## Unstructured problems

Currently I’m in the process of completing a dissertation based around problem solving in A level mathematics and how this can be improved. This is a focus as in our setting students have struggled with this in the past. It was timely, then, that the article picked for this week’s Maths Journal Club was around the same subject.

The article was by Sheila Evans and was entitled “Encouraging Students Formative Assessment skills when working with non routine problems”. Available here.

The article itself was interesting, it set out an approach to teaching based around these unstructured problems and designed student responses to get students talking and thinking about the way they are approaching the questions. The article seems to suggest that students with an instrumental or procedural understanding are less likely to succeed at this type of problem than those with a relational understanding, and that is something I’ve been thinking myself.

I think the approaches mentioned in the article sound interesting and I am going to tailor them to students and trial them in my own context to see if there is an effect.

It’s certainly an article that has got me thinking and has given me ideas for things to investigate in teaching, as well as signposting a raft of other pieces of literature that I want to investigate further too.

## How are we questioning our students?

This month’s maths journal club is based in the article “Contrasts in mathematical challenges in A – level mathematics and further mathematics, and undergraduate mathematics examinations.” By Ellie Darlington

I found the article quite interesting overall. It looks at the differences in examination questions between A level mathematics and undergraduate mathematics. It starts off with the idea that A level mathematics is tested in a manner that involves routine questions and that as such this doesn’t prepare students for undergraduate mathematics, which it presumes is tested in a higher level. I think this is one of the issues with A level mathematics and I hope that when the new curriculum appears this will have been addressed. The problem is even worst at the transition point between GCSE and A level though, but again, I have hope that the new specification will address this.

Interestingly, my own experience of undergraduate mathematics was that there were a lot of courses that were tested in a routine manner, and that learning the lecture notes by rote and practicing the past papers for a course could allow people to score well despite not understanding what was going on and not being able to apply their knowledge in other contexts. There were some of my peers who had no conceptual understanding of some of the modules yet still scored high enough to achieve firsts.

That said, I still feel that the procedural nature of the GCSE and A level papers is a massive problem. In recent years we have seen a change in the A level papers towards questions that are not answerable in a routine manner, but it needs to go even further.

There are many problems with these procedural questions. My main issue is they allow students to score well without understanding the mathematics behind the questions. This in turn can allow teachers to skip teaching for a relational understanding and just teach an instrumental or procedural understanding, which lets down the learners, especially if they are hoping to go into mathematics or another mathematical based subject at higher education.

**So what can we do?**

Well, rather than waiting for the changes we can be implementing these questions in our classrooms, ensuring that we are teaching for relational, or conceptual, understanding rather than teaching purely procedures. Take the time to ask the questions that require application in new contexts. Take the time to teach the concepts, the why behind the what. Enrich the curriculum with tasks that involving thinking outside the box and questions framed in a way that the correct method isn’t always immediately obvious, perhaps try some of these puzzles?

**Other points in the article**

There was a lot early on that I thought I already knew, but it was nice, and useful, to see references and studies to back up some of the ideas.

The MATH taxonomy in this explicit form is new to me and I’m interested to look further into it and see how I can apply it myself.

I was a little purplexed to see that the article stated that questions can change their position on the MATH taxonomy with time, but then have no explanation of how these questions were classified in the research.

*All in all a very interesting read that I will re-read and digest in more detail later. I’d love to hear your thoughts on it also.*

*This post was cross posted to the BetterQs blog here *

## Teaching to understand

I’ve been thinking a lot this week about understanding. There are different levels of understanding, there’s the deeper understanding (often referred to as Conceptual understand or relational understanding), then there’s a surface level understand (usually called instrumental understanding). The deeper understanding implies that someone understands why things are happening and a surface understanding implies that someone understands what to do in a given situation. This paper by Skemp (1976) is excellent on this topic.

A simple example might be if you have a student a simple equation: 3x + 1 = 10 . I’d the student is working with a surface understanding they may be able to solve it because they recognise that you “undo” the terms to find x. While a student with a deeper understanding would understand the role of the = sign and know why the inverse operations work.

As far as mathematical knowledge goes, a deeper understanding is far preferable to a surface understanding, but as far as passing a GCSE exam goes both of these levels of understanding will yield the exact same results, and as the surface understanding is quicker and easier to achieve a lot of teachers focus on this method to get their students their target grade or above. For many this is down to the pressures crated by the system of league tables and Ofsted gradings that drives a lot of the people involved in this profession. This is one of the problems with the system we currently have, and certainly suggests to me that many people have the wrong focus. Education should be about educating, not just teaching students to algorithmic answer questions that come up on a paper.

The current GCSE is set up in a way that you can get a B, or even an A, with poor algebra and trig skills if you are excellent at the statistics side of it and have memorised enough algorithms for the algebra, this means many with these grade struggle immensely when they attempt mathematics A level as they don’t have the pre requisite ability. The new GCSE spec will go some way towards combating that, but the focus on instrumental understanding to achieve the grades will mean that students are unable to make the links between topics and are unable fully grasp the follow on concepts as they haven’t fully understood the basic ones, and through no fault of their own.

This week I spoke to a head of maths from another part of the country and he was talking about how he’d improved his schools results by 25% at GCSE A*-C by introducing a scheme of work that focused on the big mark questions and a pedagogy that made the algebra easy- when he described said pedagogy it was purely algorithm based and promoted no deeper understanding. He was at a point where he could not understand why his AS results were poor, despite the cohort having strong GCSE grades. It’s a situation I’ve been in and seen before. The grades are a reflection of what questions they can answer, not what mathematical concepts they understand.

