Explaining what maths is in a tweet…

During this weeks #mathschat @LearningMaths posted something like “it comes down to what we think maths is”.

This gave me the idea of trying to crowd source a definition based on peoples tweets. I follow a wide selection of mathematicians, maths teachers, computer programmers so I think I could get some interesting responses.

Please tweet to me (@DrBennison) your definition of what mathematics is. The only rule is that it must be no longer than a tweet, i.e. 140 characters.

My attempt is the following

A creative discipline with problem solving as a focus; to expose truth and beauty in the world around us in a rigorous, reproducible way.

Sounds a bit ….. I’m sure you can all do a lot better than me! I look forward to seeing your tweets.


Everything We Know Is Wrong

On Tuesday 26th August 2014 BBC Radio 4 presented a programme looking at the validity of scientific research. This programme is Everything We Know Is Wrong, available on BBC iPlayer here.

Jolyon Jenkins presents, among others, the results from some research done by Dr. John Ioannidis of Stanford University. His paper “Why Most Research Findings are False for the journal PloS Medicine surveyed 50 of the most cited (more than 1000 times) papers in medical research. For each paper, he tried to find if there was a subsequent study that was larger to back up these cited papers. 5 out of 6 of the papers whose results had been based on non-randomised data had been proved to be wrong, yet these papers were still being cited. In addition, about 25% of the claims based on randomised data have been shown to be wrong or widely exaggerated.

This issue of reproducibility is important to me, and something I feel is a major problem for educational research. As a numerical analyst, generally when results are presented in a paper it is possible to go away, write some computer code and reproduce the computations / results, checking their validity. This does not seem possible with any of the educational research papers that I have read. For a start data is often anonymised (even if it is from the public domain) and so cannot be verified. Not only this, there are so many factors at play in a classroom it is hard to strip everything out apart from the factor the researcher is interested in. Another problem is mentioned in this programme: many papers seem to use very small samples, to me this throws into question the power of the test. Has the conclusion presented in a paper really been found, or is it just by chance.

Of course, their are ethical considerations (leading to the use of anonymity) and cost considerations at play here, the larger a sample, the larger the cost of the study and os the less likely it is to be funded, especially if the outcome is unclear.

John Ioannidis also discusses the influence of the culture of scientific research in what is published. Having spent a few years of researching a topic, it is not ideal to have nothing to publish, and so, if the original outcome hasn’t been realised researchers may start looking for something in the data to publish, even if the study was not designed to investigate this. This I am sure happens in the field of education.

Academic researchers, in all fields, build their career on their publications. This leads to a pressure to publish results (potentially limiting the time spent verifying a theory) and also means that previously published results are very rarely checked – You are not going to get a blockbuster publication by re-doing the work of others and verifying it. Even if some work has been checked and shown to be wrong, this fact may not be published. Suppose that a new researcher, R, has shown that a result due to a big name in the field doesn’t hold. How would publicly criticising an acknowledged expert in the field affect this researcher’s career? The big name may even be on the refereeing panel of R’s paper – is this going to be fair?

All of these issues should be considered when reading a piece of educational research. It is important to not take research at face value without first passing a critical eye over it.


Integration by a flowchart.

After years of dealing with any integrations that I had to do by thinking “let’s load Mathematica” I thought I should probably practice doing some by hand as I will be teaching Further Mathematics this year and will need to be able to integrate by hand.

As I was practicing a few I had a vague memory of a flow chart that my A Level teacher had given me (a long time ago) and thought it could be useful to try and produce something similar.

I had also just downloaded the iPad Grafio app so I decided that producing a flow chart for integration would be a good way to get used to Grafio. Overall a fairly easy app to use, sometimes a bit fiddly though to place content where you want.

The flow chart is shown below:


And a high resolution pdf can be downloaded at the following link.

Anyway, I would be very grateful if you commented and let me know what you thought of the flowchart. Do you work through a similar process when integrating by hand? Have I missed anything that you think is important?


The Educators – Episode 2 John Hattie

This Wednesday (20/08/14) Episode 2 of the new BBC Radio 4 series The Educators aired and is now available on iPlayer. In this episode Sarah Montague interviews John Hattie, Professor of Education at the University of Melbourne about his work analysing a wide range of educational research.

John talks about his work analysing over 60000 studies (encompassing a quarter of a billion students) to find out what has the largest impact in improving the educational outcomes of students. This work took him over 15 years and is probably familiar to many through his books on Visible Learning.

