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Me and my PhD – an Embarrassing Video

Back in 2012 I was asked to take part in some videos promoting PhDs in the maths department at Nottingham. I didn’t realise it was online until some people in one of my classes found it…

So for your enjoyment here is the awful video!!!

 

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Proof at Secondary School

On Tuesday 23rd of February I am hosting the NCETM’s #mathscpdchat on the topic of “Proving: how do pupils learn to do it?”. This post is a collection of some thoughts and opinions that I have had over the last week or so….

A month or so ago I watched this fantastic video from the Museum of Math in New York on the subject of “Proofs from the BOOK”, if you have an hour or so to spare it is well worth a watch:

To me the concept of proof is what sets mathematics apart from all the other sciences and gives me more faith in anything proved mathematically than a theory from another branch of science. As such I think it is crucial for students to be exposed to proof as early as possible (even if it is just by giving them exposure to the question “How can you be sure this will always work?”). To reduce mathematics to various procedures for performing calculations of one sort or another is a depressing idea.

Typically geometric proofs are see as a nice way to introduce proof, for example this proof of the expansion for \((a+b)^2 = a^2+2ab+b^2\) Screenshot 2016-02-22 22.30.01

Personally I question whether geometric proofs are the best way to introduce proof to students as sometimes they struggle to see what the diagram is showing, don’t necessarily appreciate that some diagrams are only sketches etc.

I thought it would be interesting to see what some of my old colleagues (3 still in academia, one in teaching and one in industry) thought about teaching proof at secondary level – this is what they had to say.

Person 1:

“Yeah. I’m definitely for the idea but it’s about mathematical rigour and I’m not sure a lot of students would appreciate it at that age. At an advanced level, proofs are associated with theorems and methods. I think there’s very few topics you could apply it to, which make it accessible. Pythagoras’ theorem is one that comes to mind or other geometry related topics (angles, shape and area) are probably the most obvious and intuitive topics – but how do you make it interesting. I think it’s one of many advanced concepts that becomes harder to integrate into the early curriculum. Maybe with operations on fractions too?”

 

Person 2:

“I think it is a good idea to teach proof at secondary school level (although maybe KS3 is too early). The problem is at Uni level and higher proofs are important – if students haven’t learnt and understood them from an earlier age they do struggle. However, students who don’t go onto Maths (or other subjects where proofs are useful) probably won’t need/understand proofs – and you risk not engaging them. I’m not sure how to aim proofs at lower levels.”

Person 3:

“Good question, and in an ideal world, of course we should be teaching mathematical proof alongside everything else. Two problems: our “pure” idea of mathematics in terms of proof being the final word seems to conflict with others’ views of mathematics in terms of methods and formulae used to work out solutions to “real-world questions” – such as voltage in a circuit. So would people accept the idea that maths is actually all about proof, rather than bare calculations? The other is that proof does rely quite a bit on logic, and so you then have the question of how to introduce rigorous logic at KS2 or KS3. In particular, why proof by contradiction or counterexample works at all.”

Person 4:

“Regarding teaching proof to secondary school children, I think it’s important to introduce the idea to children at a much younger age than A Level.  I’m not sure how much better the new A Level is, but it’s barely covered there, especially without Further Maths. I think this serves two purposes. It better prepares those students that go on to study maths at undergraduate level, which is the obvious benefit. The second advantage is that younger students will be exposed to another side of maths that is different and hopefully more interesting than just calculating things and doing sums. Other subjects, such as English, history, human geography start to get analytical from around Year 9 onwards and this typically engages students rather than putting them off, so why should maths be any different?

I think the way to do it is perhaps introduce the ideas first in the current way, such as Pythagoras’ Theorem, triangle inequality etc. and get them used to using the result so they are fairly convinced it works. Then motivate the question “how can we show that this works in ALL cases” to get them used to the idea of general ideas. Some may attempt to just do lots of examples, but will hopefully realise that no list is exhaustive enough and they need to try something else.”

Person 5:

