Author: tombennison

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

As written about here, on Monday 11th January at 8pm there will be a special #mathsjournalclub chat to discuss two articles from the special assessment edition of ATM’s Mathematics Teaching Journal.

Here are a couple of things from each article that I think could make discussion points and may provide a bit of focus as you read the articles.

1. “Encouraging students’ formative assessment skills when working with non-routine, unstructured problems: Designed student responses” by Sheila Evans 
   1. Do you use exemplar student responses to develop mathematical thinking in your classes?
   2. Do you think this would work for your students?
   3. The responses in this particular example are designed to stimulate particular ways of thinking – do you think this is a good thing? Is it better than sharing actual pupils work?
2. “Assessment: Beyond right and wrong” by Matt Lewis  
   1. The authors make the point that levels are only proxies for mathematical competence – could we test in a different way to get more meaningful data.
   2. Should we do more teacher assessment in later keystones?
   3. Have you had experience of using a comparative judgement methodology when marking tests?

I really hope you can join us tomorrow – I’m looking forward to a good discussion.

As a quick post for New Year’s Day I thought I would share my top 5 blog posts of the previous year.

1. Maths Journal Club – The first post about #mathsjournalclub. Get involved in the next discussion on the 11th January for the ATM special edition chat, details are here.
2. John Mason – Another ATM Session – Here I wrote about a fantastic morning that i spent in Leicester at the joint ATM/MA session led by John Mason. He presented some great ideas for use in the classroom, I was particularly interested in his ideas for developing mathematical thinking in those students with weak numeracy skills.
3. Another MyMaths Post – This post proved very popular over a couple of nights in September as it detailed a particularly bad piece of content on a MyMaths lesson. To MyMaths’ credit, they then did change it within a day!
4. Carnival of Mathematics 127 – In October I hosted the 127th Carnival of Mathematics that is organised by the Aperiodical. There are some really interesting links in this, so take a browse if you haven’t already.
5. An A-Level Calculated Colouring – In the run up to Christmas I shared a resource for A-Level that I had created. All of last year my classes had been asking to do a calculated colouring so for the last lessons this term I produced an A-Level calculated colouring. This has been downloaded many times and I am very grateful to the people who pointed out slight mistakes.

Over the course of the year I have really enjoyed posting on this blog, and it looks like I have been fairly regular:

For the #summerblogchallenge I managed to post every day for 49 days which I am quite impressed with – I’m going to try and do this again next year. I am hoping to post more next year, and definitely share more resources which I haven’t been that good at doing this year.

Thank you to everyone who has taken the time to read my posts during the last year – it means a lot.

Yesterday I noticed on Twitter that Springer have made available quite a few classic books from their “Undergraduate Texts in Mathematics” and “Graduate Texts in Mathematics” freely available as PDFs. Some of these I paid good money for as an undergraduate and still have on my shelves and dip into occasionally.

A GitHub user “bishboria” has usefully made a hyperlinked list of all the books from these series that are available for free as a GitHub Gist here. If you are so inclined further down the page are shell scripts that you can run to automatically download all of them.

There is a great variety of books available, but if you only want to download one, in my opinion a book that is definitely worth having is Donald Estep’s “Practical Analysis in One Variable”. This is an amazing book that covers topics ranging from the natural numbers and the invention of negative numbers up to topics in analysis such as Weierstrass’ approximation theorem. Donald Estep is also interesting to follow on Twitter at @donestep1. As an aside he is also one of the authors of the books for the “Applied Mathematics: Body and Soul” project.

Following on from Stuart’s (@sxpmaths) excellent blog post looking at the changes coming to A-Level mathematics in 2017 I thought I would take a bit more of a detailed look at the prescribed content of Further Mathematics than I did in my old post here. Since I wrote my original post there have been a few changes (notably clarification of how the prescribed content for AS-Level is decided) between the draft content and the finalised content document that was published in December 2014..

This post is my interpretation of the content contained in the content document  and I would encourage everyone to look at the original!

