Given the positive response to my recent blog-post on the subject of mathematical fluency, I thought a few people might find this useful…
I wrote the last blog-post because I’d been considering the areas of mathematics that were (a) most frequently relied upon in later learning[i], (b) often better taught in a few minutes each day over an extended period, and (c) absent from too many children’s repertoire of fluency. With this in mind, I began working on a list of fundamentals to be taught to the point of fluency in each year group.[ii] The point of the list was to indicate which areas of the curriculum might be best tackled using a ‘little and often’ approach after initial teaching. These are indicated in italics. (Alongside these, I included some conceptual fundamentals that I thought needed to be revisited throughout the year. These are indicated in bold.) The list is very much a work in progress, and I am open to feedback. The position of each component in a given year group matches fairly well to the national curriculum, but the idea is that this list only indicates where the ‘little and often’ teaching of each component should begin; where each component should end, however, is entirely dependent on how long it takes a class to reach fluency; this means that a component might continue into the following academic year if necessary. I post the list here merely in the hope that it might be useful to others as a jumping off point for further discussion of this topic:
- primarily ‘conceptual’
- primarily ‘mental fluency’
(a) Understand and use various concrete and pictorial representations of numbers up to 100 (including all of cubes, dienes and number lines)
(b) Understand and use bar models and part-part-whole diagrams as representations of number bonds inside 20; use these to know whether an unknown is the part or whole in equations such as ? – 8 = 4 and 5 + ? = 11.
(c) Recall number bonds up to 10 (i.e. without a counting strategy)
(d) Find half of even numbers up to 20 and double of integers up to 10
(e) Count up to 20 in 2s, up to 50 in 5s and up to 100 in 10s
(f) Understand different interpretations of addition (collecting similar objects, counting on and extending) and subtraction (removing objects, counting back, shortening and finding a difference)
(g) Understand rules of commutativity for addition, subtraction, multiplication and division (i.e. addition and multiplication – order doesn’t matter; subtraction and division – order matters)
(h) Recall addition and subtraction facts inside 20 (i.e. not using a counting strategy; e.g. 7 + 6 –> double 6 then add 1 = 13 or 7 + 6 –> 7 + 3 + 3 –> 10 + 3 = 13; ideally, pure recall from practice)
(i) Use recall of addition and subtraction facts inside 20 to calculate mentally TO + O and TO – O (i.e. without a counting strategy)
(j) Using note-taking (without counting) calculate O + O + O
(k) Use concrete objects, pictures or mental strategies to calculate T + T (e.g. 70 + 20 = 90), T – T (e.g. 90 – 30 = 60), TO + T (e.g. 56 + 40 = 96) and TO – T (e.g. 89 – 40 = 49)
(l) Recall (i.e. not count up) multiplication and division facts for 2x, 5x and 10x including missing number questions (e.g. 5 x __ = 35)
(m) Understand different interpretations of multiplication (repeated addition, increase in dimension, change in the counting unit and the scaling of a value) and division (sharing and grouping)
(n) Understand and use various concrete and pictorial representations of numbers up to 1000 (including all of dienes, place value counters and number line)
(o) Calculate mentally (without a counting strategy) HTO + O and HTO – O
(p) Count in 10s up to 200 and count in 100s up to 2000
(q) Recall (without counting) multiplication and division facts for 3x, 4x and 8x, including missing numbers (e.g. 8 x __ = 56)
(r) Use number bonds inside 20 knowledge to calculate addition and subtraction facts for tenths inside 2.0 (e.g. 0.8 + 0.7 = 1.5)
(s) Use place value knowledge (not adding/subtracting zeros) to mentally multiply and divide by 10 and 100
(t) Recall all multiplication facts up to 12 x 12
(u) Round numbers to the nearest ten and hundred using number line representation.
(v) Recognise the decimal equivalents of tenths up to 9/10 and hundredths up to 99/100, and vice versa.
Understand and use various concrete and pictorial representations of numbers including up to 1 000 000 and numbers including tenths, hundredths and thousandths (including all of dienes, place value counters and number line)
(x) Mentally multiply multiples of 10, 100 and 1000 by other multiples of 10 and 100 (e.g. 400 x 80 = 32 000)
(y) Mentally divide multiples of 10, 100 and 1000 by single digit numbers (e.g. 3200 ÷ 8 = 400)
(z) Calculate equivalent fractions using all multiplication facts
That’s all folks. I hope it is useful.[iii] Let
me know if there you consider there to be any glaring omissions or things on
the list that need not be there.
[i] Will Emeny’s blogpost showing the GCSE maths curriculum in a visual model is a fascinating way of showing the areas of the curriculum upon which other learning relies most often. In the model, each area is a node and the number of connections between each node is shown by the size of the node, making it clear which areas of the curriculum are relied upon again and again. My description doesn’t do it justice, so I’d highly recommend you follow this link:
[ii]Advocates of a genuine mastery approach will likely take me to task over this implication that there are particular components of the curriculum that require fluency, rather than the whole thing; I absolutely understand their point and agree. However, many – if not most – schools teach the curriculum in a way that Mark McCourt would describe as a ‘conveyor-belt’ approach. This being the case, identifying some areas of the curriculum where fluency is at its most important may be of some use, though I recognise the pragmatism in it.
[iii] Those of you who are familiar with Edutwitter might notice some similarities between this list and a much-reduced curriculum presented in one of the thought-provoking blogs of Solomon Kingsnorth (@solomon_teach). The difference between the two that makes this post worthwhile (I hope) is that this list by no means constitutes the breadth of the curriculum as I would like to see it. It is merely a list of the stuff that is essential and yet can slip between the cracks when teaching is seen in terms of lessons, topics and year groups.