Beautiful Knowledge – A great BBC article

On the 21st April the BBC published this great article containing extract’s from David McCandless’ book “Knowledge is Beautiful”. There are some great infographics in this article that could be used in the classroom to promote a lively discussion about presenting data, how to present data to increase understanding and how to use design to increase clarity. I particularly like the graphic showing common passwords and their realtive strength

I’d certainly recommend taking a look at the article and the book Knowledge is Beautiful as well as David McCandless’ earlier book Information Is Beautiful.


My Mathematical Journey

I recently read Manan’s post and learnt that in the States April is Mathematics Awareness Month. Manan had written about his mathematical journey so I thought I would do the same.

I remember always liking maths and number stuff, but apparently for the first couple of years at primary school I wasn’t seen as being good at it because I didn’t put in a lot of effort. I can remember having very boring SPMG (I think that was the acronym) work books, where you had to do things like “colour 12 balloons” – I was never good at colouring (I still can’t really colour in between the lines to be honest) and so I just didn’t do it. I probably wasn’t the best kid to teach, as far as I was concerned I could count and I didn’t need to colour things to prove it. 

I loved primary school, and it was my Year 2 teacher who probably first made me “love” maths by giving me harder problems to solve than the rest of the class where I had to think more. This primary school also first introduced me to programming – first with logo and the big black “turtle” that moved on the floor and then with BBC BASIC on a BBC Micro and Acorn A3000 – though I didn’t know then that programming and maths would converge an become very important to me. 

Whilst I was at school, I was also lucky enough to have lots of time with my grandmother, who had been a maths teacher (mainly at the Royal College for the Blind) who set me maths problems and sent me coded messages to crack which I always enjoyed. I can also remember doing maths to calm down when I was stressed or apparently hyperactive. 

After having some good KS2 primary teachers I moved to secondary school, where I had a great teacher in Year 7, who again let me do work of my own choice most Friday lessons, including plenty of investigations. I still have a book on the history of Pi that he gave me when he left the school. People are often surprised that I didn’t enjoy the UKMT Junior Maths Challenges when I was at school – I didn’t really understand the benefit of them, and to me at the time they were puzzles that I didn’t really see the point of (I now think this is something that often needs addressing with KS3 students). I admit that I was probably seen as a bit of a geek at school – I once measureddhundred of blades of grass in an effort to determine if the grass around the school was all of the same type. I took my GCSE in Year 10 and then starter A Level Maths in Year 11. For me this was a good idea as it meant I could complete the A Level alongside starting Further Maths in Year 12 – this meant I avoided the problems of needing Year 13 content for Year 12 Further. My A level teacher was fantastic, always willing to help, or discuss things off the course. I appreciated how he always tried to show things from first principles, not just give us a result and then expect us to use it. At the end of Sixth Form I was pretty sure that following a maths degree I wanted to go and do a PhD – my uncle was (and still is) a lecturer in applied mathematics at Durham University – but in Pure mathematics, I was adament I didn’t want to do applied mathematics.

I went to study a 4 year Masters course at the University of Bath, in the end I chose a slightly odd mix of modules. I did a lot of Pure mathematics (group theory, number theory, measure theory etc) and lots of numerical methods, but very little fluids and mathematical modelling. This means I had a slightly odd background to choose (ironically) a PhD in applied mathematics. I had developed a love for the finite element method during some final year numerics modules. Such a simple idea but very powerful and infinitely less frustrating to analyse than finite difference methods with their taylor expansions! I applied to work with Paul Houston at the University of Nottingham, and following a brief meeting In the january of my final year was accepted to study under him and Andrew Cliffe. 

I moved to Nottingham and initially I was meant to be working on anisotropic adaptivity for fluid problems, but about half way through my first year II changed topics and focussed on applying the Discontinuous Galerkin method to the Neutron Transport Equation – this turned out to be significantly harder than we anticipated. A lot of time in my PhD was taken up programming in Fortran. At some point I will probably write about this more…. 

To cut along story short, during my PhD I discovered that I felt more rewarded when I was teaching others – either tutoring undergraduate, lecturing postgraduates from other faculties in statistics or going into schools and running GCSE or A Level revision sessions or doing outreach events – than I did when I was doing my own research. 

Following my PhD I applied for a School Direct place with the University of Nottingham. 

That’s a fairly short run down on how I ended up in teaching. I’d love to hear the stories of other UK based teachers.


The problem with Triangles

Earlier this week one of my classes said something that I had heard can happen, but hadn’t happened to me yet: 

We were discussing angles in a triangle and I drew a triangle like the one below

One of my students then said the fabled “but that isn’t a triangle Sir”. I questioned why they didn’t think it was a triangle and I was told that it “just doesn’t look like a triangle”. We then had a good discussion about what makes defines a triangle, before settling on it being a three sided shape, at which point they accepted the shape I had put on the board was a triangle.

I think this statement highlights the over-use of similar questions. I don’t understand why so many questions in text books have triangles with at least a horizontal base, if not right angled too!