Unlike the 100% prescribed core of the standard A-Level only 50% of the content for Further Maths (FM) is prescribed with awarding bodies able to determine the other 50%. I’m not sure how I feel about this: part of me likes the freedom that this gives the exam boards to potentially include novel content and explore different areas of mathematics. The other part of me would prefer a 100% prescribed A-Level as this levels the playing field for students who (at the moment) are often at the mercy of their schools in terms of what they study to bolster results, not necessarily preparing them as well for future mathematical study. I can see this kind of “game playing” potentially occurring if there are vast differences between individual boards.

In this post I will look at the prescribed content of the AS-Level further mathematics qualification

AS-Level:

There is some prescribed core content for AS FM however a smaller proportion is prescribed than that for the full A-Level as explained in the original document.

• “At least 30% (approximately) of the content of any AS further mathematics specification must be taken from the prescribed core content of A level further mathematics.”

This 30% is composed of two parts. 20% of the overall content must come from the following areas

• Complex Numbers including:  the solution of quadratic, cubic and quartic equations (with real coefficients), arithmetic operations on complex numbers including the use of the complex conjugate. Students are to be able to represent complex numbers on an Argand diagram and know that complex roots of a polynomial (with real values coefficients) occur in complex conjugate pairs.  Manipulation of numbers in modulus argument form. Construction and interpretation of simple loci in the Argand diagram.
• Matrices including: Knowledge of the zero and identity matrices. Multiplication of a matrix by a scalar and addition, subtraction  and multiplication of conformable matrices. Use of matrices to represent single linear transformations in 3D (some restrictions apply) and successive linear transformations in 2D. Finding invariant lines and points of linear transformations. Determinants and inverses of \(2\times2\) matrices.
• Further Algebra and Functions including: The relationship between the roots and coefficients of polynomial equations, up to quartic equations. Forming a polynomial whose roots are a linear transformation of the roots of a given polynomial equation, where the polynomial is at least of cubic degree.

Another (approximately) 10% of the overall specification must be taken from the rest of the prescribed content of A-Level FM, but it is up to the boards to decide what they take.

A few comments on the above

• When manipulating numbers in modulus-argument form knowledge of radians and compound angle formulae are assumed. Depending on how schools teach the AS this could prove problematic as radians are not introduced until A2.
• I like the inclusion of loci, in some boards this isn’t currently tackled until FP# and consequently may not be encountered by students at all.
• There seems to be more of a focus on finding invariants under linear transformations than at present.
• It’s nice to see the transformation of polynomials in there as it has wide ranging applications.

Essentially, only 20% of an AS qualification in further mathematics is necessarily common to all boards.

Come back early next week for my take on the rest of the prescribed content for the full A-Level further mathematics qualification.

Apologies that this announcement is later than planned, but I still think there is time for people to read these articles for a discussion on Monday 11th January at 8pm.

The ATM Mathematics Teaching journal produced a special assessment focussed edition this November that has completely open access until the end of January and for this reason we decided to run an extra maths journal club chat looking at two articles from this issue.

The winning article was:

• “Encouraging students’ formative assessment skills when working with non-routine, unstructured problems: Designed student responses” by Sheila Evans  – Effective assessment is necessarily composed of numerous strands and strategies. Some strands will be more preeminent than others, but sound assessment depends on data derived from multiple sources. Designed student responses have the potential to gather new insights into student achievement and understanding. Here the case is well made, and students respond positively to the introduction of this type of assessment activity.

The second placed article was:

• “Assessment: Beyond right and wrong” by Matt Lewis – This is an account of an on-going evidence-based project in a number of schools in London. The professionals involved seek to set the complex process of assessment of learning in mathematics in the context of its educational and social purposes. The article draws on examples
  from recent work on approaches to the assessment of reasoning and problem-solving. The challenges are not underestimated given that we seem to be experiencing a culture that might be regarded as valuing measurement of performance regardless of the quality or meaningfulness of the data generated. The debates will continue, but the enduring need for statutory assessment should not dumb-down the process.

