In a video that I made recently about ways to improve your QTS Numeracy I mentioned things like using a range of different resources, using things like apps to help you with your addition and multiplications. Drawing on my personal knowledge that I’ve acquired over the last few years as a professional maths tutor in Leeds and successfully tutoring loads of people for their QTS Numeracy test, here are a few free online resources that I think are really useful for those of you out there preparing for the numeracy test. Remember that there are no shortcuts when it comes to learning what you need to know for the QTS – it all comes down to how much you practice and the quality of the practice.

Online Arithmetic Drill – This is a great little tool for improving your multiplications, additions, subtractions and divisions – there really is no excuse for not knowing your multiplications when you’re going in for the QTS Numeracy test! You set what you want to do whether it’s just multiplications that you want to practice or a combination of additions, subtractions etc. set which numbers you want to be asked to multiply, set the time that you want to practice for and away you go – do as many as you can in the time that you have. I personally have this bookmarked on my computer and is one of my personal favourites! My personal best is 107 multiplications in 120 seconds up to 12×12. Try to do 15 to 20 minutes of practice a day for a couple of weeks and you’ll see a huge difference.

Worksheet Generator 1 – This site can produce a never-ending supply of worksheets (dynamically generated) for you covering additions, multiplications, simplifying fractions, addition and multiplication of fractions, rounding, percentages …. the list goes on. Not all of the stuff that’s available will be relevant to the QTS Numeracy test as the site is not designed with QTS specifically in mind but there is some good stuff there that you can really take advantage of.

Worksheet Generator 2 – Another brilliant site for worksheets. This looks like it could be aimed primarily at American schoolchildren so is, again, not specifically designed for QTS. BUT….there are just tons of topics covered. I think the ones that would be most useful for QTS Numeracy would be the sections titled

  • Elementary Math
  • Fractions
  • Decimals
  • Ratio, proportion and percent

Just have a play around with the settings until you get used to it – definitely a recommended site

 

I sometimes get asked by my students if I can do some worksheets for them to practice with – I have done this in the past but if I’m completely honest with you there is really no need for me to do so. I don’t want to sound like I’m just trying to find a way to get out of doing a bit of work, but using these websites you can generate hundreds, if not thousands or even tens of thousands, of questions in the time that it would take me to write out a mere twenty questions and that’s not including the tea-break that I’d probably take half way through and then the time that you would have to wait for me to email it through to you! How can I compete with a computer program that is specifically designed to mass produce questions! And what’s more is all of these worksheets are free, created at the click of a mouse and often have the option for the answers to be generated as well. Maybe it would be better for me to stick to what I’m good at which is the tutoring!

 

 

Here is my latest video which is giving my top tips and advice for A Level Maths and Further Maths students who are looking to do one or more of the Sixth Term Extension Papers (or STEP).

These exams are ones that really get people worried – don’t get me wrong, they are tough exams – but they’re not impossible. You just have to prepare well and not get overly nervous about them. Easier said than done I suppose. Well, as a private maths tutor over the last several years I’ve tutored many people for these exams (mostly STEP I and STEP II but occasionally for STEP III) so I’ve come across a lot of things that work and a lot of things that don’t work when preparing to take the exams.

This video will hopefully give you a nudge in the right direction when it comes to getting ready for the STEP exams. It’s a bit longer than I had originally planned but I think it was important for me to be quite thorough so have a bit of patience because there might be something that you can take away that really helps you – after all I want to see you succeed as much as anyone!

STEP tuition is a favourite of mine – I love tutoring for these exams because even now the exams are a good challenge for me and there’s always something new to learn. Some of the questions are just so imaginative and really test your knowledge and ability to think carefully through unfamiliar problems.

If you are preparing for the STEP exams then I wish you the best of luck. Please feel free to send me an email or to get in touch with me if you have any questions about the exam – I’ll be glad to help if I can.

Here’s the next video in my Ten Ways series – this video is for those of you out there looking for ways to improve your QTS Numeracy. I know that there are lots of people out there who are looking to do their QTS Numeracy test who feel that they’re going to have a rough ride. Well the good news is that as a professional maths tutor in Leeds over the last several years I have done loooooads of tuition for QTS Numeracy so I’ve got a few tips here in this video that might help. It is quite a long video – I didn’t actually expect it to be as long as it was but at least you’ll be getting your money’s worth!

