Author: tombennison

This morning I made the journey down to the University of Leicester’s Fraser Noble Building for the latest ATM/MA East Midlands branch session. 

Today it was John Mason – a big name in the Mathematics Education world. His session was entitled “Teaching More by Teaching Less: Getting Learners to Make Use of their Natural Powers” and I had a great time at this thought provoking session and I was lucky to sit with Pete (@MrMattock), Mel (@Just_Maths), Em (@EJMaths), Andrew (@ApApaget), Mary (@PardoeMary) and Andrew Price (@ColonelPrice) where we had many great conversations. It was also good to see a strong contingent from The University of Nottingham’s CRME in attendance. 

For this post I am going to share some of the tasks we looked at and my thoughts on them, but I’m going to start with a few nice quotes from John Mason himself.

• “Mathematics is working on problems.”
• “Learning to read graphs is much more important than learning to draw graphs.”
• “My job is not to choose questions kids find attractive, it’s to choose questions with the most potential for learning maths.”

Sadly I think these three highlighted thingd probably happen far too little in the classroom today with all of the current assessment and accountability pressures.

One thing from the sesssion that I liked straight away was the use of magic squares without numbers.I love using magic squares in lessons, but I admit I tend to use them as a way of practising procedues, I hadn’t thought of using them to “tease out” students’ reasoning. John pointed out that there was lots of research that students who have been deemed “low achievers” in mathematics can reason mathematically, it is their low level of numeracy that gets in the way of them demonstratng this in the classroom – so why not use tasks that don’t require numeracy. I firmly believe that improving numerarcy is important, but so is developing the mathematical reasoning skills and so I like the idea of using tasks that develop this without requiring a high level of nmeracy. Indeed, I suspect that developing reasoning will improve their numeracy anyway as the students will be able to reason more about the number system. Fo those of you who aren’t familiar with magic squares, they are arrays of numbers (commonly \(3 \times 3\) such that the column sums, row sums and leading diagonal sums are all the same. John suggested removing all the numbers from a magic square and using just mathematical reasoning to show certain properties. He started by asking us to reason out that the blue coloured squares in the magic square below add up to the same value as the red coloured squares. 

  Of course this is relatively clear, but you can develop this idea into some harder examples 

 Or even extend the idea to larger magic squares  

 
I am definitely going to try this in the classroom. 

This session has also inspired me to use more prompt photos and as students to formulate their own mathematical questions. Of course it is important for students to develop fluency, but I also think that to become a mathematician they need to be able to pose interesting mathematical questions.

John showed a vey cool applet that he has written using Cinderella. It contained a straight line of fixed length divided into sa blue section and a red section. The blue line segment was the perimeter of a square and the red line segment was the perimeter of a triangle and as the joining point moved from left to right the square gre and the triangle shrunk as you would expect. One interesting question to pose is “At what point is the perimeter of the square equal to that of the triangle”. Of course this is true when the join of the red and blue line segments is in the middle, but originally I (along with many others) made the mistake of thinking that we had to work with he ratio of the side lengths. The app allows you to plot the area and perimeter of the two shapes as the joining point moves, as shown in the video below.

This is a fantastic visualisation of something that I actually found quite counter-intuitive until you think about it in terms of the underlying maths. There are many other questions that you could pose about this visualisation.

We then looked at some more tasks around the topic of area and perimeter from this excellent sheet that John provided us with 

I particularly like the “More or Less Perimeter and Area” and “Shape Signature” tasks and would quite like to work them into lessosn this year at some point. I really want to find the time to explore the questions posed by the “Shape Signature” task.

 Towards the end of the session John talked about “Multi-Level Initiating of Tasks”:  
 Between the three there is clearly an increase in the structure and scaffolding given to the student. At what point does this turn from beig a help to leading to a dependence on the part of the student and reduce their ability to reason mathematically?

