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An Evening of Nerdy-ness

Yesterday evening I went to the excellent Festival of The Spoken Nerd in Derby, but before thsat I had the pleasure of seeing Alex Bellos speak at a school in Leicestershire. This post is just going to mention a few interesting things from both of these events.

Alex Bellos was a very entertaining speaker to listen to as he is evidently so passionate about people understanding the beauty of mathematics.  

 Alex opened his talk with a logic puzzle that he had featured in one of his early Guardian Puzzle blog posts: Find the odd one out in the symbols below 

 This puzzle was orignally due to Tanya Khovanova who had written about it here. It is intended as a piece of fun to emphasise the intrinsic issue of odd-one-out puzzles. Namely, that they tend to be focussed around a particular way of thinking (Alex provides a nice example in his post) when in fact many could be seen as the odd one out for different reasons. In the puzzle above, the odd one out is actually the one on the left hand side by virtua of not being able to be called the odd one out – an interesting philosophical dilemma there!
One thing I found particularly interesting in Alex’s talk was his discussion of the Sieve of Erastothenes. If I do this in school I tend to have a 10 by 10 grid of the numbers 1 to a 100. I had never thought of arranging them instead in 6 rows, as shown below. This leads to some very interesting patterns when you cross out numbers. 

He then described the Ulam Spiral, devised by Stanislaw Ulam where numbers are arranged in a spiral and primes highlighted generating a pattern where primes lie on diagonal lines. I’m going to write some MATLAB code to generate these I think and write a bit more about them in the future.

 

Alex also talked about the results of his internet survey to find the world’s favourite number; which turns out to be 7. He showed a nice annimation about this which is available on Youtube

Following Alex’s session I had a fairly mad dash to Derby in horrendous weather to go and see Festival of The Spoken Nerd – if you haven’t seen them live before I can’t recommend it enough. It was a great evening. I really liked the visual demonstration of the modes of vibration of a metal plate. These are known as Chladni figures after the scientist Ernst Chladni who first published his work in 1787.

   

  
 

Festival of The Spoken Nerd have plenty more dates in their UK tour, I urge you to see if there is one local to you on their webpage and if there is go. You won’t be disappointed, I’ve seen them a few times now and it is always a very entertaining evening – writing about the show can’t do it justice.

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#mathsjournalclub Third Poll

 Following on from our second successful discussion last week it is time to vote on the article for the next discussion. The next discussion will take place at 8pm on Monday the 7th of December. I know everyone gets busy in the run up to Christmas but I hope that you can all still take part.

Three article from the last poll have been rolled over to this one as two of them were tied in the number of votes. As usual, the titles and abstracts are below and the poll is available here

  • 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‚Äô 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.
  • “‚ÄėAbility‚Äô ideology and its consequential practices in primary mathematics” by Rachel Marks (Proceedings of the BSRLM 31 (2)) – ‚ÄėAbility‚Äô is a powerful ideology in UK education, underscoring common practices such as setting. These have well documented impacts on pupils‚Äô 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‚Äô 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.
  • “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.
  • “Relational Understanding and Instrumental Understanding” by Richard Skemp (Mathematics Teaching 77)
  • “Knowing and not knowing how a task for use in a mathematics classroom might develop” by Colin Foster, Mike Owlerton and Anne Watson (Mathematics Teaching 247) – Participants at the July 2014 Institute of Mathematics Pedagogy (IMP14) engaged in a wide range of mathematical tasks and a great deal of pedagogical discussion during their four days last summer. Towards the end of IMP14 a conversation began regarding how much knowledge about a task a teacher needs to have before feeling comfortable taking it into the classroom.

I’m looking forward to seeing which article is selected as I haven’t read all of these yet!

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A Pi Curiosity

I’m quite sad that I missed this on pi day this year but I thought I should share it now..

For this years (special) pi day Wolfram Research have produced a web page mypiday.com that enables you to find the location of your birthday in the digits of pi – it is well known that any date will appear within the digits. Here is the location of my birthday: 

 Stephen Wolfram also wrote a fairly interesting blog post about the creation of this site using the capabilities of the new Wolfram Language.

