## Student Led Learning in Maths

Student led learning seems to be a bit of a hot topic at the moment. I’ve seen these two graphics making the rounds on twitter, I can’t find them now, but one was a slide proclaiming why student led learning was better and one was the same slide but altered to say it was worse. This of course came with great debate from all quarters.

It is also certainly a much talked about issue in the teaching and learning meetings we are having in my school.

This morning we had a great presentation from a food tech colleague who described a fantastic lesson where students had been allowed to lead their own learning on the function of eggs in cooking. A number of different recipes had been provided (as diverse as Egg Custard and Chick Pea Burgers) and students were given the choice as to what to cook and asked to investigate.

This sounded interesting, so I began to think about the applications this may have in a maths classroom. While studying for my Masters I read a lot about group work and other pedagogical approaches to the teaching of mathematics. I found that there was a lot of evidence to suggest that, on the teaching of new content, “whole class teaching”, i.e. direct instruction, was the most effective method (Reynolds and Mujis, 1999). However, this approach can often lead to students being proficient in algorithmically following a process to achieve and answer – ie they can have an instrumental understand of the topic but not a deeper understanding of the underlying concepts. This can lead to issues when students encounter a question that is phrased in a different way or that requires a variety of mathematical topics to solve. (e.g Avital and Shettleworth, 1968, Davis, 1984 and Skemp, 1976)

This was an area that interested me and my dissertation focus was using group work and other problem solving ideas to deepen conceptual understanding at A Level maths. I found that with my cohort explicitly teaching problem solving approaches and then setting problems that required a variety of approaches to be solved in groups to be effective. Some real success was had when I used problems I had not encountered and as such was able to act like a member of the group while bouncing ideas around.

My findings backed up the work of others who had suggested problem solving as a good tool to deepen conceptual understanding. (e.g. Avital and Shettleworth, 1968, English and Halford, 1995, Hembree, 1992, Karp, 2004, Silver and Marshall, 1999, and Zeitz, 2006)

In the new maths GCSE we are seeing questions that are focussed on testing a deeper understanding using problems that require thinking about and often require a number of mathematical techniques to solve. This is a move away from predictable questions and as such, teaching methods aimed at giving algorithms to students to solve types of questions will no longer work.

One simple example is questions based on ratio. Previously ratio questions usually took one of two forms, use a ratio to scale up a recipe or split this amount into this ration. Both are easily solvable by an algorithm and I’ve seen this taught this was and correct answers given by students who don’t really know what a ratio is. Now we are seeing ratio questions that include other areas of maths, such as densities, as well as questions where the language is quite important and a better understanding of what is going on is required.

i.e. A student who is taught, “When you see a ratio you add, divide then multiply”, will get full marks on a question asking “Sana and Jo split £110 pounds in the ratio 6:5, how much does Sana get?” but may get nothing if the question asks: “Fred and Nigel split some money in the ratio 6:5, Fred gets £10 more than Nigel. How much does Nigel get?” Even though there is a comparable level of mathematics used.

This, I feel, is where group work / “student led learning” could be very effective in maths teaching. Once content has been taught students need to practice that content in new setting and to mix it up with other content that has been learned. Tasks need to be set and students need to be given adequate time to get stuck and struggle. This will build resilience and problem solving skills as well as allowing students to see where various strands of maths can be applied.

This ties in with something I read recently that Kris Boulton (2017) had written about the use of learning objectives. Kris argues that sometimes it is important not to use learning objectives as this tells students exactly what maths they need to be using to solve a problem. This is a big factor in this idea around problem solving and I would go further and say that it’s important not to set problems that involve topics you have taught in the last few lessons as this will have the same effect as having an objective such as “use Pythagoras’s Theorem to solve problems involving areas.”

I hope to write more about this in the coming weeks as I look to further apply the findings of my dissertation to KS3 and 4. My thoughts at the moment are that this “student led” approaches are good for the development of these skills once the core content has already been taught. There are, of course, many draw backs to group work and other student led approaches, but they are for another post for another day.

