Formal Column multiplication from concrete beginnings
Russell Timmins, February 2017
The formal column method of multiplication is required as a statutory element of the National Curriculum. Pupils can and arguably should begin by looking at how and why this procedure works. There are many different types of tactile mathematics equipment but the best one for this, because it models the final abstract procedure exactly, is Dienes.
Start with a problem like 11 x 13 (chosen for this example as it is not a multiplication that pupils should know by year 4 and relatively straight forward to compute.) Dienes has single cubes for units or ones if you prefer, strips of 10 cubes for tens and 100 ‘mats’ which contain 100 unit cubes. The problem should be set up without writing anything. A pre formatted grid for laying the Dienes blocks on is a necessity however.
On the outside of the grid the Dienes representation of 11 takes the horizontal position as this is the number being multiplied ( the first number in 11 x 13) and the multiplier (13) takes the vertical position. Pupils should now consider the Dienes shapes that fit in the four grid sections beneath and to the right of the Dienes equipment set out already. Note that there is no computation being asked of pupils just logical application.
By the time pupils are taught this they should be familiar with most of their times tables and will know that 10 x 10 is equal to 100 and that Dienes has a shape that represents this number (often referred to as the ‘100 mat’) and they will also be steered towards ‘fitting’ strips of 10 and unit cubes in the appropriate space as in the diagram above. They will then be asked to mentally calculate the sum of all the shapes inside the grid ( ‘100 + 10 + 30 + 3 = 143’) This can be repeated with increasingly challenging multiplication but do not go above 19 x 19 as the imagery loses its effectiveness when more than one 100 mat is needed.
The next stage takes the pupil into the semi-concrete or visual representation which again requires no calculation, just a transfer of the Dienes representation onto a number grid. We will stay with 11 x 13 for this example.
You will notice that every element of the Dienes model has been translated to numbers in the grid to its right. This is ideally taught at this stage with both Dienes equipment and a number multiplication grid. As the understanding progresses the grid alone should be used as the multiplications will never be more difficult than the 9 x 9 family. For example when pupils are confident with the grid alone they can be asked to solve the multiplications 14 x 14, moving on to problems like 21 x 12.
You should notice the addition of the sums of the horizontally aligned boxes and their respective sums to the right. When confidence is sufficient pupils are now ready to complete the last learning objective, that is solving by using a formal column method. As with the link from concrete to semi-concrete, the same tactic is used here. Ask pupils, for example, to use the grid to solve 17 x 12 but at the right of the grid this time they must write the multiplication as we see it in a column display.
They now complete the grid as before and find that they have what looks like the column method of multiplication.
Questions to be asked should be ‘Tell me what I calculated to get 170 in the column method.’ If necessary at this stage pupils are able to refer to the grid at the side to see it is the result of the 17 being multiplied by the 10 part of 12 and from this it will become increasingly less difficult to explain where the 34 came from. From here we want pupils to use this understanding to solve problems like 15 x 13.
What is noticeable about arriving at this point in the learning process is that there has been no laying down of noughts when multiplying by a 10 and consequent carrying over of numbers greater than 9 in any result. There is also a reliance upon pupils using mental methods in the middle stage of the calculation. However, conceptually the journey from concrete to abstract understanding has not diverted at any stage, as the numbers written down in the final abstract column method, are exactly the same as you would see if you went back and solved with Dienes.
Russell is the author of our Maths mastery with greater depth Year 6 Teacher Resource. Learn more about the series!
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