Author: tombennison

So the day to announce the next #mathsjournalclub article as come, and it was a landslide victory, garnering over 52% of the votes.

The article you have chosen is “Mathematical études: embedding opportunities for developing procedural fluency within rich mathematical contexts” by Colin Foster, as published in the International Journal of Mathematical Education in Science and Technology.

The abstract is reproduced below and the article can be downloaded by clicking on this link.

• In a high-stakes assessment culture, it is clearly important that learners of mathematics develop the necessary fluency and confidence to perform well on the specific, narrowly defined techniques that will be tested. However, an overemphasis on the training of piecemeal mathematical skills at the expense of more independent engagement with richer, multifaceted tasks risks devaluing the subject and failing to give learners an authentic and enjoyable experience of being a mathematician. Thus, there is a pressing need for mathematical tasks which embed the practice of essential techniques within a richer, exploratory and investigative context. Such tasks can be justified to school management or to more traditional mathematics teachers as vital practice of important skills; at the same time, they give scope to progressive teachers who wish to work in more exploratory ways. This paper draws on the notion of a musical e ́tude to develop a powerful and versatile approach in which these apparently contradictory aspects of teaching mathematics can be harmoniously combined. I illustrate the tactic in three central areas of the high-school mathematics curriculum: plotting Cartesian coordinates, solving linear equations and performing enlargements. In each case, extensive practice of important procedures takes place alongside more thoughtful and mathematically creative activity.

This looks a really interesting article and I hope that many of you will join us (despite it being term time) for the discussion on Monday 19th October at 8pm.

The second and third place articles will now go through to the next poll, along side some other suggestions.

I hope you enjoy this article!

One of the many jobs that I had wanted to do over the summer was to create PRET homeworks for the A-Level modules that I was going to be teaching this coming year. Along with the website design this didn’t really happen, so now I am doing them throughout the year as I go.

I thought I would share one I made this week as I am trying to do at least one resource post and one more general math/education/math education post every week.

Jo Morgan (@mathsjem does a great job collating all the PRET homework that people contribute at her PRET homework site, so make sure you take a look to see all the wonderful homework for KS3,KS4 and KS5 that are available. The one I contributed this week is for “The method of differences” which is in the Edexcel Further Pure 2 module.

The use of the method of differences to sum infinite series isn’t often touched upon in A-Level and so I included that in the research part. For the skills section i tried to make most of similar to the ones in the official A-Level text book, apart from Question 4 which is harder than you tend to get for the current FP2 syllabus. It is only harder as I don’t give the function you require to apply the method of differences; the students only need to remember that the function tends have a power that is one order above the terms you are trying to sum and it drops out fairly easily. The stretch questions I have taken from the old Rostock and Chandler books that I love, these are harder than the ones typically seen at FP2, but definitely not insurmountable.

This sheet is available here or in the Algebra section of Jo’s site.

Today I am hostingthe first #mathscpdchat of the new academic year. Between 7pm and 8pm we will be discussing the following topic: “How do you demonstrate to your pupils that you have a personal love of learning maths?”

Since being asked to host this, I’ve been thinkning quite a bit about this and this short post briefly distills some of that thinking.

Of course there are some areas of math that I have a personal leaning towards, but I would say that I love learning about any are of maths. How to demonstrate this to a class at school is somewhat harder to pin down. Rightly or wrongly, teachers are often perceived by their students as all knowing and so as far as they are concerned there is no more learning for us to do – I beleive it is important ot challenge this perception.

Below are some things that I think can help us show a love of learning.

• This is probably easier with an A-Level class, but I will normally choose exercises to do on the board that I haven’t looked at before. This serves two purposes in my opinion;it slows me down so that I don’t gloss over any exposition that may be key to someone else’s understanding and it also demonstrates that I need to try different approaches, or learn a new technique in order to be able to master it.
• Visibly reading books about mathematics. My desk will often have a book that I am currently reading, often a popular maths title (so that if a student asks me about the book I can talk about something fro the book in an accessible way. If I am trying to promote persistance with an A – Leve class, sometimes reading during the lesson can also disuade them from asking me a question too early in addition to showing that I am actively seeking out new knowledge about mathematics.
• If possible talking about current mathematics research that has made the news is good for many other reasons than just showing that I enjoy learning about mathematics. For example, the new result about tiling pentagons is accessible to anyone with a basic understanding of interior angles of a pentagon. 
• Promoting the love of learning of any subject by being interested in discussing any topic or subject with a student. I think the love of learning is infectious and so showing that you enjoy learning and see it as something rewarding in itself has to be a good thing. 
• Being positive when teaching any topic. Students seem to be scarily perceptive of whether we like (or value) a particular topic – I have definitely made this mistake before. 

