Ed Hall, the other editor and I presented two topics from the A-Level course that have increased prominence in the new syllabus: Differentiation from First Principles and the Newton-Raphson Method.

The Prezi that we used is embedded below.

As part of the workshop we discussed four Geogebra apps that will be embedded into the digital version of the textbook. Links to these are below:

- Differentiation from First Principles
- Approximating the Derivative
- Newton-Raphson 1 (Cobwebbing)
- Newton-Raphson 2 (Convergence)

Please let me know any of your thoughts and comments about the presentation and the Geogebra files.

]]>It is the tradition that this carnival is always started with some interesting number facts, so here are some facts about the number 140:

- 140 is a Harshad Number. A harsh ad number is simply a number (in a given base) that is divisible by the sum of its digits in that base. As 140 is divisible by 10, 140 is a Harshad number in base 10.
- 140 is the sum of the squares of the first seven integers.
- It is a square pyramidal number – these numbers are the number of spheres in a pyramid with a square base of a given size (think cannonballs).

On to the posts… It’s been a bit of a slow month but there have still been plenty of great things happening in the Maths Blogosphere.

Manan Shah (@shahlock) sent in this post about the concept of “no solution” problems which was prompted by a post he had read concerning the classic “Shepherds Age” problem. If you aren’t familiar with this problem it is a classic problem where useful information is given to enable it to be solved, yet a staggeringly large amount of people still try to solve it by making use of prior knowledge, adding spurious deductions etc. As a teacher, I see this kind of behaviour in the classroom often and I have often wondered if this would be as bad if we exposed children to problems with no valid solution sooner than an undergraduate linear algebra course, as is typically the case in the UK.

Matthew Scroggs (@mscroggs) submitted this post written by Steven Muirhead entitled “Problem Solving 101“. I first read this article in the print version of Chalkdust magazine, and thought it was excellent. It is a very well structured introduction to some typical strategies for mathematical problem solving, namely:

- Get some intuition.
- Find and exploit the structure of the problem.
- Look for quantities that do not change.
- Consider the extremes.

Chalkdust is – if you haven’t already come across it – an excellent, relatively new magazine for the “mathematically curious”. It is available online, but they also produce a printed version which you can order if you pay for postage. In the run up to Christmas they are producing an Advent Calendar – of the days so far I particularly like the mathematical Christmas carol.

I was excited to see a blog I wasn’t aware of in the submissions list this time. It is called Tony’s Maths Blog. Sadly I’m not sure who is behind this blog (all the bio says is that they teach at a University in London) but it has some interesting short posts, such as this one about a book where the murder victim is a mathematician.

The always entertaining mathwithbaddrawings by Ben Orlin (@benorlin) has a great, thought provoking, post about possible ways to arrange the school mathematics curriculum. I’m really not sure how I think the curriculum should be arranged, but I do think something should / needs to be done to stop the large amounts of young children who don’t see the beauty of mathematics and become very disaffected.

Not strictly a blog post, but I wanted to include it as I found it fascinating, is this article on probabilistic programming from Cornell University. It’s fairly maths and programming heavy but a nice, accessible introduction to this research area.

James Hunt has shared an article looking at the frequency distribution of colours of smarties on the site MentalFloss. Many teachers will have done lessons with classes based around a tube of smarties!

Recently Ed Southall (@solvemymaths) has been posting some excellent area puzzles, such as this one.

At the end of November an excellent article was posted on to the AMS Blogs page about the teaching of inverse functions. I’ve hated the whole “swap the variables and solve” approach as students often have no real understanding of inverse functions if they have been taught this way and instead just see it as a black-box algorithm to apply. This article discusses this problem in great detail with a few alternative proposals.

Jemma Sherwood (@jemmaths) posted about Francis Galton’s Wisdom of the Crowds observation here – I’m certainly going to steal this idea for an activity in school. A reproduction of Galton’s original article is available from here.

Stephen Cavadino (@srcav) has shared a question from brilliant.org in his post where he discusses his solution method . Stephen has also fairly recently written a blog post on the use of mnemonics in maths teaching – I’ve never been a fan of these, but they do seem popular. I’d love to know your views.

That brings us to the end of the carnival. The next one is hosted at Ganit Charcha (who incidentally have an interesting article on population growth)- make sure you check it out!

]]>The great A-Level resource site integralmaths have recently announced a new competition for Sixth Form students called Ritangle . As part of this competition 23 questions are going to be released on 23 consecutive days and you need the answer to all of them to be able to work out the 23 character string to be submitted. You can download the poster for the competition (shown below) from the IntegralMaths website

Ritangle have released 5 preliminary practise questions, the last of which went live today:

I like this question; it is fairly accessible but sufficiently challenging to be interesting to solve. My students loved solving the first 4 preliminary questions. I say they are for practise, but correct answers to the first 5 will unlock a clue for the main competition.

