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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…

Last year my predictions of grades for my A-Level were unfortunately a little off. Since then I have been thinking about what I did wrong or whether it was purely a consequence of the exams changing slightly and exam pressure….

I do think the exams were slightly harder the last time round, but I have also decided that I should make any end of chapter tests (I also do full past paper mocks) a bit more rigorous. As an example of my slightly improved tests I have uploaded the one I used for Taylor Series approximations here. I particularly like Question 9,

The easy way is to apply Leibniz’s formula for derivatives, but of course A-Level students won’t have come across this explicitly and so it relies on them spotting a pattern in the derivatives to answer the question successfully. I’d love to hear your views on this assessment!

Predicting A-Level achievement seems, to me, harder than predicting KS4 achievement. I know that over a country wide cohort predictions based on previous attainment prove to be accurate but anecdotally the spread away from predictions is significantly higher at KS5. I think that part of this is due to the increased importance of students actually working at A-Level. In maths, for instance good students can get an A* at GCSE without doing any work outside of the classroom and then can sometimes fail to recognise the importance of hard work to achieve similarly at A-Level. I often tell my students that the step up from GCSE to A-Level is harder than the step up from A-Level to degree level and I firmly believe this to be true: We expect a lot more independence, tenacity and perseverance from an A-Level student than we tend to during previous Key Stages and if they can crack this then the transition to university study shouldn’t be too hard as they will have already developed the skills required for success.

How do you predict A-Level grades (particularly in Maths and Further Maths)?

So my fellow #summerblogchallenge writer @missnorledge alerted me to the #29daysofwriting challenge organised by staffrm and I thought I might as well have a go. I will be cross posting from my own blog at www.blog.ifem.co.uk (if you haven’t already check it out, there is a wide mix of stuff on there!) I think these posts will be different to my usual posts because I don’t have the flexibility with the staffrm editor that I do with my own hosted wordpress site – for instance I don’t believe I can insert LaTeX into my posts easily on this.

I was struggling to choose a topic to write about for this first day until a conversation on Twitter earlier this evening reminded me about an article that I really enjoyed reading last week. The article was from Naveen Rivzi and published on the TES entitled “Why new teachers should not have to plan lessons. They should just get on with teaching.” I found this article very thought provoking, and whilst I may be still unconvinced by all of it – I think I would miss planning my own lessons – there are some points definitely worth thinking about. Subject knowledge is important and it’s natural that pedagogic knowledge develops with time so anything that supports new teachers or teachers with weaker subject knowledge in certain areas (we all have those areas!) gets my vote.

I am all for debate about maths education, and this is actually one of the things I love about Twitter – many differing views to digest and think about; however this article (for some reason) seemed to generate a huge amount of negative publicity. Of course there is nothing wrong with disagreeing with people’s views; it would be a very boring world if we all agreed with each other! In this case, however, I feel that the response took a more personal line, with some tweets I saw appearing to call into question the professionally of teachers who weren’t planning their lessons. To me this is un-called for, an open honest debate about the points put across in an article is one thing  but to dismiss it out of hand and ridicule the ideas in it isn’t constructive. I don’t think many teachers would allow that kind of response to a suggestion made by a child in the classroom, I know I wouldn’t!

I’m nearing the end of 29 minutes and the word count….

To sum up my views: I feel their is a danger of the maths ed debate becoming a polarised “them versus us” discussion – I don’t think this is helpful, and I feel that over my time on twitter I have learnt lots of things from people with very differing views. To close yourself off to viewpoints that you perhaps hadn’t considered before is, in my mind, fool hardy at best, and dangerous at worst.

It is my turn to host the Maths Teachers at Play blog carnival this month (check out the last edition hosted by Manan at MathMisery).

Tradition dictates that I start the post with some cool facts about the number 94, so here are a few:

• 94 is a Smith number. This means that its digit sum is the same as the sum of the digits of its prime factors. \(94 = 2 \times 47\) and \(9+4 = 13 = 2+4+7\).
• The aliquot sum of 94 is 50 since 94 has 3 divisors excluding itself, namely 1,2 and 47.
• 94 and the integers either side of it are all semi-prime.
• It is the 4th 17-gonal number, and so is a number of the form \(\frac{1}{2}(15n^2-13n)\).
• 94 is non-totient.

There were some great posts submitted this month, and in no particular order here they are (with a few additional posts that have read and thought were pretty cool!)

