Compare the following:
“First you add 4 and 20 and you get 24. Then you halve it and you get 12. Then you times by 8 and you get 96. Then put centimetres squared.”
“To get the area of a trapezium I’ll use . a and b are the parallel lengths, so I will use 4 and 20 as I can tell they are parallel because of the little arrows. I will use 8 for h as it is perpendicular to 4 and 20. The lengths are in centimetres, so the area will be in centimetre squares.”
If you are a maths teacher, you might have noticed they are both ‘correct’ answers to the same question: how do you get the area of this trapezium?
One of the ways we have been trying to increase the level of rigour in maths is to shift the focus of our questions and the focus of pupils’ answers. In particular, we aim to increase the ‘transfer’ from specific examples in lesson to being able to see how an idea, concept or example applies to other cases, or how a strategy can be deployed in novel situations.
There are four main strategies:
- Explanations should mostly use general steps and reasoning, rather than specific values
- Clamp down on pronouns
- Teach the pupils what they need to know to be able to answer the question “How did you know that…?” and “How did you know to…?”
- Help them to build up expectations
- General Terms, not Specific Values
A lot of the explanations we give to pupils focus too much on the specific example and its values (e.g. 3cm, or y = 6) and not on the general underlying structure. Consider the difference between:
“To get the perimeter add 6 and 7 and 6 and 7 and don’t forget the units.”
“To get the perimeter I need to add the lengths around the outside. Some of them are blank: I will fill them in. I know that opposite lengths in a rectangle are equal; that is why these lengths are also 6 and 7. I will now add all the lengths. The lengths are measured in millimetres, so I will give the answer in millimetres.”
The second is longer, but focuses more on the general strategy of perimeter of any shape (find all the external lengths, calculate their sum, and use the information in the question to determine the units). Not only is the first explanation restricted to being an explanation and strategy for perimeter of rectangles, it is restricted to the specific rectangle in the question.
Here are some guiding principles:
|What?||Why?||Instead of this…||Try this….|
|1||Restate the goal of the steps||This means that the subsequent explanation is linked to a specific outcome, rather than feeling like an arbitrary collection of ‘steps.’||“To find the equation of the line, we will first…”||“So, first you need to…”|
|2||Make your explanation as generalised as possible, focusing on the names of parts||This approach increases ‘transfer’, and allows them to learn a strategy or process as a general strategy, rather than as a single response to a single question.||“subtract 53 from 360, so x is 307”||“Use the fact that angles at a point sum to 360 to find x”|
|3||Used technically accurate terminology||This will, in the medium-term, clarify their thinking and reduce the chance of them confusing ideas and concepts. It also allows you to find woolliness in their thinking or where they have ‘folksy’ understanding.||“missing number” or “the letter”||“move around to get y on its own”||“the unknown [value]”||“rearrange to isolate y”|
|4||Be specific about what you expect to see||Sometimes we are tempted to give simplified explanations and steps. This may seem less daunting for pupils to hear, but lowers their chances of eventual success.||[to find gradient from a graph] “draw a triangle”||“draw a right-angled triangle”|
|5||Make the implicit explicit, and only accept ‘complete’ explanations||Sometimes we hope the most subtle ideas will become obvious, or gloss over aspects of a concept …and then are baffled when pupils lack flexibility. If there is underlying or implicit knowledge…share it with them!||“because corresponding angles are equal”||“because corresponding angles in parallel lines are equal”|
|6||Have the same standards for their answers and you have for your explanations||If they can’t say back to you the accurate and ‘general’ form of a strategy or approach, they have not understood it or cannot remember it. This is crucial feedback for you! It also allows them to rehearse the general form, which is valuable for their retention, and puts pressure on them to attend to the most ‘high level’ version of what you say, not just the question they are looking at.||“I need to do 50 divide by pi and then square root it”||“To find the radius of this circle, I will use the formula and substitute the values I know. I know that A is 50, and pi is a value, so there is only one unknown. I can isolate r to find the radius.”|
Tips for implementation
- You need to change before the pupils can change! Explain to them what is changing in your modelling, and why, and what they should focus on.
- Ask smaller questions to check they can repeat back the steps. This is a mimicry stage, of course, but it is relatively easy and low-stakes, and allows them to rehearse using more mathematical ways of speaking. It also shifts their expectations of what they will hear from each other and what ‘sounds about right.’