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A Nice Trapezium Rule Question

I’ve been doing preparation for the Oxford MAT exam with a couple of my Year 13s and in the 2008 paper I came across this very nice question about the Trapezium Rule.

Screenshot 2015-10-24 22.30.01

I really like this question as to answer it you need to have more understanding about how the trapezium rule actually works than standard A-Level questions on this subject. Invariably they are just pure “plug some numbers in and crunch them” questions which lead to the wide perception that numerical methods are boring. Of course being a numerical analyst I really don’t agree with this perception, but agree that their presentation in the current A-Level course doesn’t help with this.

The above question is nice in that it combines knowledge of the performance of the trapezium rule with graph transformations.

I will leave you to work out the answer, the small Geogebra file I have written may help you visualise what is happening…¬†Screenshot 2015-10-24 22.36.09

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#mathsjournalcub Discussion Number 2

Last night we discussed “Mathematical √©tudes: embedding opportunities for developing procedural fluency within rich mathematical contexts‚ÄĚ (available¬†online¬†here).

Like last time it was a very fast paced, interesting discussion. I really enjoyed the discussion and some great points were made by lots of people. Have a look through the storify below – I have hopefully put in all the points discussed.

Stephen Cavadino (@srcav) wrote a blog post after the discussion trying to categorise √©tudes which I recommend reading. James Pearce (@MathsPadJames) also wrote a post before hand (which unfortunately I didn’t see until about half way through the chat) looking at commonalities between different types of mathematical √©tudes – give that I read too!! I would have promoted it more during the discussion if I had seen it earlier.

Thank you to everyone who took part, and I hope you can make the next one which will be on Monday the 7th of December. The poll for voting will open for a week next Monday. If you would like to submit an article please tweet the details to me by this Friday 23rd October.

https://storify.com/tajbennison/mathsjournalclub-discussion-number-2

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Carnival of Mathematics 127

Following on from last months excellent collection of posts in the Carnival of Mathematics 126 from Stephen at cavmaths it is my turn to host this blog carnival.

The Carnival of Mathematics is a monthly blogging round up that is organised by  The Aperiodical (check it out if for some reason you haven’t before) and has a different host each month.

It is tradition to start the post with some interesting facts about the number 127 so here goes….

  • The most interesting fact about 127, I think, is that it is a Mersenne Prime. This means that 127 is a prime number of the form \(2^n-1\) with \(n\) here being 7. Since 7 is also a Mersenne prime 127 is known as a double Mersenne prime, that is, it is of the form \(2^{2^n-1} -1 \).
  • 127 is a number that appears in the list of centred hexagonal numbers. Coincidentally I have written a post concerning these numbers when I discussed nRich’s Steel Cables problem here.
  • Since 127 is a centred hexagonal number and prime it is Cuban prime.Namely it is a solution of the equation \( \frac{x^3-y^3}{x-y} \) where \(x=y+1\) and \(y>0\). This can be simplified so that Cuban primes are prime numbers of the form \(3y^2+3y+1\).
  • 127 is a cyclic number. This means that the Euler totient of 127 and 127 are co-prime. (Clearly all primes are cyclic.)
  • 127 is the 8th Motzkin number. For a particular value of \(n\) the \(n-\)th Motzkin number is the number of different ways of drawing non-intersecting chords between \(n\) points on a circle.
  • 127 is a polite number as it can be written as \(127 = 63 + 64 \).

To start on the posts that make up this carnival I thought I would start with the following comic produced by Manan of MathMisery. This made me laugh a lot!Screenshot 2015-10-17 22.14.23

His blog is one that I routinely keep an eye on and I also enjoyed his recent post entitled Executive Education

Ganit Charcha submitted this really interesting article on Montgomery Modular Multiplication. I hadn’t heard of this algorithm before and I need to get round to coding it up soon – this article is very clear and written in a style ideal for being able to code it up.

Shecky R shared a short post looking at a curious finding (made without modern calculating machines) of Pierre de Fermat in his post Get A Life!

I enjoyed reading Tracy Herft’s post Unique Lesson… Polar Clocks. In it she describes a lesson she has developed for Year 7 students which she uses to link the topic of angles with fractions.

