3rd May 2017 - Lynne’s logic - Division
Welcome back to my blog and today’s topic - Guzintas. Though not the spicy cuisine they may sound like, they can raise some learners’ temperatures as high as the hottest chilli. So what are these mysterious Guzintas? Division calculations. (As in ‘two guzinta six, three times’!)
They are a necessary evil, I’m afraid, as an ability to divide is essential to calculate with fractions and percentages, amongst other things.
Learners normally first meet division in KS1 but might still be experiencing difficulties well into Key Stage 3. So why the problems? I believe there are a number of issues responsible and I apologise here for the length of this blog, but it reflects the profound difficulties that are experienced by many learners.
Firstly, there is an assumption that learners will automatically understand and utilise the relationship between multiplication and division. This can often be seen by an emphasis on multiplication in commercial maths schemes, with a much-reduced lesson time spent on division, and this could be one cause of the problems experienced with division. It can be a dangerous assumption, particularly for the learner for whom multiplication is still causing problems. (Remember that multiplication uses numbers as a process, rather than as a finite quantity.)
Secondly, there is not an equal weighting given to the different concepts involved in division. At the primary stage, division calculations should reflect both the ‘sharing equally’ concept and the ‘grouping’ concept. However, there can be a tendency for an overemphasis in the early years on sharing equally. If asked to form a word problem from the number sentence 12 ÷ 4, the vast majority of learners would give you a ‘share equally’ scenario. For example, “If you share 12 sweets equally amongst 4 children, how many will each child get?” – an answer of 3 for each child. However, 12 ÷ 4 can also translate into “I have 12 sweets. If I give 4 to each child, how many children can have sweets?” I would suspect that only a minority would suggest a ‘grouping’ scenario independently.
Thirdly, the same calculation (12 ÷ 4) also reflects finding a quarter of 12, with division directly relating to a fractional part – a further complication for some learners.
Fourthly, the move towards written division calculations can rely heavily on remembering a process, without always developing the understanding needed to relate it to the specific layout. This problem can be compounded by the differing approaches of the chunking method and traditional short and long division methods. Care must be taken that the accompanying vocabulary reflects the written process used.
So what can be done to help?
- It is very important that younger learners understand both the sharing equally and the grouping concepts. Both of these can be shown with apparatus in the numeracy activity, but ensure you make it clear how the answer can be found. If the linked recording is a horizontal calculation, the learner will see that both concepts share the same number sentence but can look very different represented with apparatus.
- Ample time needs to be spent unpicking the concept of division using apparatus, which will develop full understanding so that - when ready - the relationship between multiplication and division can be fully understood.
- When appropriate, widen the learner’s understanding by including the relationship between division and fractions. Using apparatus will clearly show that when finding 1/3 of 15, you are sharing 15 equally between 3, or finding 15 ÷ 3. Support learners fully with the vocabulary used.
- If we are aiming to ‘catch up’ learners as close as possible to their peers, then older learners will benefit enormously from the one-to-one support which Catch Up Numeracy can provide with linked recording of division calculations.
So what about written calculations? A quick look at the two written methods…
Some schools will still be using the chunking method of division, which does have some advantages:
- It is ‘relatively’ easy to understand, as it uses a repeated subtraction of the ‘grouping’ structure.
- When you ask, for example, “How many 7s do you think you can get from 86?” this is what the child actually looks for.
- The learner has the choice of the groups subtracted. Usually, they will use 10 lots, 5 lots, 2 lots or 1 lot (of 7 for example) as they are more confident with these tables. This will result in a more fluent calculation.
- The process is the same no matter the size of the numbers – there is no short and long format as in the traditional algorithm. So a learner could find the answer confidently to a division involving large numbers, such as 3718 ÷ 17- though it might take a lot of paper!
So what are the disadvantages of chunking?
- The main disadvantage, and it is an important one, is that the quotient (the answer) if not exact, will result in a remainder. It does not move into decimals as easily as standard long division does.
So now a look at the traditional methods of short and long division, particularly the long method which has made a return to primary education in some schools.
Firstly, the advantage - only one, but it is major:
- It extends very easily into decimals so that the learner can achieve a quotient without a remainder, or can see that there is a recurring element (as in 10 ÷ 3, for example.)
Now the disadvantages:
- It is usually taught as a process, which needs to be memorised. For struggling learners, this can be a problem.
- The language used can cause confusion. Consider 154 ÷ 6, for example. Often the adult will support with “How many 6s in 1, there are none.” But probably - for their entire school career - the learner has been told that it is not 1 in 154, it is 100, and they might be questioning internally that rather than no 6s in 100, there are, in fact, many.
- There is no option but to use the multiplication table which is the divisor; 6 in the example 154 ÷ 6.
- Getting the correct answer depends totally on there being no errors in the written layout, and here can be the issue.
So what can you do to support written short and long division?
- To avoid using language which might confuse (see above,) then the language should be based on the sharing concept. I have supported the calculation with play money in £100s, £10s and £1. In the example 154 ÷ 6, you would need 1 x £100, 5 x £10 and 4 x £1. “Can we share the 1 x £100 equally between 6 people? No, then change the £100 into 10 x £10, so we now have 15 x £10. Can we share these 15 amongst 6, yes they will have 2 each with 3 left over. Now change the 3 x £10 into £1s. Added to the 4 we have already that will give 34 x £1. Shared between 6 people, they will have 5 each with 4 x £1 left over. That gives a total of £25 each with a remainder of £4. So 154 ÷ 6 = 25 r 4.”
- When the learner is ready to move into decimals, the £4 left over can then be changed into 40 x 10p to be shared amongst 6. And so on.
- Many adults use the DMSB acronym to prompt the correct layout for long division. Divide, Multiply, Subtract, Bring down. The DMSB can stand for anything which is relevant to the learner, such as ‘Dad, Mum, Sister, Brother’.
- I believe that you need to include after ‘Divide’ an A for Answer, as it is vital that a number is placed in the answer box above every number being divided. This could of course be a 0, (as it would be above the 1 in 154 ÷ 6.) So, for me, DAMSB is a better acronym. So you could use, for example, ‘Daffodils Are Mainly Spring Blooms’ or ‘Double Algebra Makes Super Brains’!
And breathe. That is, if you are still with me!!
Division is a tricky beast to master and does need lots of practice if it is to be embedded. I often hear, “Just teach the method,” but - from experience - I know that some learners also have memory issues and will not remember the process. So ample support with apparatus is essential to try to achieve understanding, and in getting the learner to understand that the two concepts of sharing equally and grouping are interchangeable, and will give you the same answer.