The video, as with my GCSE Maths and A Level Maths videos, is not just giving a list of topics to revise for the test although it does mention a couple of topics that might be particularly useful. I didn’t want to make a video just giving all of the topics to revise because I think there’s already a ton of websites and other videos out there already doing just that. What I did want to do was to bring something unique and new; to share my personal experiences, observations and advice that I’ve learned over the last half a decade or so – no-one else can give you that.

I hope that this helps you out a bit – I’d love to hear any of your comments; your own advice that you can pass on to others preparing for the test and your own personal experiences of the test itself.

So here is my first monthly review video – I decided to make this video just to share with you what’s been going on over the last few weeks in my life as a professional maths tutor. Sometimes I get some really interesting questions from my students during lessons and I thought that it would be a good idea to share some of those questions and their answers with you.

In this video, for example, I discuss weak and strong forms of proof by induction – usually A Level Further Maths students will only really get familiar with weak induction but strong induction is a lifesaver in some cases where weak induction just wouldn’t cut it. I also talk about functions as I was asked some really good questions about functions in one of my lessons when I was doing transformations of graphs with a GCSE student of mine.

I made a couple of videos over the last month – one was about how to improve your GCSE Maths and the other was how to improve your A Level Maths. There will be another couple of videos over the next month so keep an eye out for them.

Other things that I touch on are a couple of books that I read over the last month. The first was Alan Sugar’s Autobiography – I know, I know, it’s not exactly a maths book but I read it anyway. I also read Surely You’re Joking Mr. Feynman by Richard Feynman, which is another autobiography, and I started reading, or should I say working through, The Works of Archimedes (that’s the famous Greek scientist Archimedes).

And lastly I introduce my new friend the Soroban (Japanese abacus) which I’ve started to learn to use. I’ve only been learning for a weeks or so but things are going in the right direction. I’ve found a link that is really useful for learning – it doesn’t tell you how to use a soroban but it throws up random addition problems so that you have to try to figure them out on the soroban; it’s something called Flash Anzan and has been really useful so far in getting me used to the various combinations that you need to know to use the soroban.

Following on from my previous video Ten Ways to Improve Your Chances of Success in GCSE Maths here is the next video in my Ten Ways series for A Level Maths and Further Maths.

As with my last video I’ve tried to focus more on the habits that I think a good A Level student needs to have to have a better chance of being successful in their studies. I didn’t want the video to just be a list of what I think are the most important topics to revise for A Level maths because I’m sure that there are a ton of videos already out there that do just that – personally I don’t think that you should necessarily prioritise any topics over others as they could all come up in your exams as ‘big questions’.

Also as with my last video this video has come about through my own personal observations as a professional maths tutor in Leeds of my A Level maths students and noticing common themes (I didn’t want to say denominator!) amongst those who do well and get the grades that they want and those who unfortunately don’t do so well. Believe me when I say it isn’t just down to knowing the topics – you have to have the right attitude to the subject to be successful at A Level maths.

I hope that you find the video useful – it’s a bit longer than I expected it to be but if you think it’s too long then … well … I guess just don’t watch it to the end, though I really hope you do watch it to the end because there might be some useful stuff for you. I’d love to know if you have any of your own tips and advice that you’d like to share with other A Level maths students. If I get enough suggestions then I might make another video to share some of your tips!

I never really used to be interested in history in school – in fact I couldn’t wait to drop the subject. I just couldn’t see the point in thinking about the past when it just didn’t seem relevant to me or anything for that matter. Historians out there will be pleased to hear that I take a very different view of history now and, although I wouldn’t say that I’m a fully fledged historian, I do enjoy reading the odd history book.

My interest in mathematics and history come together when I read mathematics books written by some of the greatest mathematical thinkers in history. It’s amazing to see how problems were solved by the ancient Greeks using the technique of application of areas which is mostly a lost art now and has been replaced by algebraic techniques; or to see how philosophical issues have shaped the development of mathematics such as whether the Axiom of Choice is valid as an axiom.

