# What Matters (Mathematically) the Most

Sometimes when maths teachers visit I feel frustrated that I can’t make suggestions to them of what they could change (or keep) to improve how their pupils do in maths, as the holistic approach of the school can make it feel as everything is inextricably tied together. However, there is one strategy that has made a big difference for us that can, crucially, be replicated in other settings:

• Identify ‘high leverage’ topics
• Teach them as early as possible in KS3
• Give them lots of time when they are initially taught
• Interleave them into every subsequent topic, whenever possible
• Frequently revisit and retest them as stand-alone topics

1. Identify high-leverage topics

These are the topics that:

• Can come up anywhere (e.g. fractions in a perimeter question)
• Can derail a lesson if they’re not in place (e.g. dividing by 10)
• Tend to terrify all but the strongest pupils (e.g. calculating -629 + 433)
• Are counter-intuitive (e.g. adding fractions)
• Are prone to ‘System 1’ (i.e. unthinking) errors (e.g. squaring, order of operations)
• Are the typical building-blocks we don’t have time for in KS5 and are the bane of A-level teachers’ lives (…..all of algebra?)
• Are easy ways to gain (and lose) simple, predictable marks in exams (e.g. rounding)
• Are easily confused: what Bruno Reddy describes as ‘minimally different topics’ (e.g. three measures of centrality (mode/median/mean), or perimeter and area)

The ones we’ve identified are:

• Automaticity with simple calculations (adding and subtracting small values, halving and doubling small values, times tables (including division), adding 10 to any value)
• Some aspects of place value (multiplying and dividing by powers of 10, in particular)
• Efficient and reliable written methods for the four operations, including with decimals
• Directed numbers (particularly the four operations)
• Fractions (pretty much everything about them: simplifying, forming equivalents, comparing, ordering, the four operations…)
• Simplifying expressions, especially when the variables look scary
• Rearranging expressions, especially when there are negative coefficients
• Solving equations (linear)
• Rearranging equations (both linear and polynomials)
• Substitution
• Order of operations, particularly the importance of leaving addition and subtraction to the end of the calculation
• Rounding (including to significant figures)
• Area and perimeter (simple cases, focus on not muddling procedures)
• The three averages (simple cases, as above)
• Simple proportional reasoning (e.g. in an equation, being able to multiply both sides by 5, or doing opposites to both parts in a product (e.g. 12.5 x 16 = 25 x 8), etc. Obvious cases are finding the ‘best value’ product when differently-sized packets have different prices, but it also comes into percentages of amounts, etc).
• ‘part + part = whole’ (e.g. in an L-shape, the two lengths on the right-hand side must have the same length as the height on the left-hand side), typically represented on blank number lines
• Key vocabulary (integer, associative, inverse, eliminate, variable, etc)
• Key number facts and relationships (the first 10 primes, 15 squares, 10 cubes, fraction-decimal conversions, the effect of dividing by 0.5 or 0.1, etc)

1. Teach them as early as possible in KS3

That list is a lot! It takes up almost all of Y7 and Y8. Even then, some are only covered to the extent that they allow us to continuously drill, quiz and probe how well they differentiate ideas. For example, our Y8s frequently do questions with perimeter and area, but haven’t been taught anything beyond area of rectangles and perimeter of ‘any shape’. Similarly, our Y8s and Y9s are pretty good at applying the three averages as procedures, but have few insights (yet) into what they measure. That will come in Year 10 (I am convinced it’s hard to meaningfully understand statistics before then, so am hoping the gamble of separating procedures and understanding will pay off).

We put directed numbers as early as possible in Year 7, once their written methods with decimals are sound. This is followed by ‘all of fractions’ and then…Year 7 is almost over! Angles are studied to a limited extent, but as a vehicle for practising written methods (e.g. practising subtracting by finding the missing angle on a straight line).

Year 8 could probably be summarised as ‘algebra, all year.’

In Year 9 we go a bit wild and do percentages and then shape until the end of the year (angles, with understanding what an angle actually is, then Pythagoras, trigonometry, transformations, etc).

1. Give them lots of time when they are initially taught

Teach every variation that you can think of

Take the simplest aspect of solving equations. 4a=12, or a+5=11 might spring to mind. That’s barely scratching the surface. These cases are relatively obvious to all but the weakest quartile. The examples below need to be shown to pupils.

2a + 2 = 12 (simplify first)

12 = a – 10 (unknown on the RHS)

7 = 3a (result is a fraction)

5.7 = a + 7.8 (decimals AND a negative solution)

3/4 + a = 9/10 (fractions requiring LCD)

10 – a = 20 (answer is a ‘surprise negative)

Failure to teach these explicitly disadvantages the vast majority of pupils. As teachers we frequently make the mistake of showing our pupils the classic examples of a case and thus never increasing the flexibility of their thinking. Inevitably, they freeze once the problem doesn’t take the form they’ve seen – it looks like a totally different problem!

Thinking of every variation is time-consuming, but also enjoyable. It forms the basis of fruitful and professional discussion with colleagues (and can be done via Twitter, I’ve discovered) and can be aided by trawling through very old textbooks and exams (enjoyable, if dusty, work!).

Plan for every misconception, and pre-empt problems by explicitly teaching about them

Many of us wait until the pupils say “but….why?” when we show them the procedure to add fractions. Inevitably, the explanation is messy and confusing, even for us. Sometimes we plan to ‘explain’ it by using diagrams. This might show it working, but still doesn’t give them the language to explain why one way makes sense and another way doesn’t. It should be planned for in advance, agreed with colleagues and scripted (to ensure clarity and economy of language). Some of my best explanations were in my PGCE year, when I used to rehearse the rationale for things; it should never have been dropped!