All in all it was a very thought provoking session and I have taken away lots of things to ponder. John has kindly made all the materials from his talk available online here; in particular his PowerPoint slides are available.

I may be wrong, but it seems un-common to introduce the Poisson distribution in S2 by discussing its original application to real data. Before teaching in schools I developed an “Introduction to Quantitative Research Methods for Engineers” course and this was how I introduced the poisson distribution in the lecture notes. I think it is a great way to bring a bit of the history of mathematics into your teaching, and in my opinion certainly motivates the Poisson distribution more than the fairly awful way it is introduced in the Edexcel text book.

The distribution function for the Poisson distribution is as follows: If \( X \sim Po(\lambda) \) then \( P(X=x) = \frac{e^{-\lambda}\lambda^x}{x!} \)

In L. Bortkiewicz’s book of 1898, “The Law of Small Numbers” he discusses the number of fatalities in 10 corps of the Prussian cavalry that were the result of horse kicks. He has 200 years worth of data, and to this data he fits a Poisson distribution after calculating an appropriate rate parameter.

From the table it is clear that a Poisson distribution, with rate parameter \(0.61\) provides a pretty good fit (of curse the goodness of fit can be calculated!).

As an aside, tabulating Poisson probabilities is extremely tedious… You can save yourself a lot of time by using WolframAlpha, especially if you know some Mathematica / Wolfram Language syntax.For example to tabulate the values for a Poisson distribution woth rate parameter 2, for example you can do the following

In the above the command “N[Table[(Exp[-2]*2^x)/Factorial[x],{x,0,10}],4]” generates the following

Here I have used the Table command to tabulate between \(x=0\) and \(x=10\) the Poisson distribution function with parameter \(-2\). I have then wrapped this with the N command so that I get values to 4 decimal places output instead of the default exact expressions.

Whilst WolframAlpha’s natural language processing is good, using the syntax of the underlying language makes it a lot more powerful.

On Saturday 7th November Ed Southall (@solvemymaths) organised a morning’s conference. This was primarily aimed at his trainees but a few tickets were made available generally. It was definitely worth making the trip up the motorway in the awful rain to go to this and was great to catch up with Ed, Hannah  (@MissRadders) and Beth (@MissBLilley) afterwards. Andrew (@ApApaget) and Megan (@MeganGuinan1) but I didn’t realise until the end.

The day started with Craig Barton (@mrbartonmaths) talking about his excellent Diagnostic Questions website. I confess that as yet I hsven’t really used this website to its full potential but it is one that I want to explore more. The GCSE Maths Essential Skills quizes are excellent (a question from one of these is shown below) as each of the multi-choice answers exposes a misconception on a common GCSE topic.

 

Craig’s weekly insight blogs are excellent and well worth a read, I found the one about writing algebraic expressionsparticularly interesting.

Another feature of this website that I like is that students have to explain their reasoning (admittedly a contentious issue at the moment) when they select an answer which is great information for the teacher to have. If they make a wrong selection they are shown a range of correct explanations from their peers.

After Craig I went to the lovely Beth’s session on using the history of maths to help teach maths.

  I learnt a few fascinating things about the Greeks that I didn’t know, including an ingenious method to measure the radius of the moon.

 

Beth has written a great (first) blog post about her presentation here and has allowed her files to be made available either on TES or here.

Following Beth’s presentation I went to Laura Hadfield’s workshop “Making Differentiation Visible Throughout the Lesson”. For me there wasn’t much new here, but I was really interested in how everything in her lessons is traffic-lighted for differentiation. She doesn’t use the usual traffic light colours instead going for blue for “Practise”, green for “Mastery” and pink for “Apply”. I do something along these lines in some lessons, but certainly not in every lesson and it was interesting to hear of a department where this is a consistent practise. Like most things, I imagine it has more of an impact when consistently done. One comment that Laura made intrigued me – she said something like “there is no problem with using textbooks as long as the instructions aren’t ‘do question 1 to question 20’.” To me this is a problem with the textbook, not the instruction. If the textbook is well designed then working through a sequence of questions should develop fluency and understanding well. Admittedly there would have to be some teacher intervention but a good textbook would have pedagogically well designed questions.

