We are coming up to the end of @staffrms #29daysof writing and for this post I have stolen an idea from Stephen Connor and am going to answer a few questions about my education.
What were you like at school?
Overall I was pretty good at school and did well academically. Looking back, as a teacher, I don’t think I would have enjoyed teaching me! If I didn’t particularly respect my teacher I think I possibly made that clear but never did anything outrageous enough to really get in to trouble. I was also pretty stubborn, and if I thought something was stupid I made it quite clear.
Primary, secondary, college or uni – which was your favourite?
To be honest it is hard to pick here as I enjoyed all of them for different reasons. I guess overall my time in Sixth Form was probably my favourite. First time I got to only do subjects that interested me (specifically I wasn’t forced to do PE!) and I got to spend lots of time in friends. I can’t remember that time being too stressful work wise and it was a good experience to be involved with organising the Sixth Form prom.
Which of your teachers do you still admire?
From primary school it has to be Miss Carpenter, I really enjoyed Year 2 as there was none of the stupid “let’s listen to a boring story while children tie up the teachers hair with rubber bands” rubbishy that I endured the previous year. Miss Carpenter was also the first teacher to really stretch me with maths, providing me with extra questions. I still admire my Year 6 teacher, Mr Massey and it was sad that he died so young. I think my love of computing technology is in a large part down to him as I was allowed to be an “IT Monitor”, which essentially meant I got to spend a lot of time outside the classroom helping other teachers with things like printing and fixing basic issues with their computer – I don’t think that kind of thing would be possible now.
From my secondary schooling there are a couple that stand out:
The first is Miss Dodwell, who I had for English from Year 9 through to Year 11. I had never been particularly keen on English as a subject before, always doing the bare minimum in terms of work but I always really enjoyed Miss Dodwell’s lessons (though I’m not sure if I let it be known that I did). I certainly don’t think I would have done as well with my GCSEs if I hadn’t have had her, and her approach to teaching has had a pretty big impact on how I want to be as a teacher. My classes now often find it strange when I say that I really liked my English GCSE classes and will quite happily discuss with them one of their current books.
The second has to be my A-Level maths teacher, Adrian Green. His lessons went a long way in leading me to choose maths at university instead of physics or engineering. I really respected how he always found the time to help with maths despite being a deputy head. He was a great A-Level maths teacher and my current teaching is very much inspired by his approach.
Which lessons do you still remember?
From Primary I remember a ridiculous craft session (in Year 1 I think) where I had to use pasta shapes to make a picture of a beach and a sandcastle. I wanted to use the same shapes for the beach and the castle but was told that I wasn’t allowed as you couldn’t make out the castle so they had to be different pasta shapes. I pointed out that it was unlikely that the sand castle would be made of different sand than was present on the beach but that point fell on deaf ears. This picture went straight in the bin as soon as I got in…
At Clarendon I can vividly remember GCSE drama lessons in the run up to performances and spending lots of time with Justin and Luke up in the Tech room on the Zero 88 Illusion 120 lighting desk and sound desk – lots of fun times were had doing tech with these guys.
For Sixth Form I had to take three of my subjects at Clarendon, and two of them at John of Gaunt. Chemistry was one of the ones that I took at John of Gaunt and I can remember doing lots of cool experiments with Mr Treble.
What are the main differences between the classrooms you were taught in and those you work in now?
I guess one of the biggest differences is how rare blackboards are now, quite a few of my classrooms at secondary had blackboards and no digital projectors. I can remember in primary school it being pretty exciting when we got an overhead projector for the hall – how times have changed.
The rise of the Internet and the fact that these days you can get a reasonable mobile data connection almost anywhere mean that collaboration with other teachers (or people in general) that are miles from where you are is a very real possibility.In the past I have always used something like Skype for this purpose, but that really isn’t very suited to the job. Yesterday I had a web conference with Jasmina Lazic, my #mathsconf6 co-presenter. We will be meeting in person for the first time on that Saturday morning and so all of the preparation for the session and communication beforehand has been through emails. The Mathworks (the makers of MATLAB) must have a corporate contract for Cisco’s WebEx product, and for me a WebEx meeting was a revelation! Cisco’s WebEx software enables sharing of files, sharing of screen etc which actually allows proper useful meetings to take place. I also couldn’t guarantee that the software would work on my school’s computers and so as a back up I installed the iPad WebEx app and I was very impressed with it. Using the app I joined the meeting smoothly, and you can even share files and screens from your iPad too which I didn’t expect to be able to do.
If you have never used it before, I would encourage you to check it out. The free accounts allow you to host meeting containing up to 3 people.
Today one of my colleagues asked me how this followed: Thinking back my response (in front of the class) was something along the lines of “that’s a fairly standard trick used when integrating”. This troubles me as it makes it seem that it has come out of nowhere by some kind of magical process and probably adds to the feeling that mathematics is a just a collection of rules to be followed.
A better description, in my mind would be something like the following: “When faced with an integral that you can’t do straight away, try to manipulate it to something that you can do by either ‘adding 0’ or ‘multiplying by 1’. That is you do something that changes the form of the integrand but which leaves it equivalent to what you started with. In this case you ‘add 0’ by adding 4 and then subtracting 4:”
Do you think that avoiding the use of the word “trick” is important?
In my experience professional mathematicians use this phrase to mean a “commonly used technique” and so it doesn’t have the connotations that the phrase can have.
Should we avoid the phrase? Or try to push for a “maths specific interpretation”?
A lot of you probably know that for a while I have had the idea of organising a CPD conference aimed at KS5 mathematics. I feel that this is an area that isn’t served as well as lower Key Stages (of course MEI, FMSP do fantastic work in this area!) and with the imminent changes to A-Level mathematics there is a need for some collaboration in my opinion.
With this in mind Tom Wicks (the University of Nottingham’s School of Mathematical Sciences’ teaching officer) and Ria Symonds (the FMSP Derbyshire and Nottinghamshire coordinator) and myself have decided to organise a day conference towards the beginning of the summer holiday. We intend for this day to provide some great CPD opportunities for teachers and plenty of time for networking.
The day will take place on the University of Nottingham’s beautiful University Park campus and we have tried to keep cost to a minimum as we realise that many teachers will be paying this themselves.
Nottingham is well placed for people travelling from further afield and if you come by train you can now jump on the tram to the University which is very convenient.
The workshops announced today are just a taster of what will be on offer – we intend to have 4 workshop slots throughout the day with a minimum of 3 options for each workshop. Our closing plenary is going to be delivered by the mathematician and author David Acheson. He regularly takes part in the Maths Inspiration lecture series and if you haven’t seen him speak before we promise that you will enjoy it!
Tickets and further information can be found on our Eventbrite page and a webpage for the conference with further details will be going live within the next week.
We very much hope you can make the time to attend!
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…
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 \((a+b)^2 = a^2+2ab+b^2\)
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.
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.
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:
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.
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.
Make the substitution \(x = \frac{\pi}{2}-y \) and so \(dx = -dy\) which gives the following
and so, by adding this expression to the original integral (and changing back the dummy variable of integration) you obtain
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):
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.
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.
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.
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.
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?
