An Integration a day Advent

I really should have shared this before the beginning of Advent! You have probably realised that I have been struggling to post on here – a New Years resolution for next year is to post more.

For my Year 13s I have made an “integration a day Advent Calendar”. Nothing fancy, just 24 integrals to find as students only get good at Integration by doing lots of them…

If you would like to download it to use please do so here.

MEI Conference 2018 – Day 1

It’s MEI Conference time of year again! Sadly as a Maths Hub Level 3 lead I am going to be unable to attend most sessions on Friday and I can’t make Saturday due to family commitments so I need to make the most of today!

Session A: Differential Equations in A-Level FM (Sharon Tripconey)

The maths for the day (well apart from Tom Button’s mathematical interlude during the opening plenary – give the question a go here.) started with a great session on differential equations with Sharon from MEI.

I loved the start of the session activity – a classic Venn diagram activity that really made you think.

Differential equations are in content areas G6, H7 and H8 in the new A-Level specification (only first order by separation of variables).

For further maths there is a lot more differential equations content in the compulsory

content than students would often meet before – including simple harmonic motion.

In the 2017 Further Mathematics Qualifications the concept of linked differential equations will be new unless you taught MEI Differential equations.

Sharon emphasised that what we do in A-Level maths should support the further mathematicians and so it is important that we expose all A-Level students to differential equations that can’t be solved by separating variables. I wholeheartedly agree with this.

A bit of time was spent looking at the integrating factor method – In my experience lots of practice of implicit differentiation needs to be done by students before tackling this topic as being able to spot the function which after application of the integrating factor method is less likely to result in mistakes than just remembering the IF method by wrote (in my opinion anyway!)

Sharon shared a lovely Geogebra worksheet from the MEI Geogebra Institute. In this you can explore how the solutions to a second order homogenous change as the coefficients change. It is also nice to be able to see the roots of the auxiliary equation plotted on a graph.

We began to look at some practical situations which lead to differential equations; possible examples are

  • Motion against a resisting force
  • Oscillations
  • Damped oscillations
  • Forced oscillations

After a brief into we performed a very easy practical involving a bottle of water, an elastic band and a plastic bag. I’m ashamed to say I have never thought of doing this in a classroom.

Once students have had a bit of a play they can easily be led through the modelling of the physical system, discussing how to model the resistive force would be particularly interesting.

The slide below has also reminded me of an old Geogebra file that I have which demonstrates oscillations for a variety of parameters. I’ll dig this out and share it on here sometime next week.

There was only one slide that concerned me in this session — however as she spoke through it she emphasised that she meant “analytically solvable”. Overall I like this slide as I think it is vital that time is taken to emphasise correct use of mathematical terminology in this topic.

Session B: Preparation for Oxbridge and Other Elite University Applications (Rachel Sharkey)

Rachel began the session by briefly discussing her research for her EdD which looked at how teacher’s cultural capital and connections impact their students’ applications to university.

She made an interesting comment that she has heard that the different questions on the TMUA are ranked differently, but this is not official. TMUA questions are intended to be like Olympiad questions, but multiple choice. I also wasn’t aware that Warwick University no longer requires STEP.

She shared her version of a personal network diagram which she advocates as a way to understand the capital that you bring to your students as a teacher.

This was interesting and provided the source of some debate in the session… I should perhaps go and think about this for myself more.

She talked about ways to help students demonstrate an interest in mathematics:

  • Structured Reading: Provide structured reading chapters from a few books on the same topic – in her case infinity – send them away to read and then discuss as a group.
  • Presentations: Students research and present a topic beyond A-Level maths. This increases student’s own capital and their peers when they listen.
  • Independent Reading: Provide help with the reading lists provided from the universities. They then write a book review as this provides a short summary of the book which they can call on during interview or when they are putting together a personal statement.

When preparing students for the admissions tests Rachel has started grouping questions by the skill/technique used including the following:

  • Completing the square.
  • It’s ok to try numbers.
  • Graph transformations.

For the longer form questions her advice is very similar to my default advice. I think one of the most important things is that students need to be very proficient with the algebra, checking every step of the working, watching out for signs and to ensure that they use previous steps of a question later on in the question.

Chris Sangwin, the admissions tutor for maths at Edinburgh University emphasised how useful it is for teachers to contact the admissions tutors in universities they often send students too. They are often happy to say things in person that can’t be written down on a University website.

Session C: Mechanics in A-Level Further Mathematics (Sue de Pomerai and Tom Button)

Sue and Tom had to deliver this session instead of Avril at short notice, but you couldn’t tell!