One aspect of this competition, that is different to other competitions, is the encouragement to use technology to help with solving the problems. Being a computer coder and someone who enjoys computational maths this excites me. I hope that during the competition I will be able to blog a couple of times about the use of technology. Of course, in the spirit of the competition I won’t be able to discuss methods to solve directly competition problems, but I will be able to talk about computational techniques in general.

I’m really excited by this competition, and am pleased that some of our A-Level students want to to take part. I would encourage you (if you are an A-Level Teacher) to see if your pupils want to take part. If the preliminary questions are anything to go by then there are going to be some great problems to solve.

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My descriptions below are really just the rough notes I made during the sessions (with a bit of neatening up) and so are my interpretation of the sessions – I may have interpreted things said differently to how the speakers intended.

**The Genius of Siegfried Engelmann in Practice – Kris Boulton**

Following Kris’ Research ED Maths and Science talk I bought and have started reading Siegfried Engelmann’s “Theory of Instruction: Principles and Applications“. This is a very interesting book, but I am finding it hard going. Kris recommended another book by Engelmann and Carmine “Could John Stewart Mill Have Saved our Schools?” Engelmanns work is logico-empirical in the way in which it was devised. The logico aspect involves deconstructing the topic to a very fine level of detail to the point that it is logically impossible for a student to mis-understand what the teacher is trying to say. The empirical aspect then tests this in the classroom and makes small changes to the programme, gradually making it better and better.

The idea of ruling out incorrect inferences made by student by deploying examples and non-examples is, when you think about it, is a blindingly obvious way to clarify a concept in the mind of a student. I think

Kris split most of maths into Concepts, Fact and Process – but emphasised that there are some aspects, such as proof, that do not fit these characterisations. His session today forcussed mainly on concepts and processes. Engelmann splits concepts into five different types (names by Kris though)

- Categorical – Is this a pentagon (the response is always “yes” or “no” in the initial questioning sequence)
- Fuzagorical
- Comparative – Did gradient go up or down? Does the speed get
- Correlated Features – You understand the concept by correlating it with others.
- Transformation

Kris explained that he would have originally thought of transformation as a process as opposed to a concept.

Kris considered an expression like \( 2^5 \times 2^4 = 2^9 \) and considered some inferences that pupils may have drawn if this was presented as an inquiry prompt:

- Only works for the numbers shown.
- Only works for single digit integers.
- Only works for positive integers.
- Only works for rational numbers.
- Only works for numbers (not letters).
- Only works for real numbers.

The inference made by any given student will be a result of that persons given prior knowledge. All too often I think I underestimate the range of inferences that can be made by pupils – this session reminded me that i need to try and think about this more.

For me the most useful aspect of this session was seeing how Kris structured a teaching sequence for indices laws. It was split up into an initial instructional sequence, an initial assessment sequence followed by an expansion sequence.

*Initial Instructional Sequence*

- \(7^{10} \times 7^3 = 7^{13} \)
- \(3^{10} \times 3^3 = 3^{13} \)
- \(3^{8} \times 3^9 = 3^{17} \)

It was emphasised that the addition in the first example was deliberately trivial so as to not cause difficulties.

*Initial Assessment Sequence*

- \(2^9 \times 2^{10} = 2^{[ \quad ]} \)
- \(2^9 \times 2^{6} = [ \quad ]^{[ \quad ]} \)
- \(12^9 \times 12^{6} = \)
- \( 193^7 \times 193^14 = \)

*Expansion Sequence*

- \(13^{50} \times [ \quad ]^{10} = 13^60 \)
- \(97^{ [ \quad ]} \times 97^{[ \quad ]} = \)
- \(3^1 = 3 \) (true or false)
- \( 3^8 \times 3 = [ \quad ] \)
- \( 2^5 \times 2 \times 2^{10} = \)
- \( (-2)^5 \times (-2) \times (-2)^{10}\)
- \( \left(-\frac{1}{2}\right)^5 \times \left(-\frac{1}{2}\right) \times \left(-\frac{1}{2} \right)^10 = \)
- \( (\sqrt{5})^7 \times (\sqrt{5})^3 = \)
- \( b^7 \times b^3 = \)
- \( 17^{[ \quad ]} \times 17^{[ \quad ]} = 17^{10} \)
- \(25^4 \times 25^{[ \quad ]} = 25^9 \times 25^3 \)
- \( 502^{5m} \times 502^{3m} = \)
- \( a^{5m} \times a^{3m} = \)
- \( a^{5m} \times a^{3n} \)
- \(a^m \times a^n \)

I need to look more into this approach I think; for one reason or another I haven’t had much time to read up on this stuff sadly.

Kris gave us a glimpse of the planning process behind a series of lessons on simultaneous equations and how topics are returned to in different lessons.

