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Fluency in Mathematics: Part 1

October 4, 2014

I gave a talk on fluency in mathematics in March at Pedagoo London (my first public appearance) and again last weekend at the La Salle Education maths conference. This post is based on those talks and so, inevitably it is long enough to take several posts and revisits some old ground.

pedagootalk

My talk at Pedagoo London

I started by asking the audience the following questions (yes I know, exciting start), and giving them a couple of minutes to work them out:

  1. 7×8
  2. Simplify 49/84
  3. Find √729
  4. Solve 3x+40=19
  5. Write √7/√175 as a decimal

At both talks there was somebody who took not much more than a minute and others who struggled. It is possible to answer every one of those questions in a few seconds if you have memorised the correct basic facts, such as times tables, and how negative numbers and surds work, and can recall them fluently. Both times though, I appeared to be the only person in the room to have memorised the first 27 square numbers.

As well as being useful for solving problems, fluency is now one of the major aims of the new maths National Curriculum, quoted below:

Aims

The national curriculum for mathematics aims to ensure that all pupils:

  • become fluent in the fundamentals of mathematics, including through varied and frequent practice with increasingly complex problems over time, so that pupils develop conceptual understanding and the ability to recall and apply knowledge rapidly and accurately.
  • reason mathematically by following a line of enquiry, conjecturing relationships and generalisations, and developing an argument, justification or proof using mathematical language
  • can solve problems by applying their mathematics to a variety of routine and non routine problems with increasing sophistication, including breaking down problems into a series of simpler steps and persevering in seeking solutions.

To have fluency as the first of three aims is a big change compared with the old national curriculum, the aims of which seemed to include everything else but fluency:

Curriculum aims

Learning and undertaking activities in mathematics contribute to achievement of the curriculum aims for all young people to become:

  • successful learners who enjoy learning, make progress and achieve
  • confident individuals who are able to live safe, healthy and fulfilling lives
  • responsible citizens who make a positive contribution to society.

The importance of mathematics

Mathematical thinking is important for all members of a modern society as a habit of mind for its use in the workplace, business and finance; and for personal decision-making. Mathematics is fundamental to national prosperity in providing tools for understanding science, engineering, technology and economics. It is essential in public decision-making and for participation in the knowledge economy. Mathematics equips pupils with uniquely powerful ways to describe, analyse and change the world. It can stimulate moments of pleasure and wonder for all pupils when they solve a problem for the first time, discover a more elegant solution, or notice hidden connections. Pupils who are functional in mathematics and financially capable are able to think independently in applied and abstract ways, and can reason, solve problems and assess risk. Mathematics is a creative discipline. The language of mathematics is international. The subject transcends cultural boundaries and its importance is universally recognised. Mathematics has developed over time as a means of solving problems and also for its own sake.

Which to me, now, really sums up an era where you could have every variation of progressive education together in one document, while missing any mention of what is worthwhile knowledge.

One of the main reasons I think this has now changed, is a change in the understanding of how we think. One of the more popular diagrams in education today is this one from Dan Willingham’s book:

It’s a very simplified, but uncontroversial, model of how we think. We have a limited working memory, where conscious thought takes place, and a potentially unlimited long-term memory.  To use working memory effectively, we draw on information that is already in long-term memory. To get things into long term memory we have to overcome the limitations of working memory. Having useful information in long-term memory, and being able to recall it without difficulty, makes thinking easier. In maths it is useful to be able to fluently recall a lot of knowledge, particularly basic number facts, rather than work everything out from first principles. The question at the start about simplifying 49/84 was an example of this, as fluency with the 7 times table makes the question trivial.

In order to build fluency we need to acquire knowledge and to learn to remember it without effort (automaticity).

pedagooworkingmemory

See? I did say this

And if that is our aim then, in maths, the best method is to tell kids what they need to know and set them lots of questions where they practise recalling it.

Continued in Part 2

21 comments

  1. Not sure why I would want to memorise the first 27 square numbers? Why not the first 127 or 1027? Root 729 is not difficult to find without memorising more than how to multiply 2 x 2 and 3 x 3 and knowing place value in decimals. 2 x 2 is 4 so 20 x 20 is 400 and 30 x 30 is 900 so root 729 lies between 20 and 30 and nearer 30. It ends in 9 so which digit >5 multiplied by itself ends in a 9? 7 so its going to be 27 if its a whole number.