**Reference List:**

Avital, S.M. and Shettleworth, S.J. 1968. *Objectives for mathematics learning; some ideas for the teacher*. Toronto: Ontario Institute for Studies in Education.

Boulton, K. 2017. Whywe need to get rid of lesson objetives. 17^{th} April. *To the real*. [online] accessed 13^{th} July 2017. Available: https://tothereal.wordpress.com/2017/04/17/why-we-need-to-get-rid-of-lesson-objectives/

Davis, R.B. 1984. *Learning mathematics: The cognitive science approach to mathematics education*. London: Croom Helm.

English, L.D. and Halford, G.S. 1995. *Mathematics education: Models and processes*. New Jersey, United States: Lawrence Erlbaum Associates.

Hembree, R. 1992. Experiments and relational studies in problem solving: a meta analysis. *Journal for research in mathematics education*. **33**(3), pp.242–273.

Karp, A. 2004. Conducting Research and Solving Problems: The Russian Experience of Inservice Training. In: Watanabe, T. and Thompson, D. eds. *The Work of Mathematics Teacher Educators. Exchanging Ideas for Effective Practice*. Raleigh, NC: AMTE, pp.35–48.

Reynolds, D. and Muijs, D. 1999. The effective teaching of mathematics: A review of research. *School Leadership & Management*. **19**(3), pp.273–288

Silver, E.A. and Marshall, S.P. 1990. Mathematical and scientific problem solving: Findings, issues and instructional implications. In: Jones, B.F. and Idol, L. eds. *Dimensions of thinking and cognitive instruction*. Hilsdale, New Jersey, United States: Lawrence Erlbaum Associates, pp.265–290.

Skemp, R.R. 1976. Relational understanding and instrumental understanding. *Mathematics Teaching*. **77**, pp.20–27

Zeitz, P. 2006. *The art and craft of problem solving*. USA: John Wiley.

## An interesting area puzzle

Regular readers will know that I have a tendency to collect puzzles and I like to have a go at them. This evening I had a crack at this while my daughter was playing before bed.

It looked interesting when I saw it on Facebook a while ago and so I thought I’d have a crack. If you haven’t yet, do it now. I want to know if you took the same approach!

I wasn’t too sure where to start, so I drew it out, labelled some stuff and came up with some equations:

I thought if I multiples two of my equations together and rearranged I could get the yellow area as the subject:

Then I needed cx:

I thought “now I need ay”, then realised I had it:

This meant I could.sub back in through the equations:

So the area:

I thought 27 was a nice answer, and I’m fairly sure it’s correct, however I have a feeling that I may have missed something blindingly obvious that would have gotten me there much quicker.

*If you did it, I’d love to hear your approach, especially if you spotted something I missed!*

## A little circle problem

I’ve just seen this post from Colin Beveridge (@icecolbeveridge) answering this question:

Naturally I had a go at it before reading Colin’s solution. When I read his I found a lovely concise solution that we slightly different to mine, so I thought I’d share mine.

I started by just drawing a right angled triangle from the centre of the circle like so.

I seem to have cut off the denominator of 6 on the angle there. I know that the hypotenuse is 12-r (where r is radius) and the side opposite the known angle is r.

This means I can use the sine ratio of pi/6 to get

r/(12-r) = 1/2

Which leads to:

2r = 12 – r

Then

3r = 12

r = 4

Which is the same as Colin got.

I’ve seen questions like this on A level papers before and I know they often throw students, so I make sure I explore lots of geometry based problems and puzzles to combat this. I’d be interested to know which way you would approach this. Colin and I used a very similar approach, just differing in the point at which we introduced the 12. Which way did you do it?

## Proving Products

Just now one of the great maths based pages I follow shared this:

So naturally I figured I would have a go. I thoughts just get stuck in with the algebra and see what happens, normally a good approach to these things.

My first thought was that if I use 2n – 3, 2n -1, 2n + 1 and 2n +3 then tgere would be less to simplify later. I know that (2n + 1)(2n – 1) = 4n^2 – 1 and (2n – 3)(2n + 3) = 4n^2 – 9 so I multiples these together.

