Saturday, 13 July 2019

Teaching Exact Trig values

***Warning - untested idea alert***

With the advent of all GCSE pupils needing to know exact trig values for 30, 45 and 60 as well as 0 and 90, a lot of people have been searching for a way to make these accessible for Foundation pupils. This came to the fore again on Wednesday prompted by this tweet from Drew Foster:

Now this is interesting for me. I have typically taught this using the two standard triangles:
(Image from Don Steward - Median Blog)

However, I have been know to resort to some of the "tricks" contained in the original tweet, particularly for Foundation pupils or pupils closer to the exams. I know that understanding on its own is not enough to lead to memory, and so I justified this in terms of helping remember. This was particularly needed because, even with kids who understood how the triangles worked, is that they often forgot to draw the triangles, or what the triangles looked like. Instead, they would supplement my teaching by endlessly drilling themselves on the values, usually found in table form.

Now I don't necessarily have a huge problem with this, as it will allow them to access most standard questions. Of course, these days, the concern is around the non-standard questions. When one of those pops up, pupils will likely struggle if they don't have the necessary flexible knowledge with working out exact trig values from what they already understand about trigonometry.

I was mulling this over, and a possible approach came to me. Because it literally only came to me on Wednesday, I haven't had time to trial it yet, but I will be teaching Year 10 set 2 next year, and I think I have it straight enough in my head to try it with them (hopefully this blog will help with that, and I welcome feedback).

The idea centres around motivating the drawing of these triangles by linking them back to a unit circle type definition of the two major trig functions. I will start with a type of inquiry prompt based on this triangle:

And the prompt will be, "What angle is needed, with a hypotenuse of 1, to make the vertical side equal to ½?"

I expect the pupils' first wrong suggestion will be 45˚. At least I hope it will be. Because I want to use it to motivate looking at 45˚ later on. For now, when/if it comes up, I would want to deal with this through isosceles triangles, and the triangle inequality - something along the lines of "If θ is 45˚, then the other angle will also be 45˚, which means both would have to be ½. What does that mean for the triangle?" How much of that I will tell pupils, and how much I will prompt/look for pupils to recognise I will decide in the moment.

Having dispensed with this, the problem becomes a bit more interesting. We could do a bit of calculator trialling, I haven't made up my mind yet. Whether we do or not though, and whether pupils find the result or not, I want to move to justification. The justification for this would come from reflecting the triangle in the horizontal side, giving this picture:

This of course leads directly to θ being 30˚ , as the triangle is an equilateral triangle.

Now of course this is very similar to the way that others would introduce the same idea. I think the difference is the focus of the approach. Previously I would have introduced this by first introducing the triangle, and using it to prompt find sine, cosine and tangent of 60˚  and 30˚ . The question of course is "why these angles?". Just because they are the ones that arise from an equilateral triangle? I think this is difficult for pupils because right-angled triangles and equilateral triangles are not well associated prior to this, particularly in the area of trigonometry. Instead, this seems a more natural question to ask - what angle makes the opposite half of the hypotenuse? It means not having to remember to draw an equilateral triangle of side 2, and then generating something useful,but rather actually drawing what you want to find, and then deducing it.

From here it would seem natural to ask about the length of the horizontal side (knowing it can't also be ½), and what that implies for the trigonometry.

But what about the other angles of 45˚ , 0˚ , and 90˚ ? Well I feel like this can lead naturally to those as well. As I alluded to earlier, I suspect the issue of 45˚ , will have come up already, and so it would be there for us to go back to: "What if the hypotenuse stayed 1, but the angle became 45˚ ?"

Which can be approached in the usual way using a bit of Pythagoras to show that sin 45˚  = cos 45˚  = 1/√2, and that tan 45˚  = 1 (because it will be something divided by itself).

As for 0˚  and 90˚ , I honestly can't see a way of arriving at them naturally, unless we first want to explore 15˚  and 75˚ . This I can see being justifiable, to continue the pattern that 30˚ , 45˚ , 60˚  would suggest. Unfortunately, I can't see a nice way of arriving at these without at least knowing some stuff about trig for non-right triangles. However,  I think the same triangle with a hypotenuse of 1 can at least be used to give an intuitive understanding of the exact trig values for 0˚  and 90˚ .

