We have learning walks at work where everyone (I think!) is encouraged (it’s probably ”expected” but I haven’t done it!) to pop into other people’s lessons on a regular basis – to be honest it sounds worse than it actually is and ever since my NQT year when I hated people coming in I don’t have an issue with it .. they will see what they will see. I’ve almost become a bit immune to it but I also think, for me anyway, it’s been about conditioning myself to accept it. My classrooms have always been next to, or opposite Seagers and he was constantly in and out and I now teach in what is called “the learning hub” (it’s awesome!!) which is right next to reception and so is very visible. If people want to see a “showboating” and “jazz hands” lesson then they’ve come to the wrong place … I can deliver that if I have to but personally I think that it’s not a good measure of what makes good teaching. I genuinely believe that being able to deliver consistently good lessons is outstanding … that’s also not to say I think “outstanding” is a decent measure because measuring teacher effectiveness based on a ticklist is very subjective and almost impossible IMO.

In the Autumn of my NQT+1 the school I was at, was inspected (it was also a “national challenge” school so the stakes were quite high!) and I remember talking to someone about what I’d be teaching on day one of the inspection and was given all sorts of advice (at the time I thought that they felt I was going to be the weakness in the department!). I ignored all of it and went with what I would usually do in my lessons with my groups. In those days, lessons were graded by inspectors, so at the end of the day I went to get my feedback and the guy asked me what I thought of the lesson and I’d said “yeah it was OK, the kids got what I wanted them to, but there’s always something that I’ll change next time I do that topic” and the inspector then said “I wouldn’t change a thing … it was outstanding” …. I’m not sure he expected what happened next when I threw my arms around him and hugged him!! I’m not regaling the tale to boast but telling you because I still chuckle at this poor guy getting hugged by me. I think I did it because it was such a relief to know that I hadn’t let the school down. I’m not saying this pressure is right (it’s not!) and looking back, I now know it was stupid thinking it but at the time there was soooo much pressure to show that the kids were getting a decent deal. Having gone through another inspection in Jan (not sure I’ve even written about that!!) I can say that the process was different in my current position – as a classroom teacher it was almost painless!

Bloody hell … I’ve gone right off on a tangent … The point about changing things for next time I teach a topic is still true today … and every day! Each time you teach something the dynamics of the group are different, their prior knowledge is different, their ability to recall the essentials are different but also the misconceptions they will unearth/reveal are also different and it’s not until you are in the moment that some of these become obvious. Yesterday’s lesson was one of those times … and yes I was “learning walked” (I’m sure it’s not really a verb!) … but it was also a lesson which triggered me thinking about the language we use in maths and it being so obvious as to why students get certain terms confused.

The lesson was about fractional indices (and maybe negative fractional if I was going to push my luck!) and during the starter we’d recapped some basic “sums” and also applied the laws of indices we did last week so I could check the recall of previous work, specifically relating to the lesson. I then used several examples like the ones I’ve shown below … trying to make the link that 4 to the power of one half is the same as the square root of 4.

We got this quite easily and also then moved onto powers of one third (i.e. cube roots) … it’s quite an abstract concept but it was accepted that powers of 1/3 and cube roots are the same thing but the kids struggled with relating cube numbers back to cube of that number. The basic recall of cube numbers and their respective roots just wasn’t automatic (as I expected it not to be … I know these kids pretty well now) and we spent some time looking at square and cube numbers and their roots as inverse operations ( I even threw in a little game of hangman to unveil the word “inverse” .. nice little literacy touch!)   … there were several things that I was able to draw out of the lesson in terms of sources of confusion and it got me thinking about a conversation I had with Steve Lomax (of Kangaroo maths/Glow maths hub fame!) many moons ago about the numbering system and the use of twelve, thirteen or even twenty-four etc. The fact that the English language means that students must acquire additional terms for these numbers is so very different to some other languages where the same numbers can be literally translated as: one ten and two ones, one ten and three ones or two tens and four ones. Whilst this makes sense when you think about it – this extra language acquisition takes time – and in fact has been a discussion point for over 200 years, when Edgeworth and Edgeworth (1798) suggested that “English-speakers might be at a disadvantage compared with speakers of other languages due to the relatively irregular English counting system.” More recently, (Fuson and Kwon, 1991; Miller et al., 2005; Ng and Rao, 2010) have suggested that the superior arithmetic performance of Chinese and other Asian students could be explained by the relative linguistic transparency of many Asian counting systems, and … well blow me down!!! .. it has even got a name – > Chinese Number Advantage.

In the name of offering a balanced view, let’s look at a more recent study (2015) by Winifred Mark  and Ann Dowker called: Linguistic influence on mathematical development is specific rather than pervasive: revisiting the Chinese Number Advantage in Chinese and English children. In this study they looked at Chinese and British primary kids in Hong Kong and the UK (n = 49) but what’s interesting is that in Hong Kong they were able to add a different facet to their study by having two groups due to the fact that kids can be taught maths either with regular counting systems (i.e. Chinese, n = 47, referred to in the study as HK-C) or irregular counting systems (i.e. English n = 43, referred to in the study as HK-E). If you want to check out the full paper (its here and I have to say it’s a relatively easy read!) In summary the results “indicated that students in HK-C were better at counting backward and on the numeric skills test than those in HK-E, who were in turn better than the UK students. However, there was no statistical difference in counting forward, place value understanding, and a measure of arithmetic”.

They concluded that “children who were learning mathematics in Chinese were better at manipulating the number line than those learning mathematics in English, whether English be their first or second language.” However, whilst “linguistic transparency in number representations might facilitate place value learning in young children” they go on to say that “such an advantage is neither sustainable nor necessarily translated to better arithmetic performance in older children.”  They are also very pragmatic and suggest that the evidence is not sufficient to demonstrate that CNA can explain the cross-national differences in arithmetic consistently demonstrated across age groups and even go onto suggest that “general educational and cultural differences are at least as important as linguistic differences”. They openly discuss the fact that the HK-E students could also very well have been “advantaged by their exposure to Chinese counting at home.”

Arghhh … there’s that bloody tangent again!!. Back to the lesson … all of this has got me thinking about the use of the language in my lessons, specifically about powers and roots and this in itself being a cause of confusion:

  • We use “squared” instead of power of 2 and “cubed” instead of power of 3 yet when we get to higher orders we use “to the power of … “. Similarly, we use “square” roots and “cube” roots and then nth roots for higher orders. I get why we do this and the links to area and volume or being able to show “dots”, visual images or even manipulatives to make up the square and cubed numbers is all very useful but I do wonder if we make the jump to using “squared/cubed” when it comes to powers/indices too soon. I’m not saying we shouldn’t use these terms but I think that maybe consideration should be given to when the transition to using them happens and that shouldn’t be until the relationship between raising a number to a power and its inverse of taking roots is fully understood.
  • We use the radical symbol for taking roots but convention means that when taking square roots we don’t need to write the index number “2” and I’m suggesting that maybe we should make sure that we do … especially in the early acquisition phase! By excluding it we are suggesting that there is an “irregularity” to maths … i.e. in some circumstances we do “this” but in others we do “that” and then we wonder why our students get confused. It’ll also remind students that they are looking for a number that multiplies by itself two, three or four etc times.

The language we use in the classroom is so important and I just think that we need to think carefully about whether or not we are adding to the confusion some students experience – it needs discussing! . So this post is just something for you to think about! I know I am!

PS: I’ve done some reading about the origins of the radical symbol and WAS going to write about that … I obviously haven’t! that means that my “unposted” blog count has just gone up by 1! Bugger! Bugger! Bugger!