The mathematician Nira Chamberlain has shared his poster about the black heroes of mathematics. He has created this for Black History Month (which happens to be October).Screenshot 2015-10-17 22.29.08

I’m ashamed to say that I hadn’t heard about any of these before!

This post by Evelyn Lamb (@evelynjlamb) concerning cutting letters of the alphabet featured a video by Katie Steckles of The Aperiodical. I love how Evelyn describes the “peculiar laziness” of mathematicians – it is certainly true!!

R.J. Lipton has this great accessible post “Frogs and Lily Pads and Discrepancy” discussing Terence Tao’s recently announced proof of Paul Erd√∂s’ Discrepancy conjecture. If you fancy some more of the detailed maths take a read of Terence’s post announcing his papers (and the other related posts). Tao is pretty rare amongst professional mathematicians in that as well as publishing in academic journals he also discusses his research in an open blog.

Herminio L.A. submitted an interesting discussion of a calculation paradox in “Sabotage in the Stores”.

I had recently seen this video on Youtube and it was nice to learn that someone had submitted it for inclusion in this carnival (I would have myself if no one else had). The animations that go with some of the (many) patterns in Pascal’s triangle are fantastic.

Edmund Harriss has this post of beautiful animations of eigencurves of various matrices that are dependent on parameters in one entry. Edmund has also produced a colouring book with Alex Bellos titled “Snowflake, Seashell, Star” that taps in to the colouring craze at the moment. I can’t colour at all, but I am tempted to get this book, if only for the mathematical notes. He discussed the book here.

Christian Perfect (@christianp) shared this  article by Lior Pachter. This article concerns something that I am very passionate about – the discussion of unsolved mathematical problems at school. Lior has taken each year in the American Common Core and selected an unsolved mathematical problem whose description is accessible to students of that age. I particularly like how each unsolved problem is accompanied by a starter problem that is accessible by the students. I think discussion of unsolved problems in school is very important as far too often students think that everything in mathematics is known. It’s sad that students s of school age don’t typically get exposed to current research or unsolved problems. 

Diane has shared this problem from part of The Center of Maths’ Advanced Problem of the Week series. It’s quite a nice problem involving partial differentiation. This series provides a nice selection of problems to try, however some do require more area specific knowledge than others. 

Peter Rowlett (@peterrowlett) recently wrote this fascinating post about Mathematical Myths. E.T. Bell did much to create a series of myths about mathematicians and some of these feature in this post. There are extensive links in this post so it may take a while to read. 

During this month I enjoyed reading “The Nuts and Bolts of Writing Mathematics” by  David Richeson and Cleve Moler’s blog (the first in a series of 4) on Dekker’s Algorithm. More recently Kris Boulton (@Kris_Boulton) had a great article about the importance of memorising times tables published in the TES. 

Personally my most involved blog post in the last month was the write up of my NQT advice workshop at #msthsconf5. 

This brings to an end this issue of the Carnival of Mathematics. The next edition is being hosted by Mike at Walking Randomly. If you haven’t before check out Mike’s blog, he regularly posts good articles about mathematical software. 

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Happy Ada Lovelace Day

Today is Ada Lovelace Day everyone! 

I think it is great that Ada Lovelaceis beginning to get the recognition she deserves. I can remember reading Ada’s notes on Babbage’s difference engine when I was in sixth form and being completely fascinated by them. 

The University of Nottingham have produced this great video about her:

And there is a good article on the Guardian about why Ada Lovelace is important for women in STEM subjects. 

Hannah Fry (@FryRsquared) has also written a nice article that goes alongside her programme about Ada Lovelace. If you haven’t seen that yet it is definitely worth a watch. 

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#Mathsjournalclub Next Week

Next Monday (19th October) at 8pm we have the second #mathsjournalclub discussion. This time (in case you don’t already know) we are looking at Colin Foster’s paper on “Mathematical √©tudes: embedding opportunities for developing procedural fluency within rich mathematical contexts”. 
I read the article through again on the train on Saturday night and enjoyed the read. 

From reading it here are a few things to think about in advance of next Monday. 