I spoke to another teacher this week from a different school in a different area of the country, and she had found that her students all got part b this C3 question wrong last year.

She had completed her question level analysis and decided that it was due to the students not being able to recall the rules involved and had set about creating mnemonics to help them. I disagree with her analysis. I think the inability to answer this question comes from only having an instrumental understanding of the concepts. If you understand that the range of a function is the “y” values it produces, that the nature of a turning point means it’s gradient is 0, and that the derivative of a function at a point gives the gradient of a tangent at that point then this is an easy 6 marks.

She went on to discuss a number of mnemonics shed come up with for C3 trigonometric identities. This seems particularly daft to me. Why would you create extra things to remember? If you have a deep understanding of the unit circle and right angled triangle trigonometry then the two main trig identities are obvious. *Sin^2 (x) + cos^2(x) =1 and sin (x)/cos (x) = tan (x) *

Once you know the reciprocal trigonometric ratios then the rest fall out from there.

I’ve seen A level students able to differentiate and integrate perfectly, but not be able to find the area under a curve or the gradient of a curve because they’ve never been told what these algorithms they are performing do.

I feel we need to move away from this, we need to encourage a deeper understanding or we are setting out students up to fail in the future.

*What are your views on this? I’d love to hear them. If also be interested to hear if this is a problem in other subjects or just maths. I’d imagine that it does occur elsewhere, but I’d love to hear confirm2 (or otherwise) of that, and specific examples of where it occurs if it does. *

**Reference**

Skemp, R. R. (1976). Relational understanding and instrumental understanding. *Mathematics Teaching* available here.

## A lovely trigonometry question

*This post is cross – published and can be seen on Betterqs here.*

I’ve written before about textbooks, they can be troublesome if used incorrectly but they can also provide a good amount of questions to allow students to get their teeth into without having a ridiculously high photocopying budget. A good textbook, in my opinion, is one with great questions covering a range of difficulties to allow stretch and differentiation when used properly.

Currently I have been struggling to find good textbooks for the A level syllabus, I think the majority of the textbooks out there for the current syllabus are a little rubbish and I hope that the ones being produced for the new board are better. *(If anyone wants to pay me a load of money I’ll write you a belter….)*

The ones we use are produced by our exam board and are one of the better ones out there but are still lacking. The questions tend to be straightforward, testing skills and not understanding and not really differing in difficultly. Imagine my surprise, then, when I came across this little beauty:

It was in an exercise on addition formulae and it really threw a number of my students. I loved the question and in the end we worked through it together on the board:

Once I’d railed at them about the importance of sketching they agreed that would be the best place to start. I made them sketch the triangle on mini whiteboards first to ensure they could then I sketched it in the board. They worked out the angle to be 60 minus theta and came up with the sine rule the selves, and then it was just a case of simplifying with the addition formulae.

This question doesn’t involve any overly taxing mathematics, but it does mix the skill being learned in with prior knowledge and as such serves to enhance the relational understanding of the students. They can see where the links are and they can build those links themselves. I think that this is a superb question and we need to be using questions like this regularly to build that relational understanding, helping our learners become mathematicians, and not just maths exam taking machines.

## Isosceles triangles and deeper understanding

When marking paper 3 of the Edexcel foundation Sample Assessment Materials recently I came across this question that I found interesting:

It’s a question my year tens struggled with, and I think it is a clear marker to show the difference between the current specification foundation teir and the new spec.

The current spec tends to test knowledge of isosceles triangles by giving a diagram showing one, giving an angle and asking students to calculate a missing angle. This question requires a bit of thinking.

To me, all three answers are obvious, but clearly not to my year 10s who do understand isosceles triangles. The majority of my class put 70, 70 and 40. Which shows they have understood what an isosceles is, even if they haven’t fully understood the question. They have clearly mentally constructed an isoceles triangle with 70 as one of the base angles and written all three angles out.

What they seem to have missed was that 70 could also be the single angle, which would, of course, lead to 55 being the other possible answer for B. One student did write 55 55 70, so showed a similar thought process to most but assumed a different position for the 70.

I already liked this question, and then I read part b:

Now students are asked to explain why there can only be one other angle when A = 120. Thus they need to understand that this must be the biggest angle as you can’t have 2 angles both equal to 120 in a triangle (as 240 > 180), thus the others must be equal as it’s an isoceles triangle.

The whole question requires a higher level of thinking and understanding than the questions we currently see at foundation level.

In order to prepare our students for these new examinations, we need to be thinking about how we can increase their ability to think about problems like this. I think building in more thinking time to lessons, and more time for students to discuss their approaches and ideas when presented with questions like this. The new specification is going to require a deeper, relational, understanding rather than just a procedural surface understanding and we need to be building that from a young age. This is something I’ve already been trying to do, but it is now of paramount importance.

There is a challenge too for the exam boards, they need to be able to keep on presenting questions that require the relational understanding and require candidates to think. If they just repeat this question but with different numbers than it becomes instead a question testing recall ability – testing who remembers how they were told to solve it, and thus we return to the status quo of came playing and teaching for instrumental understanding, rather than teaching mathematics.

What do you think of these questions? Have you thought about the effects on your teaching that the new specification may have? Have you any tried and tested methods, or new ideas, as to how we can build this deeper understanding? I’d love to hear in the comments or social media if you do.Further Reading:Teaching to understand – for there thoughts in relational vs instrumental understanding

More thoughts on the Sample assessment materials available here and here.

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