I had an interesting chat with a friend recently who’s just done a PGCE as a maths teacher. He’s been trained to build understanding through plenty of problem solving tasks.

The discussion made me reflect on the stark difference between the way I’ve taught maths to my own children at home, with the lion’s share of time spent learning to fluency, and the focus in schools on exercises to build understanding. After all, I reflected, the progress of my children has stunned even me. How is it they missed out on SO much work on understanding while accelerating far ahead of their peers?

It isn’t that I don’t appreciate that children need some degree of understanding of what they are doing. I remember when I discovered that the reason my friend’s daughter was struggling with maths at the end of Year 1 was because she had failed to grasp that crucial notion of ‘one more’. Her teacher had advised that she needed to learn her number bonds (and indeed she did) but while she did not grasp this basic notion the bonds were gibberish to her. What we call ‘understanding’ does matter (more thoughts here).

I’ve realised the reason I’ve never had to invest significant time in exercises to build understanding. It is because when my children are given a new sort of problem they can already calculate the separate parts of that problem automatically. All their working memory is focused on the only novel element of a procedure and so it is very quickly understood. Understanding is just not a biggy. Identify the knowledge necessary to calculate the component parts of a problem and get fluency in those and generally activities for understanding become a (crucial but) small part the maths diet.

The degree of focus on fluency that my children were given is highly unusual. I have huge piles of exercise books full of years of repeated calculations continued a year, two years, after they were first learned. My children learnt all possible addition and subtraction facts between one and twenty until they were known so well that recall was like remembering your own name. I did the same with multiplication and division facts. There were hours and hours and hours and hours of quite low level recall work.

Generally the the focus in schools is the opposite and this creates a vicious cycle. Children are taught more complex problems when they are not fluent in the constituent parts of the problem. Therefore they struggle to complete calculations because their working memory breaks down. The diagnosis is made that children don’t ‘understand’ the problem posed. The cure is yet more work focused on allowing children to understand how the problem should be solved and why. The children may remember this explanation (briefly) but it is too complex to be remembered long term as too many of the constituent elements of the problem are themselves not secure. When the children inevitably forget the explanation what is the diagnosis? – a failure of understanding. Gradually building ‘understanding’ eats more and more lesson time. Gurus of the maths world deride learning to fluency as ‘rote’ but perversely the more time is spent on understanding instead of fluency, the harder it is for children to understand new learning. By comparison my children seem to have a ‘gift that keeps on giving’. Their acceleration isn’t just in the level of maths proficiency they have reached it is in the capacity they have to learn new maths so much more easily.

*Fluency… the gift that keeps on giving.*

I’ve not got everything right but I’ve learned so much from teaching my own children including that the same general principle is true of understanding maths and understanding history. If understanding is a struggle it is because necessary prior knowledge is not in place or secure.

Go back – as far as you can get away with.

Diagnose those knowledge gaps.

Teach and secure fluency.

You’ll find understanding is no longer the same challenge.

Great post–we will be quoting it on social media.

We cover much of the same ground in our latest policy paper: http://parliamentstreet.org/research/2017/free-schools-free-society/

Hmmm…I see the principle, but the pain of getting my kids to do that kind of maths just wouldn’t be worth it. Do yours enjoy it? Mine love problem solving but just hate number manipulation. Maths fact are less interesting to them than history facts. I personally excelled at maths – A grades at A level without breaking a sweat – but because we didn’t do maths in primary school I never learnt times tables or long division, etc. Maybe you can be taken by the logic side of things while finding numbers unfriendly.

Kids tend to accept/like what they’re used to and not to like failure. If we want our children to be successful at maths I don’t think there’s much point doing things we think they like that don’t work well.

Brigid–We trialled our programme to teach automatic recall of number bonds with a class of Yr 9 pupils with mild behaviour problems, and they loved it. We gave them all a pre-test where they were timed to see how long it took them to write the answers to 50 number facts for addition, and their times ranged between 1 minute 51 seconds to 8 minutes 30 seconds (1 minute 30 seconds would be considered evidence of automatic recall). After ten sessions of approximately 10 minutes with flashcards and workbooks– the times ranged from 47 seconds to 4 minutes 2 seconds. The secret was that the kids worked in pairs closely matched for times, and they timed each other on a written exercise every session. Of the 23 pupils, 8 had achieved scores < 1.30, and 2 more were within a few seconds of doing so. The pupils were unanimous: they all enjoyed the course, and wished they had learned their number bonds in primary school.

Admittedly, it's a lot harder to get that keen competitive instinct going at home when your child is tired after school (and you're tired after work!), but what the kids really liked was seeing their times improve. You can get stopwatches on Amazon and Ebay for under £2–an amazingly good investment in your child's future!

Your programme interests me greatly! Is it available commercially?

Reblogged this on The Echo Chamber.

For a study of the effect of fluency (or choosing not to teach fluency) in math, from a cognitive science perspective, see http://www.ChemReview.Net/MathAndChem01.pdf . The data cited is mostly US, but US and UK practices I think have been similar, and the cognitive science will be relevant to both. The article is also posted at

http://arxiv.org/abs/1608.05006 . An article on the US Common Core math standards is at

http://www.ChemReview.Net/CCMS.pdf

Dear Heather,

I have a 4 year old whom I self teach at home for the past 1.5 years even thou he attends preschool. And unlike you I am not a trained teacher but through my observations of my child’s learning, I have adopted teaching of ‘fluency’ in reading, writing, spelling, and Maths with him as oppose to concepts. I have not been able to find other like minded people either in life or on the internet as you probably know, as the most recent concept of teaching is via ‘play’ and ‘child led’, therefore I am extremely surprised that your entire blog regarding your teaching of your own children almost completely mirrors mine! Anyways, I thought I drop you a note just to say good job because I know and understand how difficult it is to convince a child to master fluency. Please write more blogs regarding your teaching especially of your own children as I personally find them extremely interesting to read. 😄

Thanks for such a lovely comment. It is really great to discover a likeminded parent. I’d love to hear more about what you do. Are you on twitter?

Hi,

Thank you! Nope unfortunately I don’t have twitter. But you can email me. Are you able to see it on this account?

I agree with you completely! Interesting thing happened this week. My son is a 10th grader nearing the end of his Geometry course. He attends a charter school with students in grades 7-12. A few days ago I received an email, sent to all parents, that as an extra challenge they were putting the students into teams and having them compete against each other on the math-facts website Xtra math. While I feel that it’s sad that high school students need this, I am also impressed that the school was actually seeing the lack of basic skills and addressing it. My son is actually pretty decent with the basics but I have noticed that as he’s been doing more math online he tends to use the calculator far more than he should–he gets lazy. I’ve tried to prove to him that it’s actually much faster in many case to solve it in your head or to quickly jot down a 2-3 digit calculation than to use the touchpad to manipulate the computer’s calculator. As a Spanish teacher I feel exactly the same about building fluency with basic vocabulary and conjugation. Cognitive Load Theory needs to be more widely known and discussed!!!