Author: tombennison

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… 

A while ago I posted about “Increasing Cognitive Load at A-Level” where I discussed removing the leading steps from some FP1 questions. I think this is an even more powerful technique for the mechanics modules as these questions can be made significantly more difficult by simply removing the diagram that is attached to the question.

I have always thought that at least half of a mechanics paper should have questions where the student needs to draw the diagram themselves.

As an example, consider Question 1 from the Edexcel M1 Paper, June 2014  

  The first part of the question is just a guide to finding the answer to the second part so removing that and the diagram makes the question a bit more challenging.   

As another example, consider question 6: 

 and my increased difficulty version:  

 My full increased difficulty version of this June 2014 M1 paper can be downloaded here.

Last night at 8pm I was sat at home nervously awaiting the start of the first #mathsjournalclub discussion. I really wasn’t sure if anyone would take part – especially after I had found the article relatively hard going! The article didn’t turn out to be what I had expected, but once I had got through the first few pages I found it very interesting and it gave me lots to think about. 

Thankfully, lots of people took part and I really enjoyed hosting the discussion (even if it was surprisingly tiring!). There were some great points made and lots of excellent discussion- see below for some of these. 

If you didn’t take part this time please consider it next time. I will be releasing the poll to choose the article next week so if you have a suggestion send it to me or to the @mathjournalclub account. 

To try and distill the discussion I created the following storify… Please don’t be mad if all of your tweets didn’t make it – there were so many of them!

On the 24th August between 8pm and 9pm the first #mathsjournalclub discussion took place. We were talking about the article “A Glimpse into Secondary Students’ Understanding of Functions” by Brendefur, Hughes and Ely. Discussion was frantic – this is an attempt to provide some kind of record.

On the 24th August between 8pm and 9pm the first #mathsjournalclub discussion took place. We were talking about the article “A Glimpse into Secondary Students’ Understanding of Functions” by Brendefur, Hughes and Ely. Discussion was frantic – this is an attempt to provide some kind of record.

On the 24th August between 8pm and 9pm the first #mathsjournalclub discussion took place. We were talking about the article “A Glimpse into Secondary Students’ Understanding of Functions” by Brendefur, Hughes and Ely. Discussion was frantic – this is an attempt to provide some kind of record.

https://storify.com/tajbennison/getting-started

Very short post today to just promote the first #mathsjournalclub discussion tonight between 8pm and 9pm. We are going to be talking about “A Glimpse into Secondary Students’ Understanding of Functions” by Brendefur, J, Hughes, G and Ely, R which is available here if you haven’t read it yet.

Last week I posted some possible themes for the discussion, for convenience I have reproduced them below:

• What did you think were the key points of the article?
• Do you agree with the four ways given that “students typically represent functional relationships: graphs, tables, verbal descriptions and equations” ?
• Has this article impacted on how you introduce and teach the understanding of functions?
• What are the limitations of the article/research?

The more people who can take part the better the discussion will be so I really would love it if you could make it tonight.. Don’t forget to use the hashtag #mathsjournalclub in all your tweets so that I and others can follow the discussion.