I have been floating the idea of treating key mathematics ideas as handrails for teachers and students. Recently several positive comments about this perspective have floated back to me, so I am sharing the idea with you.
In mathematics, teaching key ideas could be handrails. Handrails can be held, used as guides or supports, or gripped onto by teachers and students across all ages so that, as they are hanging onto the same handrail, they can communicate meaningfully across phases and between each other. We can see both ways from our own position, so we can sense something about how the same basic idea is manifested at different levels of mathematics, and how our own part of it, as teacher or learner, relates to its development towards higher levels of mathematics and arises from lower levels. We can think about how gaps in, or limitations of, earlier experience affect the learners we teach. It is an aspect of our teacher-knowledge to understand the whole handrail to some extent, and to think about how smooth the handrail can be, and where it might be fractured.
In some countries secondary teachers have to follow the early years and primary training courses in mathematics pedagogy before they follow a secondary level course, so that they understand the foundations of their students’ learning. To us in UK that may sounds excessive but on the other hand it is not helpful for secondary teachers to think of primary as where all the basics are taught ready for secondary, without some understanding of the complexities of learning. I often hear teachers saying ‘I like them to have done P, Q, and R so that I can teach them X, Y or Z’ without comprehension of what it takes to learn P, Q. and R. I hear this most often from secondary and tertiary teachers but perhaps KS2 teachers say that as well.
After a lot of thought, reading and observation about this, and about learners’ difficulties with secondary mathematics, I think there are a few key ideas which provide handrails for teachers to talk with each other about what could make teaching more coherent across years and phases, and lead to better understanding of each other’s work.
Several groups of educators worldwide are thinking about this too, but their perspective is on learning trajectories, i.e. on improving coherence in learning mathematics, whereas mine is about ‘what can teachers talk to each other about?’ We are looking at the same threads, but for different purposes.
The number handrail
At every level of learning, understanding number comes from connecting representations, images, actions, words, quantities and the ways they are combined and communicated. The big shifts come when moving between
- counting discrete objects and whole numbers to understanding number as continuous,
- thinking of number as quantity to understanding its abstract qualities such as negative number,
- thinking of decimals and fractions as different ways to represent the same numbers to making distinctions between rationals and irrationals,
- number as unidimensional to bi-dimensional and imaginary, and more shifts beyond school.
Teachers can all appreciate that there are shifts in the meaning of number throughout school, even if they do not teach those shifts themselves, and understand that some representations, images, actions and words fit with one meaning but not another. Teachers can therefore anticipate the variety of limitations and extensions of understanding number that might be present in the class they teach. It is still the case, however, that with any group of people the most likely response to ‘give me a number’ seems to be 7!
The division handrail
I have talked and written about the division handrail at length, and it was this that gave me the handrail idea because most shifts and extensions of meaning take place before the end of KS3, so involve conversations around transition to secondary, and also cause problems if the handrail is fractured. We build a handrail by organising a card sort, left to right, according to progression of ideas that mostly contribute to division in some way. Some parts of the progression end up following the national curriculum but others do not, and some of the ideas seem applicable the whole way along the progression from understanding multiplication to ratio and proportion. Warning: these cards are not printed in any sensible order – that is up to you. You may need to reject some cards and create new ones.