    So which demonstrates fluency? Learning all the square numbers off by heart or learning what a square number is and strategies to solve such numerical problems? Of course that also assumes just typing square root of 729 into Google doesn’t count ;-). I agree with fluency and feel for number but I’d draw the line at memorising arbitrary number facts that are a) pretty obscure as far as everyday use is concerned and b) discourage the development of strategies to solve numerical problems.

    The argument here is not that learning stuff and filing it in long term memory is unimportant, more about what is worth learning and storing and what is not. Is it better to learn every single square number to infinity or a principle and strategy and store those in long term memory? In computational terms do we learn all the data outcomes or the algorithm that will produce them? I once wrote a program to teach young children how to form letters. The BBC B only had 32k of memory so forming all the letters from two procedures, one to draw curved lines and one to draw straight lines with different parameters to define position shape and size was a lot more memory efficient than storing 52 patterns for all the upper and lower case letters at all possible sizes. It’s the use of Bezier curves in vector graphics that enables infinitely scalable diagrams from very compact data. I think education could learn lot from these computational principles. If mathematics is about anything it is about elegance, efficiency and optimisation not just memorising obscure things indiscriminately.


    • “Not sure why I would want to memorise the first 27 square numbers?”

      The first 20 are part of the GCSE spec. Up to 25 gives you access to an easy mental method for calculating them up to 75. Knowing 26 squared is easy because it is 100 more than 24 squared. 27 squared should be known because it is the square of 3 cubed, and therefore also the cube of 3 squared, i.e. 9.

      “Why not the first 127 or 1027?”

      Why not?

      “Root 729 is not difficult to find without memorising more than how to multiply 2 x 2 and 3 x 3 and knowing place value in decimals. 2 x 2 is 4 so 20 x 20 is 400 and 30 x 30 is 900 so root 729 lies between 20 and 30 and nearer 30. It ends in 9 so which digit >5 multiplied by itself ends in a 9? 7 so its going to be 27 if its a whole number.”

      Yes, but that won’t tell you how easy it is to factorise “x squared – 729” or to simplify the square root of 7290.

      “So which demonstrates fluency? Learning all the square numbers off by heart or learning what a square number is and strategies to solve such numerical problems?”

      Off by heart. The whole point is that you free up working memory from having to use those mental strategies.

      “Of course that also assumes just typing square root of 729 into Google doesn’t count ;-). I agree with fluency and feel for number but I’d draw the line at memorising arbitrary number facts that are a) pretty obscure as far as everyday use is concerned and b) discourage the development of strategies to solve numerical problems.”

      I haven’t actually told anyone to memorise square roots up to 729, but having a bank of number facts does make life easier and increases the available mental strategies.

      “The argument here is not that learning stuff and filing it in long term memory is unimportant, more about what is worth learning and storing and what is not. Is it better to learn every single square number to infinity or a principle and strategy and store those in long term memory? In computational terms do we learn all the data outcomes or the algorithm that will produce them? I once wrote a program to teach young children how to form letters. The BBC B only had 32k of memory so forming all the letters from two procedures, one to draw curved lines and one to draw straight lines with different parameters to define position shape and size was a lot more memory efficient than storing 52 patterns for all the upper and lower case letters at all possible sizes. It’s the use of Bezier curves in vector graphics that enables infinitely scalable diagrams from very compact data. I think education could learn lot from these computational principles. If mathematics is about anything it is about elegance, efficiency and optimisation not just memorising obscure things indiscriminately.”

      Who says they have to be obscure or that memorisation is indiscriminate? I’m mainly pushing for number bonds, times tables, and the most useful “special numbers”, i.e. squares, cubes and primes.


      • I’m saying the example you used is obscure so in the absence of any other evidence we have to assume you are promoting learning obscure facts by heart.

        Firstly I do think we should ensure in primary schools there is emphasis on fluency in the numbers 0-10 with all operators and results on those numbers. That is because we work in a decimal number system.

