# New AS level maths SAMs

Having spent most of my Saturday this weekend working through the new AS level sample assessment materials for each exam board I thought I would briefly summarise my thoughts so far. Being a little rusty on statistics and mechanics I have brushed up mostly on M1 and S1, so all the binomial distribution and hypothesis testing questions I haven’t touched for now.

As I am currently teaching the Edexcel specification and so decided to attempt their materials first. They are the only exam board who have proposed a 2 hour core pure paper and a 1 hour applied (statistics and mechanics) paper. The material in both papers seemed reasonable and at similar level to the current qualification.

I then looked at the AQA material which seems to be ever so slightly ‘easier’ than the Edexcel papers. AQA, like OCR A and OCR MEI, have gone with two 1 hour 30 minute papers split into pure and applied. For some of the new material, particularly applicable to differentiation from first principles, AQA have designed a ‘guided’ question (see question 8 in paper 1). In contrast, a question on this topic when assessed in all other boards is just asked up front, e.g. Differentiate $f(x)=3x^2$ from first principles (Edexcel paper 1 question 9). There are also some multiple choice questions which, following a brief conversation on Twitter with Tom Bennison, I am informed is similar to their GCSE papers. This, I think, could be seen in both positive and negative views. It’s quite nice to be able to get students thinking about some of the misconceptions and really thinking about their answers in the exam, however some students could get frustrated over the quantity of work they might need to do for a single mark (I make this statement with respect to my year 12 class who showed me their AS physics sample paper in which they sometimes needed to do large amounts of work to find the correct choice).

OCR A were the next papers I looked at and I very much enjoyed these ones. Some more interesting and different questions cropping up and the statistics element taking up what felt like half the first paper. I particularly liked the trigonometry and vectors questions in paper 1 (question 5 & 6) and the proof question in paper 2 (question 6). The papers were slightly more difficult that the Edexcel papers in my opinion, though with some similarities.

Finally, I looked at the MEI materials. The first paper opened up with a nasty looking question in comparison to other papers, but most of the content again seems to be on par with the OCR A sample materials. Interestingly, MEI are also the only exam board to mix the pure and applied questions together, rather than separate the sections.

I still need to look over the physical specifications for each board, but at this stage I think my preferred set of sample materials for the AS in mathematics are those provided for the OCR A.

Next, I think I will look over the A level mathematics materials before I look at some of the further maths. But before that I need to revise some mechanics and statistics!

Edit (12/06/16): Note also that OCR A has included binomial expansions as part of their statistics section, whilst Edexcel and AQA included this within pure.

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# The NQT year is almost over…

It has been an interesting first year in teaching to say the least. Aside from development in school I have particularly enjoyed engaging with the Twitter community in the sharing of ideas and resources, as well as getting involved with chats and attending CPD events. I have also had the privilege to be involved with the mathematics magazine, Chalkdust.  In this post I hope to document some of the things I have made use of or got involved with during this year.

One of the first Twitter chats I was able to get involved with was #mathsjournalclub which is hosted by Tom Bennison. A new chat which also started last year, the discussions have a mathematics education research paper as the focus, though discussions have also developed around ideas and suggestions which are raised during the chat. Past discussions can be found on Tom’s Storify page. The sixth chat is up for voting until the 9th June here and will take place on Monday 11th July at 8pm. If you don’t already take part, the discussions have all been very interesting and it would be great if you got involved!

I have also tried to get involved with the weekly #mathscpdchat and #mathschat discussions and have had the pleasure of hosting two #mathscpdchat discussions this year (Wild Maths and Marking at A-level). These discussions always have an interesting topic of focus with lots of fruitful discussions. Following a break for half term both of the discussions should be very exciting next week with a special #mathschat webinar and #mathscpdchat focusing on working collaboratively on teaching, learning and assessing mathematics.

I have attended ChristMaths and MathsMeet Glyn (organised by Jo Morgan), MathsConf6 (La Salle Education) and Maths in the Sticks (organised by Stuart Price). All of these event provided excellent presentations and provided me with lots of things to take away and think about. My next planned event is Tom Bennison‘s East Midlands KS5 Mathematics Conference which already has an excellent line up! I would highly recommend getting along to any good CPD event, especially when they are free!

I now frequent many maths teaching resource websites. For A level I tend to find myself looking for something on Integral Maths. For homework, Jo Morgan and Kathryn Forster’s Pret Homework website provides quality and worthwhile worksheets designed by teachers. In addition Jo Morgan’s Resourceaholic website provides access to some outstanding resources which can be used from KS3 to KS5.

I have without a doubt used an extensive list of other resources, however these have been of particular help when struggling to find something good to incorporate into a lesson.

