Friday, 31 January 2020

Session for NW3 Maths Hub in Liverpool

On Thursday 30th January I had the privilege of leading a session for the NW3 Maths hub around making sense of maths concepts. There was some very positive feedback following the event both through the feedback forms and from Twitter.

The link to download the presentation from the evening is here.

Wednesday, 30 October 2019

My #VisibleMaths tour!

Over the Leicestershire half term I went on quite a tour around Britain, talking about some of the ideas from my book #VisibleMaths. Starting in Peterborough on the first Saturday for the always fantastic Complete Mathematics Conference (#mathsconf), I was speaking about threading an idea through the curriculum in a coherent manner - in this case the idea of factorisation.

From there I had a couple of days at home before travelling up to North Yorkshire on Tuesday ready for working with teachers from the Esk Valley Alliance. We had a good time looking at addition, subtraction and division using representations. I particularly enjoyed this trip because my fiancee Rowan came with me and we had a lovely night away and a bit of time in York together.

A day off on Thursday was then the lead in to my really busy time. A short hop across to Birmingham on Friday lunch time for a session with maths teachers from Niksham High School, which was followed by jumping straight on a train to get down to Farnham ready for the excellent ResearchEd Surrey. Many people have commented that this ResearchEd was one of the best local versions for planning and organisation, and I must say I agree! I did a morning session there focusing on the idea of addition and how, with just two ways of making sense of what it means to add, all additions from Y1 to Y12 work in the same way (which was also mentioned in my latest TES blog). The session went down very well, and it was great to then spend the day listening to others talking about education.

Having arrived back in Leicestershire at 11pm on Saturday night, one week after my tour of England started, I then had the thrilling experience of my first ever trip to Scotland on Sunday and Monday. I must admit the sights walking out of Edinburgh Waverley station were simply breathtaking - I am not generally one to take photos of my surroundings, but even I had to capture some of those amazing visages.

The tour finished with my biggest ever audience - a little over 200 Scottish teachers of maths as part of the South East Improvement Collaborative joint INSET day. Having looked at the importance of making sense in different ways (using one of my favourite games!) we looked at making sense of addition and subtraction, and useful models for both of these operations.

A lot of people have suggested they would find a copy of the presentations helpful, so here they are:

Complete Maths Conference Peterborough (Saturday 12th October)

Esk Valley Alliance (Wednesday 16th October)

Niksham High School (Friday 18th October)

ResearchEd Surrey (Saturday 19th October)

SEIC INSET day (Monday 21st October)

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 90˚ . 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.