Introducing Proof at A-Level

We are almost one year in to the teaching of the new A-Levels in Mathematics and Further Mathematics. The first overarching theme of the new A-Level (as identified in the subject content guidance from the DfE) is “Mathematical Argument, language and proof” as shown below.

Screenshot 2018-06-18 10.26.51

There is a greater focus on this than there used to be and it is something students often struggle with.

To begin with I normally try to link back to the kind of thing they may have seen at GCSE Higher, for example proving properties of products/sums of even numbers. A typical question on this at GCSE would be something similar to “Prove that the sum of four consecutive numbers is always even”.

To move on from this I often use the card sort resource below.

Screenshot 2018-06-18 10.52.49

The idea for this is that students work in groups to discuss the 12 statements and sort them into always, sometimes or never true. Some of these are harder than others, and listening in to their conversations is particularly interesting and can provide a good idea of how quickly to move on with the class.

This activity can be downloaded here.

This activity is one of many included in the new book on proof that I have co-authored for Tarquin Group. The book is called “Understanding Proof” and is available from the publisher here.

Screenshot 2018-06-18 13.40.06

If you are interested in this book the voucher code PROOF718 will give you a £5 discount when ordered from Tarquin direct (it should work till the end of July. This code will infect work for anything in the Tarquin A-Level range – I hope this tempts people. We are proud of this book and I would love to hear any readers’ thoughts.

Sorry that this post has ended with a bit of a plug for a paid-for book…. I’d love you to comment on how you introduce the concept of proof at A-Level.

Anscombe’s Quartet – Again

Back in 2015 I posted about Anscombe’s Quartet here. I have finally gotten round to doing something that I had planned to do back then.

I have written a small Geogebra applet that allows students to visualise the four datasets that comprise Anscombe’s Quartet and change the number of decimal places displayed.

The applet is hosted on my website and can be accessed here.

I hope it is useful..

Happy Christmas

Happy Christmas everyone!! I hope you have all had a fantastic day 😉

Inspired by James Tanton’s (@jamestanton) “Personal Polynomial” I have a small Christmas message encoded in the polynomial below:

The file available here contains the definition of the polynomial which would certainly be helpful.

Christmas Calculated Colouring

I know this will be too late for some people to use – sorry I am struggling to stay on top of things at the moment.

Here is this year’s Christmas calculated colouring. It’s slightly easier than the previous one with less regions so that it should be able to be completed in a single lesson.

26FF3C85-F3B7-4A2E-A3BA-9219A4102F9A

Here is the image and questions to download:

Questions

Image

I hope you enjoy it 🙂

Learned Societies and Professional Associations

Recently the Mathematical Association (MA) have proposed the merger of the MA, the Association of Teachers of Mathematics (ATM), the Association of Mathematics Education Teachers (AMET), the National Association of Mathematics Advisors (NAMA) and the National Association for Numeracy and Mathematics in Colleges (NANAMIC).

The MA is inviting comments on this here.

Please, please make your voice heard!

Below, are a few personal thoughts – this started as a comment but then seemed too long.

I am completely in favour of the associations merging. A few thoughts are below.

