by Matthew Leitch

This article is about specific techniques to learn maths at school with less effort and frustration, regardless of whether you think you are naturally good at maths or not. You can benefit from upgrading your maths learning skills if studying maths feels like this:

In maths classes you often feel confused, frustrated, bored, or sleepy. Often you cannot understand what is being explained and hope to sort it out later. If you try to read a maths book or website then you just feel sleepy and your mind wanders in a fog. None of it makes sense and you don't know what to do. You have no feeling of learning from doing homework and just want to get it done as quickly as possible. Unfortunately, homework often takes ages because you get stuck. In tests and exams you are often stuck, make mistakes, and struggle to decide which questions to attempt.

In contrast, there is little you can gain from this article if your experience of learning maths feels like this:

In maths classes, when teachers explain something you listen carefully and notice their mistakes. Lessons often feel slow because the teacher is repeating things others have forgotten. You often read a maths book or website for half an hour on your own and find it relaxing. You often enjoy homework problems, especially new or difficult ones. Every problem you tackle feels like another opportunity to increase your mental powers. In tests you just relax, do maths, and often perform better than when doing homework.

The rest of this article explains efficient learning techniques for mathematics at school. You could browse through to get an idea of the possible value to you and maybe pick some ideas to try yourself. There are tips to do all these things:

You should study maths for the right reasons. This helps you push through disappointments and obstacles.

Sadly, much of the maths taught from about year 10 is not useful in the real adult world. Very few people have to solve quadratic equations and even adults who like maths rarely get an opportunity (unless they are teachers or tutors). Those skills are not a good reason for studying maths.

However, the *learning* skills you need to master maths are highly relevant to many things you might want to learn to do in the real adult world. Maths is a problem solving skill that lets you solve a wide range of complicated problems. If you can learn maths then probably you can learn to build a house, design a bridge, apply the law in cases, do tax calculations, invest money, or program a computer. Maths also requires learning to think accurately, logically, and with no mistakes. If you can learn to do that then probably you can learn to do adult jobs that require complete accuracy, such as medicine, accounting, programming, building a house, and many more. In the real adult world giving customers something that is wrong is often a serious error.

(Whether you actually do apply your maths learning skills to other things is largely up to you. If you know what maths learning skills you use and *deliberately* apply them to other things then you are much, much more likely to benefit.)

Maths at school is a great opportunity to develop those learning skills and show you have done so. No other school subject requires as much problem solving and correct thinking as maths.

Also, success with maths in school shows people you have the skills to learn to solve problems and become accurate and correct. This is why maths is so often a requirement for later courses and looked for by employers. A person who cannot learn maths has very limited options compared to someone who can.

Another practical reason for developing good maths learning skills is that, if you struggle with maths, your school will not let go. They will pester you with interventions such as extra sessions after school, meetings with teachers, sessions in holidays, target setting and reviews, meetings with your parents, and exam retakes. Staying on top of maths is the easier option.

This is the most important section of this article. It is about what *exactly* your mind should do in maths lessons, when reading about maths, and when doing homework. Put an end to vague, sleepy, mind wandering. You don't have to concentrate fiercely – just stay focused, relax, and think the right types of thought.

Learning maths requires you to take in and master a huge number of specific pieces of information, many of them seemingly too small to bother with. Your brain cannot do that quickly. In a 30 minute period of studying maths, working in the efficient way described below, you may feel you have hardly covered anything at all. How can you possibly make progress when it seems to go so slowly?

The logic is simple. The material must be mastered, one way or another. You can either settle down and patiently work through the details, sorting out each little thing, one at a time, or you can flick through, skimming for something that makes sense, and hoping that in your desperation a light will suddenly come on. It won't. If you are patiently focusing on specific points then you are probably making progress. If not, you aren't.

Be patient with the details of each new topic and you will soon realize that teachers often repeat points and problems are often very similar to problems you have done before. There is no need to patiently notice all the details of this repeated material. You will also be able to solve the problems sooner and will get stuck less. So the ultra patient, seemingly slow approach is really a quick way to make progress.

The key is to notice specifics systematically. Make observations consciously and follow a pattern so that you don't miss details. Don't just let your eyes wander aimlessly across text or a diagram. Pick out specific details and patterns. Be clearly aware of each item so you can remember it later. You don't need to talk to yourself but it might occasionally help. Here are some examples to show the right level of detail and typical patterns. The first example assumes you are reading about mathematical functions for the first time:

**Words in the book:** ‘A function takes an input and gives you back an output.’

**What you could notice:**

(1) the word *function* being used in a special way

(2) the phrase *takes an input* which also sounds like a special phrase

(3) functions can be given an input

(4) the phrase *gives you back an output*, another special phrase

(5) functions give outputs

(6) this is in response to an input

(7) *function* is defined or explained in this sentence

(8) the definition of function is as something that you can give an input to and it gives you an output.

Typical systematic patterns to use are: (1) in reading order, (2) visible details then on to connections and inferences, and (3) 'outside-in' for formulae, which means describing the overall form of a formula before drilling into it, layer by layer. (See also the example below about memorizing formulae.)

