H T U
x 40 0 4 0 0
10 400 0 0 0 0
5 200 0 2 0 0
0 0 0
6 0 0 40 x 15 = 600
With the whole multiple of ten ending in zero, the grid and column method produces two partial calculations of zero for the addition. Some children may be able to disregard this if they recall that multiplying by 40 is the same as multiplying by 4 and then by 10. Or, as Suzy described in an earlier lesson, multiply by 10 and ‘add a zero’. This example therefore introduces some difficulties regarding place value conventions which are sometimes overcome by colloquial phrases such as ‘add a zero’ which will fail to work with decimal numbers.
The second example is: One mug has 22 spots. How many spots have 34 mugs got? It could be argued that the context for this question is spurious, since mugs are not generally distinguished by the number of spots on their external pattern. However, from the point of view of the numbers to work with for the grid and column method, this example provides all non-zero digits, giving a set of four non-zero partial calculations in the working:
22 x 34 Th H T U
x 20 2 6 0 0
30 600 60 6 0
4 80 8 8 0
8
7 4 8 22 x 34 = 748
The third example is as follows: One banana costs 10p. How much do sixteen bananas cost? This appears to be a backward step by providing a much easier and more trivial example than the first two. Multiplication by ten has been introduced as a step along the way to using the grid and column method, making this example one that can readily be solved mentally. This can be done either by recognizing the column movement of digits when multiplied by ten or by using the ‘add a zero’ technique which Suzy has previously alluded to.
16 x 10 H T U
x 10 6 1 0 0
10 100 60 6 0
0 0 0 0
0
1 6 0 16 x 10 = 160
The fourth example is as follows: There are 45 trees in each of the 5 local schools. How many are there in all 5 schools? This example moves back to using non-zero digits, enabling the grid and column method to be used less trivially than in the previous example, but with only a units digit as one of the numbers in the product, this again represents a simpler example than those earlier in the lesson.
45 x 5 H T U
x 40 5 2 0 0
5 200 25 2 5
2 2 5 45 x 5 = 225
The fifth example is: There are 35 people in a running team. How many people in 17 teams? On examining the numbers used, this appears to be the most complex example yet encountered. There are four non-zero digits used in the calculation and none of them are repeated. Further, the digits include a 7 which represents one of the more difficult multiplication tables. The grid and column calculation is as follows:
35 x 17 Th H T U
x 30 5 3 0 0
10 300 50 5 0
7 210 35 2 1 0
3 5
5 9 5 35 x 17 = 595
The final example is as follows: Apples cost 89p each. How much for 34 apples? The lesson ends with another example using four non-zero digits and also the digits 8 and 9 which are amongst the more difficult multiplication tables. The calculation is therefore the most complex and perhaps appropriate for the final example.
The calculation is as follows:
89 x 34 Th H T U
x 80 9 2 4 0 0
30 2400 270 2 7 0
4 320 36 3 2 0
3 6
3 0 2 6 89 x 34 = 3026
This final example has a money context and so the final answer needs to be interpreted as a total amount, which is £30.26 rather than the 3026 which is the total in pence.
Knowledge Quartet Coding Commentary
Contributed by: Ray Huntley, Brunel University, UK
Knowledge Quartet Dimension: Transformation
Knowledge Quartet Code: Choice of examples (CUE)
Scenario: Suzy teaching multiplication methods
The teaching of grid multiplication, as discussed here from Suzy’s lessons, resonates with findings by Rowland (2008) in which he describes a lesson by ‘Laura’, who offered pupils a series of multiplication problems involving multiplying a 2-digit number by a single digit. Laura’s examples were intended to become progressively more challenging, however, it was remarked that many of her examples could have been solved using mental methods such as doubling. If there was an element of increasing challenge, then Laura should have been providing examples which required recall of products or examples where ‘carrying’ from units into tens or tens into hundreds was part of the calculation. Suzy’s examples above seem to provide evidence of carrying in many cases, but she randomizes the level of challenge which may not have been helpful for the children.
Summing up the examples used by Suzy, she appears from the plans to have an approach to choosing examples which is to some extent quite random, with examples moving from simpler to more complex and back again with no apparent purpose. The subsequent range of examples which she uses to demonstrate to the children shows that perhaps she is unclear about the purpose of demonstration examples in her teaching.
Suzy had stated in interview that she likes to; ‘pick out certain questions for them to do rather than just work through’, and then later she also said: ‘I’d try to mix it up sometimes’ and the evidence from the worksheet above seems to demonstrate that she does indeed mix her examples. In this case, it is important to note that:
It is very common for learners to identify concepts with one or two early examples that they have been shown by the teacher. Because these early examples are often simple ones, the learner is left with an incomplete and restricted impression of the concept (Watson and Mason, 2005).
References
Rowland, T. (2008) ‘The purpose, design and use of examples in the teaching of elementary mathematics’, Educational Studies in Mathematics, (69) pp.149 – 163.
Rowland, T. (2008) ‘Researching Teachers Mathematics Disciplinary Knowledge’, in Sullivan, P. and Wood, T. (eds.) International Handbook of Mathematics Teacher Education: Vol. 1, Knowledge and Beliefs in Mathematics Teaching and Teaching Development, Rotterdam, Netherlands, Sense Publishers, pp.273 – 298.
Watson, A. and Mason, J. (2005) Mathematics as a Constructive Activity: Learners generating examples. Mahwah, NJ, Erlbaum.