“Proofs are fundamental throughout the whole of mathematics and they are what make mathematics so different to others sciences. Theories in other sciences are based on our best knowledge at a certain point in time, they can be backed up with evidence, but could be disproved in light of new experiences. Mathematical theorems, on the other hand, once proved, are fact. As an example, consider Newton’s laws of motion, for 200 years these were assumed to be fundamentally true, until Einstein (and others) showed that they did not hold at small scales and high speeds. In contrast, Pythagoras’ theorem is as valid today as it was 2,500 years ago in ancient Greece. For those wishing to study mathematics past A-Level, understanding how to prove things is essential; indeed, a large part of a professional mathematician’s work is taken up in the quest for new proofs. Exposure to as many proofs as possible will leave students with the building blocks to develop their own novel proofs. For those inquisitive students, who do not with to pursue mathematics further, attempting to prove a theorem can lead to better understanding of that theorem, as well as teaching the student the crucial skill of logical thought. Of course, for some students, proving theorems is not required, a firm grip on how to use the theorem is the most important thing, and this is OK too. By teaching students why we need proofs and how useful they can be, hopefully we can avoid situations similar to the one I faced when lecturing engineering students. I was told that ‘I did too many proofs’, when, in fact, I had done none. It has been a while since I was at school and I cannot recall exactly what exposure to proofs I had. I have this vague idea that I may have been proving things, but I didn’t know it. I am not convinced that developing proofs amongst KS3 students is required, except for those with a definite aptitude for the subject. For older students, I think it is vital they are told why proofs are so important in mathematics. If a theorem is given, and its proof is feasible, then I think it should be explored in class. Geometry and number theory seem to me the most accessible areas where theorems could be proved. Later, trigonometric theorems could be proved, as well as those fundamental to Calculus. Students should also be encouraged to come up with their own proofs, after all, there are often multiple ways to prove something.”

I think the quotes above throw up some very interesting ideas that could be discussed in our twitter chat, for example:

  • Should all students of mathematics be exposed to proof?
  • How should we develop the logical thinking required to prove something?
  • Does not teaching proof disadvantage children?
  • Do we need to reformulate the idea of “what maths is”  for many people to understand the importance of proof?
  • How do we make developing proofs engaging?

Last week I also ran a poll on twitter asking the question “When you teach ‘angles in a triangle’, do you demonstrate it or prove it? If you prove it please reply with how.” 61 people voted and the results are shown below.

IMG_0213

To be honest I was a bit surprised by the outcome as I expected a significantly higher proportion of respondents to say that they proved it. Of course a demonstration of the result by either tearing the corners of a triangle and sticking them on a straight line or folding of the triangle is popular and a nice active way for students to visualise why the rule works. But one of the proofs is a classic application of the angle rules for parallel lines, as nicely demonstrated by Dawn (@mrsdenyer) on the back of an envelope, and so I expected a higher percentage of teachers to go through the formal proof with their students. angle_proof

Some time ago Professor Smudge (@ProfSmudge) shared a file of proof questions that were used with year 10s in the “Longitudinal Proof Project” research study and I can remember being particularly struck by question 4:Screenshot 2016-02-22 22.04.01

This is a really nice collection of likely responses you would get in the classroom if you asked this question – Eric’s is particularly interesting as I can imagine many students being convinced by this because of its algebraic nature, despite being nonsense. I haven’t managed to find the time to read through all the reports from this research project and the subsequent “Proof Materials Project” which are all helpfully available online on the mathsmedicine website.

Finally, for tonight, Mark Greenaway (@suffolkmaths) shared with me a fantastic page on proof from his sufmolkmaths website which lists many activities designed to promote students thinking about proof, including this one from Mark Dawes about the prime numbers. Showing that all primes (greater than 3) must be of the form \(6n\pm1\) is a particularly nice thing to discuss with students as you can develop their thinking from an intuitive grasp of the result to a proof involving modulo arithmetic.

Anyway, enough of my rambling – I am looking forward to the discussion tomorrow.

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An Integration Trick

Before I started teaching A-Level I hadn’t done an integral by hand for about 5 years – I would always use Mathematica when I had to evaluate an integral symbolically.

I’ve actually enjoyed re-familiarising myself with some of the techniques required for some of the more difficult integrands and fancied looking at some more challenging integrals and so I bought the book “Inside Interesting Integrals” by Paul J. Nahin. This is a great book packed full of clever tricks to evaluate integrals and I thought as it is the last day of the holidays (and so don’t want to write a long post) I would share one of them here.

It was nice to be reminded of the “flipping the integral’s variable’s direction” trick which works in a lot of places. For example consider the integral below.FullSizeRender-4

Make the substitution \(x = \frac{\pi}{2}-y \) and so \(dx = -dy\) which gives the followingFullSizeRender-3

and so, by adding this expression to the original integral (and changing back the dummy variable of integration) you obtainFullSizeRender-1

Hence, the original definite integral is equal to \(\pi / 4 \). I think this is pretty neat!

Here are some for you to try (you may need to pick a substitution as well as using the trick described above):FullSizeRender-2

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Debbie Morgan and Primary Mastery

Last year I was lucky enough to go to one of my local MathsHub’s Secondary Curriculum Development sessions which Debbie Morgan, the NCETM director of primary was leading.

I really enjoyed this session and now the NCETM have helpfully put up a video of a similar session online. Even though it is targeted at primary I think it is definitely worth a watch by secondary teachers.

Here are some points, that I feel are important, made in the video.

