Skip to main content
GlossaryA Mathematics Teaching & Learning Resource

The idea behind this site is to allow students and teachers to access explanation of concepts, fully worked examples with explanation of methods, exercises for practice with immediate feedback, essays on history of mathematics and other educational research ideas. I want to make it possible for anyone who wishes to learn about mathematics, up to university entrance level (at the moment), to do so for free and entirely online. There is no fee for accessing these materials and I do not collect your data or surveil you in any way.

Throughout the site, you will come across words or short phrases which are underlined with a grey dotty line. These are created dynamically and access brief definitions of mathematical ideas, terms, people or concepts you may or may not be familiar with. These will pop up in a green box when you run the mouse over the word. This is how I have decided to implement a glossary of terms and these will grow as the site grows. The idea is that these will give you a quick reminder if you’ve forgotten what words like quotient, triangle or derivative mean. If you don't require the assistance, don't run the mouse over such words.

Problem based learning

Each topic here should be seen as a response or technique to a difficult problem. All problems are difficult until you find the right approach, after which they often become quite easy. The topics, therefore will begin by suggesting one or more problems which demonstrate the need for the techniques we are about to discuss. I am persuaded that it is very hard to learn any idea until you are able to identify a question or problem which clearly requires such techniques.

A technique whose usefullness is not obvious, is not going to inspire the graft required to practise it and make it part of your working model. I take much of this from my experience of playing video games. One of the best examples in recent years, is Zelda, Breath of the Wild, which is a Nintendo Switch title. In playing the game, you have a great deal of choice in how you deal with the situations, the game puts you in and a lot of the fun is in finding the best way to successfully navigate the game. When you start playing, you assume that Guardians are simply to be avoided in the early stages of the game and most people deal with them by running away as fast as possible. However, there is a technique called a Perfect Guard, which is a shield technique. The first time I managed to take out a Guardian with a pot lid (the weakest shield in the game) was a major moment for celebration. Now if I were to give a context free lesson on the Perfect guard, it would be considered boring and irrelevant, but once we have a problem which cannot be solved using currently available techniques, we have a motivation for getting good at the techniques and the learning is fun because we can see an advantage to learning it.

  1. Identify a problem (Dealing with a Guardian early in the game, without dying)
  2. Learn the technique (Perfect guard)
  3. Don’t die!

Many games offer techniques which are complicated while it is unclear how they are going to help. These techniques are often not learned at all and thus the player runs into problems later on in the game. So learning techniques should only occur when we perceive a need.

Other sites

There are many sites which deal with the topics I am concerned with here, but most are either very simple indeed or as complicated as all get out. For example, Wikipedia has brilliant mathematics articles, but they tend to get very opaque very quickly and are all but useless to the layperson. Others, like BBC Bitesize or MyMaths, deal very well with simple ideas well but often miss out on the rigour as they are entirely designed for students. I aim to try to find a balance between these extremes, producing resources which I hope will be useful as much to teachers as to students. I intend to look at the structure of mathematics, building the techniques and concepts which make up the subject. I want to attempt to make the more difficult ideas as accessible as possible while maintaining a decent level of rigour. My models here are books like "Mathematics for the million" by Lancelot Hogben, though I shall also discuss problem solving ( Heuristic) and attempt a demystification of much of the language and notation associated with mathematics.


  • The hamburger icon at the top right of the title bar (by default) brings the menu in and the image is changed to a close icon.
  • Next to the menu burger are the ‘back’ and ‘forward’ buttons. Because browsers don’t give developers easy access to the browser’s buttons, I’ve done my own which work with the technology on which the site is based. (AJAX, for those interested in geeky stuff.)
  • If you prefer the menu over on the right, then go to the Settings at the bottom of the menu and check the checkbox there.
  • The system will keep track of whether you like the menu to be open or closed, or to the right or left, when the site loads.
  • Darker green entries in the menu connect to short essays.
  • Lighter green entries in the menu connect to mathematical topics.
  • The bold italic rows in the menu open a folder containing other items.
  • Normal typeface entries are links which open pages or topics.
  • Words/phrases with a dotty underline has definitions which pop up when the mouse goes over them or the user touches them, if you are using a touch screen.


Every topic on the site contains one or more of the following sections. Much of the text features underlined non-bold "links", which don't go anywhere but they will display a pop up when the mouse hovers over it. Moving the mouse away, closes the tooltip popup. There are a couple of examples of this in the penultimate paragraph of this page.

An introduction, giving some basic information or history pertaining to the topic.
A set of worked examples which are used to talk about the underlying principles. Clicking on the list will open the example so it can be followed line by line by the student with accompanying notes as a guide.
Eg list Eg open
A set of exercises for self-assessment with an 'A' image on the right of the page, to click on to reveal the answer.
Interactive activities related to the topic. This Pythagoras jigsaw let's the student play around with the squares and triangles to demonstrate to themselves that areas of the two smaller squares are equal to the big one.
Some puzzles, which relate to the topic. (These are currently in short supply due to time constraints, but they are going to happen, so long as I don't drop dead any time soon.)
problem Hint Solution Explanation

Many sections also have Articles, each of which deal with a discrete part of the topic. I have not attempted to write fully fledged lesson plans because this will instantly restrict how useful the text can be. My notion of a good lesson is one which I organise around how I do things best. Other teachers use different methods and those are often just as good, if not better. Teaching is an art/craft and similarly to how I don't expect David Gilmour to make the same kind of noise as Frank Zappa or Albert King, different teachers engage students best if they use techniques which work for them. That is not to say that I don't offer advice about delivery, just that any such can be happily ignored, if it doesn't work for you.

Some of these articles will have to be ignored first time around, as (and I'm really talking to teachers here) there are multiple dependencies at this level of mathematics which need to be covered and understood before many others. The approach is to talk about topics repeatedly, but each time at a slightly higher level. A bit like climbing up the slope of a helter skelter. We meet the same stuff repeatedly, but at each time around we have more background knowledge so we can go into a bit more depth. For example, in the Differentiation topic, you will see an article which proves the derivative of sine from first principles. This proof requires that the student understands the Compound angle formulas and also knows how to deal with the sine of tiny angles. It is perfectly fine to say to a student, “Don't worry about Article X, we will come back to it later.”

My expectation is that students will be guided (if possible) through the introduction, articles and/or worked examples. You should play with the activities as you will. However, it should be fun, not torture! You should then try the exercises, in which the questions are largely ordered according to difficulty. You must then check your answers after each and every question you attempt. This is something which many students of all ages are pretty bad at, as it takes additional time. It is true that it takes longer in the first instance, but I would argue that the advantages outweigh the disadvantages and even the time argument gets stood on its head. The method which most students (and even some teachers) adopt is to do an exercise and then mark it, which in my opinion is worse than pointless, as all it achieves is to make the student practise their errors until they become second nature. Just as when playing music, practising your mistakes makes you good at your mistakes. This is not a desired outcome. Now, the authoritarians out there will be tearing their hair and bellowing about cheating and such. The only cheating that I know about is the pretense that you understand something which you do not. (It is also important to note that the argument against anything because it “might be abused” is profoundly wrong at every level and is usually just a word salad which politicians use to make sure they are the only members of society who can get something for nothing.) Do not practise your mistakes. Please! Check your answers every time. That way, you don't need to waste time doing questions which you have already mastered and can move on to the fun stuff later on. It is important to note at this point that mathematics gets more fun as you progress.