In the programme, which is well worth a listen, following interviews of parents discussing what they look for in a school, Hattie makes the comment that they focus on what they can observe; things like class sizes, leadership of the school, Ofsted reports and whether children appear happy in the school. Hattie points out that these are poor proxies for what is really important – the quality of teaching. I, as a teacher (and I’m sure mos teachers too) agree that it is the the teaching that makes the difference between success and a pupil not achieving their potential.

Hattie lists many things that actually have zero or little effect such as class size, homework (greater effect in secondary schools than in primary) and the type of school. All things commonly held to be important. This is all discussed in more detail in his books, with proper references to the original studies and comments on the validity and quality of the studies looked at.

On the subject of the quality of the study he made one comment that I found fairly strange:

If the effect is quite large, then the quality of the study is not that important

I really don’t agree with this, I could make up any old junk and get a large effect size, but surely the validity of the method used in a study is important to whether the study and it’s conclusions have any value! Not being a statistician, I don’t feel qualified to critique Hattie’s methodology, though I would be amiss to not point out that such criticisms do exist. Some of these, including the use of the concept of effect size are discussed in the blog of @OllieOrange2. As I say, I am not a statistician and am also a big fan of Hattie’s books and research, but some points raised in this blog do raise concerns that at some point I should look into in more detail.

Hattie makes two points that struck me in this programme: 1) The biggest predictor of future health, wealth and happiness in the future is the number of years in schooling, not achievement at school. 2) In excellent schools there is a dialogue about the teaching and the impact that teaching is having.

I would recommend this radio programme and John Hattie’s books to all teachers, wishing to investigate what makes a difference in schools. However I would echo Hattie’s remark that he is not giving school leaders a recipe for what they should do, rather he is giving them a way to think – to investigate the impact teaching and decisions have in their school.


Thoughts on the draft maths A Level content

So, before I had even read through the Government’s documents outlining the proposed changes to the A Level Mathematics and Further Mathematics courses I came across an excellent post by @srcav discussing the changes – I really need to learn to stay more up to date.

The Department for Education and Ofqual are currently in a period of consultation and seeking input on these proposed changes. The documents are available here, just scroll down the page to contribute to the consultation.

In this post I am going to outline my initial reactions to the changes and the proposed content upon reading the documents.

General comments:

I think the linear model for maths, while perhaps being more challenging for the students, will be an improvement as it will emphasise the connected nature of the subject. I did like the element of choice in the previous A Levels, but prescribing 100% of the content of A Level Mathematics and 50% of A Level Further Mathematics should make it easier for university lecturers to understand where the students are if they decide to study mathematics at university. Like the new GCSEs there seems to be a greater emphasis on proof of various kinds. The lack of any content from the Decision modules surprises me and is a shame given the massive use of mathematics in Computer Science. I would have liked to have seen at least some discussion of algorithms and their computation cost, though perhaps this could be included in the numerical methods topics.

This draft document also says that the specifications must encourage students to

Read and comprehend articles concerning applications of mathematics and communicate their understanding.

This interests me; are we going to see some kind of mathematical comprehension element to the assessment of the new A Level courses?
The fact that 100% of the A Level is prescribed surely means that schools will be free to choose the exam boards for A Level Mathematics and A Level Further Mathematics independently. If there is a genuine difference in content between the boards for further mathematics I definitely think this is a good thing.

A Level Mathematics<:/span>

As mentioned above there is a greater emphasis on proof, and I like the explicit inclusion of the proof of the irrationality of \(\sqrt{2}\) and the infinite number of primes. The omission of proof by induction is a surprise though (this is surely a mistake?!). The use of set theory notation is also mentioned as is the use of Venn diagrams – this must be a good thing for students going onto further study of mathematics.

I feel that it is a shame that no numerical methods are included in the AS content, but I am happy that one of the bits of knowledge in OT2 is for students to understand that many mathematical problems cannot be solved analytically. This is something that I did not appreciate fully when I was an A Level student, and as a numerical analyst by training believe is a key bit of knowledge.

I am intrigued by the requirement that students must

become familiar with one or more specific large data sets in advance of the final assessment (these data must be real and sufficiently rich to enable the statistical concepts in the specification to be explored.

This, I am sure, will place a greater load on the exam boards to provide meaningful examples for the students to explore in the statistical aspects of the course. Of course, this should hopefully also lead to these aspects being a bit less “dry” than the present S1 content.

Content A9 is nice as it explicitly links transformation of graphs to the transformations of the graph of the Normal probability function \(N(\mu,\sigma^2)\) – this is something that in my experience, very few undergraduate and postgraduate students appreciate, let alone A Level students.

