Author: tombennison

I’m a little over halfway with the @staffrm #29daysofwriting challenge

Yesterday I was puzzled to see lots of “Halfway” posts as to me a half of 29 is 14.5 and so surely if we only allow full days halfway would round to day 15… Anyway I liked the idea of answering some of the same questions as everyone else so here are my responses.

Why are you doing the challenge?

Why not?! I quite fancy a mug and last summer I was one of the people to complete the #summerblogchallenge (as was @missnorledge ) so 29 days seemed quite doable, especially with the 29 minute rule

Where and when do you post?

Generally at home once I have done Jessica’s (my daughter) bath and bedtime things.

What have you enjoyed the most about taking part?

I’ve enjoyed having something to motivate me to share ideas and my thoughts – sometimes it can be very hard to find time to do this after a busy day at work. I’ve also enjoyed reading lots of posts about a wide range of subjects that I wouldn’t have necessarily come across on Twitter without coming on staffrm.

What have you least enjoyed about taking part?

I’m not a fan of the simplistic editor provided by staffrm! I’m used to the more sophisticated  editor on my WordPress site where I can include LaTeX, code snippets, use lists etc. Later on in the challenge I may have to resort to handwriting some posts and having my staffrm post consist almost entirely of pictures, or have the staffrm post just provide a link to the post on my own website.

Have you picked up any ideas?

I wouldn’t say I have picked up any concrete ideas but many posts (especially ones by @missnorledge , @mrbenward , @towens , @becskar ) have been thought provoking and given me lots to think about.

Only a very short post today as it is Valentines day and I want to devote my time to my family.

I have put up on my website a Geogebra file that allows you to view 2 heart-shaped parametric curves. Which do you prefer?

The original Geogebra file is also available for download here.

Desmos have also produced some cool Math-o-grams including this very cool heart inspired Sierpinski triangle! I particularly like that you can see the underlying equations.

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…