Include lessons that focus on addressing cases where pupils muddle concepts and are likely to make errors

In the past, I have finished each year thinking “I really should have done a lesson to get [class] to practise the difference between finding a percentage of an amount and reverse percentages” and then not done it. Because it is hard, and it is scary. The Y9 teachers are mentally preparing themselves for it: we’ve planned our percentages lessons in anticipation teaching what will undoubtedly be one of the toughest lessons (or 2 lessons…or 3 lessons…) of the year. It is really, really hard to help pupils see the difference between these two types of questions. This is why we have to force ourselves to do it; praying they’ll see the difference ‘on the day’ – what I’ve always done in the past – is to abdicate responsibility at the point we know they need us most.

There are many topics like this: order of operations questions with a negative result (e.g. 3 – 10 x 2) or questions that combine surface area and volume.

Do lots of drills, both on decision-making and on procedures

I was slowly losing my mind as I taught some of the weakest pupils to calculate with directed numbers. Showing them quick work-arounds for some questions (e.g. -4-5 = -9 can be quickly calculated because it is a total distance, or -3 + 7 can be quickly calculated by rewriting as +7 – 3 = 4) was causing as many problems as it solved as they were confusing when to apply them, even though they understood the ideas in principle. A teacher in the department asked me what drills I’d used to help them spot when to use them. It was a blinding flash of the bloody obvious: I hadn’t done any. I’d focused on calculation drills – completing those calculations – but not on decision-making drills – identifying the questions where those work-arounds are applicable. Drills aren’t mindless if they’re helping pupils to focus in on critical decisions, or improving pupils’ fluency and accuracy.

Do lots of extended and applied questions where they have to make decisions

Once they know what they’re doing with the basics, make the questions TOUGH! It’s no good learning to multiply two negatives if they can’t extrapolate to four negatives. It’s easy to simplify the signs in -2 x -3; it can quickly become mindless and ineffective to practise. A better test of application is -2 x -3 x 4 x 0.5 ÷ -10. Once the pupil has to think about other things – such as how to multiply by 0.5, or divide an integer by 10 – we can see how well they really recall and apply the rules of negatives.

1. Interleave them into every subsequent topic, wherever possible

Answering this question requires much more thought than “Expand 7(2a-3)”

“Form an expression for the perimeter of a regular heptagon with a side length of 2a-3.”

And

“Calculate the area and perimeter of a square with a side length of .”

Requires much more thought, recall and organisation than simply presenting the underlying calculations.

Simplifying expressions is relatively easy. Simplifying this expression is not:

7/8ab + 2/3a -1/2b + a/2 -ab

Interleaving in this manner, in every topic, communicates to pupils that everything they have been taught is relevant and important. They can’t decide “I’m bad at fractions, but that’s ok.” They’ll realise they have little choice but to improve and stay good. Similarly, their teacher will face the same reality: they can’t afford to give up on the topics that are most challenging to teach.

I have this checklist in my mind when I design questions for any lesson:

Could we include….

• decimals?
• fractions?
• directed numbers?
• the order of operations?
• perimeter, area or angles?
• averages?
• indices?
• more challenging language?
• Technical vocabulary (write ‘variable’ or ‘unknown’ instead of ‘letter’)
• Technical syntax (“A number is picked such that…” “Demonstrate that, for all integers…”)
• An opportunity to include some ‘scary’ generalist words (writing ‘nasturtium’ instead of ‘flower’ and ‘yacht’ instead of ‘boat’ is another way to bring valuable difficulty to routine practice and teaches them to be comfortable with not knowing every noun they see in questions)?

1. Frequently revisit and retest them as stand-alone topics

A third of our weekly quizzes is given over to explicitly testing pupils on these high-leverage topics. This gives us valuable information about their retention and growing misconceptions, and forces us (and our pupils) to give over regular revision time to them.

All of this takes a lot of planning up-front, unsurprisingly. It also demands a pretty rigorous mindset when planning. Thankfully, it can be introduced gradually and has a snowball effect as pupils become accustomed to regularly revising tough topics. Additionally, this strategy can be applied in almost any setting, regardless of your school’s meta-approach to teaching, learning and behaviour, so can be operated without too much interference.

Think this sounds interesting? Come visit! We love having guests. It challenges our thinking and it boosts the pupils’ confidence to have people come in to see them.

Think it sounds wonderful? Apply to join our team of enthusiastic maths nerds! We are advertising for a maths teacher, starting in September (or earlier, for the right candidate). Closing SOON: https://www.tes.com/jobs/vacancy/maths-teacher-brent-440947

For those who spotted it, the title is ripped off from one of those poems that we fall in love with aged 15 and meet again 15 years later…

Filed under curriculum design, lesson design, pedagogy

### 4 responses to “What Matters (Mathematically) the Most”

1. Mike Patrick

If only my undergrad business students, who have GCSE Maths, could actually do all this then I could spend more time addressing business problems, rather than revising basic maths. Also we wouldn’t have to graduate students, who employers complain, lack the numeracy skills they require.

2. Gary

A lot of this is applicable to Primary as well. Obviously, the concepts are different but approach is similar.

3. Pingback: Starting at Michaela | mathagogy