Author: tombennison

In December last year I had a review of the MEI/Sigma Network’s game published in the IMA magazine Mathematics Today. I later published an expanded review on my blog in December. Check it out if you haven’t already. 

Richard started  by talking about himself and saying that his first experience with computers was when he was bought a ZX81 for Christmas in 1982. As an aside my first experience with computers was on a BBC Micro at primary school. Richard explained how programming increased his enjoyment of mathematics – an experience that certainly resonates with me. 

Richard completed a PhD at Warwick and after lecturing for a while became involved with the Further Maths Support Programme, before in 2014 taking a job at MEI developing online resources for teaching and learning maths.

There are many great reasons why games are good medium for learning mathematics, Richard highlighted that when gaming you expect to make mistakes and it creates a non-threatening environment for students to try things out. He also highlighted the reasons shown below: 

 What does a good maths game look like?! Richard’s criteria are these

• Encourage students to practise lots of examples
• Provide a representation of mathematical problems in a more accessible way.
• Have a mechanic which has a “physical” connection to the mathematical idea.
• Give players lots of choice about what they can do at any given moment.

If you look on the App Store the vast majority of maths games fall into the first category, promoting practise of key skills such as multiplication facts, basic numeracy etc. 

I hadn’t seen Keith Devlin’s game Wuzzit Trouble before (even though I had heard about it) – I feel that I need to investigate this some more now. Richard also completed a secondment developing some games as a consultant to Manga High and he discussed some of the games he helped develop while there including Ice Ice Maybe and Algebra Meltdown – again Manga High is something I really should explore more..

It was fascinating to hear about the development of Sumaze and how it developed out of this initial idea which Richard felt didn’t have enough variety as a game: 

 Following this he began to consider the idea of moving through squares to get to a target value as a way of teaching something, specifically he looked at developing some of the ideas that led to the logarithm levels in the online version of Sumaze.

The Sigma network then funded the production of a web-based version covering more mathematical topics and for an App version. To develop the app a maths startup company MathsCraft was engaged to provide the graphics, sound and menu version. 

I hadn’t thought too much about the game mechanics before so it was nice to be told a bit more about that. Specifically only nested expressions with only one variable are permitted and so the game can’t test things such as \(x^3+6x\) or \(log(xy) = log(x)+log(y)\). Richard explained that the decision was made to not include things like memory blocks or multiple blocks which would have allowed this kind of thing as it would detract from the gameplay. I think this was probably the right decision – the inclusion of these would have made the game seem a lot harder to play, and probably not add too much in terms of mathematics. 

Richard then talked about the mathematics underlying a couple of the levels:

Negatives 13 -Gridlock Here, if you think mathematically you need to find the fixed point of the function \(f(x) = -3(x+12)\) to get the correct value once you have passed through all the modifier blocks. I certainly hadn’t thought of it like this when I played this level.  

 
 Powers 10 – Binary Finery Here the level is asking you to find the binary representation of 61. 
 
As of 8th February 2016 Sumaze has had over 17,000 downloads since the 20th October and another version dealing with lower level numeracy topics is planned.

There was a very good question about the game mechanic versus the conventional order of operations. For example the operations are applied consecutively and don’t follow the conventional rules, for example in Sumaze 3 + 2 multiplied by 3 would give 15 as opposed to 9. I, like Richard hadn’t considered this before as I had seen the blocks as functions that are applied to an input but I guess this could be an issue for students. What do you think? 

Some people on staffrm may not know that I host the #mathsjournalclub discussions on Twitter. In these some of us discuss a research paper (normally chosen by an open public poll) on a mathematics education topic. The summaries of the discussions of three recent discussions are available here (I still have one to put online!):

• Discussion 1 – A glimpse into students understanding of functions
• Discussion 2 – Mathematical études: embedding opportunities for developing procedural fluency within rich mathematical contexts
• Discussion 4 – ATM MT Assessment Special

With one thing and another I didn’t get a poll up to choose the next article – I’m really sorry about this and this way of choosing articles will return so please send me any recommendations, ensuring that they are open access and not behind paywalls.

However, I don’t want to go too long without having a #mathsjournalclub chat and so I have picked an article that i think everyone will enjoy reading. The first page is shown below

The article is by John Mason, Hélia Oliviera and Ana Maria Boavida and entitled “Reasoning Reasonably in Mathematics“. In this article they discuss some student responses to the Magic Square task, for example the one shown here

I found this article fascinating, especially after seeing John Mason speak towards the end of last year, and I am sure you will too.

I propose that we chat about this on Monday the 11th April at 8pm which is the usual time. I very much hope that you can join me.

The discussion after that will be based on an article that has been voted for 🙂

As it is Shrove Tuesday I thought I would briefly introduce the computer science problem of pancake sorting, which incidentally is the subject of Microsoft’s Bill Gates’ only (I believe this to be true anyway) academic paper. If you are interested, this paper is available online and is relatively accessible for an academic paper.

