Scenario: Naomi teaching multiplication
Grade (student age): Year 4 (age 8-9)
Contributed by: Ray Huntley, Brunel University, UK
Context – national, curricular, professional, other
Naomi was an undergraduate pre-service teacher, and the lesson took place in a school-based placement towards the end of her three-year teacher preparation. She was working from lesson plans from the Hamilton Trust, a UK based group which provides training and curriculum materials in English, mathematics and science, including detailed unit plans for mathematics with objectives, teaching ideas, resources and so on.
In her lesson, Naomi sets out a number of examples so that children can use a grid method to multiply a 2-digit number by a single digit. The starter example is 34 x 2 which is presented in grid form so it can be calculated in a similar way to the doubling and halving of the previous lesson, that is, by partitioning into 30 x 2 and 4 x 2 then recombining. The main part of the lesson offers the children differentiated sets of multiplication problems, starting with this set for the lower attainers: 42 x 2, 42 x 3, 56 x 2, 56 x 3, 28 x 2, 28 x 3, 62 x 2, 62 x 3.
These examples offer some calculations with no regrouping, some with carrying from units into tens, some with carrying from tens into hundreds, some with carrying from units to tens and tens into hundreds. However, the sequence of the examples does not present the various alternatives in increasing difficulty. The common features to this set of examples are that they each have one of four 2-digit numbers (42, 56, 28, 62) multiplied firstly by 2 and then by 3, giving lower attainers practice with their 2 and 3 times tables.
The middle attaining group was given the following set of calculations:
42 x 3, 42 x 4, 42 x 5, 56 x 3, 56 x 4, 56 x 5,
28 x 3, 28 x 4, 28 x 5, 62 x 3, 62 x 4, 62 x 5.
The numbers used are the same as before, but this time each is to be multiplied by 3, 4 and 5. In this set of calculations, every example requires carrying. The highest attainers were given the following set of calculations:
42 x 6, 42 x 7, 42 x 8, 56 x 6, 56 x 7, 56 x 8,
28 x 6, 28 x 7, 28 x 8, 62 x 6, 62 x 7, 62 x 8.
The same set of numbers are used as before, with the multipliers being 6, 7 and 8 this time, which again means that every calculation includes some carrying from one column to the next, both for units to tens and tens to hundreds. The selection of numbers in these calculations again demonstrates a repetition of particular digits or numbers. This seems to be a feature of the examples provided by the Hamilton Trust, although it is not the scope of this work to analyse that scheme in depth. However, Naomi draws from it regularly for her examples and missed the opportunity to vary her examples.
Further work for the children came in the form of two differentiated worksheets, each containing a number of word problems with a multiplication calculation embedded within the worded context. The first example on the lower attainers’ sheet asks for the number of pencils in a cupboard if there are 9 packs of 5 pencils.
This is a realistic context and given that the numbers to be used are printed as digits in the question, it is likely that many children will be able to deduce that the solution is to be found by multiplying 9 x 5 to get 45 pencils.
The last two examples on the first worksheet change to a different style of question which is used throughout the second sheet, aimed at higher attainers, and so will be examined from that perspective. The higher attainers were given a sheet of examples that were of a more open nature.
For example, the first example is as follows: ‘In a classroom there are __ rubbers on each table. There are __ tables and 28 rubbers altogether. What different combinations of rubbers and tables could there be?’ The context is seeking a solution in which the missing numbers form the product 28 and so this could be 1 and 28, 2 and 14 or 4 and 7 (and with the corresponding commutative versions). However, the context is likely to be realistic for only some of those combinations. The 1 and 28 pairing cannot be correct since the question gives both tables and rubbers as plural quantities. The possible solutions are then:
2 rubbers on each of 14 tables – unlikely for a typical primary classroom
4 rubbers on each of 7 tables – possible if desks are grouped as tables
7 rubbers on each of 4 tables – again possible in a primary classroom
14 rubbers on each of 2 tables – not a likely situation in a classroom
Question 3 is based on the plants and leaves example from the previous worksheet, but expressed in the open-ended format as: ‘A school garden has __ plants and each plant has __ leaves. There are 30 leaves in total. What different combinations of plants and leaves could there be?’ This question again implies an equal number of leaves on each plant which is perhaps unrealistic and given the plural nature of each, the possibilities are as follows:
2 plants with 15 leaves each
3 plants with 10 leaves each
5 plants with 6 leaves each
6 plants with 5 leaves each
10 plants with 3 leaves each
15 plants with 2 leaves each
The question provides an opportunity to explore factors of a given number, although this is not the purpose of the worksheet from Naomi’s perspective.
Question 5 offers another example where factors could be explored but the context within the worded question is very unrealistic. The question is set out as follows: ‘At a fish and chip shop there are __ portions of chips sold and __ chips in each portion. 120 chips were sold in a day. What different combinations of portions and chips could there be?’ The factors of 120 provide the range of combinations, but many of these are likely to be unrealistic in terms of numbers of portions of chips sold and the number of chips in a portion.
Knowledge Quartet Coding Commentary
Contributed by: Ray Huntley, Brunel University, UK
Knowledge Quartet Dimension: Transformation
Knowledge Quartet Code: Choice and Use of Examples (CUE)
Scenario: Naomi teaching multiplication
The lesson in which Naomi is teaching multiplication seems to demonstrate that whilst she has a clear purpose in mind for her lesson, she is unaware of the limitations she puts on the possible learning by restricting her choices of examples. Naomi is using the grid method for multiplication, an area-based model which became a key strategy in the National Numeracy Strategy in the UK. In the opening examples, she draws on a limited number of possibilities for the children to practise the skill of multiplying a two-digit number by a single digit and whilst there is some useful practice of multiplication facts, the breadth could have been developed. One of the issues for Naomi to consider here would be the notions of ‘dimensions of variation’ and ‘range of permissible change’ (Watson and Mason, 2005). Naomi could have used a series of two-digit numbers that covered the full range of digits from 0 – 9, and this would have ensured that the children worked with the full range of multiplication facts for each single digit multiplier. However, within the possibilities, she would have been restricted in certain ways; for example, she could not have used a zero as the first digit of the two-digit number as this would effectively reduce the number to a single digit. This may have been a possible variation she could make at a later stage when the children were more secure with the process she was teaching them.
It is questionable, however, whether the children will really be demonstrating an understanding of multiplication from some of these examples, simply an ability to interpret word questions in such a way as to anticipate the likely calculation. This is another area where other research has been carried out, but this does not feature within the scope of this study.
As the examples continue on Naomi’s first sheet, some of the contexts appear to be either contrived or unrealistic. For example, question 3 describes a context where there are 3 plants in the school garden, each with 7 leaves. This suggests perhaps a very poorly stocked garden and an unusual collection of plants, given that the number of leaves on the plants is exactly the same in each case which is rather unlikely. Her choice of example could lead to some discovery learning by the children, outside the scope of the lesson objective, but if Naomi allowed this, she may have found that some children benefited enormously from the possibilities afforded by her choice of example.
Question 6 also raises issues about context reality and also solvability, presenting a situation as follows: ‘There are 12 boys and 5 girls in each class in a school. How many children are in the school?’ There are two issues here; firstly it is most unlikely that each class would have exactly the same gender split of 12 boys and 5 girls. Secondly, the example does not state how many classes are in the school and so it is impossible to calculate the total number of children.
Watson, A. and Mason, J. (2005) Mathematics as a Constructive Activity: Learners
generating examples. Mahwah, NJ, Erlbaum.