Author: tombennison

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

Following last weeks successful #mathsjournalclub discussion it is now time to choose the article for the next discussion which will take place at 8pm on Monday 19th October.

The poll is at this link, below are the titles and abstracts of the suggested articles:

• “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.
• “Symbol Sense: Informal Sense Making in Formal Mathematics” by Abraham Arcavi (For the Learning of Mathematics)
•  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.
• “Mathematical études: embedding opportunities for developing procedural fluency within rich mathematical contexts” by Colin Foster (International Journal of Mathematical Education in Science and Technology) – 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.
• “‘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.

The poll will be live till the 14th September – please make your choice.

Today was the first #mathsTLP chat of the new term, so I thought I would list the great maths CPD chats that all maths teachers (and especially NQTs) should get involved with. I have learnt so much from all these chats over the last year.

#mathsTLP : This excellent chat organised by Jo Morgan (@mathsjem) and Ed Southall (@solvemymaths) is dedicated to teachers sharing ideas and plans for lessons and takes place on Sundays between 7pm and 8pm. Since it has been running I have got some great lesson ideas. Join in and share your ideas, there is always someone you can help and always someone to help you when you are stuck.

#mathscpdchat: This is a chat organised by the NCETM every Tuesday 19:00-20:00. A wide variety of topics are discussed and this is a very popular chat with lots of people contributing. I have been asked to host the first one of the new term, so this is a bit of a promo – take part!!

#mathschat: Weekly chats organised by @BetterMaths and AQA that take place on a Wednesday for an hour at 8pm. You can vote for the topic of the next week’s poll at www.bettermaths.org.uk. These chats are starting back this Wednesday (2nd September) and I am looking forward to taking part again.

#mathsbookclub: This is an excellent chat organised by @MathsBookClub on a bi-monthly basis. This was the brainchild of Hannah (@MissRadders) and Victoria (@sundayteatime) and a great book is always chosen. The current book is “How Not To Be Wrong” by Jordan Ellenberg (I wonder if @JSEllenberg will take part) and the chat is on Tuesday 13th October at 8pm. I’m enjoying re-reading bits of this book in preparation, get your copy now so you can take part! They also have a separate hashtag that acts as a slow chat #mbcslowchat if you want to discuss the book as you read.

#mathsjournalclub: This is a new chat organised by @mathjournalclub (well me in disguise…) where every two months we select a journal article to read and then discuss during an hour one Monday evening. The first one of these took place on Monday 24th August – check out the Storify here. Check out more details of the discussion in my post about it. The next poll to select the article will go live tomorrow, 31st August and the chat will take place towards the end of October.

Some general guidance on how to take part in a twitter chat is contained in this great post by Danielle Bartram (@missbsresources) on social media and blogging in general.

Yesterday we went to London Zoo with some friends of ours and aside from having a great day I noticed some great mathematics in the architecture of a few of the buildings. 

The one I found most fascinating was the old penguin pool by Berthold Lubetkin, which is now a Grade 1 Listed structure.  

 The great design of these intersecting curved pathways is shown even better in the original architectural cross section  

 When this was constructed in 1934 reinforced concrete was new and exciting, this kind of mathematically intricate design wouldn’t have been possible in such a clean way before.  

 I also liked the geometric construction of the Snowdon Aviary which was designed by Lord Snowdon. It’s use of tension as a key force to keep the building up was revolutionary at the time.  

This structure is composed of tetrahedrons between some giant V shaped columns.   

I’m contemplating designing some mathematical tasks around these buildings…