Do all this thinking in a precise, unhurried way. Stay relaxed. Be clear rather than fast. Noticing specifics is also crucial to understanding questions.

**Words in the book:** ‘A rectangular swimming pool contains L litres of water and is 1.2 metres deep at all points.’

**What you could notice:**

(1) the word *rectangular* to describe the pool

(2) presumably this is its shape seen from above because the pool is in fact a 3D object with depth also

(3) the letter L is used

(4) it is a capital L

(5) it represents the *volume* of water in the pool

(6) the units of volume here are litres

(7) the depth is given

(8) in metres

(9) as 1.2

(10) the depth is the same everywhere

(11) which is unusual for a pool.

What you notice will change as you get more knowledgeable. For example, when you look at some algebra for the first time in your life you might notice like this:

**What you see:** 3x^{2}

**What you notice:**

(1) this is a cluster of characters

(2) the first is a 3

(3) there is an x

(4) it is just to the right of the 3

(5) there is no gap between them

(6) the x has a 2 next to it

(7) the 2 is smaller

(8) and positioned high and to the right of the x.

Once you know more that simplifies:

**What you see:** 3x^{2}

**What you notice:**

(1) an x squared term

(2) with coefficient 3.

With even more skill you will spot larger patterns almost instantly.

**What you see:** 3x^{2} + 2x + 1

**What you notice:**

(1) a quadratic expression

(2) coefficients 3, 2, 1 – cool!

It is typical that you have to notice details first to have any chance of working out what a mathematical explanation means. Imagine you are reading about sequences. The idea of a sequence has already been explained but now the explanation continues:

**Words in the book:** ‘One way to define a sequence is with a term-to-term rule.’

**What you could notice:**

(1) the phrase *one way* implies there are others

(2) the sentence is about how to define sequences

(3) the phrase *term-to-term rule*, is a technical term

(4) it is written with hyphens

(5) it contains *term-to-term*

(6) and *rule*

Sometimes symbols and words are used without being explained first. You have to notice details in order to make sense of them. Imagine that the explanation of term-to-term rule continues:

**Words in the book:** ‘For example, the sequence 3, 6, 9, 12, ... might be defined using the rule U_{n+1}= U_{n} + 3.’

**What you could notice:**

(1) an example is being given to explain the phrase *term-to-term rule*

(2) it involves a typical number sequence

(3) as usual with sequences, check the differences between the numbers. Notice each term is 3 more than the last

(4) the rule involves some new symbols

(5) there are two with *U* in them

(6) they have suffixes

(7) the suffixes are different

(8) but both include *n*

(9) they are *n* and *n + 1*

(10) 3 is added to one of the U terms – aha! this is the difference from one term to the next

(11) U_{n} is probably the nth number of the sequence, 3 is added, and that gives the next number

(12) this fits the phrase *term-to-term rule*

(13) the U is a letter for numbers in a sequence, probably a traditional choice

(14) the suffixes are a way to label items in a sequence in a generic way, making a rule connecting one term and the next.

If you are reading an explanation and realize that you already know what they are explaining then you may be able to accelerate or skip a bit. When you see something new, slow down again and take in each detail carefully.

You may know someone who is extremely good at maths who seems to read maths very quickly. This will mostly be because they know more and so there is less new for them to take in. They will still notice specifics carefully when they need to, though in a skilled and quicker way.

(I sometimes find it helps me to count how many specifics I have noticed. This reassures me I have done some work and helps to keep me doing the task. You can count up to 100 on your fingers if you use both hands. Use your thumbs to point to your finger tips and knuckles in order. This gives you 12 places to point on each hand. I only use 10, and then use my left hand for units and my right hand for tens. With practice, counting on your fingers does not interrupt your thoughts.)

Chunks are fragments of memories. They are not usually facts or processes; they are usually just parts of those. Chunks include terminology, notation, fragments of algebra, steps within larger procedures, typical elements within a type of question, Greek letters, and their pronunciation.

They become recognizable, familiar, and available as building blocks to make bigger memories. If you increase your mental stock of chunks then he will be able to understand questions and procedures more quickly and easily, and learn new mathematics more easily.

You can and should recognize useful chunks as you learn and spend a few moments dwelling on each one so that it becomes familiar.

E.g. The expression (x+a)(x−a) is built from the tiny chunks (, ), x, a, +, and −. But these are put together to make slightly bigger chunks such as x+a, x−a, (x+a), (x−a), and finally (x+a)(x−a). A more abstract chunk is the form that these have, which is the sum and difference of two terms multiplied. It appears in other ways, such as (x+2)(x−2) and (y−x)(y+x). This pattern is so famous it has a name: conjugate binomials.

If your memory skills are just developing or what you have just tried to notice is important then check your memory almost immediately. Give yourself a few seconds after noticing the specifics (e.g. of a formula, procedure, or question) and perhaps distract yourself with some other task for a minute then try to recall every detail you just noticed. For example, you might try to write down or say the formula you tried to learn, or you might try to recall the details of a question you have just read.

If you missed something when initially noticing then you will find you cannot recall it when you test yourself. Sometimes, even one gap causes general confusion. This may be because the thing you missed was the first part of the memory. Go back and notice the missing details again. Then wait and check yourself again.