  1. The current situation results in a pretty fragmented state of affairs. There is lots of overlap between the aims of the associations and I think having multiple organisations put out responses to consultations is resulting in a diluted voice. Since organisations such as the NCTM in the US show how successful a large representative organisation can be.
  2. There is currently limited “brand recognition” of the main subject associations. I suspect that many teachers would not know who the ATM or MA are for instance. A recent Twitter poll (sorry I can’t remember who by) also highlighted the widespread confusion between the subject associations and the NCETM. Many people thought the NCETM was a subject association – a single subject association with consistent branding and corporate message would go some way to combating this.
  3. A new subject association would be an ideal time to produce a new, modern website. When compared to websites built from the ground up on a modern technology stack the subject association websites feel incredibly dated (personal opinion I know) and aren’t fully responsive to different devices. It is well known that the first few seconds on a website are incredibly important in driving traffic and engagement – something I think the current websites don’t do terribly well.
  4. There are obvious economies of scale by combining forces, times are hard for many organisations, and associations with relatively small memberships are certainly susceptible to financial pressures. The screenshots below are taken from the Charity Commission website.
  5. I have been involved with 3 associations since being a student. My Granny’s collection of 30+ years worth of Mathematical Gazettes prompted me to join the MA when I was an A-Level student. Following my undergraduate and doctoral study I joined the IMA and when I became a teacher I joined the ATM. Being a member of all 3 is costly, and this can put people off. When it isn’t clear which is best to join, the decision can become “I can’t join all of them so I won’t join any”.
  6. If they do merge I hope the journals remain. The Mathematical Gazette is amazing, and the archives are a great source of mathematical nuggets. The other magazines of the MA and MT by the ATM all have slightly different qualities, it’s hard for me to imagine them not existing in their own right in the future.
  7. I think it is a bit sad that the IMA and LMS haven’t been included in the merger proposal. Not just are they much larger organisations, do we risk missing out on the chance to unify the representation of mathematics and mathematics teaching? The IMA have many members who are teachers, but it often isn’t seen as an association for teachers. It makes me sad that there is often this implicit distinction.
  8. The associations already do much in partnership – how much more powerful could they be if everything was done together? The fact that conferences are sometimes on the same dates for instance seems to be close to an act of self mutilation on the part of the associations.

Anyway, probably enough of my views. Please go and make yourself heard an comment on the consultations.

Two More A-Level Topics – A #MathsConf13 Session

On Saturday Ed Hall and I led a session at MathsConf13 in Sheffield.

As usual the whole conference was great, well attended and we managed to fit in a meeting with our publisher too. The keynote this time around was particularly funny as it was delivered by Matt Parker (@standupmaths)

The Prezi from our session is available here.

Screenshot 2017-10-02 06.31.09

The resources used are all available from the links below:

These resources have all been developed to support the teaching of A-Level Mathematics and are especially useful for those using the Tarquin Group textbooks. The resources listed above, however, will always be available for free.

During the session we asked delegates to show the derivative of the naturla logarithm function using the figure below.

Screenshot 2017-09-18 21.59.29

This can be done in the following way:

Screenshot 2017-10-02 23.37.22

We hope you found the session useful. Please get in touch with any comments, especially if you use the resources.

Mathsconf 9 and Two A-Level Topics

You may have noticed that I haven’t been posting on here much over the last few months. This is because I am co-editing the new Tarquin Group series of A-Level Mathematics and Further Mathematics textbooks. These books are being written completely from scratch for the new A-Level 2017 syllabus and I will share more of the philosophy behind these books over the next week.

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:

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

Carnival of Mathematics 140

This post is (very late) the 140th Carnival of Mathematics. This blog Carnival is organised by The Aperiodical and is hosted by various maths bloggers from around the world. The last Carnival was hosted by Manji Saikia at Gonit Sora.

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.

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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 Integral Maths Ritangle Competition

There’s a new kid on the school mathematics competition block, and I am pretty excited about it.

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 screenshot-2016-11-01-22-10-01

 

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

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.

 

#Mathsconf8

After a very early drive to Kettering (via Leicester) I was ready for my busiest Mathsconf yet. As well as running a workshop on Geogebra, I was representing the East Midlands West MathsHub (@EM_MathsHub), and launching a new A-Level textbook for which I am one of the editors with Tarquin Group (@TarquinGroup). Because of this I didn’t manage to get to go to the opening or the speed dating segments sadly but I did manage to make the main sessions.

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.

  1. 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
  2. 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.
  3. 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. 
  4. 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.
  5. 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:

  1. The language may change, but the misconceptions remain the same. Students’ understanding of fractions does not improve much in 5 years.
  2. The more skills and concepts students encounter the harder it can be to retrieve the correct one. Depth of understanding is so important.
  3. 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.
  4. 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.
  5. 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.