  • Would you be comfortable with removing all “drill” practice and relying on richer tasks to develop procedural fluency?
  • In what other settings could you use a task similar to this one?  
  • I feel that Colin makes an interesting point regarding students defaulting to  a favoured approach when tackling “problem solving” type questions. Do you agree? 

There is still time to give it a read and take part of you hadn’t considered it before – you can get the article online here

I hope you can get involved in the discussion on the 19th. 

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A Strange Step-by-Step Solution

Earlier this evening I saw the following picture posted by Mathster (@mathstermaths)  

 
As I tweeted I am really not keen on the use of “move all terms to the left” as I don’t think it is clear what this is meant to mean mathematically. Indeed, it could easily lead to misconceptions with people thinking that you just move terms over an equals sign with no conceptual understanding of what should be happening. Stephen Cavadino (@srcav) has written about this here

I thought I would try going online and doing the same equation as is solved in the example. The step by step solution I obtained was even stranger: 

 
I have no idea why there are steps 1 and 2 simplifying the right hand and left hand side before the pointless step of rearranging to obtain everything equal to zero…….

Christine (@MissNorledge) also pointed out the strange behaviour this app shows when rounding. According to the example above 8/3 rounds to 2.666666666666665. To me this indicates poor use of floating poor arithmetic. 

Jasmina then showed something that is clearly a bug in their parsing library  

 I’m curious as to how this app is working, specifically

  • How the user input is being parsed?
  • Are they using a JavaScript symbolic algebra library (and if so which one)?
  • How is the “maths” implemented?

Unfortunately at a first glance the JavaScript viewable using the Safari developer options isn’t particularly illuminating. As I’m away for the weekend I haven’t been my able to delve very deeply into the code with a good editor like Sublime and work out what is doing what. 

The app can’t solve quadratics with complex roots or cubics: 

  

 All in all I want to know more about this…

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How to Enjoy Your NQT Year

I’m finally getting the material from my session online – if you want an overview of #mathsconf5 last sturdy check out my post here.

In this post I am aiming to share the materials I handed out in my session “How to Enjoy (and Succeed at) Your NQT Year”

I’ve embedded my Prezi below, if you were at my session you would know that I didn’t get anywhere near through it.

[prezi id=”http://prezi.com/-dz30jtfabtw/?utm_campaign=share&utm_medium=copy&rc=ex0share” align=center width=600 lock_to_path=1]

All the materials can be accessed on my website at the links below:

I started the session by finding out the strengths and weaknesses of attendees. In my opinion you will have a good understanding of these by the time you start your NQT year (even if you are an ITT student I think you often know what they will be before you have spent much time in the classroom). For me, my strength is that I am pretty good at stretching the high attainers whereas I see behaviour management as a weakness as this is something that I have had to work quite hard at to get where I am now. Being frank with yourself about your personal qualities is very important if you want to improve and develop your practice.

Following this we talked about behaviour management strategies; I was very interested in some of the conversations happening about the use of reward lessons in the computer room for good behaviour amongst other things. My top tips for behaviour management is to ensure that you are always consistent (however tired and fed up you are) and follow through with any sanctions you hand out. It is very easy to let things slide as term goes on – especially in the winter months when it is dark and cold – try your best to not let that happen.

Moving on to lesson planning I talked about my five key things to consider when planning a lesson:

  1. The big picture.
  2. The key learning objectives.
  3. 5 key questions.
  4. Any possible misconceptions.
  5. How will you assess learning.

For me the questioning used with the class and individual learners is the most important thing to bring on the mathematics of students by prompting them to think critically about it. This fits in nicely with a school focus on questioning in my school which is good. Our new Year 7 scheme has questions for most lesson plans. I think considering questions before the lesson means that I think more in depth about the maths that I am presenting and how it will come across to learners. The same can be said for considering misconceptions; some of the best questions in my opinion will draw out misconceptions and enable us to tackle them head on as a class. I was possibly a little controversial in this section by saying that I don’t really like the assessment being on lesson plan pro-formers. This is not because I don’t think it is important – it is of course crucial to know where your students are with the topic of a lesson! My point here was that I don’t want how I am going to assess my students (which is really my problem to resolve) to affect how they experience the mathematics that we are doing that lesson and so my mode of assessment shouldn’t be a focus of the lesson planning process.