This technique is incredibly powerful. It allows you to build the perfect memories you need for mathematics. It also shows you powerfully the difference between just looking at something and noticing it specifically. If you struggle to remember mathematics then this technique will show you that your eyes have been wandering across the page but your brain has been sleeping. Switch on your brain and in just a few minutes and you may well notice an extraordinary change.

You should try to learn which specifics are important. This is vital. At first you will probably notice specifics as if you were going to repeat the material verbatim. With growing skill you will soon focus on the specifics that matter.

E.g. A beginner might look at 3y + 2 = 4 and think ‘There's a 3 next to a y, and then a bit of space, and then ...’ The expert will look and notice there is one equation, it is in one variable, named y, and that it is linear.

What to notice depends on the topic, the problem, and the specific item. For algebra the key points are usually the number of equations, the number of variables and their names, the powers of the variables, other functions appearing (e.g. sin, cos, log), specific forms (e.g. x^{2} + y^{2} = r^{2}), and constants represented by letters (usually identifiable from the traditional choices of letter).

Flawed explanations by teachers, tutors, websites, videos, and books (even the official textbook from your exam board) are common. Sometimes, problems are badly explained. In comparison, mathematical errors that lead to the wrong answer to a question are usually very rare, except when teachers try to solve problems for their class. Watch closely for flawed explanations at all times and deal with them.

If you are patiently noticing specifics as you read then you will be able to spot many defects in explanations immediately. Typical errors include:

- Easy to spot:
- using a technical term, variable, symbol, or other notation without explaining it, or before the explanation is given
- not explaining clearly what types of problem can be solved with a particular method
- not explaining why a particular solution strategy was chosen when it is not obvious
- not explaining the purpose of a solution step before going into its details
- Usually harder to spot but revealed by contradictions with your existing knowledge:
- using a technical term, variable, symbol, or other notation in a different way to usual without explaining what has been done or explaining later
- using a technical term, variable, symbol, or other notation incorrectly
- using two technical terms for the same thing without explanation, creating the false impression that they refer to two different things.

E.g. Imagine you are reading a Further Maths A level chapter about how to make equations from other equations. (You will not need to know this maths to understand this example.) The chapter makes a point about ‘transforming’ equations but has not previously explained what *transforming* means for equations and this new sentence is not an explanation. Obviously this is an explanation error. Later in the same sentence it talks about a transformation of the roots (i.e. solutions) of the equation using the transformation ‘*y* = *mx* + *c*’. This is another blatant explanation error because none of these symbols have been introduced. The *m* and *c* are probably constants but the variable *y* is particularly mysterious. It is also a contradiction because the letters the chapter has been using for roots so far are the Greek letters α and β. Surely if a transformation is of the roots then the roots should be mentioned.

One great advantage of spotting flawed explanations is that you know your struggle to understand is not your fault. The explainer has done something wrong. Another great advantage is that you can respond to flawed explanations instead of just feeling confused and giving up.

Useful responses to flawed explanations include:

**Ask for missing information:**With a teacher or tutor you may be able to get them to stop and answer your question. For example, ‘Sorry, but can we just go back a moment? You used the word “operator”. What does that mean in this context?’**Search another source:**That might be a mathematical dictionary or just a search of the internet.**Mentally note the problem:**You can notice what information was missing, or what contradiction you found, and then read on. Often you will get a clue later that helps you fill in the gap or correct the mistake. If not you can try other sources or ask.**Make an educated guess:**When you do this, make a mental note that you are using this educated guess and stay alert for clues that might confirm it or show your guess was wrong.**Create a better explanation for yourself:**This takes more effort but is incredibly powerful. Often it is a way to develop a more educated guess.

E.g. In the example above about transforming equations, you might have to think and read for quite a while before starting to fill in the gaps and correct the explanation. A transformation seems to create a new equation from an existing one, but it's not an equation with the same roots (i.e. solutions). In fact, it doesn't have the same variable. The variable in the new equation is linked by a rule to the variable in the first. This rule is also called a transformation. (This agrees with knowledge of transformations learned for GCSE, where points in the plane were mapped to new points by a transformation.)

If an equation has roots *α* and *β* then we might name the roots of the new equation *α'* and *β'*. If the variables in the two equations are linked by a rule then the roots are linked by the same logic. For example, suppose the variable in the first equation is *x* and the variable in the second equation is *y*, and they are linked by *y* = 2*x*. This means that each root will be linked by *α'* = 2*α*. You can find the second equation from the first by substituting for the variable in the first equation. In the example, that would involve rearranging the transformation into *y*/2 = *x* and then replacing each *x* by *y*/2 throughout the first equation.

Maths rewards you for going slowly and noticing details. Often this builds memories you will later find you need to understand something. Often it is impossible to understand maths until you have learned the tiny elements that make up the bigger ideas. Instead of straining to understand something, spend a few minutes just noticing details so that your mind is prepared to grasp the bigger things. There is little waffle in mathematical writing so beware of skipping ahead.