The trigonometrical content of the A level appears to have an increased focus on the geometrical proof of trigonometric identities – I guess this is in common with the new GCSE syllabus.

As a former numerical analyst I am sad to see no numerical methods included in the AS Level, however the inclusion of the Newton-Raphson method is encouraging as we could discuss some interesting properties of this method and its application. To me, it seems a shame that numerical differentiation isn’t included (though it is included in the Further Maths content) as this can be linked nicely to differentiation from first principles.

Those that know me, will know that one of my “pet niggles” with A Level is the explicit teaching of the quotient rule for differentiation so I am disappointed to see that this is included. I’m not sure I know any practicing mathematicians who would (if they had to differentiate by hand) use the quotient rule: Instead for example, they would consider \( f(x) = \frac{x^2 + 3}{(x+1)^3} \) as \( f(x) = (x^2+3)(x+1)^{-3} \) and use the product rule. The use of the quotient rule seems more error prone, I imagine due to the negative sign preceding one of the terms, than the product rule. The resulting expressions also tend to be more difficult to simplify.

In the mechanics component of the A Level Course it is encouraging to see the use of calculus for kinematics in 1 dimension. It’s a shame that currently this tends to be in M2 or higher and so isn’t often studied by pupils taking just the core A Level as M1,S1 and D1 are commonly chosen from to build up the “applied” component of the A Level. I feel that this will prepare students better for the applied aspects of university level mathematics.

Further Mathematics A Level:
The content for the 50% prescribed component further mathematics A Level seems pretty much the same as contained in FP1-FP3 currently.
I am pleased to see the inclusion of Viète’s formulae relating the roots of a polynomial to its coefficients as I think these are really neat results that can be explored in a nice way (Could the students derive these?).

The section on matrices I think is disappointingly standard. I had hoped that the content on matrices would be expanded due to their importance in computational and applied mathematics. I am always disappointed that the “matrix of minors” approach is taught to find the inverse of a 3 by 3 matrix – this is almost never used in practice and doesn’t enhance understanding in any real way, that I can see anyway. The row-reduction approach is easier, less prone to errors and can be used to build up to a practical method for solving linear systems of an arbitrary number of unknowns. The omission of eigenvalues and eigenvectors is also sad. I hoped these would have been prescribed as core content since some interesting results can be obtained.

I am pleased with the general emphasis (throughout both courses) of mathematical modelling and this is reflected in the prominence given to differential equations in the prescribed content for the Further Mathematics A Level. This should prepare students well for further mathematical study.

If the exam boards are prepared to be brave and have significant differences between them the 50% unprescribed content could allow students to study some more unusual and topical areas of mathematics such as mathematical biology, coding theory, cryptography. These topics could be introduced at the required level, though I guess this would put additional pressure on teachers who may not have studied these areas themselves.

Final comments:

As is usual in a post of this nature I have expressed my own personal opinions and I would be very interested to discuss these with others. I would love it if you left any thoughts in the comment section below this page!


Why teach?!

So, I thought as a good way to introduce myself to people who may not know me that way it would be good to describe why I have gone into teaching…

During my School Direct year I had to write a short assignment on “My Starting Point as a New Teacher”, I gave this a quick re-read before writing this post. Whilst finding it, on a re-read, all a bit woolly it did convey the main reason why I feel the teaching of mathematics is so important.

To me, mathematics is the truly universal language that can be used to explain the world and establish truth amongst conflicting media reports. I am passionate about equipping students with the skills necessary to use mathematics productively in their every-day life as well as seeing the inherent beauty of mathematics.

I studied for a PhD in Applied Computational Mathematics at the University of Nottingham and during this time I found the time I spent teaching and running outreach events with local school students far more rewarding than my actual research (I was looking at numerical methods for the Neutron Transport Equation – I may do a post about it one day!). Because of this I decided to apply for a school direct post at a school in the East Midlands, where I have been lucky enough to obtain a post from this year too.

From my experience of tutoring undergraduate students and lecturing postgraduate students (across all disciplines) in quantitative methods one problem that you notice is that British students are not as prepared for a mathematics degree as their international peers. This is of course a sweeping generalisation, however, as a teacher I want to work towards developing the independent study skills required for students to successfully engage with a mathematics degree course if that is where their studies lead them. I believe, very strongly, that mastery of the basic mathematical skills is very important for all students, I aim to develop this mastery in my lessons while still maintaining interest.

Having said all the above, I guess the main reason why I have decided to go into teaching is that I love it! I hope this blog will reflect my love of both teaching and mathematics. I also hope that it will prove useful to other people as well as being a development tool for me.