a decade of maths

I started off my three year old….yes you read that correctly….three year old….doing worksheets which I studiously made up for her. Fortunately, she enjoyed it! Unfortunately I didn’t really know what I was doing!!!
Before she was even four years old I had three other little ones and I was wondering how I would ever keep up the pace! I got hold of a maths book, which had LOTS of exercises as I thought kids needed HEAPS of practice to learn anything. How could they possibly learn otherwise? Fortunately again, she enjoyed that too.
I did a bit of reading to find out what I didn’t know about teaching maths and ?unfortunately? discovered I was supposed to have been giving my dear daughter concrete experiences. OOPS! Hoping it was never too late, we put the book away and I started playing games and looking for real life maths. A friend challenged me to “live” and let the learning happen. I was sceptical, but J4 was already two years "ahead" of herself, I was pregnant and very sick with number five so I decided to take six months off to see what would happen (and the worst case scenario was that she would only be one year “ahead” if nothing happened!). So sceptical was I, that I decided to record everything mathsy my son did during that six months (I wanted to compare what he learnt with what J had done at the same age). Within a month I had stopped recording – there was soooo much.
Although I stopped writing every little thing down, I did notice some things.
Firstly, HEAPS of maths happens in everyday life.
Secondly, each of the children learnt in their own way and at their own pace (not exactly rocket science, but the reality was so different to anything I had ever known. I had come across the concept and thought it meant that some kids would finish the textbook sooner than others!)
J would spend a long time “playing around” with a concept. For example, he got a stopwatch for Christmas when he had just turned five. This stopwatch hung around his neck for two years straight – he used it all the time to work out how long it took us to do the shopping, how long car trips were, how long sermons lasted, even how long it took to fill a vial of blood at the doctor’s!!! It seemed he was always checking the time and by the time he was seven, he had a very thorough understanding. The textbook I had used with J devoted one week’s worth of lessons to the concept and once we had filled out the pages I considered we had “done” time. In fact, I thought we had done very well because I got a “real” clock out to use! But J’s understanding at that stage was far deeper than J’s… J became interested when she was eight years old and with no apparent effort “got it” almost overnight. Seconds, minutes, hours, days, weeks, months, seasons, leap years, twenty-four hour clock, digital, analogue. They have both picked it all up in real life…and both of them in different ways. K (just turned 7) and K (5) are currently learning by reading out the numbers on the clock and having me tell them what it means. Even three year old L is on the act! (By the way, I should have listened to my friend who told me not to give my kids the same intitials!!! At least we have a boy J and girl J, boy K and girl K!!)

Another example. As dedicated interest in the stopwatch started waning, I noticed a tape measure in J’s pocket! For about six months he carried that round and if something could be measured he probably tried! Everything from insect wing spans to the length of an Olympic running track in Ancient Greece.
Measuring, measuring. The front seat in the car became a favourite because you could see the odometer from there!
J picked up measuring through an interest in making things. At first she made all sorts of “creations” out of scraps, giving no thought to sizes. Then she discovered books in the library with instructions….including measurements. As she developed sewing skills her understanding of measurement deepened. Again, no textbooks.