I first came across this problem in 2014 when Simon Singh published an article in the Guardian – as Simon Singh another famous person, David S Cohen, a co-creator of Futurama also has a published paper on a harder varied of the problem known as the “Burnt Pancake Problem”

I guess, I should probably describe the problem now…. In it’s traditional form, it was stated by the original proposer of the problem, the mathematician Jacob Goodman (writing under a pseudonym at the time) as follows:

“The chef in our place is sloppy, and when he prepares a stack of pancakes they come out all different sizes. Therefore, when I deliver them to a customer, on the way to the table I rearrange them (so that the smallest winds up on top, and so on, down to the largest at the bottom) by grabbing several from the top and flipping them over, repeating this (varying the number I flip) as many times as necessary. If there are n pancakes, what is the maximum number of flips (as a function of n) that I will ever have to use to rearrange them?”

This is one of those classic problems that is deceptively easy to state but very hard to solve – in fact Wikipedia tells me that last year a team of scientists determined the minimal number of flips required was proved to be NP complete but i haven’t read that paper yet.

Conceptually some rough estimates on the upper bound for the number of flips required for flipping n pancakes are quite easy to arrive at – e.g. it can not require than 2n flips. I think it would be fun to talk about this with a school class – I certainly wish I had thought about it earlier today and spoken to my A-Level class about it.

 In January I noticed that Steve Blades (@m4thsdotcom) was posting some interesting problems, many designed to stretch and promote thinking in students studying for their GCSEs.

I was particularly drawn to this question in the middle of January 

Initially I thought this was harder than it was and was intending to use the information about the interior and exterior angles to form two simultaneous equations in \(x\) and \(n\) where \(n\) is the number of sides of the underlying polygon. However, there is of course a much simpler way if you use the fact that the interior and exterior angles of the polygon must add up to \(180^{\circ}\). 

Using this fact you can obtain \(7x+1+28x+4 = 180\) and so \(35x = 175\) giving \(x = 5^{\circ}\). Hence the exterior angle of the polygon is \(7 \times 5 + 1 = 36^{\circ}\) meaning that the polygon has \(10\) sides and is a decagon.

I would be fascinated to see how students go about solving this question.

Steve has also collected a paper worth of challenging problems in a “Grade 9” paper available here. I particularly like this question about functions: 

For a while I’ve been thinking about how maths teachers see themselves, in fact I wrote about it in March last year in my post “Mathematician or Educator”.

Last week I decided to re-open the debate with one of the relatively new twitter polls asking whether maths teachers saw themselves as mathematicians, educators or both. The final results are shown below:

I’m very pleased that 50% of the respondents saw themselves as both a mathematician and an educator it does concern me that 45% of people said only an educator.. I’m very interested in to why some maths teachers don’t see themselves as mathematicians. mrvman (@vahorai) suggested a possibility:

I think more generally a big reason is the common belief that to be a mathematician you must be researching mathematics or using mathematics professionally was opposed to teaching “elementary” mathematics to school children. I think this is a sad way to think –  I’m sure many teachers will write things such as “.. is an exceptionally gifted young mathematician” so why wouldn’t a teacher see themselves as a mathematician?

Personally it is important to me to feel that I am a mathematician and a teacher of mathematics if I am to truly show a love of mathematics.

I’m not sure how to change this, I guess in a way we perhaps need to fight against some intellectual snobbery!

I’d be really interested to hear your views on this topic…

Last year my predictions of grades for my A-Level were unfortunately a little off. Since then I have been thinking about what I did wrong or whether it was purely a consequence of the exams changing slightly and exam pressure….

I do think the exams were slightly harder the last time round, but I have also decided that I should make any end of chapter tests (I also do full past paper mocks) a bit more rigorous. As an example of my slightly improved tests I have uploaded the one I used for Taylor Series approximations here. I particularly like Question 9,

The easy way is to apply Leibniz’s formula for derivatives, but of course A-Level students won’t have come across this explicitly and so it relies on them spotting a pattern in the derivatives to answer the question successfully. I’d love to hear your views on this assessment!

Predicting A-Level achievement seems, to me, harder than predicting KS4 achievement. I know that over a country wide cohort predictions based on previous attainment prove to be accurate but anecdotally the spread away from predictions is significantly higher at KS5. I think that part of this is due to the increased importance of students actually working at A-Level. In maths, for instance good students can get an A* at GCSE without doing any work outside of the classroom and then can sometimes fail to recognise the importance of hard work to achieve similarly at A-Level. I often tell my students that the step up from GCSE to A-Level is harder than the step up from A-Level to degree level and I firmly believe this to be true: We expect a lot more independence, tenacity and perseverance from an A-Level student than we tend to during previous Key Stages and if they can crack this then the transition to university study shouldn’t be too hard as they will have already developed the skills required for success.

How do you predict A-Level grades (particularly in Maths and Further Maths)?