E.g. A circle theorem might be explained as follows ‘Inscribed angles subtended by the same arc and in the same segment are equal.’ Unless you can recall fluently the meanings of *inscribed angles*, *subtended by*, *arc*, and *segment* you cannot understand this sentence. You need to have built the memories that let you do that by studying the definitions of those terms and practising using them.

Some material on a maths course is obviously suited to straightforward memory cramming, perhaps with flashcards. It will often be in boxes or a summary page of your textbook. This material includes:

- definitions
- formulae including identities
- theorems
- proofs of important results (if they are examinable)
- common differentials and integrals.

There is an effective technique for memorizing formulae that most people do not know. It involves noticing details systematically, outside-in. This means you start with the overall form of the formula and gradually get into detail. Once you have done that, cover up the formula and try to recall it. Be patient and unpack it mentally just as you did when memorizing. You should be perfect first time, or very nearly. Then test yourself later that day, the next day, and at intervals in future so that the memory does not fade away. If you can think of reasons why the formula is the way it is then you may find it easier to learn.

E.g. Like most formulae, the Pythagorean theorem for right-angled triangles has a diagram to define the variables used in it. You need to spend some time on that first. Then you can start on the formula itself, which is stated as something like h^{2} = a^{2} + b^{2}. Working outside-in, the things to notice, in order, are (1) that this is an equation, (2) that the left hand side has one term, (3) that the right hand side has two terms, (4) that the operation between them is a +, (5) that all the terms are something squared, (6) that the squared thing on the left hand side is h (meaning the length of the hypoteneuse of the triangle), (7) that the first thing squared on the right hand side is a (one of the shorter side lengths), and (8) that the second thing squared on the right hand side is b (the length of the other shorter side). You might also realize that there is nothing special about the two shorter side lengths so they should be interchangeable in the formula, which they are. The length of the hypoteneuse is treated differently because it is special. When you cover the formula and test yourself you will mentally look up the elements as you need them, burrowing into the detail, and say ‘h squared equals a squared plus b squared.’

It does not matter if you will have a formula sheet in the exam. Memorize these items anyway. It's not hard and makes you much more fluent when problem solving.

Since learning maths is mostly learning to solve problems, learning from worked examples of problems solved is crucial. It is one of your best opportunities to learn important things quickly.

Of course, just analysing how someone else has solved a problem does not in itself give you the ability to do the same, any more than analysing how someone hits a tennis serve gives you the ability to serve. However, analysing problems and how they have been solved gives you the information you need to build the skill in yourself, and fragments of knowledge that can be used in that skill. With maths this works so well that you can often go from reading a worked example to immediately and successfully solving a similar problem. What do you need to think for this to happen?

In overview, you are trying to learn to use solution procedures in the appropriate problems. So, you need to identify important characteristics of problems and types of problem, identify solution procedures, and then link them.

E.g. The teacher shows two calculations involving ‘standard form’.

(5.4 × 10^{5}) × (2 × 10^{-2})

= 10.8 × 10^{3}

= 1.08 × 10^{4}

(5.4 × 10^{5}) + (2 × 10^{-2})

= 540,000 + 0.02

= 540,000.02

At first glance you might think these are two different solution procedures for the same kind of problem, but they are not. The first one is only a good choice if the two numbers are multiplied or divided. If they are added or subtracted then this cannot be applied. The second method is the way to deal with standard form numbers that are added or subtracted. It also solves multiplication and division problems but is much, much more work in those cases than the first method.

The first and most important part of the example to study is the problem. You want to learn to spot and understand problems of that type. Read it carefully and learn its key points, just as if you were going to solve the problem yourself. If you want to check that you have paid enough attention to the problem then cover it up after you have studied it and see if you can recall all the details.

Also, try to summarize the elements of the problem and which characterize that particular example. In other words, what type of problem, exactly, are they going to solve?

E.g. A probability problem might feature drawing coloured balls from a bag at random. It gives you some details of the balls but not the total number of them. It tells you the probability of one possible combination of outcomes and then asks you to show that a quadratic equation featuring the total number of balls is true. Most GCSE level students are confused when they first see this type of question but once you have seen it and learned about it you can easily recognize it in future, and recall how to solve it. Looking at a worked example you should notice the elements of the problem: a random drawing problem, coloured items selected at random, without replacement, two stages, an unknown number of balls, a given probability of one type of outcome, and an equation to be derived. You will need to take some time to identify points like this.

To solve maths problems you must learn to apply the right solution procedures in the right problems. That requires you to notice things about the problems that are important. You might not know in advance what is important so just absorb all the mathematical details carefully.

But just noticing the elements of a problem type is sometimes not enough. Sometimes you need to develop a more refined skill where you notice some distinctive characteristic of a problem and that leads you to check for other points before knowing what the problem is and taking in all the important points for that type of problem.

E.g. The Circle Theorems taught at GCSE can only be used if you learn to spot that they are applicable in a problem. You have to learn to look at a geometrical diagram and identify which Circle Theorems you could use. This is much easier once you realize that certain easily-spotted features appear in some of the Theorems: a triangle inside the circle, a diameter, a tangent, a tangent to a point that is also the corner of an inscribed triangle, an angle inside the circle, an angle to the centre of the circle, a radius, and a radius through a chord. Many of these have a distinctive look that is much easier to spot than the full Theorem configuration. Once you spot one of these tell-tale features you can check for other features of each of the Theorems that might be relevant. Having learned the Theorems this way it usually only takes half an hour of practice with suitable geometry questions to become quite fluent at spotting which Theorems you could use.