Sadly I couldn’t go to his and Bruno’s later session on planning more than a lesson.

It was fascinating to watch a video of Kris teaching a mixed attainment Year 9 class about solving simultaneous equations. The precision and economy of the language was evident, with very simple sentences used to ensure students understand the examples. He gave examples of equations which could be solved (i.e. They only have one unknown) and those that can’t (i.e. With more than one unknown), before doing an assessment sequence of questions similar to the example.

We then watched a video of him adding equations together, where a student asked “What if you had 6x and 4y?” Kris explained how he doesn’t take questions until the end, as this question will be answered in the sequence anyway. He treated this as a transformation and so chose numbers where it was obvious that they were adding. Hearing about the success rates of students adding equations where the x and y terms were put in a different order I found impressive.

This session gave me lots to think about, and I think I will be thinking about these for a long time to come.

**What’s the Point of Maths? – James Valori**

In this talk James gave 5 reasons for learning maths.

*So people can’t lie with statistics*: James gave us three examples of where statistics have been used to mis-inform. 80% of dentists recommend colgate where only 80% of dentists had listed Colgate amongst other brands. The reporting of the statistics about head injuries hides the fact that there are vastly more people in cars than on bikes. He then talked about Simpson’s Paradox*It Rules the World*: As an example of this, James talked about a situation where many people witnessed a crime and each statement rinfluenced other people’s statements – this of course then related back to Google’s PageRank model.*Understanding the world and modelling*: James talked about modelling the situation “should I buy or rent a house?” And considered various factors Roth considering. I think this would make a great Core Maths question. He then mentioned Dan Meyer and the idea of removing information to motivate why you would want to model the filling of a tank. This is one of Dan’s Three Act Maths activities.*It teaches you how to think*: I liked the activity of James Mason that he shared \(\int_0^2 (x-1) \mathrm{d}x \) where the idea was to remove the procedural aspect of showing this fact and to develop understanding and expose the creative side of mathematics.*Because it’s there*: Why wouldn’t you study it?!?

I really liked the JavaScript implementation of a Quincunx that James showed, I am hoping that he will send me the code.

**An Introduction to Geogebra for Beginners – Tom Bennison**

I could hardly not turn up to my own workshop. I shall post details of this in a future blog post.

**How Misconceptions Change Over Time – Craig Barton**

This workshop was fascinating, if a little depressing. Craig presented 5 questions that have been answered more than 5000 times on his excellent website Diagnostic Questions by pupils of a range of ages. He considered groups of 10-11 year olds, 13-14 year olds and 15-16 year olds. For the questions about calculating the area of a parallelogram, sharing in a ratio and solving equations the eldest group did poorly. For the fraction question there was only a 6% increase in in the percentage of people getting in correct over 5 years of schooling.

Craig listed 5 key takeaway messages:

- The language may change, but the misconceptions remain the same. Students’ understanding of fractions does not improve much in 5 years.
- The more skills and concepts students encounter the harder it can be to retrieve the correct one. Depth of understanding is so important.
- The older children get, the less carefully they read the question. Maybe there is also an argument that the way primary school students are being taught ratio is leading to a deeper understanding.
- Rules (inverse, change side etc) without the depth of understanding behind them, cause problems. Two biggest causes of mistakes on DQ are: 1) Confuse related concepts, 2) Misremember of misapply rules.
- Older students struggle with long division if they have not practised it in a while.

I wholeheartedly agree with Craig’s “Big Takeaway”

Thanks again to LaSalle for organising another great conference.

]]>This article is now almost 10 years old, and despite the numbers of students studying A-Level mathematics having increased since the 2002 data talked about in the report the participation rates of UK students in 16-19 mathematics is still often highlighted as a concern for the Government, being considerably leass than many high performing jurisdictions.

As teachers, I am sure that we have all heard mathematics described as “boring” and “hard” and this paper makes unwelcome reading, particularly as the sample of schools were significantly above the national average in terms of GCSE performance.

A few things to think about during the discussion tonight:

- A large dip in percentages considering further study is shown between those pupils who achieved an A and those who achieved a B. Do you think this is the same today?
- Does your school take people on to an A-Level if they got a Grade B?
- How can we, as teachers, teaching within the confines of the system, tackle the perception that mathematics is boring?
- Hatred is a strong emotion – is this to be believed?
- What kind of activities should we be using to enhance “enjoyment” of school mathematics?

I decided to attend this session as I wouldn’t describe Statistics as one of my strengths.

I use Geogebra quite a bit but I have never used the probability calculator option before. It works well on the iPad edition as you can see below, and provides a really nice interface for visualising probabilities and calculating them.