I’m looking for better ideas than this to demonstrate my love of learning mathematics. It seems very hard(to me at least) to convey the excitement I feel when working through a new problem or learning some cool new result. 

I’m looking forward to discussing it in just over half an hour. Join in tonight’s #mathscpdchat at 7pm (UK time)!

Today was an inset day for me, so technically yesterday’s post was my last #summerblogchallenge post, however I thought I would briefly round off the challenge tonight. 

Overall I have enjoyed it, but I am looking forward to relaxing a bit and not having to post every day. Some days it was hard to find a topic that enthused me to write about! Having said that, there are plenty of things that I considered writing about but didn’t get around to, and these may make appearances over the next few months. In particular, I remember saying that I will be writing about the Cauchy-Schwarz inequality so I will try to do that some point soon. 

I have loved reading all the other posts by my fellow #summerblogchallenge people and I have tried to select my favourite post from each of them. 

• Christine Norledge (@MissNorledge) – this post about the top 5 resources for rich tasks and problem solving. 
• Mark Wilson (@mwimaths) – I enjoyed reading about his joining into maths in this post. 
• Kim Thomas-Lee (@kimThomasLee) – great post about various activities at a “Numeracy Challenge day”. 
• @funASDteacher – This post has added a book to my reading list 

I’m not really sure if I have a favourite post but I did like the one about the first #mathsjournalclub discussion as I was very glad that the discussion went well. 

I’m contemplating doing the #summerblogchallenge again next year…. 

During the last week of the holiday I was inspired by some videos that Jo Morgan (@mathsjem) has produced (as seen in her great blog post on Behaviour Management) using the VideoScribe software to produce a Classroom Expectations video of my own to use.

VideoScribe is a piece of software that you can download to your Mac, PC or iPad to produce what it terms as “WhiteBoard Animations”. They have a 7 day trial licence available which is great for having an initial play. Aside from not liking some of the stylistic touches of the applications icons and having a few issues uploading my “scribe” to Youtube – it took three attempts, and the upload that did work took hours the software was incredibly easy to use. I was very impressed with the results it produced and in the future I would like to get Inkscape out of the closet to produce some .svg files and then use VideoScribe to produce some short animations about A-Level mathematics and other interesting mathematics. The monthly cost of a Pro subscription seems a bit steep at £18 but they do say that you can stop and restart that subscription when you like, which is good as this is a tool that I don’t think I will have the time to use every month. Once you have gone pro you can also export your scribe as a QuickTime video which you can then upload to Youtube separately which I think will be less troublesome than using VideoScribe’s straight to Youtube.

I am looking forward to using this piece of software some more.

I was wondering today how many people in the UK had seen the article about Proof School in the San Francisco article.

I saw this article on KQED.org towards the end of July and found it fascinating. The idea of a school dedicated to helping the most gifted at mathematics excel really appeals to me. Personally, I feel that the most gifted students of maths, particularly in state schools, sometimes don’t get the maths education that that they deserve. I remember that when I was a student, it was a widely held view amongst other students that the “top set wasn’t really cared about as they would get the grades needed for the school without particularly inspiring or good teaching”. I certainly, now that I am a teacher myself , don’t believe this is a widespread view held by teachers and generally I think teachers do the best they can for all of their students. However, it is a fact that in many schools top set classes are generally bigger than the lower attaining classes resulting in each individual having less teacher time spent on them. This means that there is, necessarily, less dedicated time spent on developing that gifted individual’s mathematics – to me this doesn’t seem that fair.

Teacher expertise is also an issue – the most gifted students at mathematics by the time they are at secondary school are probably significantly better at maths than most teachers of the subject (I’m definitely including myself here – I just know a lot about one tiny tiny area of maths!), and whilst things such as IMO and UKMT exist I think it could be nice to bring the most gifted students in a region together for sessions where they are taught (or allowed to are discover for themselves) some mathematics that is of a much higher level than they would normally be exposed to in school.