If a problem features algebra then you will need to study the algebra carefully to identify the type of thing you are looking at. Is it an equation, an identity, a function definition, or something else? If it is an equation, what type of equation is it? Mathematicians like to arrange equations in standard forms to make them easier to recognize and work with. Types of equation covered at GCSE alone include:

- Linear: e.g. 2x + 3 = 9, y = 2x + 9
- Quadratic/parabola: e.g. 2x
^{2}- 3x + 4 = 0, y = 2x^{2}− 3x + 4 - Hyperbolic: e.g. y = 3/x
- Circle: e.g. x
^{2}+ y^{2}= 25 - Exponential: e.g. y = 5
^{x} - Trigonometrical: e.g. 2 sin[x] = 1, y = 3 sin[x]

E.g. To recognize the simplest type of linear equation (e.g. 2x + 3 = 9) you have to realize that its distinctive characteristics include (1) an equals sign, (2) one variable only, (3) the variable is not raised to any power except 1 or −1, and (4) the variable only appears on the top line (as in 2x + 3 = 9) or only appears on the bottom of fractions in the equation (as in 2/x + 3 = 9). With practice you will be able to spot linear equations in their most common forms at a glance.

The next thing to focus on is the solution. You want to notice:

- how the procedure breaks down into steps, each with a simple goal
- how each step is done
- how the solution is written and laid out.

E.g. One GCSE question involves a circle with a tangent to it and the task of finding an equation for the tangent line. The solution to this usually has these steps (1) Find the gradient of the tangent line by (1.1) finding the gradient of the radius to the point where the tangent touches the circle and (1.2) calculating the perpendicular gradient, then (2) work out the equation of the line through the touching point with the ascertained gradient. At the point where this problem is first encountered, most students already know how to do each of the subsidiary steps, so the new aspect is just the idea of using the gradient of the radius.

When you have identified the procedure, cover up the page or look away and run through the procedure in your mind to check you can recall it clearly and strengthen your memories.

It may take some thinking to analyse the solution in this way but it is worth it. You need to develop the ability to plan your solutions at a higher level and then know how to do the details of each step.

The way solutions are written is surprisingly important and needs more attention than most people give it.

Some procedural skills are intricate. It is like programming your mind to follow a computer program. This is typical for arithmetic and algebra. It is like building a watch. If all the cogs, springs, and other tiny pieces are put together perfectly then the watch will tick smoothly and tell the time accurately. But if even one piece of the watch is slightly out of position then the watch will not work at all. With many of the intricate skills of algebra and arithmetic, if you put the skill together just right then you can easily get the right answers, but if there is any defect in your algorithm then you will usually be stuck, confused, tired, and wrong.

E.g. What is the algorithm for multiplying two algebraic terms? Take the example of −3a^{2}b × 2ab^{3}. These should be tackled by doing the sign, then the number, then the variables in alphabetical order, like this: The overall sign will be negative, so write down −. The numbers are 3 × 2, so write down 6 next. The a^{2} × a simplifies to a^{3} so write that down next. Finally, the b × b^{3} simplifies to b^{4} so write that down to finish. The final answer is −6a^{3}b^{4}. Done this way the arithmetic needed is easy and you hold as little as possible in your mind as you work. The correct answer is written down, perfectly formatted, first time.

If you are struggling to develop a smooth algorithm for something like this then get help from someone who can do it well, like a friend, parent, teacher, or tutor. With many of these skills you will need to use them hundreds or even thousands of times in your life.

Having created the problem and its solution in your mind you can now learn how the solution follows from the problem. Think about what elements of the problem suggest the solution or would help you recall that solution method in future at the right time.

E.g. In the problem with a tangent to a circle where you have to find the equation of the tangent, you might remember the link like this: ‘The most important thing about a circle is its radius. I can find the slope of the radius easily and it is perpendicular to the tangent.’

If all this sounds like too much work to be worthwhile, think about the logic of this. You need to know all these details to learn the skill. That is unavoidable. The only question is whether you are are going to do it in a calm, orderly, focused way as described in this article, or muddle through over several disorganized and unfocused attempts during which you might be lucky enough to learn all the details you need.

Learning by doing problems is crucial for turning your analyses of problems and solutions into smoothly working problem solving skills. This is where you transform your observations into skills.

However, it is also a way that you encounter new challenges. Problems in exercises often have variations not previous explained or featured in worked examples and some types of problem are only taught through questions in an exercise. If you struggle to solve a problem it may be because you missed something earlier or because something new has been thrown at you.

Again you need to focus on the problem, studying it carefully to notice all relevant specifics. You will then think about your approach to a solution and try it out. At the end you will check if you are correct. If you are not correct then you will usually try to correct your solution and understand where you went wrong.