They started with a brief summary of where each mechanics topic sits in the new linear Further Maths A-Levels (which strangely are described with modules)

Hopefully before students study elastic potential energy they will be familiar with Hookes law that states the force is proportional to the extension. Tom then discussed using a spring balance in a very simple way to help students understand elastic potential energy.

If you pull on a spring balance what do you notice? (Maybe you notice that the scale is linear and that if you exert a greater force then the extension is longer – what do these mean? )

From Hooke’s law, considering the area under the graph of force against extension it is clear that the area under the curve is the elastic potential energy \( EPE = \frac{1}{2}kx^2 \). Tom talked about this being a fantastic situation to draw out any misconceptions using the fact that Work Done = Force * Distance – in this case we do not have a constant force.

I really liked this question that Sue shared:

This would really check the understanding of students, especially if you asked them to explain their reasoning, possibly bringing any misconceptions to the foreground.

There was a very interesting discussion about attempting to move a heavy rock. You have done some work (and expended energy) but what if the rock hasn’t moved – with the usual definition we wouldn’t have done any work, yet energy has been expended. Sue emphasised that with any mechanics work we are working with ideal situations, have made assumptions and so the model does not always represent reality. I agree that emphasising these assumptions with students is very important – what seems obvious to us (we probably know the assumptions made and the impact they have) is likely not obvious to the students beginning their mechanics journey.

Top Tip: If you want to do practicals with balls that don’t bounce take a a ping-pong ball, poke a hole in it (with a compass or a drill). Now cut up a lot of thin elastic bands into little strings and stuff your ball with them. Keep filling them with elastic until the ball doesn’t bounce.

With the increased emphasis on mathematical modelling in the new A-Level it is important that we develop mathematical models for physical situations with our students. Sue led through modelling a bungee jumper. We started by finding the spring constant (plotting weight against extension and finding the gradient – how is this related to k?) Developing the model leads to a quadratic in the extension x which can be solved.

We then looked at a loop-the-loop situation.

Paul Glaister (of Reading University) got to have a bit of fun with this…

All of these practicals are contained in the classic “Mechanics in Action” book. I may blog about this at some point in the future.

Session D: Student Tasks for Integrating Technology in A-Level Maths (Tom Button)

This session was about the student tasks developed by MEI that are designed to expose students to the benefit of technology in lessons.

The first thing Tom showed us is the very cool graph \( y = x^5-5x^3+5x \).

There are two cool things about this graph – I will let you explore this..

We then investigated the graph \(y = x^3 + ax+1\)

  • When (what values of a) does the graph has maxima and minima?
  • What affect does a have on the intersection of the graph with the x-axis?
  • What is same / what is different?
  • What is important about the point (0,1) in this graph?
  • How does the gradient change at (0,1)?

The point (0,1) on this graph is a non-stationary point of inflexion – now in the second year of A-Level.

Tom shared a load of the MEI Student Tasks. Their are sheets for Desmos, Geogebra and graphing calculators.

As a group we discussed the task below.

The emphasis on how the technology supports the learning of this session was great – too often it can be easy to focus on the cool things tech allows you to do. I know I am guilty of this!

The structure of these sheets is also really nice.

Towards the end of the session we considered a question from the MEI SAMS Paper 1.

One interesting point made, that I hadn’t really considered before is that using technology may in itself lead to students being more comfortable with functions having more than one unknown.

Plenary (Charlie Stripp, Kevin Lord, Stella Dudzic)

The plenary today was also being live streamed as it was also serving as a launch of the Advanced Maths Support Programme. It is always nice to see positive graphs such as these:

I hadn’t seen this graph below which is at least still encouraging:

However – what will happen next year?

Kevin Lord then provided a very informative introduction to the AMSP (Advanced Mathematics Support Programme). It sounds a very supportive and exciting programme which I will be keen to be involved with. The new extra focus on the AMSP Priority areas is also welcome. These are comprised of the 12 government Opportunity areas and 20 areas that have been identified as having low Level 3 provision.

One of the key tasks for the AMSP will be to increase the percentage of institutions offering Core maths – currently only around 17% of institutions enter students for this exam.

Stella Dudzic then spent some time talking about Core Maths. I’m very excited about the Core Maths Online Teaching Platform.

Food:

Everyone knows that the most important aspect of any conference is the food and luckily the food today was as good as the sessions.

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.

screenshot-2016-12-05-06-43-51

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!