We can ask questions like "What is going to happen to the opposite/adjacent sides as the angle gets smaller?" Pupils should be able to see that the adjacent side will get close to the same length as the hypotenuse (i.e. 1) whilst the opposite will get very small. This of course is a pre-cursor to the formal idea of limits, and this can later become the limit as θ → 0˚ . Similarly, we can then switch it around and ask "What will happen as the angle gets bigger?", the limit as θ → ∞. Again, pupils should be able to see that the opposite will happen, and that the opposite will get close to the hypotenuse, but the adjacent will get very small.

Like I say, I haven't tried his yet, so if anyone wants to make suggestions for how I could make this better, do this in a way to maximise success etc then please do give me a shout.

Wednesday, 10 July 2019

East Midlands Maths Hub Joint Conference

Hi all! Lot of conferences recently! Last week I was lucky enough to present at the joint East Midlands South, West and East Maths Hub conference. I did a session on using representations (surprise, surprise!).

People at the conference suggested they would like to the slides to that session. They can be downloaded here.

Hope it helps!

Thursday, 27 June 2019

NW3 Maths Hub Conference

This Wednesday I was lucky enough to deliver the closing keynote to the Wigan NW3 Hub Conference at Haydock Racecourse. I absolutely loved the chance to mirror the development of an operation through a counters game, before exploring the importance of making sense of mathematics through the power of multiple interpretations of a concept.

The slides from my session are here (including the correct formula - well I hope so anyway!).

Thanks to Lindsay Porter for inviting me to speak (and giving my a lift back to the station) and to Jen for picking me up from Bryn station.

Tuesday, 25 June 2019

SEND Conference from LIME/Maths Hubs

On Monday 24th June I had the privilege of presenting the closing session at an excellent event hosted at the Ashton-on-Mersey school. This was primarily for teachers of pupils with SEND. The main focus was on the use of manipulatives to support mathematical understanding - a personal favourite topic of mine.

Many delegates suggested they would find the slides useful, so I have made them available for download here.

I should give a shout out and offer thanks to Louise Needham for asking me to speak at this fantastic event.

Tuesday, 18 June 2019

Putting the "Theory" into Cognitive Load Theory

These days we are hearing a lot about Cognitive Load Theory. But what does this actually mean? Well to understand this it is worth reminding ourselves about what it means to be a theory in science.

A lot of people see the idea of a "theory" as something that is somewhat uncertain. This is often the use in everyday language - if someone has a "theory" about something, it often means they have no more than a vaguely plausible explanation for it.

A scientific theory is different though (or at least a good one is). A good scientific theory should broadly aim to do two things:

1) Explain observed phenomena
2) Predict the outcomes of other observed phenomena

This is what Cognitive Load Theory tries to do. It tries to explain phenomena about how/when the brain forms memories that have been observed, and predicts what might happens in certain circumstances. For example, it has been observed that people find it difficult to remember content if they are reading text at the same time as someone is talking. Typically people in this situation will not be able to answer questions about either the text or the content of the speech. CLT explains this by suggesting that the brain processes text in the same way as speech (in a way, you "hear" the words in your head) and that the brain only has one "channel" for processing auditory input. Trying to process two inputs through your "phonological loop" results in cognitive overload.

So what happens when a prediction goes wrong? What happens when CLT predicts a different outcome? Well the same as what happens when any other scientific theory predicts something incorrectly - either the theory is modified to include the new observation, or if it can't be modified sufficiently then it is deemed incorrect. However, incorrect theories can still be useful. A prime example of this is Newton's theory of gravitation.

Newton's theory of gravitation is wrong. It definitely doesn't adequately explain how gravity works in all cases. This was known in the 1800s, as Newton's theory of gravitation was slightly out in predicting the correct orbit of the planet Mercury. Einstein's general relativity is a better model. Its predictions are more accurate, and more applicable. However, in most cases, Newton's theory is still used. Why? Because it is much simpler. The equations that accompany Einstein's general relativity are absurdly complicated. If you are talking about black holes, or getting close to massive bodies in the universe, they are essential. But for most situations, the equations associated with Newton's theory do just fine. They predict to a high level of accuracy the gravitational forces between bodies. Newton's theory was used to put man on the moon.