So my fellow #summerblogchallenge writer @missnorledge alerted me to the #29daysofwriting challenge organised by staffrm and I thought I might as well have a go. I will be cross posting from my own blog at www.blog.ifem.co.uk (if you haven’t already check it out, there is a wide mix of stuff on there!) I think these posts will be different to my usual posts because I don’t have the flexibility with the staffrm editor that I do with my own hosted wordpress site – for instance I don’t believe I can insert LaTeX into my posts easily on this.

I was struggling to choose a topic to write about for this first day until a conversation on Twitter earlier this evening reminded me about an article that I really enjoyed reading last week. The article was from Naveen Rivzi and published on the TES entitled “Why new teachers should not have to plan lessons. They should just get on with teaching.” I found this article very thought provoking, and whilst I may be still unconvinced by all of it – I think I would miss planning my own lessons – there are some points definitely worth thinking about. Subject knowledge is important and it’s natural that pedagogic knowledge develops with time so anything that supports new teachers or teachers with weaker subject knowledge in certain areas (we all have those areas!) gets my vote.

I am all for debate about maths education, and this is actually one of the things I love about Twitter – many differing views to digest and think about; however this article (for some reason) seemed to generate a huge amount of negative publicity. Of course there is nothing wrong with disagreeing with people’s views; it would be a very boring world if we all agreed with each other! In this case, however, I feel that the response took a more personal line, with some tweets I saw appearing to call into question the professionally of teachers who weren’t planning their lessons. To me this is un-called for, an open honest debate about the points put across in an article is one thing  but to dismiss it out of hand and ridicule the ideas in it isn’t constructive. I don’t think many teachers would allow that kind of response to a suggestion made by a child in the classroom, I know I wouldn’t!

I’m nearing the end of 29 minutes and the word count….

To sum up my views: I feel their is a danger of the maths ed debate becoming a polarised “them versus us” discussion – I don’t think this is helpful, and I feel that over my time on twitter I have learnt lots of things from people with very differing views. To close yourself off to viewpoints that you perhaps hadn’t considered before is, in my mind, fool hardy at best, and dangerous at worst.

It is my turn to host the Maths Teachers at Play blog carnival this month (check out the last edition hosted by Manan at MathMisery).

Tradition dictates that I start the post with some cool facts about the number 94, so here are a few:

• 94 is a Smith number. This means that its digit sum is the same as the sum of the digits of its prime factors. \(94 = 2 \times 47\) and \(9+4 = 13 = 2+4+7\).
• The aliquot sum of 94 is 50 since 94 has 3 divisors excluding itself, namely 1,2 and 47.
• 94 and the integers either side of it are all semi-prime.
• It is the 4th 17-gonal number, and so is a number of the form \(\frac{1}{2}(15n^2-13n)\).
• 94 is non-totient.

There were some great posts submitted this month, and in no particular order here they are (with a few additional posts that have read and thought were pretty cool!)

• Recently both the UK and the US have had lotteries with larger than usual rollovers. Manan (@shahlock) wrote this amusing post about the Powerball lottery to celebrate.
• Julie Morgan (@fractionfanatic) has shared this post on a “tried and tested task” as part of the #MTBoS12days blog challenge. In it she describes using RedAmberGreen tasks to level the challenge of questions on a topic.
• Lisa Winer (@lisaqt314) has shared this great post titled “Teaching (and Learning) Grit by Having Students Solve the Rubik’s Cube” where she describes watching the tenacity of students develop over solving the Rubik’s cube. I think this is a great side effect of a puzzle. In this article she also links to an article about an American teacher Dan Van der Vieren which I found interesting.
• In “Math Beyond the Textbook” Tracy Kmosko (@TrayKay11) submitted a post contains a load of links for various non-traditional materials for maths instruction. I particularly like the website about outdoor maths.
• Rodi Steinig has shared this lovely post recounting the conversations that took place during a “Math Circle” session with 15-16 year olds looking at the concepts of one-to-one correspondences and infinity. There have been a couple of other posts from her that follow on from this one here and here.
• “Learning too Teach” is the first in a series of 5 posts by Melanie where she provides a deeply personal reflection on a mathematics subject knowledge enhancement course she has undertaken during her training to become an elementary teacher. I really enjoyed reading this.
• Denise Gaskins (@letsplaymath) has shared the great animation by Tova Brown about Hilbert’s Hotel paradox. I hadn’t seen this before so thank you for sharing.
• Stuart (@sxpmaths) reposted this interesting piece on board games that hide maths well.
• Colleen Young shared two posts that have proved popular this month: one on using colour when teaching algebra (I was also interested to learn about the Trace Precedents feature of Excel) and this one on multiple choice questions.
• I found this post by John Trump on counting the legal positions in the game of Go fascinating.
• Two posts from “A Thomas Jefferson Education” have been shared. The first discusses “7 steps to Successful Math” and one more generally about teaching and learning mathematics.
• Stephen Cavadino (@srcav) wrote this short inspiring post on him giving an FP1 conics question to a non Further Maths student who successfully answered it.

That’s it for this month, make sure you check out the next one when it is posted which will be hosted at “Life Through a Mathematician’s Eyes“.