If the problem was easy then just move on to something else. If it was harder then spend some extra time learning about the problem, its solution, and how you could have thought of the correct solution given the problem. (Just like with worked examples.) This is vital for rapid progress.

Often, homework involves solving several questions of the same type one after another. You just need to execute the same solution method repeatedly and only need to glance at the problem. This can create the illusion that you have mastered the skill. When you take a test with questions of different types, at random, you will find you do not know how to read, understand, and recognize the problem, so you do not know how to recall or work out a solution method.

When we tackle a maths problem we usually start with *orientation*, which means reading the question, understanding it, learning it, and planning our solution (or at least the first part of it). We then move on to *execution*, where we carry out our plan. Typical homework gives very little practice at orientation but plenty of practice at execution.

Recognize when a homework is too repetitive and compensate as best you can. Here are some suggestions:

- Force yourself to pay attention to the problem every time. Identify its elements and type just as if you didn't already know.
- Do the questions out of order if that will introduce variation. For example, if you have 5 similar problems of one type and 5 of another type, do 3 from each block then go back and do the remaining two from each block. Or do them alternately from the two blocks.
- Do something else entirely for two minutes in between each question.

If you can then choose problems that introduce novelty at the right time for you. If you struggled with a question then pick another similar one so that you experience getting it right easily. If you get one right easily then look ahead in the exercise for a problem that introduces some new twist. During exam practice, use past papers and mixed questions to develop your skills rather than picking questions in similar groups.

It is not always necessary to write out complete solutions to problems to get most of the benefit of study. This can save you a lot of time. In particular, during revision you may find that, with many problems, it is enough to analyse the problem and decide what you would do to solve it. Often it is clear that the solution would be routine and require only skills you are already fluent with. Take care to look out for subtle traps and, if you are not sure you, write the solution out fully. Also write out solutions fully if you need to build greater speed and reliability.

Part of orientation on a problem is spotting potential complexities and reasons why you might make an error.

This is extremely important. Often we solve problems by choosing tactics that will probably work. We don't know for sure but we have learned that, in particular types of situation, some tactics are often a good choice. The rules of thumb that allow us to do this are known as problem solving heuristics. Looking at worked examples and doing problems are opportunities to learn problem solving heuristics.

For example, they may help you decide whether to multiply out some brackets or factorize, whether to combine fractions or split them, or which substitutions to make.

E.g. At A level, proving trigonometrical identities is depressingly difficult for many students because they do not have good problem solving heuristics. There are too many potential problems to learn exactly what to do with each one. Once those same students have the heuristics they need they find that the problems become simple. They just make the moves that usually work and usually they work. The seemingly impossible becomes easy.

Slips are what most people call ‘silly mistakes’. They are more likely in particular situations. Look at the slips you make, try to work out why you made them, and work out ways to reduce them. Sometimes that just means taking more care when you know you are about to do something you often get wrong. My personal list of danger situations looks like this:

- Orientation
- Missing or misreading details when reading the question. (Read patiently. Notice each point, checking against any diagram. Make some deductions from points about likely questions, solution processes, other part of the question, and potential mistakes/complications needing special attention.)
- Workings
- Copying a formula from the question incorrectly, which is more likely because the formula is still unfamiliar and there is a bigger distance between eye fixations. (Copy with extra care and check back immediately afterwards.)
- Mistakes when writing out a formula that is complex or not used for a while. (Mentally retrieve the entire formula, clearly, then use it i.e. write it down substituting from the question.)
- Mistakes in mental maths and algebra. (Recheck when tired, zoned out for a moment, or working harder than usual before writing down a result.)
- Other inexplicable slips with familiar skills. (Go at a moderate pace and take extra care when impatient, thinking it is taking too long, tired, or otherwise distracted. If in doubt, reperform and check.)
- Mistakes misreading my own handwriting. (Write characters very clearly, especially decimal points and anything that is very small.)
- Missing out some items e.g. a force in a mechanics question. (Go around systematically to get them all.)
- Common slips e.g. confusing N with kg, sin vs cos, speed vs velocity.
- Getting lost – not a common issue for me. (Write a clear explanation of your approach, signposting methods, dividing into cases with subheadings, etc.)
- Final answers
- Not answering the specific question set e.g. giving velocity instead of speed, or an elapsed time instead of clock time. (After getting to an answer recheck exactly what was asked for and give it.)

Maths classes can be a major cause of wasted time and negative feelings. If you sit in class with the teacher talking but you not understanding then you are wasting your time and not enjoying it. Teachers often say that if you don't understand then you should put your hand up and ask, but it's not practical to put your hand up often and some teachers make things worse when they talk more.

This problem can be greatly reduced by reading about the topic before the class (e.g. the evening before or during a lull in a previous class). If you feel you have fallen behind in maths and there are topics that you didn't get at all then your best first move is probably to get ahead and on top of new topics coming up. That will save the time you need to go back and sort out the problem topics.

You will need a suitable source to read, such as a book or web page. If your source is excellent then you may be able to master the material before your teacher covers it in class. The class will still be useful to you as a revision. You can listen and watch for mistakes by the teacher.

