Talking of numbers

My maths teacher used to try and make the learning of multiplication tables into a competition. Each morning he would point to a random fact written on the blackboard, and we were to see how many of the following problems we could complete before he called 'time'.

The process did two things for me. Firstly it made me wonder why none of the other kids realised that all you needed to do was find where you'd written this answer most recently and then copy all the answers that followed straight into your maths book. Secondly it made me into an incredibly fast writer.

I invariably used to win maths drill - but I never learned my tables. To this day I skip-count my 7s and 8s instead of knowing the answers by heart, because despite spending 15 minutes every school-day doing exercises like the one I've just described, I couldn't see the reason for knowing this stuff and so I didn't bother to learn it.

Nowadays there are proponents for and against rote learning of maths facts. Relaxed types say that you'll learn them by using them. They're probably right. My 7s and 8s are better now that I'm teaching them to Jemimah then they've ever been before (and can I remind you that I did University maths?). More formalised classical educators say that rote learning works and committing stuff to memory works better when you're younger so why not begin teaching maths facts to your four year old? They're probably right as well. We've had lots of success with early rote memorisation in our homeschool.

So which group to follow? Either, neither or both?

This dichotomy is not new. Back in about 1945 an Adelaide man called John Flanagan wrote a little book called Numbers - A Book of Figures. The forward, written by senior lecturer at Adelaide Teachers college, H H Penny, MA, PhD, Dip Ed (Pr), Dip Ec (phew!!) says this:

There was a time (wholly gone?) when arithmetic was the grimmest of lessons. Many will remember the dolorous chanting of "tables" day in, day out. And despite the massive weight of repetitions 9x7 would sometimes be 56 and 7x0 still oftener be 7. To the teacher of the old school every such error was an incitement to more and yet more rote work. Later came the view that children should learn their number facts entirely by experiment and use. "Away with the dulling drone of rote drill," said the New Educators. "Soft pedagogy," was the retort of the Old School-ers.

Today? Today we can see the right and wrong of both points of view. In a nutshell it is, I believe, this: Each new step, be it only the adding of 2 and 2 should be a conquest and a delight, but certainly should follow on the heels of conquest. There are so many "new steps" in arithmetic; they can be safely taken only if the child stands firmly on all the steps he has already been asked to take.

This as I see it, is the purpose and justification of Numbers - to give the child a firm grip of the basic number facts. From the very nature of the material there can be little that is new in the book. Its merit lies in the ingenuity with which the old, old facts have been grouped and displayed and in the thoroughness with which the ground has been covered.

Seems to me that in the past 65 years nothing much has changed.

Which is probably why I've found John Flanagan's little book as valuable in Jemimah's homeschooling as HH Penny predicted that it would be for kids back in 1945. To Mr/s Penny, "the aim of Numbers (was) that the learning and testing... (be)... 'thorough-going and comprehensive". I agree. To me, whether you learn your maths facts opportunistically or methodically by rote, the bottom line is that you must learn them. You can't do maths without them. First you need to learn your numbers. Next come your addition and multiplication tables, whether by rote all together, or bit-by-bit. The aim of this memory work is not an ends in itself, however. A child who can sing all of his times tables but not do a multiplication problem using these facts has learned nothing useful at all.

Second, the maths facts we learn need to build on each other. Our memory of a fact depends on how useful it is to us - and in how connected it is to other things we already know to be useful. My times tables were not useful to me during my school tables competitions, for example, so I didn't bother to learn them. I just wrote the stuff out mindlessly each day for a whole year with the sole aim of coming first in the competition.

"New steps...can be safely taken only if the child stands firmly on all the steps he has already been asked to take."

Some people will take this quote as support for a mastery based maths programme instead of a spiral programme. I disagree. I thing there are advantages and disadvantages to both of these approaches, but the thing that is important, regardless of which you chose, is that you need to ensure that the foundational skills are mastered befor you move on to something that requires that skill. Whether you learn all addition before subtraction and all multiplication before division or all basic skills in the four binary operations all at once is not the important thing here.

Understanding is what is important at each step - HH Penny called it a conquest - is the third thing. Conquer one step before moving up the ladder to the next. Got it?

Step one: Learn it. Step two: Know why. Step Three: Conquer it.

Which is where Numbers come in. As the author puts it so well,

Numbers is like your figures' gymnasium. This is the place to make your number memory strong. You have plenty of ladders to climb up and down. Be like the boxer with his punching ball. If you work you must succeed and will get your reward - speed and accuracy.

Yes, memorisation of maths facts is important to a child's later success in mathematics, but look at that memorisation as a means to an end. Look for real, relevant ways to use those maths facts repeatedly. Look for word problems that practice the maths facts. Over and over and over again.

And look to make the memorisation a conquest and a delight.

So what does your figures gymnasium look like? We use Numbers. It is worth looking out for this delightful little book during your second-hand book shop forays. Yours could be Maths games. Playing maths card games. Using Timez Attack. Making up fun and relevant word problems. The Living Math site is full of maths related activities as well. Perhaps you've discovered a way to make rote times table drill fun and exciting.

Look for ways to develop your gymnasium. Look for ways to exercise that number memory muscle. Then use your membership and don't let it lapse if the going gets tough. If your child still doesn't get it then be patient and try something else. All they may need is more time.

So finally, either, neither or both? Well it still doesn't matter much. Probably you'll use whatever works best for your students - or for you - or both! What does matter is that they learn their maths facts. Then they can move on to the good stuff that builds upon their foundation. That's what they're doing all of this for, remember, for what they can build on top!

Goodbye!

Well, every good thing comes to an end, so they say, and I have finished my explanation. I do hope you have been able to understand me. Would you mind if I told you one last short story? Two men had a saw. At the end of six months one man had a new table and some chairs; the other still had just the saw. Do you see?