Categories

## Stuart’s Puzzle

A couple of weeks ago Stuart Price (@sxpmaths) posted this question on Twitter

I hadn’t really thought about it before but it is a bit strange that we ask similar questions about the mean in Year 7 but (I at least) have never asked an A-Level student this kind of question before.

I think it’s a really nice little question, one that is harder than you expect and gives students plenty of time to practise their algebra. My workings are below; as you can see I made a few little errors that I had to correct as I went and of course I used Wolfram Alpha instead of solving the quadratic manually!

I will definitely be using this next year when I teach S1.

Categories

## Pentominoes

Since starting teaching I’ve noticed that a lot of teachers like to use pentominoes for various activities.
As a way of learning and practicing some Javascript and getting to know the Fabric.js (@fabricjs) canvas library I have produced a pentomino arranging exercise, a screen shot is shown below:

You can move and rotate the pieces with a mouse (to rotate use the rotation handle that appears when you click on a piece), it should also work on touch devices.

Feel free to have a go here if you aWnt. The objective is simple – rearrange the pentominoes to tile the rectangle on the left hand side of the screen.

Categories

## Nottingham Lakeside’s George Green Exhibition

This is a post that I should have written months ago when this exhibition opened. Unfortunately it is closing on Sunday, it’s open 12pm-4pm tomorrow and Sunday, the details are available here. If you are local to Nottingham and have some spare time it is well worth a visit.

George Green is probably one of Nottingham’s most famous residents. The science library of The University of Nottingham is named after him. He was a miller based in the Sneinton area of Nottingham, he had little formal education but published a paper (by subscription) in 1828 where he presented the following theorem, given below in modern notation.

Green’s Theorem
Let $$R$$ be a simply connected plane domain whose boundary is a simple, closed, piecewise smooth curve $$C$$ oriented counter-clockwise. If $$f(x,y)$$ and $$g(x,y)$$ are continuous and have continuous first partial derivatives on some open set containing $$R$$, then

$$\int_C f(x,y) \mathrm{d}x + g(x,y) \mathrm{d}y = \int \int_R \left ( \frac{\partial g}{\partial x} – \frac{\partial f}{\partial y} \right) \mathrm{d} A$$

This theorem has had wide ranging influences, but at the time received limited exposure due to the local nature of the publication and it not being published in a scientific journal.

An interesting fact about this paper is that Green used Leibniz’s notation for the calculus as opposed to Newton’s notation which was in common usage in England at the time.

If you want to learn more about George Green the following two videos are a good place to start:
Maths history trail of Nottingham – George Green
Sixty Symbols – George Green

Categories

## The Perils of Grading Work

Last night, just as I was going to bed I was notified through Twitter of Manan’s (@shahlock) latest blog post where he looks at the grading (by another person) of some work that had been sent to him.

As he rightly points out commenting on just the written grading is fraught with problems as you do not have any idea of what has been said in class in relation to the work. I agree completely with everything Manan has said about this particular piece of work.

The assessor has been very thorough; looking at every line and not just skipping to the final solution. This is of course a good thing and should be applauded, as should the detailed feedback given. However, my initial thought on looking at this assessment and its grading was that it had been marked by someone slavishly (is that a word?!) following a mark scheme with less knowledge of mathematics than the person taking the assessment. On reflection, this is perhaps a bit harsh, maybe the focus of this assessment is exposition and explaining mathematical reasoning – this would explain some of the comments given for problem number 7.
I don’t feel that this explanation really holds for the other marks lost though. The “order of operations” one probably bothers me the most. Working out $$3(-7+9) – 5$$ by using the distributive property gives the same answer as calculating $$3(2) – 5$$. Frankly who cares how this is worked out as long as the correct answer is given; it is a short calculation that doesn’t really require working out what is inside the brackets first! Similarly the marks lost on question 6 just don’t make sense…

I passionately believe that mathematics should be seen as a creative discipline (as well as a scientific one). This sort of grading completely goes against this belief and is only going to serve to diminish confidence and turn pupils off mathematics. We should be celebrating different ways of working things out and I think you can encourage and strive for correct written communication without being pedantic!

Please let me (and Manan) know what you think….

Categories

## Thoughts on A Level Mathematics and Making the Transition

Last night I read an excellent blog post by Stephen (@srcav) discussing the readiness of students for the current maths A Level. It turns out that this blog post was actually a response to another great post by Jo (@mathsjem) and I was going to comment on both posts, but then thought they would probably be a bit long for comments. So here are some of my thoughts…

The Importance of Algebra
As Jo points out it is possible to do well at GCSE without much of an understanding of algebraic manipulation. Algebra is, in my opinion, the bedrock of A Level, and students need to be able to perform algebraic manipulation correctly, confidently and quickly to do well at GCSE.
This appears to come as quite a shock to many students, who have got used to algebra questions being one of 4 main topics, and normally quite distinct from others. I have only been teaching at school level for a short time, however, to me a diagnostic algebra test at the beginning of Year 12 seems to be a very good predictor of A Level grades (far more so than GCSE grades) if the student doesn’t put work in to practice the algebra skills that they may be potentially lacking.