• Recently both the UK and the US have had lotteries with larger than usual rollovers. Manan (@shahlock) wrote this amusing post about the Powerball lottery to celebrate.
• Julie Morgan (@fractionfanatic) has shared this post on a “tried and tested task” as part of the #MTBoS12days blog challenge. In it she describes using RedAmberGreen tasks to level the challenge of questions on a topic.
• Lisa Winer (@lisaqt314) has shared this great post titled “Teaching (and Learning) Grit by Having Students Solve the Rubik’s Cube” where she describes watching the tenacity of students develop over solving the Rubik’s cube. I think this is a great side effect of a puzzle. In this article she also links to an article about an American teacher Dan Van der Vieren which I found interesting.
• In “Math Beyond the Textbook” Tracy Kmosko (@TrayKay11) submitted a post contains a load of links for various non-traditional materials for maths instruction. I particularly like the website about outdoor maths.
• Rodi Steinig has shared this lovely post recounting the conversations that took place during a “Math Circle” session with 15-16 year olds looking at the concepts of one-to-one correspondences and infinity. There have been a couple of other posts from her that follow on from this one here and here.
• “Learning too Teach” is the first in a series of 5 posts by Melanie where she provides a deeply personal reflection on a mathematics subject knowledge enhancement course she has undertaken during her training to become an elementary teacher. I really enjoyed reading this.
• Denise Gaskins (@letsplaymath) has shared the great animation by Tova Brown about Hilbert’s Hotel paradox. I hadn’t seen this before so thank you for sharing.
• Stuart (@sxpmaths) reposted this interesting piece on board games that hide maths well.
• Colleen Young shared two posts that have proved popular this month: one on using colour when teaching algebra (I was also interested to learn about the Trace Precedents feature of Excel) and this one on multiple choice questions.
• I found this post by John Trump on counting the legal positions in the game of Go fascinating.
• Two posts from “A Thomas Jefferson Education” have been shared. The first discusses “7 steps to Successful Math” and one more generally about teaching and learning mathematics.
• Stephen Cavadino (@srcav) wrote this short inspiring post on him giving an FP1 conics question to a non Further Maths student who successfully answered it.

That’s it for this month, make sure you check out the next one when it is posted which will be hosted at “Life Through a Mathematician’s Eyes“.

A short post tonight with a Geogebra resource that I used when teaching the Normal distribution to my year 12 further mathematicians.

I find that the hardest thing when learning the normal distribution and the linear mapping to the standard normal is that students can’t visualise the areas they are trying to find and how they relate. Because of this I made a small Geogebra app that shows how the areas from an arbitrary normal distribution correspond with the areas on the standard normal. It also proved useful for visualising the effect of changing \(\mu\) and \(\sigma\).

This is available as a webpage and you can also download the geogebra file from this link.

Following on from Manan’s great last issue of the “Maths Teachers at Play” blog carnival I am hosting this months issue, and will be publishing next Saturday.

Please submit any posts of your own (or others that you have enjoyed reading) either through this form or to me directly either by email or through Twitter. Technically the deadline for submissions was yesterday but as I am a little un-organised at the moment I will take submissions right up till when I publish 🙂  As it says on Denise Gaskins’ site (she organises the carnival) “We welcome entries from parents, students, teachers, homeschoolers, and just plain folks”.

I look forward to reading your submissions.

For most of January people have been talking about the new Times Tables tests being introduced into primary schools by the Government.

There has been much negative press about these, which personally I think is unwarranted.

For me there is no need for tests to put undue pressure on children, or increase anxiety – it is all about how they are presented to the children by parents and teachers. I can remember being told by my Granny (who was a maths teacher) to “go in and enjoy it” when I talked about tests and I can never remember hating maths tests. Of course I realise that some children may not particularly enjoy tests, but the “children hate tests and they make them hate maths” talk that is common is a massive stereotype and not universally backed up with any evidence. I believe a more pressing issue is the projection by teachers of their anxieties about tests onto their pupils; understandable since they are often judged these days on their pupils performances on high stakes national tests. I certainly don’t see why a new times table test will lead to children not enjoying mathematics! In addition, I feel that not expecting all children to be able to know their times table facts by the end of primary school is just symptomatic of having low expectations.

But, all of these problems are to do with how tests are interpreted or the results used, not with the tests themselves. This distinction is important to me as I always enjoyed doing tests in maths lessons – they were a time where I could just do maths as opposed to being bothered by other things. A nicely designed test is an opportunity for a child to express themselves mathematically – sadly this seems to be a rare thing…

However, I am a little unsure about each individual question having a time limit. If a student is anxious about maths then their performance in this test is likely to be an underestimate due to the anxiety getting in the way. I’ve only really just started thinking about this issue and came across this paper which is interesting reading – I will add it to the next #mathsjournalclub poll.

Apologies for the slightly rambling nature of this post – it’s more an attempt for me to put some thoughts down for myself than anything. For me fluency with multiplication facts underpins so much of later mathematics, even A-Level students who are weaker at these basic skills struggle.

As a final aside, I was discussing this with an old colleague and we think it should be known as “The Multiplication Matrix” instead of “times tables facts”.

On Monday the 11th of January we discussed two articles from the ATM’s Mathematics Teaching journal special edition on Assessment.

A storify of the discussion is embedded at the end of this post.

@PGCE_Maths shared a great article from nRich which though aimed at Primary teachers I think is worth reading regardless of the phase you teach in.

I am taking suggestions for inclusion in the next poll (which will be going live on Friday 22nd January) – please get in touch with articles (that aren’t behind paywalls).

On the 11th of January we discussed two articles from the ATM Special Edition of MT on assessment (available at http://www.atm.org.uk/Special-Edition-MT249)

https://storify.com/tajbennison/mathsjournalclub-discussion-4-an-atm-special#publicize