This year I have the privilege of teaching top set year 11 – a group of very bright mathematicians. Occasionally though, they’ll come out with something that makes me cringe. This week it was their method for adding fractions. At the start of the lesson I put this on the board:
and asked the students to add it together. Very quickly they held up their whiteboards with the correct answer:
‘Great’ I thought – no problems with that, and just to check I asked a student how they’d done it.
“Easy,” he replied, “just use the upside-down picnic-table”.
For those who are unfamiliar with this idea, the upside-down picnic-table is the algorithmic method of adding two fractions together by simply multiplying top-left by bottom-right added to bottom-left multiplied by top-right, all over the product of the denominators – so-called because when you draw what you’re multiplying, you end up with what resembles an upside-down picnic-table; we’d probably refer to it as cross-multiplying!
Now, I despise this method, and told the students as much. They asked why (fantastic – I like students to question what I’m telling them!).
I explained to them that I disliked it for a couple of reasons – one, that students often don’t understand why it works, and two, that it often results in students doing more work than necessary. They weren’t convinced by the first reason – they seemed happy that they knew how to add or subtract fractions, so I gave them an example and asked them to add:
They soon showed me whiteboards with:
I then asked them to make sure they simplified their answer. A couple of moments later 29 students held up whiteboards with this answer:
When I asked them if they could simplify it, they said “no, because we can’t factorise it”.
I then suggested that they looked at the original question and thought about what they needed to do to the two denominators to make them equal. One student very quickly pointed out that they right-hand denominator was the difference of two squares, and so if he multiplied (x+1) by (x-1) the denominators would be equal.
I then asked them to do that with the left-hand term:
They then all simplified it to:
We then did several questions similar to this, and within ten or fifteen minutes the students were convinced that this method was the best method.
Cross-multiplication, the upside-down picnic-table, whatever you call that method, is an easy method to teach, but does it promote understanding? When taught that way, students can very quickly gain procedural understanding of how to add fractions, but I don’t think they truly understand what they are doing – instead they are simply applying a method that they have learnt parrot fashion. This method will very quickly fade with many students – and with those where it doesn’t, they’ll find themselves getting stuck when faced with harder questions, as illustrated above.
The same is true for other “quick” methods, for example:
Methods like this do have a use – for last minute revision when all other methods have failed. However, when teaching fractions lower down the school – in year 7 and 8 for example, I think it is really important to teach methods that develop the understanding.
I would start off using diagrams and a fraction wall and making sure that students were equally happy seeing fractions such as one half being equivalent to two quarters, or three sixths etc, before showing them an addition method that relies on them finding a common denominator first.
Whichever methods you do use – ask yourself the question: “Am I teaching for understanding, or am I teaching a quick fix?”