At school I always found mathematics easy, and fluency with numbers came easy to me, as a consequence I find it hard to understand how students can come to secondary school and not be flent with manipulating the number system, especially the integers.

Earlier this week I asked a KS3 student to tell me how many 20s were in 120. In answer to this they said seven and when I asked how they arrived at this they counted up in the following way: 20,40,60,80,91,102,120. I later went back to them and asked them to count up in 20s and they gave me the same list of numbers. I am struggling to explain this and wondering about the following:

- Why did they count up in this way?
- Is there anything special about the numbers 91 and 102?
- What barriers are there that mean they haven’t developed this number processing fluency?
- How can we develop this?
- Can we tackle this quickly?
- How can we develop the fluency so that the student recognises that 120/20 can simply be calculated by working out 12/2?

I’d love to know your thoughts on these questions, or anything else for that matter.

*Related*

Hi Tom and thank you for asking me to read this.

My first though was, yes, this was similar to the kind of reaction I had when doing some one-to-one teaching back in 2010 and I wrote about this in MT243: http://www.mikeollerton.com/pubs/atm_pubs/ATM-MT243%20At-homeness%20with%20mathematics.pdf

I have come to believe that developing fluency, with anything, whether it is computing with numbers, using GeoGebra or changing a bike wheel inner tube, is to do with a) the pleasure of using any skill, b) the confidence to perform a task and c) being able to carry out a task almost without consciously thinking about it; on auto-pilot. For young children in KS1 and KS2, I believe the ethos of the school and the culture within each classroom must be supportive of:

a) it being okay to make mistakes b) the only ‘dumb’ question is the one we don’t ask, c) not pressurising children to give instant answers or within a given, brief time frame d) enabling children to gain a sense of numbers, that for example 4×5 = 20 and 5×4 = 20 so 20 ÷ 4 = 5 and 20 ÷ 5 = 4 and the guzintas of 20 are {1, 2, 4, 5, 10, 20} and 4×5 can easily be transformed to 2 x 10 (halving the 4 and doubling the 5) and every multiple of 5 always ‘ends’ with a 5 or a 0 and…

I notice you use the phrase “What barriers are…” and this is interesting – because such barriers emerge from a variety of places, such as:

a) some parents (“I could never do maths…”)

b)some teachers who do not understand why some children just cannot compute with, so-called, ‘simple’ numbers

c) an education system which piles pressures on teachers to, in turn, feel they have to train their children to pass tests, to meet their targets.

My solution, for what it is worth, is to actively slow learning down. By this I do not mean to offer less challenging work for learners to do, but I do mean to take time to analyse answers, to see how different children have arrived at a common answer, to constantly help learners see connections between skills and concepts, to get learners to reflect upon and write journal entries about what they have understood from some work they have been doing and to write about (or make a PowerPoint) of the best problem they have solved in the past two or three weeks… okay time to get off my soap box.

Regards Mike

I agree with Mike’s perceptive comments and suggestions above.

I would like to add the following observations derived from many years of teaching in secondary schools and also from working with teachers in primary schools.

I agree with Mike that teachers often feel under pressure to ‘cover’ content, which may contribute to their paying too little attention to noticing/discovering and dealing with pupils’ misconceptions. A misconception may develop when a pupil focusses on some aspect of a diagram or of a statement that the teacher regards as ‘incidental’ – which the teacher does not intend to be ‘central’ in the pupil’s current learning. For example a teacher reported to me recently (I hope she won’t mind my mentioning it here – she could describe it much more powerfully herself!) a lesson in which most of the pupils in her class (who had all been at the same primary school) INSISTED that a diagonal of a polygon must be ‘a slanting line’ – they would not accept that, because they were horizontal and vertical line segments drawn on the board, the diagonals of a rhombus actually were its diagonals – when asked to draw a diagonal a pupil joined the centre of a side of the rhombus to a corner thus creating (on the board) ‘a slanting line’ across the rhombus! Possibly in the past, when talking about the diagonals of a square or rectangle oriented with sides horizontal and vertical, the pupils’ teacher had repeatedly said something like ‘look, the diagonals are the two slanting lines – here and here’.

It is so easy unintentionally and ‘innocently’ to ‘seed’ a misconception which is revealed only when pupils are given many and frequent opportunities to articulate their thinking. Although it is helpful for all teachers to be aware of common misconceptions so they can look out for them, I do not think that it is essential to know them all; what matters is providing enough opportunities for pupils to express THEIR THINKING, not for the teacher to keep repeating, or even trying to perfect, THEIR EXPLAINING.

What can be done to avoid pupils developing wrong, or ‘slightly distorted’, ideas that they carry around in their heads for years, and what can be done to remediate their thinking when that has happened? These are big questions that many people have, and are currently, tackling. It is obviously better to reduce in the first place, as far as possible, the wrong ideas that intelligent pupils develop – it is vital for teachers of young children to understand why it is, and how important it is, for them to create lots of opportunities to ‘hear’ and understand their pupils thinking. Once wrong-thinking is revealed, as in your example, it is usually necessary to probe further and further back until you reach sound conceptions from which to develop new sound ideas.

Your pupil’s strange thinking was revealed when you asked the pupil to explain their thinking. Unfortunately too often pupils are only asked to ‘explain their thinking’ when they give a wrong answer to a closed question. If it is expected that they will always try to ‘explain their thinking’ whenever they are talking about mathematics (to each other or to their teacher) fewer ‘wrong ideas’ would pass unnoticed for years.

These are very general comments, Tom, and so not very helpful in dealing with your particular example! But I don’t think it can be said too often how vital it is for mathematics-learning atmospheres to be such that pupils are naturally frequently expressing their thinking – particularly when they are unsure – that they are not afraid to do that, but WANT to do it /can’t help doing it. John Mason’s conjecturing atmosphere – ‘everything we say is a conjecture’ is so important!