I will be returning to my second PGCE placement school for my second year of teaching and I am looking forward to a change in dynamic and continuing to make use of the outstanding resources that have been made available by fellow mathematics teachers.

# “Discourses of Assessment – Discourse of Mathematics” (Morgan 1999) | LMERG reflections

LMERG: London Mathematics Education Reading Group.

Original post and links available on the LMERG blog.

Paper selected by Lisa Chalmers – “Discourses of Assessment – Discourses of Mathematics” (Candia Morgan, 1999).

Justification: Interest in the development of viewpoints on assessment from 1999 in comparison to now. What is the goal of assessment and what should we be doing?

Reflections on the discussion (R. Beckett)

Assessment is something we all have to do and the focuses of assessment have changed–now if a student doesn’t ‘make the grade’ then it is deemed the teacher’s fault. Is the goal of assessment really to just get students a better grade in their examination? What are we doing it for?

It was suggested that Candia Morgan, in this paper, is evaluating the […]

https://lmerg.wordpress.com/2016/03/22/march-discussion-2016/

# #ChristMaths15

On the 21st December 2015 Jo Morgan hosted the first ever ChristMaths party event with CPD, networking and of course alcohol! This post contains some of my reflections after reading what I wrote down during the talks.

Strategies for teaching previously ‘Grade C and beyond’ topics to Foundation students – Mel Mundowney (@Just_Maths)

Mel delivered some interesting points in her talk, particularly addressing how many students are now playing ‘catch up’ with the curriculum changes due to different demands and teaching. One of the key things I noted were that there is a need to ‘keep things fresh’ as, more recently, there has been cyclic and repetitive content being taught in a disjointed curriculum. Another was that it’s ‘all about the questions!’ In particular how these differentiate tasks and how it can help develop pupils.

Flexible maths – The Michaela Community School Maths Team (@BodilUK, @danicquinn & @naveenfrizvi)

The Maths team from Michela school delivered a session discussing their approach so far (being a new school they have only year 7 and 8). They showcased their knowledge booklets which have been used to aid lesson teaching and have a strong focus on developing mathematical vocabulary and knowledge through ‘drilling’ the basics. It was interesting to hear their approaches and the booklets that were shared at the event will probably be a valuable resource!

Developing problem solving skills – Colleen Young (@ColleenYoung)

Colleen Young’s talk had a focus on developing the problem solving classroom. She suggests that the teacher-student relationship is a key aspect of this development. Additionally our use of vocabulary and students understanding of this could develop. When tackling ‘problem solving’ questions students should not fear just trying something; there should a resistance in the urge to rely on the teacher as ‘mathematical guru’. She provided lots of resource suggestions and her whole talk can be found on her blog!

A five year GCSE – Kris Boulton (@Kris_Boulton)

Kris talked about how we can better our own teaching, in particular discussing examples where he had not explained or taught topics well and had then adapted or radically them at the next time of teaching. He also identified that there is plenty of time to teach the new curriculum, but we need to focus on teaching well in the first instance and sequencing the content better. He concluded his talk by saying “Mathematics is mathematics!” This was a suggestion that we should be teaching students maths from (at least) year 7 through to the end of (at least) year 11 and we shouldn’t ‘start teaching GCSE in year 9 or year 10’.

Closing remarks – Jo Morgan (@mathsjem)

Jo was the concluding speaker raising some of her talking points and concerns about the new GCSE. One that stood out to me was the mention of Gove claiming the new GCSE would include fundamental mathematical content – but who decided what is fundamental (e.g. trig ratios in foundation)?

Evening networking

Having recently joined the Chalkdust magazine team Jo kindly allowed me to distribute copies around the room for people to take. It was really well received and I hope everyone from #ChristMaths15 enjoys reading issue 2 as much as I did!

The evening of the event saw maths teachers talking, completing Jo’s quiz and solving puzzles from Emma Bell (@El_Timbre).

An excellent ending to a very well organised and successful event! Well done Jo!

# Mathematics in Action

On Wednesday 25th November 2015 I took 13 sixth form students from my school to the Mathematics in Action event organised by The Training Partnership. It was an excellent day out and I summarised some of my notes from the day’s lectures in a post for my students which is replicated below.

Host (Tom Evans)

The host of the day was Tom Evans. He opened up the session with two puzzling questions which the audience were asked to attempt during the breaks between speakers. The problems are below:

Happy birthday Fermat’s last theorem (Simon Singh)

The opener was Simon Singh talking about Andrew Wiles and Fermat’s last theorem. He opened up the talk with the start of the BBC Horizon episode (see here) he directed on the same topic. In the video Andrew Wiles expresses an analogy for mathematics whereby it can, for a time, feel like you are walking around a dark house stumbling into things all the time. Then, one day, the light switches on and suddenly you see. Wiles becomes very emotional when talking about Fermat’s last theorem, which as Simon explained in his talk is due to the sheer length of time and determination spent solving a problem which had become a childhood dream.