To end the day we had the legend that is Don Steward present some excellent things. This was genuinely the best hour and a half of CPD that I have had.

Don opened with the following picture (I think the inquiry maths prompt has been inspired by this) as an investigation to do with Year 7 – “Do we get the same a nswer if we go either way round the diagram?”. Following a discussion students can then substitute some small numbers in and  investigate what happens: Is there a pattern? Can this be generalised? What if we change the numbers/operations  on the branches? I really liked his suggestion of how to introduce algebra into this task and it is definitely something I am going to try in the classroom. The idea being that when asked “how do we show this for all numbers?” many students will suggest trying a big number such as a million, but we can’t fit all those zeros into the circle and so we write ‘m’ and work through creating an algebraic expression. I think this is a fantastic way to introduce algebra in a non-threatening way. He then moved on to suggest that a student should think of a number  (Beth must have been very excited to be included in Don’s presentation here!) but not say what it is, so we have to give it a letter, ‘b’ say. We then pass this through the operations deriving the same expressions, but as we don’t know what this letter stands for we have shown that the pattern always holds. Don has also written this activity up back in 2013 in his excellent post “Both Ways”.There was a large emphasis in Don’s session on visual representations for proofs, and Ed has posted the visual proof of this pattern in his write up of the session.

Don then talked about some great proofs that made use of the following diagram

 I particularly liked the one for the arithmetic-geometric mean inequality (or two numbers) shown below:

 Don then discussed some investigations involving relationships between the area and perimeter of rectangles – he has recently written about these on his blog here.

I was also pretty excited to see an alternative construction of an angle bisector that is is considerably easier than the one traditionally taught. 

The session ended on a mathematical approach to Magic squares, but Ed has written an excellent post about this, so head over to solvemymaths.com for more details.

Overall I had a fantastic morning in Huddersfield, thank you Ed for organising this.

On the 2nd November 1815 George Boole was born and in celebration he has been honoured by a Google Doodle.

In addition the University of Lincoln’s mathematics and physics department organised a public lecture. to be given by Evegeny Khukhro

George Boole is credited with many things, including

• Founding invariant theory.
• Invention of boolean algebra.
• Intorduction of mathematical probability.

Boole’s first mathematical interest was is the realm of mathematical analysis and differential equations. He published two textbooks that also included soem original research contributions concerning thr algebraic approach to differential equations. Broadly speaking we denote the derivative as the application of an operator \(D\). So that if \(D(f) = u \) then \(f = D^{-1}(u) = \int u \). For example we could have the following Differential equation (DE) \( \frac{\mathrm{d}^2f}{\mathrm{d}x^2} + 3\frac{\mathrm{d}f}{\mathrm{d}x} + 2f = \sin (x) \)  can be written in operator notation as \( D^2(f) + 3D(f)+2f = \sin (x) \) or \( (D^2+3D+2)(f) = \sin (x) \). For his contributions in this field he was awarded The Royal Medal of the Royal Society.

In 1841 Boole published one of the very first papers on invariant theory, this paper is credited by Arthur Cayley in his paper of 1845 which is often credited as being the start of invariant theory.

Of course Boole is most famous for his contributions in the world of logic. He first pulished a book on logic in 1847 whilst he was still in Lincoln, before publishing again once he had moved to Cork.

I was interested to learn that Boole only allowed the operation of Union when the sets were disjoint, Evegeny then presented a nice table comparing modern notation with that of Boole.

 Boole is also famous for his contributions to mathematical probability, most notably his name lives on in Boole’s inequality.

In conclusion “Boole made a giant step towards mathematics as a truly abstract discipline, causing a paradigm shift, giving mathematics enormous scope and potency” (Khukhro, 2015). Evegeny has made his PowerPoint slides available here.