Timing
The choice of student’s applied modules seems to be important, and to me this choice means that an A Level grade is not really fit for being used by universities to compare students. M1 is a tougher module for a student to complete than either S1 or D1 and so an A Level where M1 and S1 have been chosen isn’t directly comparable to an A Level where the student has “chosen” (or pushed towards by a school looking for improved results) S1 and d!. The new 100% prescribed content of the new A Level should remedy this and enable the students to see maths as the connected subject that it is without the arbitrary distinctions placed on it.
Timing is also a problem for those students who are taking Maths and Further Maths. Ideally, a student could complete all of the standard A Level in Year 12 before studying Further Maths in Year 13. Where this isn’t possible students study Further alongside the standard A Level. Working on FP1 content whilst working on C1 content is obviously problematic as students don’t know all the techniques (polynomial division being one of them) that is expected for the Further Maths content. They also haven’t had the practice, or necessarily developed the mathematical maturity that is expected of them. This has obvious effects on confidence levels of the students, as well as creating time pressures when delivering the content.

Type of Exam Question
As Jo mentioned there is a large change in the wording of exam questions at A Level. I think that there should be greater parity between GCSE and A Level mathematics in this. I understand though, that this may go against some of the GCSE reforms to push questions towards simpler wording and grammatical constructs (as AQA definitely seem to be doing) so that students with weak language skills aren’t at a significant disadvantage in their mathematics exams. If we are taking students onto A Level who have achieved a grade B with less than 50% of the paper correct, it is entirely possible that students with weak language skills will end up taking A Level maths where understanding the exam questions could be problematic.
Students also sometimes find the use of more formal mathematical language and symbols confusing. This could be remedied by introducing some of this into the higher GCSE classes; for example sigma notation could be introduced when computing the mean of a set of numbers.

Managing the Transition
I believe that a mathematics A level is the most rewarding A Level a pupil could take, and so I, like all teachers I’m sure, want to make the transition from GCSE to A Level as manageable as possible.
In my school we set some summer work which is essentially revision of GCSE algebra, though it is clear that not all students have revised this sufficiently when they come to do their initial algebra assessment. This year I also sat the Further Maths students some harder work that was more open ended and of a problem solving nature. I wanted to stretch them and also get an idea of how their brains worked when faced with a challenge. These questions than formed the first lesson of the year. On the whole they were well received by the students with them saying that they enjoyed having a go at them even if they did not correctly solve them. This experience of struggling towards a solution is important. Many Further Mathematics students have probably never struggled with a maths question before and the increase in difficulty can sometimes be uncomfortable and hard for them to handle. Learning to fail and accepting that they will learn an awful lot in the process is an important thing to experience on the road to becoming a mathematician.
I’ve also found encouraging group work important too as this forces them to discuss the maths instead of working in isolation. Being able o discuss the questions and work through problems is important as there often isn’t enough time to talk through problems with all questions in class.

Off Syllabus Mathematics
Last year I was lucky enough to have some timetables time to work with the most gifted of the year 11 students, looking at mathematics that wasn’t on either the A Level or GCSE Syllabi. Due to my interests we looked at, among other things, floating point representation of numbers and round off errors, formulae for calculating Pi and iterative schemes for solving linear systems. Many of these topics are typically studied in Year 2 of an undergraduate degree, yet they were accessible to students who had just completed a GCSE course (of course some of the rigorous analysis was left out….). I am passionate about exposing students to maths outside of the syllabus as I think background knowledge often helps them see how to approach a question in a different light, or at the very least provides a motivation for studying what we do at school which is sometimes not explained enough.

A Final Niggle
One thing that I have always loved about maths is the “universal truth” that maths brings. A mathematical concept is always true no matter what, and something that you learn early on cannot be then shown to be rubbish. This is in contrast to the other sciences, for example in Chemistry when you learn about chemical bonding you seem to be forever told to forget what you have been told previously! For this reason it always pains me when teaching the quadratic formula that students begin to believe that getting a negative discriminant must mean they have gone wrong! I can’t see any reaso. Why a higher tier GCSE student couldn’t handle a question such as the following:

I certainly don’t understand why complex numbers are not in the standard A Level as they are so fundamental.

Anyway, I’m looking forward to joining in with Jo’s chat tomorrow and looking forward to hearing other people’s views.

Categories

## Why teach?!

So, I thought as a good way to introduce myself to people who may not know me that way it would be good to describe why I have gone into teaching…

During my School Direct year I had to write a short assignment on “My Starting Point as a New Teacher”, I gave this a quick re-read before writing this post. Whilst finding it, on a re-read, all a bit woolly it did convey the main reason why I feel the teaching of mathematics is so important.

To me, mathematics is the truly universal language that can be used to explain the world and establish truth amongst conflicting media reports. I am passionate about equipping students with the skills necessary to use mathematics productively in their every-day life as well as seeing the inherent beauty of mathematics.

I studied for a PhD in Applied Computational Mathematics at the University of Nottingham and during this time I found the time I spent teaching and running outreach events with local school students far more rewarding than my actual research (I was looking at numerical methods for the Neutron Transport Equation – I may do a post about it one day!). Because of this I decided to apply for a school direct post at a school in the East Midlands, where I have been lucky enough to obtain a post from this year too.

From my experience of tutoring undergraduate students and lecturing postgraduate students (across all disciplines) in quantitative methods one problem that you notice is that British students are not as prepared for a mathematics degree as their international peers. This is of course a sweeping generalisation, however, as a teacher I want to work towards developing the independent study skills required for students to successfully engage with a mathematics degree course if that is where their studies lead them. I believe, very strongly, that mastery of the basic mathematical skills is very important for all students, I aim to develop this mastery in my lessons while still maintaining interest.

Having said all the above, I guess the main reason why I have decided to go into teaching is that I love it! I hope this blog will reflect my love of both teaching and mathematics. I also hope that it will prove useful to other people as well as being a development tool for me.