I would urge anyone interested to take a look at the Proof School’s website. I particularly found the academic information interesting (additional languages offered are Latin and American Sign Language – no foreign language which is a bit strange). The structure of the mathematics curriculum is very much more like a University level structure with some really interesting topics included. I like their description of a “maths kid”

Their characterisation of a maths kid as someone who loves maths is really nice, I firmly think love of maths not ability should be celebrated. If someone loves maths then they will soon be able to excel at maths I think. 2.5 hours of maths in the afternoon would have been a dream for me when I was at school.

The school has axioms that are used to define the school 

Number 3 is something that I think should be talked about more – mathematics is a social and creative subject, not just an intellectual pursuit!

I wonder if something similar could work in the UK – I would possibly be interested to be part of it if it was tried – the closest that I can think of are academies with a specific STEM focus, such as the newly opened NUAST Academy in Nottingham.

What do other people think of the idea of a school where maths is given much greater prominence (though other subjects aren’t neglected) than it is in most schools?

Yesterday Luciano (@DrTrapezio) tweeted asking if anyone had a document comparing the A-Level content across different exam boards. I’m convinced that I have seen exactly this in the past but I can’t find it now. Sue de Pomerai (@SuedePom) mentioned that she had a copy of one for further maths and emailed it to Luciano, who has asked me to put it up and share with people. So this was a nice easy #summerblogchallenge  post today.

This file is now available on my web site here and looks a bit like this

The file is ordered around the MEI further mathematics modules and then the location of each topic in the specification of the other boards is given. It is pretty much complete, but I think there are a few AQA topics missing (for example Viète’s formulae) and don’t we cover simultaneous equations with matrices in Edexcel FP1 and FP2.

I think this document is very interesting and I wish more boards did some of the topics that MEI do, for example the Cayley-Hamilton theorem and Lagrange interpolating polynomials.

Short post tonight….

Next year I will be moving about classrooms a fair bit, not least because I will be frequently moving between main school and sixth form. As we don’t have movement time, on those occasions I will sometimes want to have a starter that I can get the students working on quickly whilst I wait for the ‘computer to log on etc.

Because of this I have produced the sheet below with a few options to go on the back cover of the exercise books, with the intention that after a while i will be able to just say, for example,  “Task 7 and write a few numbers on the board. 

 There is nothing new or revolutionary here, but I thought I would share it in case it is useful for anyone. A pdf is available here.

You may notice that i have also put a RAG123 key at the bottom. After being inspired by many people on Twitter, I’m excited to be trying this properly for the first time this year.

Yeaterday I picked up my free copy of The Times with my My Waitrose Card and on page 3 it had this puzzle: 

 The rules are deceptively simple

• You must place the numbers 1-9 in the 9 squares, using each number only once. 
• The number in each circle should be equal to the sum of the four surrounding squares. 
• Each colour sum is correct. 

This puzzle turns out to be trickier than it looks, and this was the intention according to this little bit of history. The puzzle was created by Jai Gomer of Kobayaashi Studios. 

The three pictures below show my workings to solve this puzzle.  

    
 To start with we have 9 unknowns and 7 equations so clearly an indeterminate system; hence brute mathematical force alone won’t be sufficient. Applying a bit of logic we can deduce two of the numbers. At this point I thought great, now I have 7 unknowns but 7 equations. However this was very foolish of me, as in fact we only really have 5 equations for our remaining 7 unknowns. And so, I had to make a few educated guesses on the likely magnitudes of some of the unknowns based on the totals that they contributed to. Once I had done this, the other unknowns dropped out fairly easily and a quick verification at the end showed that I had all the values correct. It could have been different though if I had been incorrect with these educated guesses. 

So I have a few unanswered questions

• Have I missed something? Could I have done it without these educated guesses?
• Could I remove one condition and the solution still be unique? 
• Would colour totals split into 3,3,3 squares instead of 2,3,4 lead to easier or harder puzzles?

I shall ponder these….

Inspired by Kim’s (@kimThomasLee) post about maths quotes and needing to fill a bit of space on a display board I thought I would see if I could quickly produce some posters of quotes using the Retype app for iOS.

Here are the results:

You can download the full resolution images from the links below.

• Paul Halmos
• G. H. Hardy
• Carl Friedrich Gauss