So what does this mean? Well if we apply the same sort of ideas to Cognitive Load Theory, what it means is that CLT may well make incorrect predictions, particularly in extreme cases, but that doesn't necessarily mean that the other predictions it makes are automatically wrong, or that they can't be useful. But it also means that if you are going to try and apply the ideas within Cognitive Load Theory then it might be useful to remember the following:

1) CLT may well not a complete theory of cognition, and it may well produce incorrect predictions. This doesn't make it worthless.
2) If you are applying CLT, make sure you read information about the studies that supported aspects of the theory. This will give you a greater appreciation for how useful/accurate its predictions might be for your context.
3) Cognitive Load Theory may well support in your pupils converting more of what you teach into long term memory, but that also means you have to make sure that the memories you are getting your pupils to form are the right memories. CLT can't tell you how to teach the content of your subject so that the connections between topics become apparent - that is part of the knowledge/skill (contentious!) of the teacher.

Monday, 17 June 2019

Research Ed Rugby - Mathematics Teaching for Mastery Using Rosenshine's Principles and Cognitive Science

This Saturday I attended the truly excellent ResearchEd Rugby. Despite all sorts of problems with trains from London to Rugby, Jude Hunton (organiser) got a fantastic line up of speakers together for sessions on Research, Leadership, Maths, English, Science among others.

As part of the maths strand I was speaking in the afternoon. The thesis of my talk was that the NCETM approaches outlined in their Teaching for Mastery program, Rosenshine's Principles of Instruction, and some of the effects proposed by Cognitive Load Theory are the same ideas, discussed using different language.

It is worth reminding ourselves about how these three came into being. The NCETM ideas of Teaching for Mastery came from looking at "high performing jurisdictions" and their practice, as well as the research that underpins their approaches, and suggested what might have the most impact in mathematics education in this country.

Barak Rosenshine derived his Principles of Instruction by examining individual high performing teachers, and the common practice that they share. This of course was not specific to mathematics teaching.

Cognitive Load Theory is a theory for how the brain forms memory, and things that support forming of memory, based on experimental data. Of course, the point of a theory is that it explains observed phenomena, and predicts the outcomes of future phenomena. So CTL aims to explain things that have been observed about memory formation, and predict what might help memory formation in the future.

The three "competing" theories can be shown using these three images:

I contend that many of the ideas from these three sources actually significantly overlap, and in some cases are indistinguishable. Take this example from my presentation:
On the left hand side is a picture of the expression x2 + 5x + 6, which then shows how this picture can be rearranged into a rectangle, which shows the factorisation. This can be physically shown using concrete or virtual manipulatives. Then there is an expression on the right hand side which is intended for the learner to factorise (given access to the concrete version of the manipulative).

The title for the slide was:
The proposition here is that this could be considered Modelling, if thinking about Rosenshine's principles, or it could be considered Representation if thinking about NCETM Teaching for Mastery  approaches, or could be considered the use of the Worked Example Effect if thinking about Cognitive Load Theory.

I highlighted several other examples throughout the session:

This activity is one that can be used to support developing Fluency, which is also an example of independent practice and also uses the Goal Free Effect.

This activity prompts Mathematical Thinking, it can be seen as using the Expertise Reversal Effect, and can be used to provide Scaffolding for Difficult Tasks. Note; if you are going to use this task, an important point is that learners should aim to make the minimum change possible from the starting shape in the middle.

The point of course is that these are not competing ideas at all. There is something to be gained from all of them, particularly where they actually say the same thing.

*Hat tips to the Learning Scientists and Oliver Caviglioli for the posters, Jonathan Hall and his website for the virtual manipulatives, Open Middle for the open box problem, and Professor John Mason for the Area/Perimeter activity.

*The full presentation can be downloaded by clicking this link.