Well here are my top historical mathematics books with a brief explanation of what’s inside

  • Principles of Mathematics – Bertrand Russell Bertrand Russell is one of my favourite philosophers and mathematicians in history. Not many can match Russell for his depth of knowledge – even today. Don’t be confused by the title – this is not a book on simple mathematics; it’s certainly not for the faint-hearted and will take a good chunk of time to plough through. Most of what is inside the book would now be considered a bit dated but at the time this was ground-breaking stuff.
  • Elements – Euclid Probably one of the most famous mathematics books ever written. I still think it’s incredible that this book was still the standard geometry textbook in most schools up until the 19th Century – about 2000 years after it was written in ancient Greece. Nowadays most textbooks are thrown out after a year to bring in the next lot of ‘updated’ textbooks. If any book shows how timeless mathematics is then this is it.
  • Treatise on Conic Sections – Apollonius of Perga This is another book by an ancient Greek but is nowhere near as well known as Elements. It’s a shame because inside this book are some of the most inventive uses of the technique known as application of areas to prove various properties of the conic sections – circles, ellipses, parabolas and hyperbolas. Again, although the technique has now been replaced with algebraic techniques the solutions are nothing short of beautiful.
  • Contributions to the Founding of the Theory of Transfinite Numbers – Georg Cantor Cantor is the father of the infinite in mathematics. His works took mathematics in a whole new direction – a very controversial direction at the time it would seem. When you read this book it all seems so straightforward to deal with the infinite but the imagination required to come up with some of the arguments and proofs is off the scale.
  • The Continuum – Hermann Weyl This is an unusual book as Weyl decided that he wasn’t happy with the way that mathematics was working at the time – he felt that mathematics had inadvertently created different ‘levels’ and there was a number system on each of these levels that comes about through the logic used and numbers on different levels were being combined when they shouldn’t be. He aims to demolish these levels and create a single number system but his logical system pays a price for this. It sounds like the plot to a novel! Weyl’s philosophy changed several times throughout his life and this gives a bit of insight into his personal philosophy at the time it was written. He later abandoned this work in favour of a diffeent philosophy but then, after a few years, he changed his mind and thought it was a good work after all.

So there you have it – some of my favourite maths books from history. I love reading these historical maths books because it feels like you’re reading the minds of some of the greatest mathematicians, philosophers and scientists to have ever lived. I’m sure there’s loads more of these books for me to read – I’ll be on the lookout for some good ones to read. By the way – some of these books are in the public domain now and you can download some of them for free from the internet. I prefer to buy copies of the actual book but just so you know.

This video shows a few clips from a presentation that I gave almost two years ago now for Leeds Skeptics (thank you to Chris Worfolk for inviting me to give the presentation and to follow in the footsteps of some fantastic speakers that the club has had over the years). The room was quite dark and so the video may not be very clear at times but hey, you get to watch it for free anyway!

The presentation came about through my annoyance with the huge amount of crappy statistics that float around everywhere (and I mean everywhere!) we go; in newspapers, on television, on the internet, advertisements, in our mail, on food packaging and of course the inevitable stream of carefully chosen (but often misleading and in many cases very suspect) statistics that flows out of any mealy-mouthed politician or pressure-group leader that gets a few minutes of airtime on the telly.

Although statistics and probability has good intentions and is a tremendously fascinating area of study (according to recent polls at least) it is wide open to abuses of all kinds. What a surprise! I’m sure most people have come to realise this over the years and may even have become quite passive about it and just accept it. Most of the time I do in all honesty! At the risk of sounding a bit negative, there is very little that can be done to stop statistical misuse; the only way that it can really change is if people choose (all by themselves) to be clear and transparent with the basis of the statistics that they produce. However, it does pay to be more familiar with the sorts of things that go on so that you can make a more informed decision as to whether the statistic that you are given is reasonable or is a big steaming pile of … (hold it right there!!) So I decided to familiarise myself with what goes on a bit more behind the scenes. Although I am quite well acquainted with various statistical methods learned in the academic bubble of university I was really quite surprised at how statistics can be manipulated so easily and so irresponsibly.

Here’s a few ways that everyday statistics might be fudged;

  • Choosing a biased sample – this may be deliberate or not in some cases.
  • Omitting certain undesirable outcomes – almost as if they never happened.
  • Moving the goalposts – changing the significance level of a statistical test after an experiment has been carried out to give the desired result.
  • Using a sample size that is too small which increases the chances of skewed reults.
  • Incompetence – You might think that it’s obvious that someone who doesn’t know how to deal with statistics isn’t going to be very good at producing reliable statistics.