The poll is now closed and the winner is…… “Contrasts in Mathematical Challenges in A-Level Mathematics and Further Mathematics, and Undergraduate Examinations” by Ellie Darlington, and published in the IMA Teaching Mathematics and Its Applications journal. This article achieved almost 50% of the votes, and I am really looking forward to discussing it.

The article is available here.

  
We will be discussing this at 8pm on Monday the 7th of December. I know it will be the run up to Christmas, but I hope lots of you will still be able to make the discussion.

As usual, about a week before I will post some possible topics of conversation or things to think about. I hope you enjoy the article.

Yesterday evening I went to the excellent Festival of The Spoken Nerd in Derby, but before thsat I had the pleasure of seeing Alex Bellos speak at a school in Leicestershire. This post is just going to mention a few interesting things from both of these events.

Alex Bellos was a very entertaining speaker to listen to as he is evidently so passionate about people understanding the beauty of mathematics.  

 Alex opened his talk with a logic puzzle that he had featured in one of his early Guardian Puzzle blog posts: Find the odd one out in the symbols below 

 This puzzle was orignally due to Tanya Khovanova who had written about it here. It is intended as a piece of fun to emphasise the intrinsic issue of odd-one-out puzzles. Namely, that they tend to be focussed around a particular way of thinking (Alex provides a nice example in his post) when in fact many could be seen as the odd one out for different reasons. In the puzzle above, the odd one out is actually the one on the left hand side by virtua of not being able to be called the odd one out – an interesting philosophical dilemma there!
One thing I found particularly interesting in Alex’s talk was his discussion of the Sieve of Erastothenes. If I do this in school I tend to have a 10 by 10 grid of the numbers 1 to a 100. I had never thought of arranging them instead in 6 rows, as shown below. This leads to some very interesting patterns when you cross out numbers. 

He then described the Ulam Spiral, devised by Stanislaw Ulam where numbers are arranged in a spiral and primes highlighted generating a pattern where primes lie on diagonal lines. I’m going to write some MATLAB code to generate these I think and write a bit more about them in the future.

 

Alex also talked about the results of his internet survey to find the world’s favourite number; which turns out to be 7. He showed a nice annimation about this which is available on Youtube

Following Alex’s session I had a fairly mad dash to Derby in horrendous weather to go and see Festival of The Spoken Nerd – if you haven’t seen them live before I can’t recommend it enough. It was a great evening. I really liked the visual demonstration of the modes of vibration of a metal plate. These are known as Chladni figures after the scientist Ernst Chladni who first published his work in 1787.

   

  
 

Festival of The Spoken Nerd have plenty more dates in their UK tour, I urge you to see if there is one local to you on their webpage and if there is go. You won’t be disappointed, I’ve seen them a few times now and it is always a very entertaining evening – writing about the show can’t do it justice.

 Following on from our second successful discussion last week it is time to vote on the article for the next discussion. The next discussion will take place at 8pm on Monday the 7th of December. I know everyone gets busy in the run up to Christmas but I hope that you can all still take part.

Three article from the last poll have been rolled over to this one as two of them were tied in the number of votes. As usual, the titles and abstracts are below and the poll is available here