|Counting in twos, threes, etc. from zero||Counting in twos, threes, etc. from a given starting number|
|Counting in tens||Counting on repeatedly|
|Counting in hundreds||Repeated addition|
|Sequences that go up in equal steps||Sequences that go down in equal steps|
|‘Gazzupins’||Counting the number of steps|
|Enumerating completed arrays using repeated counting||Recursive description of additive sequences, i.e. how to get the next term from the previous term|
|Multiplication of single digit whole numbers||Multiplication facts written several ways|
|Multiplication facts known as times tables||Undoing multiplication facts (backward tables)|
|Multiplying by ten||Multiplying by a hundred|
|Enumerating completed arrays using multiplication||Area of rectangles with squared paper|
|Enumerating completed arrays using counting||Enumerating incomplete arrays using multiplication|
|Area of composite rectilinear shapes||Multiplying n to get a sequence value|
|Double||Double and double again …|
|How many parts?||How much is in each part?|
|Unit fractions of stuff||Unit fractions of cake|
|Unit fractions of length||Unit fractions of number|
|Non-unit fractions <1 of stuff||Non-unit fractions <1 of cake|
|Non-unit fractions <1 of length||Non-unit fractions <1 of number|
|Half of||Half of half of …|
|Dividing whole numbers that go exactly from times tables||Dividing whole numbers that do not go exactly but are within times tables scope|
|Fractions of length||Multiplying and dividing lengths|
|Fractions as division||Division as fractions|
|How many … go into …?||How much more than?|
|How multiplying the unit has the same effect as multiplying the whole measurement||Changing units|
|Grouping non-countable stuff||Grouping countable objects|
|Sharing countable stuff||Sharing non-countable stuff|
|Measuring growth||Shrinking and growing as multiplication / division|
|Comparing lengths||Comparing liquids (water, lentils)|
|Per cent||Rates of change|
|Speed as distance / time||Growth over time|
|Unequal sharing of countable objects||Fractions with the same denominator that add up to one|
|So much per so much||Conversion rules and graphs|
|‘This corresponds to that’ problems|
The full size cards are available at www.pmtheta.com/publications.html#multiplicative reasoning. One big shift is between division as sorting equal piles of discrete stuff to undoing multiplication of continuous quantities. This shift involves fractions as an expression of division. Another shift is between the actions connected with division and the calculations that give answers where we need to do them. It is so easy to say that ‘3 cakes shared between 7 people means they each get 3/7 of a cake’ but so much more complicated to ask for an answer in decimals (and so unpractical when cutting a cake!). The discussions teachers can have while sorting the cards are illuminating because while they are doing it they are all holding onto the same handrail.
The abstraction handrail
I was affected greatly by a teacher I saw once talking to year 1 children about how, if they know that two numbers add to give a third number, they can find any missing number of the three if they have the other two. And she then said: ‘you are doing algebra’ and they all solemnly repeated ‘algebra’ with the seriousness that young children give to strange new words. I have often thought of those children and how naming what they were doing as ‘algebra’ would influence how collecting its extended meanings in their later experience. In the primary curriculum non-statutory guidance you will find the statement: “Pupils should be introduced to the use of symbols and letters to represent variables and unknowns in mathematical situations that they already understand” and this accords with international research about early algebra, and was indeed what this year 1 teacher was doing but without yet using letters. In KS3 curriculum the idea is extended in the statutory requirements:
- model situations or procedures by translating them into algebraic expressions or formulae and by using graphs
- interpret mathematical relationships both algebraically and graphically,
In my view these requirements are of major importance because, without them, what is the point of algebra? However, they get slightly lost among requirements about the symbol system and relationships between representations – important, but lacking meaning without the purpose. The handrail that extends from early years to KS3 is about the ways in which we express generalities and relationships that arise in number, measure, patterns and puzzles.
The ingredients for teachers to discuss the algebra handrail are lurking throughout the non-statutory guidance and come blaring out in a crescendo in the statutory art of KS3, but using the handrail to inform teaching could mean that the examples given in the primary curriculum might be used as a foundation for KS3 work, because they relate to what learners have already met.
Here are some; there are many more:
- Pupils use and explain the equals sign to indicate equivalence, including in missing number problems (for example, 13 + 24 = 12 + 25; 33 = 5 x what?).
- Pupils describe the properties of shapes and explain how unknown angles and lengths can be derived from known measurements. These relationships might be expressed algebraically for example, d = 2 × r; a = 180 – (b + c).
- Pupils draw and label rectangles (including squares), parallelograms and rhombuses, specified by coordinates in the four quadrants, predicting missing coordinates using the properties of shapes. These might be expressed algebraically for example, translating vertex (a, b) to (a – 2, b + 3); (a, b) and (a + d, b + d) being opposite vertices of a square of side d.
- Perimeter can be expressed algebraically as 2(a + b) where a and b are the dimensions in the same unit.
- Missing measures questions such as these can be expressed algebraically, for example 4 + 2b = 20 for a rectangle of sides 2 cm and b cm and perimeter of 20cm.
Anne Watson has two maths degrees and a DPhil in Mathematics Education, and is a Fellow of the Institute for Mathematics and its Applications. Before this, she taught maths in challenging schools for thirteen years. She has published numerous books and articles for teachers, and has led seminars and run workshops on every continent.