Planning and Workload

A mastery approach requires one very good lesson; you shouldn’t be planning loads of different lessons as a way of differentiating. She also said that there is no point writing 32 “next steps” in books as the next step in a mastery approach is just the next lesson.

Shanghai Maths and Rules

Shanghai maths is about rules but it isn’t about rules without reason. Memorising things after understanding something leads to more secure learning. Shanghai students are asked to give answers in full sentences such as “The whole is divided in to 4 equal parts, one of those parts is one quarter and so the shaded portion of the circle is one quarter.”These sentences are useful as they get students used to taking and understanding the mathematics – it gives a “context to hang the mathematics on”.

Variation Theory

Debbie talks about this example from Mike Askew’s book “Transforming Primary Mathematics” and discusses the differences between the two sets of questions. Set A is good for promoting just an algorithmic approach to subtraction, but set B (which contains exactly the same questions ) is ordered in such a way to promote students engaging with some mathematical reasoning. I don’t think Variation theory is talked about enough to be honest – I know I should think about it more when designing questions for my classes. 


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Women in Maths Videos

I studied for my PhD at the University of Nottingham in the School of Mathematical Sciences and there was lots of discussion about women in mathematics whilst I was there.

As a department they are very supportive of female mathematicians and assisting them in building careers in mathematics. They have a bi-weekly meeting for all female members of staff and postgraduates.

Recently they have produced a series of 13 videos – Women in Maths – that feature some of my friends / ex-colleagues as well as people who have started in the department since I left. I think some of these videos have some very inspirational messages about mathematics and the enjoyment of mathematics. They would make great short videos to show to all sixth form mathematicians (not just the female ones!)

I have embedded a few of them below:

I particularly like the following quote from Susanne Pumpluen

“It’s as creative as if you would do arts or music and people don’t see it because it’s often, unless you have a very good math teacher at secondary school, it’s very hidden that it can be so creative and fulfilling”

You can watch the whole playlist of 13 videos here.

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Girls and A-Level Further Mathematics

I recently received my IMA member’s magazine “Mathematics Today” in the post and in this issue there is a very interesting article by Claire Baldwin, Sue de Pomerai and Cathy Smith titled “The Participation of Girls in Further Mathematics“. Luckily they have made this article available online here so if you aren’t a member of the IMA I’d encourage you to read it now.

This article draws heavily on a literature review, “Gender and participation in mathematics and further mathematics A Levels: a literature review for the Further Mathematics Support Programme” prepared by Cathy Smith and a subsequent report detailing 5 case studies authored by Cathy Smith and Jennie Golding titled “Gender and participation in Mathematics and Further Mathematics: Interim report for the Further MathematicsSupport Programme“. If you are interested in gender and the take-up of A Level mathematics I would encourage you to read them.

I think most teachers of A-Level mathematics would love to have more girls continue mathematics and in particular continue their study into Further Mathematics, I think it is important that we do everything we can to encourage this. About 20% of both my Year 12 and Year 13 classes are female and I am glad that there are female peers in these classes.

From the IMA article I particularly liked the 4 reasons they gave for Further Mathematics being valuable:

  • The increased time spent engaging with mathematics and developing greater fluency.
  • The study of important topics in pure mathematics not covered at A-Level, such as complex numbers and matrices, that are essential for anyone going on to study maths, physics or engineering.
  • The opportunity to study a broader range of applications of mathematics.
  • The development of increased confidence and resilience in tackling demanding mathematical problems.

These four things very nicely sum up why Further Maths is such a good qualification.

Despite a significant increase in participation for further mathematics in general, the proportion of girls taking the qualification has stayed broadly consistent with significantly less girls taking the subject than boys. I wasn’t aware that the situation was different in the US, where participation is roughly equal at similar level optional calculus courses.

One of the interventions highlighted in the case studies to increase participation at A-Level was the provision of extra courses alongside GCSE for the high attaining students. This is a concern to me, as anecdotally I have heard of many people stopping the provision of these with the introduction of the more demanding GCSE syllabi – I think this is a shame as the AQA Level certificate in Further Mathematics is a really nice qualification.

I believe we need to do everything we can in schools to encourage students to take A-Level Mathematics and Further Mathematics. What do you do to encourage participation?

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QuickKey

The day before yesterday I wrote a post entitled “FP1 Multiple Choice Revision Quizzes” where I shared some multiple choice quizzes that I used last year with my FP1 class and marked using the QuickKey app for iOS.

At first I was quite sceptical about how good this scanning app would be when used with actual student responses, but after a few teething problems I have become very impressed with it.