        Now, take an obscure mathematical “fact”, the name Henagon. Would anyone be disadvantaged in life by not knowing this is an 11 sided shape? It is exactly the sort of thing ideally suited to Googling if you ever needed it. Knowing the name square for a 4 sided shape is in such common usage we should make sure all KS1 children know it. So just because I say we shouldn’t worry whether or not a child learns the name Henagon it does not mean we should not make sure they know triangle and square at a very young age. Learning Greek prefixes penta, hexa, hepta, might be useful as they are transferable to other contexts in common use but they are easy to find without having to learn the more obscure ones. http://en.wikipedia.org/wiki/Numeral_prefix and I’d say showing children where to find such info beyond the commonly used ones is more important. Why would anyone want to learn diacosioi- as part of a general education? So obscurity is open to some debate in itself.

        My definition of obscurity is stuff few people ever use or ever need. Now its true that some of this is inevitable in a broad education but it rather under-mines the argument for fluency in fundamental number operations if the example chosen is something obscure and that can easily be found by more generalisable strategies. The working memory argument is bogus. If it was true that it was always better to remember obscure results than the algorithms to get to them we would not bother learning F=Ma, we’d learn every possible outcome of that formula. That is obvious nonsense so there is a debate to be had about priorities. At what age/stage? In what context? Why? Not just because all facts are equally useful. With some people it might be more important to put more balance on one thing than the other but getting that wrong risks alienating a lot of people – that has happened with maths throughout history.

        So the answer to why not is because there is an opportunity cost in everything we do and we want to optimise learning for effective rather than obscure outcomes. Why teach obscure stuff that virtually no-one ever uses when a lot of people are not even fluent in the fundamentally important things and that includes the strategies to solve these problems. I can easily factorise (x squared – 729) by the same methods I alluded to in the previous post or find root 7290 because they are essentially the same problem. root 7290 = root 729 x root 10. I’d say recognising them as the same problem is more important than learning 27 squared off by heart. Since I didn’t know 27 squared off by heart but had no difficulty solving the problem quickly shows that that number fact is not that important. It would be just as valid to teach strategies for decomposition than spending the time learning all the square numbers to anything beyond 10 by rote. Ones yo use a lot you will remember in any case. Spending time with your working memory learning stuff you don’t really need when it could be contributing to a generalisable rule that works in all cases is the opposite of what is needed for optimisation.

        The essence of mathematics is to build on a relatively simple set of axioms to develop generalisable rules in order to solve mathematical problems, it is not about learning every possible result off by heart. Sure some key essentials do need to be very well lodged in long term memory. This takes time and presents an opportunity cost in an over-crowded curriculum. Exactly what gets learnt, what is practiced and how much is crucially important. It’s why a lot of these debates lack the precision of context to be useful to any real extent. Motivation is another dimension that impinges on this too.

        GCSE? AQA syllabus 4360 says squares up to 15 x 15 but it does not say you have to know them off by heart. In any case its certainly not 27 x 27 or higher. Those things might be Ok for a party piece or a bit of one upmanship in a pub quiz but I’d say they could easily do more harm than good in a mandatory general education setting bydiverting time from more important learning.


  2. Oh dear! Automaticity is surely not as important as knowing how to find the answer, even if all the facts are not one’s head?


    • It depends what you need to be automatic and what you need to be strategy. See above.


    • Have you read the post? The whole point is that knowing certain facts makes it easier to find the answer and remember how to find the answer.


  3. Eagerly awaiting Part 2 – I’ve been changing the way I teach maths (primary) to move the kids more towards automatic recall, rather than what I was doing before, which I now think was moving them too quickly through a wider range of topics without enough consolidation. Seeing good things in what they can now do; only problem is not having enough time to drill that knowledge in at a deep level with so many factors competing for time.
    Any particular tips for what works when kids are not responding well to setting lots of questions to test recall? Generally modelling it one-to-one helps but a minority of kids seem to have blockage in that space between long and short term memory. Any advice as a maths teacher? Ta.


    • Post 3 will cover suggestions. There is little alternative to practice, but practising in different ways might help.


  4. I have 10 IEPs in my year 5 class of 31, including a girl so disabled mentally and physically that she has the class TA all to herself. My background, before teaching, is in finance/accountancy so I like to think I have some understanding of maths per se.

    What I have noticed is that lower achieving children do really well on the formal methods. However, I am forced to teach these rather odd methods like ‘Half and half again’ so that children can meet the learning outcome ‘I can divide by 4 using different mental strategies’. The children hate this because they think that short division could be used for everything. I agree. ‘Half and half again’ does exist in our heads for the occasional mental calculation of 48/4 say, but it depends on how your brain works.