Simon gave us the problem and spent a large part of his talk discussing the back story and some of the people who tried to provide a proof for Fermat’s last theorem across some 300 years. He recommended purchasing his book for further reading (which I did!). If you are wondering the theorem states the following:

No three positive integers $x, y$ and $z$ can satisfy the equation $x^n+y^n=z^n$ for any integer value of $n$ greater than two.

There were also recommendations at the end, for further enriching mathematics, to take a look at Numberphile, Vi Hart and Martin Gardner. Do take a look at all if you haven’t yet!

How big is infinity? (Chris Good)

Chris focused a large part of his talk on infinity around the work of Georg Cantor. He opened the talk by asking a classic question; ‘Which is bigger 1 or 0.99999999999…?’ He used this to clarify that every number between 0 and 1 can be represented as a unique non-terminating decimal (due to one-to-one correspondence) which was introduced by Cantor first in 1874. As a result of much of Cantor’s work there was an implication that there are an infinite number of infinities (i.e. I can always choose a bigger infinity than the one you just chose!). Chris concluded his talk by discussing transcendental numbers.

Exam technique (optional session with Colin Beveridge)

I really liked the Venn diagram Colin used at the beginning of his short talk about exam technique to describe the ‘ideal’ experience maths A level students should have (i.e. extended subject knowledge and support with exam technique). One of the things I really liked, and have already used, is the ‘error log’ idea. This is where pupils are asked to write down their mistakes, take note of where they went wrong or got stuck and then include the steps they need/needed to take to rectify the problem. Colin had some really valuable things to say so I am glad I stayed to listen to this (even if my students didn’t!).

7 things you need to know about prime numbers (Vicky Neale)

Fact 1: 1 is not a prime number.

Fact 2: 2 is a prime number (and the only even prime!).

Fact 3: (Theorem) There are infinitely many prime numbers.

This was proved by both Euclid (c.300BC) and Euler (18th century). This theorem of the week blog post gives some detail on both.

Fact 4: (Fundamental theorem of arithmetic) Every integer greater than 1 can be expressed as a product of primes in an essentially unique way.

Fact 5: Every prime is one more or one less than a multiple of six if $p>3$.

Therefore, could you prove the following statement (imagine delivering your explanation to a year 9 student)?

If $p$ is a prime number greater than 3, then $p$ is of the form $6n \pm 1$, where $n$ is a natural number.

Fact 6: Let $\pi(x)$ denote the number of primes less than or equal to $x$. For example, $\pi(10)=4$ or $\pi(100)=25$.

(Prime number theorem) Then $\pi(x) \sim \frac{x}{\log{x}}$ where here $\log{x}$ is the number theory notation for $\ln{x}$.

Fact 7: (Twin prime conjecture) There are infinitely many primes $p$ such that $p+2$ is also prime.

This conjecture is yet to be proved, but there have been many recent developments!

Exit question:

If 3 and 5 are twin primes then let us call 3, 5 and 7 a prime triple. Are there any more prime triples?

Geometry and the art of optimisation (Richard Elwes)

Richard opened his talk by discussing how much of his work on optimisation is applicable in everyday life (e.g. traffic lights and train timetabling). He discussed an example of a toy factory where we needed to consider how to maximise the profit from the production line. The calculations made led to the consideration of the feasible set of values which we then needed to consider maximising. Richard then introduced us to the simplex algorithm found by George Dantzig in 1947. He concluded with the Hirsch conjecture to which a counter example was found in 2010.

Cryptography (Keith Martin)

Keith was a very engaging and entertaining speaker, and whilst his talk didn’t contain much ‘maths’ he was able to discuss the importance of cryptography. Some elements of security are ‘lost’ in the cyber world and we need to ensure our information holds its confidentiality, integrity and authentication. Cryptography is a tool used and built by mathematicians to help with encryption and security in cyberspace. Keith discussed ciphers, our data integrity (using ISBN codes as an example) and authentication:

On the internet no one knows if you are a dog.

Keith recommended taking a look at cryptool.org if you are interested in looking further into cryptography. His alternative suggestions were Piper and Murphy’s Introduction to Cryptography book or Simon Singh’s The Code Book.

# First half term as an NQT completed…

… And hasn’t it been tough?! An ‘interesting’ start to the year resulted in a number of timetabling (and therefore class) changes, which meant having to go through the introductory stage with classes again after two weeks! I have enjoyed experimenting with a number of ideas and resources I have seen this half term, and the idea of this post is to document some of these, much like my previous post ‘First two weeks as an NQT’. I shall try to ensure that these ideas are ‘new’ and that I am not repeating things which are mentioned there!