There was an emphasis on linking the teaching of the binomial distribution to the teaching of the binomial expansion. I do this, but I like the idea of making it explicit to students by getting them to do a dice experiment and relate the outcomes to the binomial distribution. The experiment suggested was to roll a dice 4 times and count the number of fives that occur. these outcomes could then be tallied up and you could also construct a tree diagram. Then you could relate the branches of the tree diagram to the number of occurrences of \(p^4, p^3q, p^2q^2,pq^3,q^4\). One resource that I particularly liked from this session was a card sort linked to the following article from the Daily Mail:

- “A study has found that 41 per cent of Britons admit to regularly using their phones, tablets and eReaders when sat on the loo.” “Smartphones were identified as the preferred gadget of choice, cited by 65 per cent of respondents”.

I think it is great that the card sort contains differing values of \(n\) and \(p\) taken from the information given.

Following looking at the Binomial distribution Terry talked about using data sets to obtain data that is approximately normally distributed. This is likely to be interesting in terms of the new A-Level, so I am going to do a bit of work on it and then blog at some point in the future.

**Session 2 – Core Maths: a Teacher’s Perspective, David Phillip and Claire Phillips**

At my school I am going to start teaching a Level 3 Core Maths next academic year and so I picked this workshop to get some ideas for classroom activities.

The initial activity of recording when popcorn pops in either a maker or a microwave was nice, but it’s hard to get accurate data due to issues with reaction times etc. Though, of course, this could lead on to nice discussions about the validity of recorded data.

I had a very interesting discussion with Mike Ollerton about whether some fermi style questions are more worthwhile than others. I am inclined to agree that students will get more out of some fermi style problems than others, but quite how to taxonomies this isn’t completely clear. Do you think some of the questions shown below are worth more (in a measure of learning) than others?

There were many other great resources shared during this session which will definitely be worth a look once MEI put all the materials up. I quite liked this matching activity:

**Session 3 – Teaching A-Level in Early Career, Cathy Smith**

I found this session fascinating (as I do with Cathy Smith’s work in general) as in my experience there often isn’t much attention given to preparing ITE students to teach A-Level, and the demands of teaching A-Level are different to teaching lower school. I can definitely identify with some of these issues in A-Level teaching that were identified in Cathy’s research.

This quote from one of her study participants was particularly intriguing to me

I can definitely see where this come from, but I find it a little sad as if you do spend time on the preparation then you tend to get an awful lot out of the lessons.

As an added bonus from this session Cathy got all the attendees to work in groups on the card sort shown below.

I really enjoyed doing this card sort and I think it would be a very valuable activity to do both with beginning teachers and A-Level students. Cathy made an interesting point about the language used in this task. She had re-written a task from nRich but been more consistent in the descriptions of the processes. For example each process starts with a description, followed by the \(y\) variable and then the \(x\) variable. This highlighted to me that getting the right amount of structure in an activity is very important. Too little is just as bad as too much! Interestingly, many people seem to start with the bacteria ones first, possibly for reasons of similarity.

**Plenary – All Change in Post-16 Mathematics, Charlie Stripp**

Charlie Stripp, the Chief Executive of MEI gave the first plenary of the conference. He gave a fairly positive talk about the state of mathematics education in the UK and the influences of all the changes coming through.

The picture of A-Level participation is very positive, though it remains to be seen how this will change in the future.

Charlie said that we have a “duty to use technology in teaching mathematics” and I completely agree with this. He also mentioned that there was research evidence that students owning their own graphical calculators helps their mathematics learning. I haven’t heard of this before, if anyone knows where this statement comes from please let me know!

**Session 4 – Further Pure with Technology, Richard Lissaman**

Being a keen programmer I was looking forward to this session with Richard Lissaman looking at the MEI unit Further Pure with Technology. A typical programming question from FPT is shown below

I think the idea of doing this kind of thing in A-Level as so much of real-life mathematics is computational, and I would love to do MEI maths so that I could do this unit.

I was quite impressed with the CAS capabilities of Geogebra. I’m not sure how robust it is, but for a free package Geogebra is very impressive.

I’m glad that they have been able to incorporate FPT into the draft specifications for A-Level 2017. It doesn’t seem to have changed too much apart from replacing the complex variables section with content on differential equations.

]]>I’m not sure now who on TES I got the template from, so if you recognise it as yours please say so that I can credit you.

The powerpoint can be downloaded here.

**UPDATE: **Solutions now available.

We will chat about this on Monday 11th July t 8pm, I really hope you can join us.

As usual, about a week before I will post some points to think about.

]]>I’m currently enjoying the AQA webinar and I love what I am seeing so far.

I have set up a Dropbox Folder where I will put all specifications and associated resources from all boards as soon as possible so that I (and you) can access them in one place!

I’m excited for tomorrow to come!

]]>One of the things that I am going to do with them is an increased difficulty version of the June 2012 paper. This is a very similar idea to my increased difficulty M1 paper I posted a while ago. Essentially, all that I have done is remove any unnecessary intermediate steps and diagrams.

You can find the paper here.

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