• Contrasts in Mathematical Challenges in A-Level Mathematics and Further Mathematics, and Undergraduate Examinations; Ellie Darlington (Teaching Mathematics and its Applications) – This article describes part of a study which investigated the role of questions in students’ approaches to learning mathematics at the secondary/tertiary interface, focussing on the enculturation of students at the University of Oxford. Use of the Mathematical Assessment Task Hierarchy taxonomy revealed A-level Mathematics and Further Mathematics questions in England and Wales to focus on requiring students to demon- strate a routine use of procedures, whereas those in first-year undergraduate mathematics primarily required students to be able to draw implications, conclusions and to justify their answers and make conjectures.While these findings confirm the need for reforms of examinations at this level, questions must also be raised over the nature of undergraduate mathematics assessment, since it is sometimes possible for students to be awarded a first- class examination mark solely through stating known facts or reproducing something verbatim from lecture notes.
• “‘Ability’ ideology and its consequential practices in primary mathematics” by Rachel Marks (Proceedings of the BSRLM 31 (2)) – ‘Ability’ is a powerful ideology in UK education, underscoring common practices such as setting. These have well documented impacts on pupils’ attainment and attitude in mathematics, particularly at the secondary school level. Less well understood are the impacts in primary mathematics. Further, there are a number of consequential practices of an ability ideology which may inhibit pupils’ learning. This paper uses data from one UK primary school drawn from my wider doctoral study to elucidate three such consequential practices. It examines why these issues arise and the impacts on pupils. The paper suggests that external pressures may bring practices previously seen in secondary mathematics into primary schools, where the environment intensifies the impacts on pupils.
• “Train Spotters Paradise” by Dave Hewitt (Mathematics Teaching 140) – Mathematical exploration often focuses on looking at numerical results, finding patterns and generalising. Dave Hewitt suggests that there might be more to mathematics than this.
• “Relational Understanding and Instrumental Understanding” by Richard Skemp (Mathematics Teaching 77)
• “Knowing and not knowing how a task for use in a mathematics classroom might develop” by Colin Foster, Mike Owlerton and Anne Watson (Mathematics Teaching 247) – Participants at the July 2014 Institute of Mathematics Pedagogy (IMP14) engaged in a wide range of mathematical tasks and a great deal of pedagogical discussion during their four days last summer. Towards the end of IMP14 a conversation began regarding how much knowledge about a task a teacher needs to have before feeling comfortable taking it into the classroom.

I’m looking forward to seeing which article is selected as I haven’t read all of these yet!

I’m quite sad that I missed this on pi day this year but I thought I should share it now..

For this years (special) pi day Wolfram Research have produced a web page mypiday.com that enables you to find the location of your birthday in the digits of pi – it is well known that any date will appear within the digits. Here is the location of my birthday: 

 Stephen Wolfram also wrote a fairly interesting blog post about the creation of this site using the capabilities of the new Wolfram Language.

I’ve been doing preparation for the Oxford MAT exam with a couple of my Year 13s and in the 2008 paper I came across this very nice question about the Trapezium Rule.

I really like this question as to answer it you need to have more understanding about how the trapezium rule actually works than standard A-Level questions on this subject. Invariably they are just pure “plug some numbers in and crunch them” questions which lead to the wide perception that numerical methods are boring. Of course being a numerical analyst I really don’t agree with this perception, but agree that their presentation in the current A-Level course doesn’t help with this.

The above question is nice in that it combines knowledge of the performance of the trapezium rule with graph transformations.

I will leave you to work out the answer, the small Geogebra file I have written may help you visualise what is happening… 

Last night we discussed “Mathematical études: embedding opportunities for developing procedural fluency within rich mathematical contexts” (available online here).

Like last time it was a very fast paced, interesting discussion. I really enjoyed the discussion and some great points were made by lots of people. Have a look through the storify below – I have hopefully put in all the points discussed.

Stephen Cavadino (@srcav) wrote a blog post after the discussion trying to categorise études which I recommend reading. James Pearce (@MathsPadJames) also wrote a post before hand (which unfortunately I didn’t see until about half way through the chat) looking at commonalities between different types of mathematical études – give that I read too!! I would have promoted it more during the discussion if I had seen it earlier.

Thank you to everyone who took part, and I hope you can make the next one which will be on Monday the 7th of December. The poll for voting will open for a week next Monday. If you would like to submit an article please tweet the details to me by this Friday 23rd October.

https://storify.com/tajbennison/mathsjournalclub-discussion-number-2