First off I want to talk about marking. Whilst I have by no means found the ‘best’ approach to marking, this sticker shared by a member of my department has helped me to share feedback clearly with students. What I really like about it is not only the ‘RAG’ style effort rating bar, but that you can invite students or peers to assess work in their books using the sticker. I am still looking for effective marking methods, so please share your strategies with me!

We have used the diagnostic questions website (@MathsDQs) this half term with year 7 to assist with placing them into ‘appropriate’ sets after half term. These were an excellent resource and I would recommend use of the website across key stages. I imagine the GCSE collections would be invaluable to assist students and teachers in becoming (more) aware of what areas still need to be addressed, and where misconceptions are most often occuring. There is also a collection focusing on the new AQA specification which would also be of benefit.

Due to our mixed ability year 7 classes there were a few occasions when the work was not sufficiently challenging for some pupils. To tackle this I looked at the ‘My Classroom’ post from @solvemymaths which I mentioned in a previous post. I had recalled seeing a section called ‘extension activities’, and when reviewing the post I found the link to resources from mathschallenge. The questions are excellent and challenging, so I printed some of these out which solved the problem of not always having sufficient material to occupy students in the lessons!

With KS5 I have been making use of MEI’s integral maths on a regular basis as well as looking at Jo Morgan’s (@mathsjem) bank of resources on resoureaholic. I am awaiting word on login details for CMEP having seen a selection of these resources at an FMSP development session at UCL earlier in October. I have also started blogging for my KS5 students (see ‘Blogging for KS5’ post) after each lesson, providing them with access to the lesson resources and some additional follow up material. After half term I am beginning a KS5 enrichment club which is starting off as training for the senior maths challenge hosted by the FMSP before developing into a less specific enrichment club. I hope to continue improving where I go to find resources so as to provide my sixth formers with engaging lessons and encourage them to see the real beauty and excitement of mathematics.

Another thing I saw via Jo Morgan is the website Create A Test (@createatest). I consequently took a look and on finding it to be free, signed the school up. It is an outstanding resource for producing assessments and exam style questions with the ability to generate variations of one particular type of question. I really like the website and have already begun to encourage other members of my department to take a look and make use of this FREE resource.

Monday 19th October saw the second maths journal club discussion on Twitter (@mathjournalclub/#mathsjournalclub). This was a nice ‘break’ from teaching, diving back into the research and taking part in an interesting discussion. This focused on Colin Foster’s paper “Mathematical études: embedding opportunities for developing procedural fluency within rich mathematical contexts”. There were a number of great ideas in the paper which I hope to now implement in my future teaching. Check out the storify of the discussion put together by host Tom Bennsion (@DrBennison) here. Also, next discussion is on Monday 7th December!

I am still developing my bank of resources and ideas and the maths education Twittersphere has been one of the effective places for me to do this. Thank you to everyone who freely shares their teaching ideas and resources, so many of us appreciate your hard work!

# Youcubed Week of Inspirational Math(s)

Since our year 7 classes are being taught in tutor groups until half term our department has been looking at finding good mixed ability lessons. We came across Jo Boaler’s (@JoBoaler) Week of Inspirational Math from the Youcubed website which has lots of tasks and resources relating to growth mindset and encouraging everyone to try their best in maths lessons.

This ‘Week of Inspirational Math’ consists of five lessons (and lesson plans) based on both thinking and group work tasks. Here I include a brief summary of the tasks involved.

The first lesson challenges pupils to share their ideas about what good and bad mathematical group work looks like before then tackling the four four’s problem (finding numbers 1-20 using four four’s only). Lesson two is called ‘number visuals’ and asks pupils to look for, share and discuss patterns in the visualised numbers provided. The third lesson encourages pupils to challenge each other through questioning and tries to help them develop their explanations and justifications. In the fourth lesson pupils are invited to look at Pascal’s triangle (with gaps) and to try and work out what is going on. They are also then expected to try and identify other patterns in the triangle. During the fifth and final lesson of the ‘week’ pupils must first think about what they see happening with the growth of a certain shape structure before then discussing ideas in their groups and then attempting to create generalisations.

I have done up to the third lesson with my year 7 classes so far, and the ‘number visuals’ lesson was particularly successful with many students coming to the front and explaining the patterns they had noticed (primes in circles, times tables, factors, etc.). These lessons seem to have been really engaging for the groups of student so far and all have been able to access it, as well as challenge themselves mathematically.

It has certainly been challenging, teaching mixed groupings, but it has also been really enjoyable to be able to support students in the same class with different aspects of the same problem.

Whilst I have used this with year 7 it is suggested on the lesson plans that it could be used right through to with year 10 groups and I can imagine it would be quite interesting to do that!