The list goes on. If you are interested in reading more about uses and abuses of statistics here is a list of some good books that I read on the subject – they are aimed at the general reader and so they are not highly technical with pages crammed with jargon and the like. They’re definitely worth a read.

  • How to Lie with Statistics – Darrell Huff
  • A Mathematician Reads the Newspaper – John Allen Paulos
  • Damned Lies and Statistics – Joel Best

After the presentation I got talking to some of the audience members (I know; who would have thought that people would come to see a presentation on statistics!) and the following website was recommended to me for a voluntary organisation called Radical Statistics that analyse and put statistics through their paces to check their credibility. I’m not currently a member of the organisation but I have been on the website regularly and there is some really good information there.

By the way – my whole presentation can be seen on the Worfolk Lectures website in much better video quality than my own recording I hasten to add.

One thing that many of my students (at all levels!) seem to have trouble with are those pesky minus signs. For some reason people get very phobic around them and very often assume that when a minus sign comes up then they must have done something wrong. The fact is that minus signs are a very normal thing to arise – but that still seems to be no consolation to many.

One question that a lot of people find bewildering is the fact that when two negative numbers are multiplied together then a positive answer results – yes that’s right, for example $-3\times-4=12$ (NOT $-12$ as I’m sure I’ve said many hundreds of times during my time as a maths tutor). How is this so? It does seem a little counter-intuitive. Well…here’s a proof of it. The proof that follows will show, once and for all, that two negative numbers (real numbers if you must know, but then does it make sense anyway to talk about negative complex numbers or quaternions or anything like that?).

Here we go…

Throughout the proof I will assume the axioms of real numbers found in Mary Hart – Guide 2 Analysis Second Edition (i.e. that the real numbers are an abelian group under addition and an abelian group under multiplication, multiplication distributes over addition, the real numbers satisfy the order axioms and the completeness axiom).

First I want to prove that $0t=0$ for any $t \in \mathbb{R}$

$0t = (0+0)t$ as $0$ is the additive identity and therefore $0+0=0$

$0t = 0t + 0t$ by distributivity

$0t+(-0t)=(0t+0t)+(-0t)$

$0=0t+(0t+(-0t))$ by associativity of addition and $0t$ and $-0t$ are additive inverses

$0=0t+0$

$0=0t$  as required (*)

Next I want to prove that $-(-s)=s$ for all $s \in \mathbb{R}$

$-(-s)+(-s)=0$ since they are additive inverses

$(-(-s)+(-s))+s=s$

$-(-s)+((-s)+s)=s$ by associativity of addition

$-(-s)+0=s$ since $s$ and $-s$ are additive inverses

$-(-s)=s$ since $0$ is the additive identity (**)

Thirdly I need to prove that $s(-t)=-(st)=(-s)t$ for any $s,t \in \mathbb{R}$

$st+(-s)t = (s+(-s))t$ by distributivity

$st+(-s)t= 0t$ since $-s$ is the unique inverse of $s$

$st+(-s)t= 0$ by (*)

$-(st)=(-s)t$

similarly $-(st)=s(-t)$

and so $s(-t)=-(st)=(-s)t$  as required  (***)

Now I need to show that $(-s)(-t) = st$ for any $s,t \in \mathbb{R}$

$(-s)(-t)+(-((-s)(-t)))=0$ since they are additive inverses

$(-s)(-t)+(-(-s))(-t) = 0$ by (***)

$(-s)(-t)+s(-t)=0$ by (**)

$(-s)(-t) = -(s(-t))$ by uniqueness of inverses

$(-s)(-t) = s(-(-t))$ by (***)

$(-s)(-t) = st$ by (**)

It may appear that I am done here but in fact I have not said that (despite appearances) $-s$ and $-t$ are indeed negative; however, if we now assume that $s>0$ and $t>0$ then by the order axioms we have that $st>0$. But is it true that $-s$ and $-t$ are negative? Let’s find out…

since $s>0$ then by the order axioms we have that $s+(-s)>-s$. But $s+(-s)=0$ since they are inverses so $0>-s$. In other words $-s<0$ and thus $-s$ is negative. Similarly it can be shown that $-t$ is also negative hence $(-s)(-t)$ is indeed the product of two negative real numbers and thus since $(-s)(-t)=st$ and $st>0$ then $(-s)(-t)>0$ and the product $(-s)(-t)$ is positive. The proof is complete.