Wednesday, 29 May 2019

Reasoning, Problem Solving, Interpretation and Fluency

So as part of my holiday I was listening to Craig Barton's most recent podcast with US educator Michael Pershan. In one part of the podcast Craig talks about his belief that reasoning required fluency before it could be conducted, but that this belief was challenged in a session with the eminent Mike Askew at the joint ATM/MA conference held over Easter. From Craigs description Mike posed a problem similar to this one:

45 × 36 = 45 × 35 + 35           True or False? 

The point was that this sort of question is one that can be considered, and the correct result arrived at, even if pupils are not capable of carrying out the calculation 45 × 36 correctly. Clearly this is true, and inspired this quote from Michael:

"Reasoning happens in the absence of fluency"

Craig reflects on this at the end of the podcast, and talks about how if you "hit a wall" with a problem, if you don't have the required tools in your toolkit (or don't recognise you do) that is when you need to reason. Craig also suggests (although he also admits that it doesn't feel right) that a possible implication of this is that teachers may hold pupils back from achieving fluency in order to allow opportunities for reasoning. Craig goes on to describe the idea of "teaching the fluency first" and then the reasoning becomes part of strategy selection - pose problems that pupils have the toolkit to solve, but pupils need to consider which strategies are appropriate. Craig offers his own SSDD problems as an example. 

It struck me whilst listening to this, that it may well be of benefit to make a careful distinction between what me mean when we talk about reasoning and problem solving. Consider this from the National Curriculum document, that says pupils will reason mathematically by:

"reason mathematically by following a line of enquiry, conjecturing relationships and generalisations, and developing an argument, justification or proof using mathematical language"

There isn't really reference in this to strategy selection, or working through a problem here. This is highlighted much more in aim 3:

"can solve problems by applying their mathematics to a variety of routine and non-routine problems with increasing sophistication, including breaking down problems into a series of simpler steps and persevering in seeking solutions"

This seems much more about what Craig is talking about with not having the necessary toolkit, and therefore not learning a lot (or simply not being able to persevere) whilst seeking solutions. This I think is where I agree with Craig - pupils need to have a secure knowledge of the mathematics underpinning the problem. The aim makes that clear - pupils solve problems by applying their mathematics. If they don't have the mathematics, they can't apply it. The purpose here is as Craig describes at the end of the podcast, to identify the mathematics required, to link the problem type to other problems they have previously encountered etc. The purpose is not actually to learn new mathematics at all. Ideally, it should not be obvious what mathematics will be needed to solve the problem, either from the context of the lesson in which the problem is set, or from the content of the question at all. What I think Craig talks about as "reasoning" about the problem is actually interpretation and strategy selection. There comes a point where we want pupils to be able to look at a problem where the mathematics required isn't explicit, and interpret the problem successfully to identify the mathematics required, before applying that mathematics through an appropriate strategy. One of my favourite areas for these is speed problems. Speed problems are a great source of both routine and non-routine problems. Some problems involving speed are solved by multiplication. Others are solved by division. And then even when a pupil can identify whether they are going to use multiplication or division, they need to choose a strategy for the calculation - for the best way to carry out the division or multiplication.

So where does this leave reasoning in comparison then? Well I actually did a session on this at what must have been mathsconf10 (by the dates of the materials) and it included this slide:
For me, the top question involves reasoning because the mathematics required is clear. That question is about highest common factor and lowest common multiple, and in particular their relationship to two numbers. The lower question is a true "Problem Solving" question (if you ignore the 'real life' context) in that it isn't clear what mathematics is going to be required. One could argue is that the last place you might want this question is in a lesson about LCM (particularly if not surrounded by others that aren't). By contrast, the reasoning question doesn't try and obscure the maths required, but at the same token doesn't just require application of a method. The reasoning question is trying to prompt a deeper consideration of the knowledge that the pupil is developing. To make them think about that knowledge in a way that, perhaps, they hadn't considered before. In the podcast Michael is therefore right, reasoning does happen in the absence of fluency, but that is because reasoning is an important part of developing fluency. To become fluent, one has to be able to approach questions like the first one above, and use knowledge flexibly to develop the chain of logic required. This of course means we can't talk about teaching to fluency before we offer opportunities for reasoning. For me, opportunities to reason are an integral part of the journey to fluency.