If you click on the article titles you will be taken to a pdf file – both are fairly short and would make great Christmas reading.

I look forward to discussing these on the 11th January 2016 for the first #mathsjournalclub of the newyear.

This post is an expanded version of a review I wrote for the IMA’s (Institute of Mathematics and it’s Applications) magazine Mathematics Today which was published in the December 2015 issue. If you are in the UK ad aren’t already a member of the IMA I really would encourage you to join, the magazine is great and there are some great regional talks etc.

SUMAZE! is an app available for both Apple and Android devices and is a completely free download (and free of adverts).

I first became aware of MEI’s new mathematics game, SUMAZE!, at the beginning of half term after seeing it mentioned on Twitter. It is available for both iOS and Android devices and “awesome puzzle game” is typical of the reviews in these stores. Indeed, SUMAZE has already entered the top 10 educational apps chart in the Google play store despite only being released on the 20th October 2015. Sumaze is based on an idea of Richard Lissaman of MEI and was then developed into the apps we have today.

Soon after downloading, it became obvious that it was going to be very addictive and consume quite a bit of my holiday.

When you start the app you are presented with the following screen:

 At first the differing levels are locked and are opened as you complete more of them.

The game is simple in concept; you, as the player, just need to move a numbered block to an exit point. In working your way to the exit however, operator blocks modify the number on your block and green blocks only let you pass if your number satisfies the condition shown. The choice of operators and conditions can make all the difference in how quickly you can complete a level. The app is very well designed, with a slick look and an intuitive interface. Six core topics are covered by the currently available levels (arithmetic, negatives, powers, inequalities and modulus, logarithms and properties of number) and these are unlocked as you complete the levels. Whilst you may think that arithmetic is a simple mathematical concept, in fact some of the levels for each section can be surprisingly (and frustratingly) difficult! I was stuck on a particular level in the Numbers section for what felt like hours and one level in the final “Fermat’s Rooms” seemed very challenging. I like to think that there is a bug in this level concerning modular arithmetic.

To me -1 should pass through the green condition block labelled “=1 mod 2” and then all the others down the right hand side. However this isn’t allowed and after forming some equations you can obtain a value in the 800s.

These mazes could be a great way for a teacher to introduce the concept of logarithms for example. Students working through the levels will develop a deeper conceptual understanding of what a logarithm “actually is” than they may have achieved through just the definition and some questions. It would be a help if there was a “teacher unlock” code to unlock levels on school devices so that particular levels could be used by the students.

The SUMAZE! puzzles also appear as resources on MEI’s integralmaths.org website for use in class. In addition, since writing the review for the IMA a web version has been made publicly available.\

In the web version all levels are unlocked and so can be used to introduce students to logarithms for example. I used this web app with a Year 7 class towards the end of the year, instructing them to do the arithmetic levels first and then allowed them to select between negatives, powers or modulus. A lot of them chose to try the modulus levels as this was something unfamiliar to them. It was really nice to watch them learn about the modulus function (a topic that they definitely wouldn’t normally come across at such an age) and be self motivated to do so.

I really hope that MEI produce some add-on levels for this game as I am now left wanting more! Similar apps for other topics would also be great, I can think of similar ideas that would be good for both younger and older target audiences.

As an aside, does anyone know who the “Dr H” some of the levels refer to is? I think I am missing some popular culture reference…..

I’ll end the review with a quote sent to me from Richard Lissaman that he had received from a user of the app

 “It happened that I was visiting with my 15-year-old grand-daughter this afternoon and mentioned the Sumaze! app. This girl is something of a mathophobe, but no sooner had I mentioned the app than she downloaded it to her iPad. Then it was all that I could do to pry it away from her to talk about adding rational functions. Thanks so much for telling me about this”

If you haven’t already succumbed to this game, download it now, I promise that you will enjoy it.