If your source is not so good or you do not have time for full self-study then you can lay valuable foundations by noticing specifics such as:

- notation (what letters and other symbols are used, how they are written, what they usually stand for)
- terminology (that terms are important, their spelling, their meaning)
- typical features of diagrams
- standard formulae
- types of problem faced in the topic
- techniques used in worked examples.

The more of these you have acquired by noticing specifics the more likely it is that your teacher will make sense to you in class. Your brain has less work to do while they talk, giving you more chance of understanding and learning something more.

Choose self-study sources carefully. Do not use videos. They can take a long time to watch, you tend to watch passively, and you have to learn at the pace of the video. Instead, choose a good book (potentially on your phone/tablet/laptop) or website. Then you can slow down for new details and skip past material you already know.

You need sources that are precise and correct but not too advanced. Fun is irrelevant. It may help to ask an expert which they think is the most precise and correct source for your level of knowledge.

I have not done a systematic study of sources but my personal views on sources I have used are:

- CGP GCSE maths revision books: Very useful but not always as precise as I would like.
- BBC Bitesize website for GCSE maths: Not very precise. A poor choice for self-study.
- Pearson A level maths books: Usually great for pure maths and mechanics and a good option for statistics.

Rely on written explanations carefully prepared by experts (e.g. a book or web page), not your notes taken while a teacher talked and wrote on a board. Even a few mistakes or omissions by you or your teacher could make those notes worse than useless. You might note useful tips that are not in your main source, but that should be all.

This has the added advantage that when you are in class your mind is not working on note taking and can focus on understanding, checking, and building memories from what is being presented. Think instead of writing.

If you are trying to stay slightly ahead on a topic then of course you are less likely to get left behind. However, you may still notice a deep error in an explanation, or slip up and get stuck, or fall behind because of illness. In that case, try to resolve your concerns or catch up as soon as you can. The knowledge you are missing might be needed for something else coming up soon.

Try thinking about it as soon as you get the chance, self-study, a teacher, a tutor, or someone else competent and willing to help you. Do not assume that your teacher is the best resource. They might be, but they might also be the cause of your problems. If your teacher is disorganized and confusing then look for someone else to help you when you are stuck.

Often, the issue is just a frustrating puzzle or worry. Maybe something the teacher said or wrote that did not make sense. Try to think it through without delay. Perhaps you can work something out yourself. You might be able to do this in a lull in the classroom, during a lunch break, or travelling home at the end of the day. Or you may have to spend a few minutes on it in the evening instead of relaxing with the internet. Just from doing this, very smart students can sometimes reach an understanding that is better than they have been taught.

More extra work?! Yes, but it will save you time and frustration overall. The biggest difference between people who struggle with maths and people who just seem to find it easy is memory. People who struggle usually cannot remember maths they have previously made the effort to learn and understand. When the teacher returns to that topic, either to build on it or revise it, the struggling students often do not even remember having studied it and they struggle to take in the new developments. The students doing well recognize the topic and know everything they have already been taught about it. They are ready to move on. Similarly, whenever they do a class test or exam the students who can remember what they have been taught benefit from answering nearly all the questions and getting most of them right. The students who have forgotten just feel miserable during the test as they struggle with the few questions they can attempt. Most of that time has been wasted and they will have to spend extra time going back over questions they could not do.

The best way to keep memories fresh and available, so that you can use them fluently, is to go over the topics frequently and systematically all year round (except summer holidays perhaps). You don't need to spend much time on each revision. Just a few minutes per topic may be all you need. The time needed for revisions will also reduce as your memories consolidate.

There are two ways to plan revision. One is to make a master plan with a calendar or other planning system. The other is to choose the next date for revision each time you revise a topic. I quite like the second approach because I can use my sense of how well I know the topic to decide how long I can leave it. Typically, the more times you have revised something the longer you can leave it before you revise it again.

Build some kind of revision folder to keep together your sources and plans for revision. You might highlight key material to help you focus.

Try to recall material rather than just re-reading it. You can often run through the points in your mind and then look at the page to see if you missed anything or made any mistakes. Usually it is not worth doing questions, but you might if there is a particularly complicated procedure you want to revise.

As you approach your exams the time it takes you to revise all your maths knowledge will initially increase because teachers are giving you more knowledge overall. However, it should then start to decrease as your memories strengthen, you need to revise topics less often, and you get more selective with the specifics you revise in each topic. By the day before an exam you may be able to refresh your memories of the whole course in just a few hours. That is what you should aim for.

On the road to the exams that really matter you will take many tests at school. The last thing you want is to sit in those tests feeling stuck and brooding about the time you are going to have to spend later going through questions you could not do.

The easiest, simplest way to minimize this problem is to refresh your memories of the relevant topics the day before or earlier on the day of the test. In a test you need your memories to be fluent and easy. If you have to reach for things you need then it will be much harder to solve the problems. Simply revising before the test will give you better scores and save you from a miserable time in the test.

Surely everyone knows this? Prior to exams your teachers will almost certainly be throwing past papers at you constantly. They will do their best to get all the teaching finished with weeks or even months to spare so that they can push you to do past papers. (It helps to compensate for the fact that their earlier teaching often lacks enough attention to the orientation phase of problem solving.)