I was the only person I knew using this kind of methodology for maths, but I believed God wanted me to trust Him with this one, so I stepped out in faith. Every so often I would get concerned and look into various curricula or start keeping records of what we were doing or even have them take a test (well, once….)……(recently!)……
But now I am confident that this is working.
J is nearly ten and we haven’t worked through another textbook yet. I don’t even plan to next year. Around about 11 or 12 we’ll crack open Math-U-See and work through that, stopping at any concepts she has missed until she picks them up. I am believing she will do it more quickly than if we had started at age five and spent half an hour a day for seven years “doing maths”! (If I am proved to be wrong, I will revise this!) (Update 2005: That was what I was thinking we’d do – now I’m wondering about using a remedial college text with the children in their mid-teens if they haven’t picked up anything more formal of their own volition before then)
In the meantime we will continue with what we have come to enjoy…..
We’ll play games like Rummy, Monopoly, snakes and ladders, chess, cards, battleships, dominoes, tiddly winks etc
We’ll plant and tend a garden – when you have planted seventeen seeds in a row you are going to want to count how many of them come up!!! Isn’t that so much more motivating than circling the plants on a page in a workbook and counting them??
We’ll bake together, doubling and tripling recipes as our family gets larger!!!
We’ll divide cakes into more and more pieces for the same reason!!!! J and J just got to know halves....or thirds if Mama wanted to share, but Baby Number Seven will know eighths very early on!
We’ll build things with lego and wooden blocks
We’ll count when we climb stairs with the nearly two-year-old
We’ll keep a finance record (well, the older ones do)
We’ll continue to do puzzles (J, J and K – ages 9,8,7 – are currently working on a 1000 piece one and they’re nearly finished! The little ones do stage-appropriate ones)
We’ll play the odd computer game
We’ll even use little activity books with number games occasionally

Sometimes it can be difficult because it is hard to articulate what we are doing. There are no tests to supposedly prove learning is happening. There are no text books completed or exercises marked. Sometimes the children are slower than people think they would be at school (that’s a really hard one).
K at age six had still not learnt to count to 100. K loves sport.
One day he was bouncing a ball next to me as I hung out the washing. I started counting out loud and he took up the chant. Very soon he said to me, “Mama I can bounce more than I can count. Can you help me?” I helped him all the way to a hundred. And we did it again the next day. Within a few weeks he was counting in twos, fives and tens….past 100!!! It was a lovely example of waiting until he was ready and trusting God would bring the need into his life. Once there was purpose he picked it up very quickly – it’s not like I hadn’t counted with him before that day! I certainly had, but it had not been his time to “click”.
One common response (amongst homeschoolers, and I’m sure those who send their children to school would echo this thought) has been: “But there are some things they just have to do. They’ve got to learn discipline.” I wholeheartedly agree. My children from a very early age do chores around the house. In fact, by the time they are old enough to enter public school, they are working at least two hours a day at home. This gives them plenty of opportunities to learn discipline and to learn to do those things they don’t enjoy! We don’t have to sacrifice the love of learning to teach those things. If we end up with a child who does not enjoy learning, then maybe I will see things differently, but so far they have all been interested in life. In fact, I am looking forward to adding examples to this piece about how God has taught them as individuals those things they need to know.

Here’s the first example:
J’s Flower Presses (by the way, this is not a maths lesson in real life, but a real life situation which used some maths along the way…and many other things too)
J expressed the desire to press a ponga frond (having been pressing lavender and pansies in a tiny flower press). He mulled over ideas about how to do it for a few days and made many suggestions. Mama directed his thinking towards making a smaller one first! (a balance between squashing his idea altogether and letting him run with his creativity). She had him closely observe the little flower press to see how it was made. She then suggested he draw a life size plan – he came up with a side view and “looking down on it view”. Drawing these necessitated learning to use a ruler properly. We went to Mitre 10 to price out the project. J took his plan and showed it to the man, doing all the talking for himself and spending his own money to buy the first piece of plywood. Because the smallest sheet of ply was still actually pretty large he worked out how many flower presses he could make and decided to sell the extras. He gave instructions to the “saw man” in the workshop. He spent many afternoons sanding. Dadda taught him how to use a drill – there were lots of holes to be made. And more sanding. Then it was time to come inside – there were instructions to write, a pattern to design and cardboard circles to cut out. Each of these steps was comprised of smaller steps:
1. The instructions
Firstly J told me how to make a flower press as I transcribed his words onto paper.
These were edited with my help into a logical sequence.
We then turned each sentence into an imperative.
J typed the instructions onto the computer, and found a suitable picture.
Dadda helped him size them to print off on stickers.
J placed the stickers on the underside of each flower press.
2. The pattern
J wanted a flower picture on the top of the flower press so he copied the flower from the instruction sheet onto a wood block. He then stuck string over the lines with PVA glue. Once this had dried, he used it as a stamp.
3. The cardboard circles
Each flower press required four circles made from cardboard. After J’s search of the house for a suitable object to trace around was unsuccessful, I introduced the compass. Discussions on radius, diameter and circumference ensued (I have not seen this in any six year old maths material I have looked at, but it was relevant to our project). A valuable lesson was learnt when J rushed through cutting out one set of circles only to discover he had done a very poor job – materials, time and effort were wasted.