To use QuickKey you need to register for an account and then use one of their answer sheets for students to record their answers to multiple choice questions on. They look like this: Screenshot 2016-02-18 14.08.23

As you can see each question has 5 possible solutions (so bear this in mind when writing your quiz – there is no point in having 6 well thought through possibilities!). Each student fills in their student ID (4 numbers) and shades in the correct ovals – this enables the QuickKey app to work out who’s solutions it is scanning.

QuickKey have produced a useful infographic as a guide to scanning the quizzes which I have pasted below and is available here.Screenshot 2016-02-18 14.18.28

This covers almost all of the issues I had when I first started using the app. Overhead lighting seems especially problematic and scanning definitely works best in natural light. I would personally avoid using pencils and emphasise to your students that they should carefully shade in ALL of the oval corresponding to what they thought was the correct answer – this will eliminate many issues and mean that you don’t spend time manually inputting results.

The app syncs nicely with the online QuickKey account, from where you can download spreadsheets and analyse class performance.

In July 2015 William Emeny (@Maths_Master) posted about a diagnostic test he had performed on his Year 7 cohort which took advantage of QuickKey to quickly obtain responses to 90 questions for all of the cohort. This is a fantastic use of the power of QuickKey and we did this with some of the classes last year. I am hoping to do the same test again this year with our current Year 7s.

I’d encourage everyone to give QuickKey a go if they are ever using multiple choice tests.

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Prompted Reflection

Yesterday morning I woke up fairly early with baby Jessica (there was quite a pretty sunrise however) and I read “Modelling in Maths“, a great blog post by Bruno Reddy (@MrReddyMaths). In this post he discusses what he tries to do “consistently with modelling” in his classroom.

This is a pretty short post but it is packed with useful tips on topics as varied as questioning, example design, structuring of class practise and the isolation of tricky steps before putting steps together. I genuinely think this post should be required reading for trainee teachers (and others really) as it prompts so much critical thinking about your own practise.

It has definitely made me think about where I need to improve with consistency. I try to think carefully about the examples I use and how I structure a lesson but recently it has been all too easy to let things get in the way of this. Having a new baby, observation lessons to think about, data points to complete and all the other associated admin that goes with being a teacher have been too much of a distraction really and this does affect the quality of my lessons. For example, with one of my classes we were looking at the quadratic formula and I know that correctly evaluating the discriminant is often a cause of mistakes, but I didn’t practise this independently first. When I have taught this topic before I have done this step separately before using the whole formula in one go and I know this works better. So, for me, there is no excuse for me not having proceeded in this way – I just let my thinking time before the lesson get distracted by other less important things.

I need to fight against this!!

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FP1 Multiple Choice Revision Quizzes

Last year I experimented with multiple choice revision quizzes for my Year 12 further mathematics class in the half term before Easter.

I’m not teaching FP1 this year so thought I would make them available for anyone to use. The students seemed surprised at how long it took them to figure out some of the solutions. Each quiz has 5 questions, as shown below, with each question having a choice of 5 answers.

Screenshot 2016-02-16 17.07.48

All  5 quizzes can be downloaded from the links below:

  1. FP1 Quiz 1
  2. FP1 Quiz 2
  3. FP1 Quiz 3
  4. FP1 Quiz 4
  5. FP1 Quiz 5

To mark them I used the QuickKey App which I was actually quite impressed with. Come back tomorrow to read about QuickKey!

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A “halfway-ish” Post

I’m a little over halfway with the @staffrm #29daysofwriting challenge

Yesterday I was puzzled to see lots of “Halfway” posts as to me a half of 29 is 14.5 and so surely if we only allow full days halfway would round to day 15… Anyway I liked the idea of answering some of the same questions as everyone else so here are my responses.

Why are you doing the challenge?

Why not?! I quite fancy a mug and last summer I was one of the people to complete the #summerblogchallenge (as was @missnorledge ) so 29 days seemed quite doable, especially with the 29 minute rule

Where and when do you post?

Generally at home once I have done Jessica’s (my daughter) bath and bedtime things.

What have you enjoyed the most about taking part?

I’ve enjoyed having something to motivate me to share ideas and my thoughts – sometimes it can be very hard to find time to do this after a busy day at work. I’ve also enjoyed reading lots of posts about a wide range of subjects that I wouldn’t have necessarily come across on Twitter without coming on staffrm.

What have you least enjoyed about taking part?

I’m not a fan of the simplistic editor provided by staffrm! I’m used to the more sophisticated  editor on my WordPress site where I can include LaTeX, code snippets, use lists etc. Later on in the challenge I may have to resort to handwriting some posts and having my staffrm post consist almost entirely of pictures, or have the staffrm post just provide a link to the post on my own website.

Have you picked up any ideas?

I wouldn’t say I have picked up any concrete ideas but many posts (especially ones by @missnorledge , @mrbenward , @towens , @becskar ) have been thought provoking and given me lots to think about.