In fact, the biggest problem with this method is that you can run out of past papers to do. If you realize that this might happen then save at least some past papers for the last few weeks before your exams. Use other materials instead that have similar properties. They should be similar questions to your upcoming exam, equally difficult, varied, and in an unpredictable order. Materials that may be suitable include:

- Mixed exercises at the end of chapters in your textbook.
- Revision questions in your textbook.
- Past papers for the same subject and level but from different exam boards.
- Papers developed specifically to mimic real past papers, either by the exam board or by teachers, tutors, or publishers.

Some students will know that they have no realistic hope of learning to answer some of the questions in their exam. If you are doing GCSE Higher and you feel you will be lucky to get a level 4 or 5 then you will only need to tackle about half the paper and gain a third of the marks, if that. Questions in exams usually start easy and get harder. So, for you it makes sense to practice only questions in the first half of past papers. Anything beyond that is likely to be wasted time for you.

Do not try to work faster in an effort to make yourself faster at maths or to simulate exam conditions. Students who finish exams with plenty of time to spare, rest for a few minutes, then check their answers, do not hurry. The secret of their speed is that they go at their natural pace, know what to do, and make no mistakes in their working. If they were to rush then they would make slips or misread questions and that would lead to delays.

You will become faster simply through expanding and refining your skills.

Part of doing very well in maths exams is learning to answer questions with ease. You should aim to refine your skills so that you save energy and avoid fatigue that might lead to errors, and so to lost time and lost marks. This means writing maths very clearly so that you can understand your own work if you get distracted and come back to it. Good writing also keeps your mind on rails, rolling towards the solution. Learn to do just the right amount in your head between each line of algebra. Too much and you will make mistakes and get tired from the memory work needed. Too little and you will waste time writing lines you did not need.

As already explained, the best approach to speed in maths exams is to know what to do and make no mistakes. That means going at a comfortable pace and taking care. Do not try to think or write faster than usual because you will make mistakes and that will cost you a lot of time.

It is hard to work with focus for more than 45 minutes without a break, even if you find easy maths relaxing. After a tough question, allow yourself 45 seconds to just chill with your mind blank. Close your eyes. Relax. Breathe. Think of nothing. Then get back to it, slightly refreshed.

Occasionally you might be working on a tough question and suddenly realize you are lost. Your mind is blank. It is like your mind has been overwhelmed with information and confusion. If that happens, stop working for a moment, close your eyes, breathe, and relax for a few seconds. Then return to the question *from the start*. Do not try to drop back into the question where you left off. If you go back to the start then you can reload the information you need. Doing it the second time will be much easier and your mind will cope much more easily.

Now that you are in the exam and have no more study to do, there is nothing that can improve your results other than doing the problems on the paper in front of you. That's it. It cannot help you to think about anything else. If you find yourself wondering why you didn't work harder, or if you have the best exam technique, or if you should compain about your teacher, or the exam, or anything else except the problem you are working on then stop that immediately. Just do the maths. That must be your best strategy now.

Be wary of skipping questions or reading through quickly to find questions you think you can do. Usually it is best to be well prepared then answer the questions in the order they appear in the paper. Only abandon a question and move on if you are sure you cannot answer it (e.g. because the topic is completely new to you as a result of a teaching or revision mistake). If you start skimming through the paper looking for questions you might be able to answer then you have lost focus on the one thing that can help you now: solving the problems. Questions usually get harder as the exam goes on so skipping ahead usually makes things worse, not better. I am not saying it is never worth doing. I am saying there is a danger of getting distracted and gaining nothing.

If the paper is surprisingly tough, perhaps with some shockingly difficult questions near the start, be happy. Why? Most exam marking involves some adjustments because it is so hard to set papers that are exactly the same difficulty each time. In effect, you are trying to do better than other students. If the paper is harder then most students will be hit by that too. Hopefully, some will not realize this and will be distracted or discouraged. Some may give up, in tears. This is good for you because their fragile thinking brings the grade boundaries down, making it easier for you to get a higher grade. Just keep trying to solve the problems using all the skill you have and do not be distracted.

This article is not long but the advice is concentrated. There are many important points here and you will not be able to remember them all. If you think there might be something you can use then I suggest you make a definite plan to try it out. Come back to the article from time to time to get more ideas and pick up details you missed the first time.

I hope you see some improvements in your maths learning soon.

You could also try applying some of these learning techniques to other activities. For example:

Noticing specifics also works well will science subjects, using software (including games and coding), playing music, and learning the rules of sports.

Focusing on the elements of problem types and on learning from examples of problems solved works well with science subjects, computer programming, practical skills in technology, and tactics in racket sports and team sports.

Frequent revision works well with lots of things, but especially learning languages.

Remember that the learning skills you develop using mathematics are much more likely to be useful elsewhere if you know what those skills are and use them deliberately.

© 2022 Matthew Leitch of The Ridgeway Expertise Company Ltd.

Company: The Ridgeway Expertise Company Ltd, registered in England, no. 04931400.

Registered office: 29 Ridgeway, KT19 8LD, United Kingdom.