Time to put it all together. Four cardboard circles sandwiched between a smoothly sanded square of ply with a picture stamped on it and another one with the instructions. All held together with four bolts and wing nuts. (As he only had enough money to buy one set at first, a lot of maths was involved, working out how many bolt/nut sets he could afford as each flower press sold, working out profits…adding, multiplying, checking).
Then came pricing and advertising (in this case, word of mouth amongst friends). The price was re-evaluated when a friend said he was undercharging and insisted on paying twice his asking price! The same friend suggested spraying the wood with varnish; a suggestion J accepted.
J took a sample of the varnished product back to Mitre 10 to show the men who had served him and had seemed very interested in his little project. One of them organised for the store to order half a dozen to sell on his behalf! Orders came in from other people. Another lady offered to sell some at a craft fair…….

Example 2:
K is five and loves shapes. She was flicking through a maths dictionary we have one day and discovered a whole page of shapes she had never come across. This sparked an interest in her making her own book about shapes. It was a project we worked on for a few days.

Example 3:
At age 8 J taught himself to skip count in 2s, 3s, 4s, 5s, 6s and 10s. Why? I don’t know! He said he wanted to. Apparently he just went over the numbers in his head until he could do it.

Example 4:
It’s half past nine at night. The older three children have just stayed up to watch a rugby match against Australia on TV. The All Blacks won. 16-7. K came and told me the score – he knew our score was bigger than the Aussies! J pointed out we won by nine points. He’d done a bit of subtraction as well. It only took a few seconds to tell me, but it was maths in action. And it happens all the time.

Example 5:
K, a few months later, is still into shapes! Just the other day she made a board game where you have to move from one shape to the next until you get to the “most-sided one”.

Example 6:

(I’ve been writing examples down on a piece of paper and can’t find the paper!! EEEK There were pages of them! I’ll have to start over)

July 2005
Earlier this week I found a piece of paper, on which were some hastily drawn lines resembling graph paper. Intersecting some of these lines were big studiously-coloured-in dots, and lines joined the dots. Hmmmmm. I had just recently been thinking about the fact that my children had not yet shown any “interest” in graphs! When questioned, J10 informed me this was one of those things they sometimes put up at the end of the news. Made mental note: watch news more closely and see what she’s talking about – it’s been at least six months since I even saw the news!!!!!
Probably due to my mounting nervousness that we would *never* do graphs, I pounced on the opportunity and gave a Graphs 101 lesson on the nearest scrap of paper to hand. J9 looked up from the writing he was doing to inform us all that he had seen those things at an Entomological Society meeting last year – it showed the numbers of apple moths going higher and higher, and then after the first spray the number dropped right down to almost the bottom line and went out straight for a while after that till it tailed off to nothing. And I thought we hadn’t *done* graphs!
Never mind what we think about the apple moth spraying ;-)
By the way, J10 was at that same meeting and only vaguely recalled the graphs. She vividly remembered thinking those meetings were not inspiration enough to stay up late during the week again! Can’t say I remember much else, except that that was the night that goes down in history as THE NIGHT I STARTED READING CERVANTES’ DON QUIXOTE!!!! (we sat in the back row, I put that “interested expression” on my face with head tilted slightly up towards the power point screen and moved my eyes imperceptibly from left to right and back again!!!)
Just goes to show, when you’re actively interested, your learning is much more deep / real / lasting / perhaps even profound.