- Use accessible ways to help them understand what to cut or keep in their answers and explanations, such as “Explain how to do it, without using any of the numbers in the question” or “Tell us how you’ll tackle it, making sure you use the words gradient and y-intercept and coefficient” or “I will be so impressed by anyone who can explain how to do ANY question where you have to find perimeter, and not just talk about this question.”
- Let them rehearse with their partners.
- If something they say is slightly off, tell them how to improve and get them to say it again (make sure you do this in a kind and supportive way!).
- Judge the room: sometimes it helps to use the easier and less rigorous approach if their confidence or buy-in is low and you need some ‘quick wins.’ If you do so, that is fine! What matters is that, in those situations, you have understood that the success you are aiming for is with their buy-in and confidence, and not necessarily with their learning.
Clamp down on pronouns
Watch out for when you – and more often the pupils – use pronouns to disguise uncertainty and wishy-washy thinking. This is usually ‘it’, ‘they’ and ‘them.’
Change “You multiply them” to “Multiply the base and the perpendicular height”
Change “You find its height” to “Find the slant height of the parallelogram”
Change “You add them and divide by how many there are” to “Add the values and divide by the number of values”
It has been fascinating how often a child has given a coherent-sounding explanation or answer but then faltered when I probed what they meant by ‘it’ or ‘they.’ In many of these cases, I think even the pupil didn’t realise the woolliness of their own thinking, possibly reflecting that I had used ‘it’ and ‘they’ so much that they (the pupil J ) didn’t notice it wasn’t making sense to them.
Teach the answers to the question “How did you know that…?”
A lot of the questions in maths classrooms have a 50/50 chance of being right:
- “Are these lines parallel?”
- “What is the value of x?” (if they need to choose between ‘subtract from 180’ or ‘subtract from 360’)
- “Should I times or divide?”
…which means we are probably getting a lot of false positives, especially considering that our body language often conspires to give away the ‘correct’ answer.
“Are these angles corresponding [shakes head slightly] or alternate [happy inflection]?”
A more powerful question – and pupils must be built up to expect this – is to follow with:
- “How did you know they were parallel?”
- “How did you decide to use 6cm as the height?”
- “Why did I subtract from 180, instead of from 360?”
If a pupil can’t answer that follow-up, then there is limited hope that they will transfer what they are learning in that question to a similar question, and even less hope they will transfer it to a superficially novel problem.
There is a big caveat:
If the ‘why’ and the ‘how do you know’ and ‘how to decide’ is not a core part of your teaching and explanation, it can’t possibly be part of the pupils’ responses!
A first step is to plan the answers to these questions when preparing your lesson:
- “How did I know to add them all up first?”
- “How did I know to rearrange?”
- “Why did I set it equal to 540?”
- “How did I know these angles would be equal? How could I tell they were corresponding? How did I remember that the F-shape gives us corresponding angles?”
Asking this has been a big shift in our practice, and it has made a huge change to the children’s confidence once they expected to be asked such questions…and to be able to answer them. Now that they expect us to ask those questions, they know to attend carefully to that part of the explanation. Even more gratifying is that they are asking questions such as “How did you know to multiply those two?” or “How did you know it would be a quadratic?” and are helpfully filling the gaps in my teaching!
Help them to build up expectations
This may be the most obvious, and I have seen many great teachers do this: make sure the pupils have some expectations around the answer before they get started! Similarly, teach them to use words in the question to begin visualising or imagining things.
For example, my Y10s have been practising the cosine rule and sine rule. One of the first things we do is build an expectation, such as
“x is opposite the biggest angle, so it should be longer than the other two lengths”
“one of the angles is 100, so my answer has to be less than 80”
“the length increased, so my angle should increase too”
“I’m looking for an angle, so I will probably have to use inverse sine or inverse cosine”
Another example: today my Y11s were learning about how solve simultaneous quadratic equations. As we looked at the pair x2 + y2 = 25 and y = 2x+1 we first made little sketches of the two graphs to set up an expectation of where they might cross, and how many points of intersection we would expect. They then set off on the question expecting a solution pair where both values were negative, and one where both values were positive. We also discussed that we expected the solutions to be two pairs in the form (x, y).
A final example: when my classes are completing transformations, they have learned to put a little arrow pointing to the ‘rough’ area where they expect the shape to end up. This helps them when they become bogged down in completing a translation or rotation, and helps them judge how well they do at expecting a particular result.
A caveat: learning ‘what to expect’ and discussing this is usually an overwhelming and slightly awkward first step. I usually help the children feel familiar with the process and then move to expectations as, by then, they have a few examples in their mind to test their expectations.
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