This proves that the fact that two negatives multiplied together give a positive is not just some rule that someone made up for a laugh to make maths difficult for GCSE and A-Level students – it is entirely consistent with the axioms of the real number system. Edmund Landau gives a much more thorough treatment of this in his book Foundations of Analysis, which I mentioned in my previous post on $2 \times 2 = 4$ in which the author builds gradually up to the real number system from the natural numbers and doesn’t assume the axioms that I have stated above. However this way of doing things is much longer to carry out taking Landau almost 90 pages to do; conciseness is desirable but must give way to thoroughness from time-to-time.

In a previous post I spoke a little bit about B-Splines and in particular, Uniform B-Splines. I didn’t really go into much detail about how they are defined, how we decide what our b-spline is going to look like or even what an explicit expression for a spline would look like given our set of control points.

I decided to type up a LaTeX document which introduces the theory of b-splines. The document looks at b-splines from a practical perspective and so it doesn’t get too bogged down with the analysis side of things but concentrates on the tools required to find explicit expressions for b-splines.

Download the full document on Uniform B-Splines here.

The document contains expressions for piecing together a b-spline. I spent a good number of hours deriving these expressions and they got very messy to deal with but, eventually, persistence prevailed. It is likely that someone, somewhere has already derived these formulae but by doing the work myself I was able to learn so much about these splines that I wouldn’t have appreciated if I had just read someone else’s work.

This is one of the most important things about becoming better as a mathematician – being prepared to spend time with a problem and being willing to make a lot of mistakes until you get things right. Persistence is rarely a bad thing. I make no apology for the length of some of the expressions – I have not made any real attempt to simplify the expressions as I feel that it would be a glorious waste of time to try and do so. If anyone out there would like to have a go at tidying up the expressions then feel free to go right ahead.

 

During my final year at Warwick University I did a project on Numerical Weather Forecasting and one of the methods that came up was the method of finite differences. I recently came across finite differences again over the last few weeks and how they can be used to solve some partial differential equations (PDEs). Here is an example that I found in one of my books that I decided to play around with

Solve, using the method of finite differences, the PDE $$x\dfrac{\partial{f}}{\partial{x}}+(y+1)\dfrac{\partial{f}}{\partial{y}}=0$$ for $0\leq x,y \leq 1$ with $$f(x,0)=x-1$$ $$f(x,1)=\dfrac{x-2}{2}$$ $$f(0,y)=-1$$ $$f(1,y)=-\dfrac{y}{y+1}$$

To solve this problem I used the central difference equations given by

$$\dfrac{\partial{f}}{\partial{x}}\approx \dfrac{f(x+h,y)-f(x-h,y)}{2h}$$

$$\dfrac{\partial{f}}{\partial{y}}\approx \dfrac{f(x,y+k)-f(x,y-k)}{2k}$$

with a step length in both the $x$ and $y$ directions of $\frac{1}{3}$. The domain can be overlaid with a grid as shown in the diagram

Domain with overlaid gridThis makes the whole problem easier to deal with because now we can see visually what is happening rather than just dealing with everything purely algebraically – there’s no need to make things more difficult than they need to be after all.

The idea is to form a system of four simultaneous equations with $A$, $B$, $C$, and $D$ as the unknown quantities. When the system is solved then the values obtained correspond to the approximate values of $f$ at each of the grid points. Here is my full solution to this problem – PDE Solution Using Finite Differences. This fills in some of the gaps in our knowledge about the function $f$ but there is still a lot of information missing – in particular the points between the grid points.

To resolve this we can make the step-lengths smaller and create more interior grid points. This gives us more information but at a price – for example with a step length of $\frac{1}{5}$ in both directions we would have a system of $16$ simultaneous equations in $16$ unknowns – I don’t really fancy sorting that mess out without the help of a computer. And what’s more – as the step-length gets even smaller we get more and more complicated systems with more unknowns so eventually we would reach a point where even a computer would struggle to do anything with the information.

Nevertheless – finite difference methods are used and are useful for all kinds of problems and can provide a very visual way of numerically solving some otherwise very difficult equations.