(4n^2 – 1)(4n^2 -9) = 16n^4 – 40n^2 + 9

I thought the best next move would be to complete the square:

(4n^2 – 5)^2 – 16

This shows me that the product of 4 consecutive odd numbers is always 16 less than a perfect square and as such that the product of 4 consecutive odd numbers plus 16 is always a square.

(4n^2 – 5)^2 – 16 + 16 = (4n^2 – 5)^2

A nice little proof to try next time you teach it to your year 11s.

## Dumb Pride

There’s this photo doing the rounds.

I think I saw it ages ago but it’s resurfaced recently and I keep seeing it and it’s been annoying me. I have no idea who this lady is, nor do I understand why she is so proud of being unable to do basic maths, but it got me intrigued about the papers so I downloaded them.

There are 3 papers. 1 arithmetic (40 marks) and 2 reasoning (35 marks each). I know what you’re thinking, I thought it too, 25 percent of 110 is 27.5 and there are no half marks so it’s impossible to get 25 percent. I’ve assumed the lady in question has correctly rounded her percentage to the nearest percent and therefore got 28 marks.

I’ve looked at what marks are given for. You get 1 mark for being able to tell the time on an analogue clock. 2 marks for being able to read a bus timetable. 2 marks for being able to shade 3/4 of a shape that is split into equal parts that are a multiple of 4. 4 marks for ordering numbers. 7 marks for being able to add and subtract whole numbers, 15 marks for being able to multiply and divide by single digit numbers. That’s 31 marks. The means she didn’t even manage to get all those marks. There’s 7 more for dividing by 2 digit numbers another 8 for adding decimals (with tricky questions such as 0.2 + 0.05) which she must have got nothing for. 2 for finding percentages of amounts, another for finding a fraction of an amount and this not to mention the marks for multiplying and dividing decimals by powers of 10.

If I couldn’t answer those questions, I don’t think I’d be shouting about it.

## Angles or Angels?

When I marked my year 11 books the other day I noticed that quite a few had been working that morning on “Angels in triangles’. This peturbed me a little, surely by Year 11 they should know the difference and be able to spell each one.

To counteract this massive literacy issue I played a game of “Angles and Angels”. I spoke to them first about the difference, then about the spelling and then did a show me activity where I showed them various pictures and they had to show me on their whiteboards if it was an angle or an “Angel”. I was impressed that they even got the picture of Kurt Angle, although none of them recognised David Boreanas…..

The activity led to a discussing with a couple of them as to why it was important to discuss these things in maths lessons. Stemming from the inevitable question “why we learning about this? It’s maths not English.”

I explained my opinion that we may be learning maths, but that literacy is important in all subjects. As a maths teacher I educate these students and literacy has to be a big party of that, as I hope numeracy is a big party of those subjects that deal with numbers but aren’t maths. I also expressed the importance of maths specific vocabulary, such as ‘angles’ and how it’s not necessarily going to be covered in English.

It is these sorts of things that we need to be thinking about, literacy wise, to ensure our students are in the best position when they leave.

## Infuriatingly impossible exam questions

Today I was working on some Vectors exam questions with my Y13 mechanics class and I came across this question:

A student had answered it and had gotten part d wrong. What he had done was this:

I have recreated is incorrect working.Obviously he had found out when the ship was at the lighthouse, instead of 10km away. I explained this to him and started to explain how he should have tackled this when a sudden realisation angered me.

Now for those if you that didn’t work through the question, here is the actual answer:

Can you see what had me infuriated?

This is an impossible answer! If the lighthouse is on the trajectory of the ship and it will hit said lighthouse at t=3 then that would stop the ship! At the very least it would slow it down!!!! In reality it would have to avoid the lighthouse and change trajectory. Meaning the second answer, T=5, would not happen under any circumstances!

My initial thought was:

“are they expecting students to spot this and discount the second answer? That’s a bit harsh.”So I checked the markscheme:

Nope, they are looking for both answers. Argh! I can understand using a real life context in mechanics, I really can. But why not check for this sort of thing!

What do you guys think? Is this infuriating or am I just getting get up over nothing? I’d love to hear your views in the comments or via social media.## Share this via:

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