At BCME over the Easter break, just before the open meeting of the Teaching Committee (TC) of the MA (oh yeah … exciting/scary news -> as of April I’m now the chair of the TC and am a little scared about following in Rachael Horsmans shadow! She’s done such fab job!!) … anyway, back to the point … I was handed a slip of paper by a guy sat opposite me. With a grin on his face, he asked me why don’t we teach dividing fractions like this? On the slip of paper was something like the below:
I initially wondered if it worked because it was a special case. It’s not! We also discussed possible reasons for not teaching it this way … one possible issue would be because of the 1/2 in the denominator and a fear of knowing how to explain it.
Several of us around the table then spent some time discussing other ways that they teach dividing fractions and it was an absolute eye opener. It really got me thinking about the way I’ve taught it in the past but also about why “we” each teach it the way we do and whether the way we choose to teach it is actually efficient or even makes sense. I googled “dividing fractions” and almost every explanation, worksheet or video makes reference to “keep, change, flip” … Have we chosen to accept that its an “abstract” topic and that students just have to accept it? I flipping well hope not! Personally, I’ve always taught it starting with the idea of 8 divided by 2 being “how many 2’s in 8?” and then with the use of pictures develop this through something along the lines of 8 divided by 1/2 means “how many 1/2s in 8?” and then introduce the idea that 8 is the same as 8/1 and lead students to the conclusion that we are effectively multiplying by the reciprocal of the second fraction. I’m not convinced it’s ideal but that’s ok right? The fact that I am reflecting on it and prepared to look to improve is ok, right?
As part of this journey I got thinking about how much we assume the students come to secondary knowing from primary schools …. check out the KS1/2 programme of study that refers to fractions …. I think its really important we understand ANY prior knowledge and what they do before they come to secondary is really important. This is really interesting but also something I’d suggest you need to be familiar with if you teach year 7 …
This whole thing triggered me to start a thread on Twitter (DO CHECK IT OUT!!) -> HERE to gather some more ideas and input from a wider audience. Thank you to everyone that replied .. I think I’ve got my head around what I’m going to try differently next time I teach this from first principles (It could be a while as I only teach years 9/10 and 11) … you’ll have your own opinions and this is based on the students I teach (ITS ALL ABOUT CONTEXT!!) but I’ll probably go with the following:
- All four operations in fractions need to be done with the same denominators initially.
- We’ll look at more efficient methods later on but initially I want to get away from the idea that for + and – we make sure the fractions have common denominators but for x and dividing we don’t do this. The idea of having different “routes” for different operations is what I want to move away from in the short term. I will however make sure they are REALLY good at finding common denominators and fraction equivalence.
- I need to make more use of manipulatives. Rather than launch into a list of rules for each operation with just my pictures of the fractions, I need to get students to be able to see what is “actually” going on when we divide fractions. What does it mean to divide? I’ve always found that division is a weakness in year 7 anyway (less so in the last two years that I’ve taught year 7 than previous years though!) so that needs to be really secure before tackling fraction division. I also think I need to slow down and allow students to picture what is going on in their own minds by carefully choosing questions that aid this. This picture from Clare Hill shows how numicon type shapes could aid in helping students to deal with this concept on a concrete level – When asking permission to use the picture Clare went onto say “the kids really get it – and can choose the right piece of numicon AND can get the right remainder but lower prior achievers find it almost impossible to move to abstract” which I think is always going to be an issue but that’s a whole other blog post! I just love this idea for introducing the “concrete” aspect of dividing fractions.
I think the approach I’ll take is pretty much summed up in the below tweet I received (adding in more use of manipulatives). Remember … you choose how to teach your students and I’ll choose how to teach mine. Don’t be hating on me!! I just think its interesting to discuss methods and find ways of refining how I teach and this post isn’t about saying this is the “best” way its just one way and the beauty of teaching is that you get to choose what works for you!