Author: tombennison

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!

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. 

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.

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?

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. 

Cedric Villani’s book “Birth of a Theorem” has recently been released and received positive revies, such as this one by Hannah Fry. I had been dubious about buying this book due to the apparent high level mathematical content of parts of it, but about 3 weeks ago BBC Radio 4 featured it as their book of the week – after hearing the extracts I have ordered the book. 

Julien Rhind-Tutt’s reading of extracts from Villani’s book is very engaging – though I think it has perhaps lost something by not being read by Villani. The episodes are available for another week or so here –  I’d encourage you to listen if you can.

Cedric Villani is a Fields Medal winning (in 2010) French mathmeatician, who works primarily on mathematical physics. The Fields Medal is highly prestigious as it is only awarded every 4 years and only to mathematicians under 40. He won the Fields medal for his work on non-linear Landau Damping and the Boltzmann Transport equation, in 2012 he wrote a book, “Théorème Vivante”,in French, describing the road to the proof of his theorem and it is this which has been translated into English and published as “Birth of a Theorem” in 2015.

Villani gives a really nice insight into the world of a mathematician, and I could definitely recognise the panic he felt when there was no tea available – “Panic! Without the stimulating leaves of camelia sinesis I couldn’t possibly face the hours of calculation which lay in store”, he then goes on to describe breaking into Princeton to procure some tea. In the first episode he paints a nice picture of how consuming mathematics can be – “While the children excitedly open their Christmas presents, I’m hanging exponents on functions like balls on a tree and lining up factorials like upside down candles”. He also describes his wife being being taken aback seeing his “face contorted by ticks and twitches” as he thought about the problem he was working on over dinner. My wife says that in the final year of my PhD she would often have to say things to me more than three times because I would just zone out into a world of my research – I think it can be very hard being with a mathematician at times! It is certainly easy to feel for him when he describes receiving the rejection email from Acta Numerica – if you have spent countless hours on a problem, to have the paper rejected is crushing.

On his website there is an introduction to Boltzmann like transport equations in his survey paper “A Review of Mathematical Topics in Collisional Kinetic Theory” despite being an introduction it is still incredibly dense. I’ve studied a particular version of the Boltzmann equation – The Neutron Transport Equation – and a few pages in I am struggling to follow this easily. I think I would have to expend an awful lot of time in order to be able to understand the mathematical parts of his book. Indeed, if I ever could, it is likely that only a few hundred mathematicians in the world understand his and Clément Mouhot’s proof!

Cedric Villani also appered on Start The Week on 9th March 2015, a podcast of which is available here which is also worth spending the time to sit down and listen to properly.

Yesterday morning on my way to school I had a nice surprise and heard Ivan Fesenko on BBC Radio 4 today talking about a new £2.3 million grant that academics at Oxford University and The University of Nottingham have won to tackle some of the greatest unsolved pure mathematics problems. 

The short interview is well worth a listen here (start 1 hour 222 minutes in). The team lead by Professor Ivan Fesenko at Nottingham will look at ways of tackling the generalized Riemann Hypothesis and the Birch and Swinnerton-Dyer conjecture. Both of these are Clay Millenium problems, and more information is available on the Clay Mathematics Institute website. They are obviously incredibly hard…..

I really liked Ivan’s description of mathematics as a “very large old oak tree, with many, many branches still developing. Most mathematicians are working and sitting on one of those new branches.”

The great site The Aperiodical maintain a monthly blog carnival, that is hosted,in turn, by other blog writers. This month the Carnival of Mathematics has been curated by Manan Shah over at the site Math Misery. 

I am privileged to have had two posts of mine included in this edition. The first concerns approximations to \(\pi\) and is available here, the second is my post about using Geogebra for self guided learning in FP1 when looking at the conic sections.

Issue 120 of the Carnival of Mathematics is here and well worth a read – I didn’t realise 120 was an abundant number! Manan has picked out some great reads, including Stephen Cavadino’s (@srcav) work through of a nice problem that eventually boils down to finding the maximum of a quadratic and a discussion of the convergence speed when computing Khinchin’s Constant by John D. Cook (@JohnDCook). 

Head over and read The Carnival of Mathematics at Math Misery and check out the rest of Manan’s blog too – there’s always something of interest there!