Many teachers now have to move rooms between lessons. When this is the case I find that it helps to have some go to starters that don’t require me to have logged on to a computer and turned a projector on. For this reason, this year I have stuck a sheet with some easy “no-plan” starters into the back cover of all of my lower school student’s books. I have written a bit more about this in my post “No Plan Starters”. With practice I find that the students don’t need too much guidance if I just tell them a starter task (and maybe write a few numbers on the board) and it means that they are practising some basic skills while I am waiting for the computer to load.

Before I was planning on discussing the advantages/disadvantages of technology in lessons I talked about homework. My guiding principle is the I want a homework that takes the minimum amount of time to prepare and mark but that gives the maximum possible benefit to the students. A type of homework that fits this bill are the PRET homeworks as shown below.Screenshot 2015-10-04 11.11.38

I particularly like the research and stretch components of this style of homework. I found that last year, being very precise with the expectations for the first couple that I handed out meant that students soon learnt what was expected when I gave them a homework in this style. Before you go away and write some you should definitely check out Jo’s (@mathsjem)and Kathryn’s (@DIRT_expertPRET homework website as there are plenty already to use!

For technology in the classroom I was going to talk about different tech tools so I think I shall write about this topic more over the next few months….

I had a large section planned entitled “Do Some Maths”. This is actually my number one tip for your NQT year! It is all too easy to get caught up in the mundane tasks of a teacher that you don’t spend time doing any maths apart from the questions you need for lessons. I’m sure that most of us chose to teach maths because we love doing maths so I think it is important to spend some time doing maths that is challenging. Here is a nice question:Screenshot 2015-10-04 21.32.04

A particularly nice source of challenging problems is the book “Calculus for the Ambitious” by T.W. Kernel. This is a book targeted at strong A-Level students giving a more university-like approach to Calculus than you see at A-Level.

Your NQT year is an ideal time to develop your subject specific pedagogic knowledge. Use those extra frees to go an observe other teachers focussing in particular on how they teach a given topic. Look for how they structure a lesson to build understanding gradually, how they tackle topics that are perceived as being hard to teach and how the activities they choose promote confidence in the learners. Another excellent way to build your subject knowledge (both pedagogic and general mathematical) is to join at least one, if not more, subject associations. Personally I am a member of the ATM and the IMA and value the access this gives me to their journals. On the education journal note it is definitely worth getting involved in the #mathsjournalclub discussions that I organise – the next one is on the 19th October, see here for more information.

On the subject of marking, which is often seen as a necessary chore, and any time you spend doing it should have an impact – it definitely should not take over your life. Kev Lister’s (@listerkev) RAG123 marking has revolutionised the way I mark. Being able to get quicker feedback from the students is really helpful, it doesn’t take that long to get through a set of books commenting if necessary. I do a deep mark in line with my schools marking policy in addition to this, but having looked at the books so much these deep marking sessions are much quicker than they would have been.

I believe that having a network of teachers you can turn to for advice outside of your own department and school is incredibly helpful. Twitter makes this easy, so spend some time on twitter and take part in as many twitter chats as you can. I provide a summary of all the maths specific chats here.

My plan was to end on giving some tips for your NQT year from people other than me:

  • Don’t do any work on a Saturday.
  • Follow up on sanctions – but make sure they are fair.
  • Always stick to class routines that you set.
  • Don’t be afraid to say when you are struggling – someone will help you.
  • Laminate resources.
  • After a bad lesson find a positive – even if it is that you kept all the pupils in the room.
  • Always remember the good pupils and how much you enjoy teaching them.
  • Write any advice given to you down.
  • Divide your time away from school into working time and non-working time and stick to it.
  • Talk to anyone and everyone.
  • Don’t overdo it

and then mention a few things to avoid including listening to people moaning all the time and working all the time.

The main thing I have learnt myself from this session is to not plan as much!! Thanks to Danny Brown (@dannytybrown) for his interesting points in the session and to Rob Beckett (@RBeckett_Yd) for having a quick look over the slides previously and giving me an NQT’s perspective on them.