22 Mar 2010

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!

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?


  1. Jeanne: as you know math is our weakest area. Neither Ditz nor I know any of our number facts ~ & not for want of trying to learn them. Ditz does a lot of concret math in her head; always has. Basic math she is actually goud at & has a sound understanding. We never had an issue untill the abstract stuff, for which we have no possible us, came into it.I know, because I once{foolishly] asked Ditz to show me how she worked number problems for which she didn't know her number facts, that she thinks in pictures mathamatically & then *sees* the answer. I suspect this is faster than many methods & as accurate but because you can't *show your working* not popular amongst teachers. Someone needs to write a math text for the random abstract non~sequential thinkers amongst us. I take your point [really I do ☺] but suspect there are other options we haven't even dreamt of because of the sequential nature of math. That of course is the thinking of a random abstract thinker because *of course* there must be another option; there always is; I just haven't thought of it yet! lol

  2. Very thoughtful post, Jeanne. I learned all my maths facts by memory. Similarly to you, the teacher would pick a number, write it on the board and surround it with the other digtits. Then we'd take turns at calling out the multiplication answer - out of order of course. I wonder if we'd learn them as well if they were always in order. I've always intended to get serious about getting my girls to learn their tables as a thing in itself, but we never got around to doing it, but just carried on with our maths books, and now, after using them so often and doing lots of mixed drills, I'm sure there are only the odd few that they don't know immediately. Once we did make up little times-table booklets, where I had them write each table out on a page. They were to use them as a bookmark in their maths text. They used them for a while to help them out with their long divisions etc, but they ended up getting lost. I guess they did their job for a while and now they don't really need them anymore.
    Maths is so important - just look at the builder, or the seamstress, or the baker! I have often used my knowledge of algebra to solve problems, often in our business to work out price increases etc.
    I also believe that the very study of maths is great training in logical thinking, which all of our children need to be able to do in this day & age.
    What a ramble!

  3. Either, neithor or both?

    Possibly a little of all of the above for us. We just hum along with our maths - I dig out some drills when I see an obvious gap/struggle in some of the facts, and then we move along to hum along our way again. And I suspect also a little of the pictorial/abstract maths going on in some of our little brains here as well.

  4. Yea, I learned by rote memorisation too but I know what you mean that you need to use it once you learn it before you learn more new things in maths!
    I like Rachel have used what I've learned in maths in various other areas of my life, it's a very useful thing - maths!:)

  5. Argh! My head hurts!
    I learned by rote and must say still know my times table pretty well. My eldest daughter learnt in a similar way (the good reader) but the others have all struggled with rote learning, skip counting etc. Just this morning the youngest dazzled me with some 8 times so I think the repetious use of them is working.
    Alas, though I knew my tables well I did not go on to be a great maths scholar in higher grades :-(

  6. As you suggest, a bit of all. I see the power of memorization, and if I understood Liping Ma correctly (more here http://educandoenelhogar.blogspot.com/2010/02/math-has-awaken.html
    I believe good math teachers would teach the skill or the WHY and HOW of what you are doing, and then they will PRACTICE (drill) to master all aspects of mathematics.
    I had to learn the tables and I always thought it ridiculous to color squares in fifth grade (as I had to do once when I was substituting in a class) to teach the tables so they understood.

  7. Maths....not my strong point that's for sure. Still, I can see the benefit of knowing our number facts, especially x tables. I try , I really do...I just don't think I really enjoy :( Still me younger two girls, and their younger brother seem to have the knack -who knew! Obviously their daddy's genes I think! Love the old book though- just gorgeous..:)

  8. Boring or not, I come down firmly on learning the times tables, doing basic addition/subtraction drills and on practicing mental math regardless of how the kids does school! For most of us this is the only math we will really use! In homeschool I also assign math and grammar vocabulary purely to get them thru the standardized tests. I also like using Living Math. I have raved many times about the How Math Works, etc, series--great photos, great activities. The Family Math books are also helpful as is Grocery Store Math--my kids really enjoyed doing these and it made grocery shopping less stressful!

  9. as usual, I totally agree.
    memorization is vital.
    we fall somewhere in the middle too.
    here's my post on the matter...


    amy in peru

  10. i have a headache, too. :)

    not a math person but found
    ways to make it fun for my

    i believe that the more they
    have memorized the more
    chances they have to make
    sense of it all.

    we loved saxon math for
    the way it kept many
    concepts before the
    children not just one at
    a time.


  11. I understood Maths much better once i began to teach it, and actually started to enjoy it once I saw it was a lot like the word puzzles I loved, only with digits.

    In my teaching, I used lots of games and physical materials. Measurement especially, I loved to get kids to estimate and heft stuff, trying to get an idea of its weight or distance etc. And patterns were always fun.


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