Slightly Harder Matrix Multiplication Problem

The lesson after I taught matrix multiplication to my FP1 guys I gave them this question as a starter


At first I gave no information of where this came from or how I expected them to solve it. After about 15 minutes had passed, and we had got past the initial “no idea” phase, many of them were solving it by just working out percentage changes of the inter-related statements. I then gave them a hint and said that I would like to see it solved using a matrix. Despite this hint, it was still hard for them to spot that it was a simple matrix multiplication problem. The majority of rose that did spotted it by calculating the final percentage market share for each company in the GCSE sense and then spotting a pattern to their calculations. 

I really like this problem, the students enjoyed it too and said that it made them appreciate the power of matrix multiplication. 

I originally found it on this Discrete Mathematics Project website. It’s worth a look, but this is the only matrix question that seems suitable for A Level. 


What do we want in an A Level textbook?

Most people I have spoken to aren’t terribly keen on the current A Level textbooks. I think that the current Pearson Edexcel books are particularly poor, worse than the Edexcel books that I had when I was at A Level and definitely not as good as the Bostock and Chandler textbooks that I love. 

I think the questions in the Edexcel books are quite samey, not necessarily challenging enough to stretch the most able and could also highlight links between topics better. The order that they tackle topics is also a bit weird, I can’t work out why the chapter on vectors is at the end of the M1 book for instance. 

I was thinknig that the 100% prescribed A Level course (and 50% prescribed further maths) that is coming in provides a good opportunity to write an exam-board neutral text book / express what we want in a textbook. 

Here is my list of things I would like to see in an A Level textbook:

  • A book written in a logical order that highlights the links between topics.
  • Rigorous derivations that highlight where methods come from.
  • Sections that discuss the use of the mathematics in the real world.
  • Plenty of exercises for pupils to practise (with worked solutions available for teachers).
  • Scaffolded exercises to develop understanding and pupil’s confidence. 
  • Off syllabus sections to help prepare students for university level maths. For example, the Fundamental Theorem of Calculus is something that could be properly discussed at A Level and not just alluded to.
  • Access to plenty of virtual manipulatives to demonstrate ideas. These should be accessible to any students that want to use them.
  • Links to (free) mathematical software pacakages, especially for investigating data in the statistics sections.
  • Explicit emphasis for multiple approaches to solving problems.
  • Longer, more free-form questions to challenge students.
  • Suggestions for self-study research projects (maybe ones that would be suitable for the EPQ).

I’m really interested to hear what other people would like to see in a text book.? What kind of things do they particularly like/dislike? 

Please comment below or comment on twitter.


Increasing Cognitive Load at A Level

I mentioned a couple of weeks ago during #mathsTLP that I sometimes adapt multi-step A Level exam questions by converting them in to a single question to challenge my students. As I used 4 of these yesterday for an FP1 revision session I thought I’d share them. 

They are all taken from Edexcel FP1 exams and the file is available here and the first page is shown below:


Students do seem to find these a lot harder, but most enjoy doing them even when I reveal that I could have given them a much easier question where they are stepped through the process. I think there should be more questions like this on exams (though I guess they would be harder to mark) as they stretch the pupils more, and test their ability to make suitable choices of approach. 

Does anyone else do anything similar?


Knowledge is Power

So, after a week away I have an opinion piece… Please give me feedback as I would like to hear other people’s opinions and I am open to mine being changed!

With the new GCSE specification coming in, I often hear how “we need to be teaching more problem solving skills”, “developing independent learners”, “using inquiry based teaching techniques”. This, along with a general focus on child centered learning makes me feel a bit uneasy….

I’m not against these techniques – I use investigations and inquiry type things to promote pupils curiosity about mathematics and give them a flavour of what being a mathematician is about – but I think they need to be used in conjunction with more traditional teaching styles to be effective. 

Most of the maths taught in schools is hundreds of years old and required hundreds of years of work to be discovered and perfected. The average child isn’t going to be like Gauss. When asked to add up the first 100 numbers I’m sure most will just add them in turn (I may try this with some of my classes), even if some notice that you can work inwards and pair up the numbers to quickly work out sum with a multiplication, I don’t think even my sixth formers would come up with the algebraic formula for the sum of the integers without any prompts! There are some things which probably just need to be learnt, so that they can be applied. After all, mathematics is advanced by people applying previous knowledge in novel situations. 

With some of the more extreme types of independent learning I question the benefit that it brings to the pupils. If they have no clue about something that has been presented to them, then even the most tenacious pupil will become demoralised. I can’t imagine a teacher that wouldn’t provide more guidance in this situation, and steer the class to the desired conclusion. I really like the unpredictability of the inquiry idea and being able to look at some maths that the class stumbles on, but for them to be able to do this they need a sound knowledge base. This is why, for students to be successful at this and other investigations time must be spent building up a good body of knowledge, that all pupils can successfully weild to enable them to make progress.

In terms of problem solving skills, I think these are probably best taught by equipping students with enough knowledge to feel comfortable that when an approach doesn’t work they can try something else. Tenacity and patience I think are the most important problem solving skills.

I guess in conclusion I don’t like the polarised investigative child centered vs didactic teaching debate – there is a place for both approaches and the good teaches will blend the methods to ensure the best progress of their pupils. But I believe a strong knowledge base is key for any success in mathematics.