    My beef is that I have to force these confused children to demonstrate various different strategies in preference to formal methods. Why? Because apparently you cannot move onto formal methods until you have ‘Shown an understanding of place value’. There it is: the teaching of these fangled ‘bunny ears’ and ‘bus stop’ methods is designed to get children to visually show, in their books (and therefore for Ofsted), that they can partition by place value.

    This is odd because I grew up with formal methods and times tables knowledge, and the practice of these built up my fluency and understanding of place value over time. Children these days are expected to do this backwards: understand place value and then go on to formal methods. The same could be said about number bonds to ten: column addition practice would make you an ace at number bonds (think about it!).

    It’s a real shame. I am forced to confuse lower achieving children (by halting their progress at the ‘chunking’ method for example) and send them off to secondary school unable to access their maths lessons.

    Furthermore, how many older people do we know who can rattle off a bit of short or long multiplication without fully understanding what is happening to each number? As their pencil dots the numbers in a set dance to produce the correct answer, they cannot explain exactly how it has happened. These people, if they were transformed back into children and put in my class, would be my ‘IEPs’. The difference is that, despite not fully knowing all the insider info on place value etc, they can still perform a calculation and come up with the goods. My current IEPs can do neither: they’re in the dark on both counts.

    Bang goes their chances of getting a job on the front desk at Barclays.


    • The obsession with place value baffles me. It’s actively unhelpful when multiplying or dividing decimals. I don’t mind teaching mental methods, that’s part of building up fluency, but written methods need to come first and there’s a certain amount of facts to be known off by heart before one can be any good at mental methods.


      • Railing against place value feels completely iconoclastic! (As in: this is blowing my mind; its importance seemed self evident). Can you elaborate, or write about it in a future post?

        It feels like a really fundamental thing for having any flexibility when manipulating numbers/calculations, or having a sense of if a calculation’s solution looks “about right” (based on a sense of how big or small the answer should be from the value of the starting numbers), or doing anything playful with them (eg maths for leisure / pleasure, along the lines of JMC questions). I know the latter isn’t as important but feels sad for some students to have limited access to things that make maths intrinsically enjoyable.


        • I think it is useful for adding and subtracting so I’m not saying don’t teach it. But for multiplying decimals, I prefer to see the decimal point as short hand for “divided by 10/100/1000 etc” rather than an indicator of place value. So 1.5 is “15 divided by 10” not “1 whole one and 5 tenths”. Most of the effective multiplying methods work on the basis of the former understanding not the latter. I also find that using the latter makes teachers obsessed with saying “you move the digits not the decimal point” when really multiplying decimals is all about treating the decimal point separately from the digits.


  5. If you want a job in processor design look at how multiplication is done in a CPU :-) Personally, if I was in a school that made me teach in a way I thought was bad, I’d leave and get another job. There is supposedly a shortage of maths teachers so it should not be that difficult.


  6. Were most of the teachers maths teachers ?


  7. “And if that is our aim then, in maths, the best method is to tell kids what they need to know and set them lots of questions where they practise recalling it.”

    Looking forward to part 2. I’ll be interested in your showing me that this is true rather than your perception of true. I’m not disagreeing, by the way, I’m just intrigued by what you mean by ‘lots of questions where they practise recalling it.’


  8. […] I gave a talk on fluency in mathematics in March at Pedagoo London (my first public appearance) and again the weekend before last at the La Salle Education maths conference. This is based on those talks and so, inevitably, it is one in a series of posts. Part one can be found here. […]


  9. This discussion is one that has been going on for decades, as part of the ‘Math Wars’. See Schoenfeld for a nice overview. The view that either ‘understanding’ or ‘procedures’ are ‘on their own’ is flawed, they go hand in hand (Rittle-Johnson, Star, Kilpatrick and more). It also is apparent in research on ‘Asia’: they also manage to do BOTH rather than one.


  10. […] talks and so, inevitably it has taken several posts and revisits some old ground. Parts 1 and 2 are here and […]


  11. […] by academics such as Jo Boaler and many teachers (and maths consultants…), and the heart of much debate. This approach argues that relational facts needn’t – and shouldn’t – be taught […]



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