July 2005 (same week as the previous example - Saturday morning this time)

The children had built huts in the dining room and were negotiating *where* they were actually located. Antarctica won out his time. J9 got a book to find out how cold it really would be there. –88.3 degrees. “That’s almost minus a hundred degrees,” he told anyone who was listening. K8 replied, “That’s more than zero” Lots of siblings just had to inform him it was “Waaaaaay less”!
Then the conversation turned to the hottest recorded temperature: 57.7 degrees. We had talked about this a few days earlier. K8 had asked and I had said I imagined somewhere around 60 degrees. J9 was adamant he had read it was 57 point something! It was obviously satisfying to him to now know exactly point what!

August 2005
Mama said, “In two months it’s my birthday!”
J9 added, “That means T will be one and a half” He’s right – she was born when I was exactly 34 and a half – to the day!
K8 thought a bit and said, “So she’ll be 18 months old in two months – that’s in eight weeks. So she’s sixteen months old today”

At eight years of age, K may struggle to write numbers, but he can “do” maths!


15 August 2005
While everyone was at the table making their New Zealand books today, I was sitting with them making a chart for the little children. I needed a circle divided equally into five segments. Knowing that no-one would be able to answer, I casually asked, “Does anyone know where the protractor is?” A chorus of “What’s a protractor?” greeted me! Inarticulately I answered, “It’s a half a shape thingy”. Before I could correct myself and speak more specifically about semicircles, three of them had jumped up to fetch the desired object and were animatedly discussing it together.
As they worked on I thought “aloud”… “If a circle has 360 degrees and I want to divide it up into five equal parts I need to do 360 divided by 5” (as I spoke I wrote down the sum on a piece of paper and noticed Jaala looking slightly fascinated at the scratchings that followed)…. “Five goes into 36…ooh let’s see, well seven times five is 35, so put 5 here…..seven fives are 35 so I write 35 down here, 36 minus 35 is one and you bring the zero down….five goes into ten twice, so I write 2 up the top…there, 72. 72 times five is 360. So I need each segment to be 72 degrees”. As I positioned the protractor over the fairly small circle someone commented that it wouldn’t fit, and they watched with some interest to see how it could be used.
No-one picked it up to have a turn, no-one was even particularly interested in my circle, but I’ll be looking out to see if anyone starts playing with protractors or division soon!

September 2005
J9 counted 110 buds on his feijoa trees.
Of course he doesn’t need any practice in counting (he’s nine years old and a more than capable counter, understanding much larger numbers), but he certainly got practice – especially as two days ago he counted 105 of them!

October 2005
Watching tv, an advertisement for Lotto came on. The kids thought it could be worth getting a ticket “just once” to see if they might win. We talked about the probability of winning (next to non-existent!) by comparing it to a jar full of sand….if you painted one grain of sand a different colour, what would be the chances of picking out that particular grain on your first try? They realised it might be at the bottom of the jar and you wouldn’t get it. Six year old K suggested you’d have a better chance if the jar was half full – missed the point a bit, but a valid observation! She also noted that if you bought one you might want to buy more and more. Right again!
The older ones then worked out how much you would pay if you bought a $15 ticket every week….$780 a year! That’s nearly $8,000 in ten years – approaching a deposit for a house!!

By the way, no-one has shown any further interest in division or protractors…yet!

July 2006
J11 and J10 helped Grandpa count the Self Denial offering at church!

August 2006
NZ coins have just been changed and this has prompted an interest in rounding numbers….which extended to rounding temperatures aswell. In fact, temperatures have been a matter of great interest this winter – the coldest winter since before I was born. At one stage J10 even graphed the hourly temperatures for a week or so!

A few months ago, L5 came to me and asked “Mum, what’s odd?” I immediately thought “odd as in queer” – but it turned out he was embarking on a fascination with odd and even numbers! That night we sat down on the floor with some duplo and I showed him the concept of odd and even when you put things in pairs. He got it straight away – in fact, half an hour later he returned saying "22 is even eh" He still talks about odd and even most days – in fact, when he does sums he’ll say something like “6 plus 5 is odd” before he tells you it’s eleven! He often says, “I’m odd” with reference to his age.
This interest has widened T2’s and M3’s vocabulary too – although, at two and three years of age, they have no idea what they’re talking about when the mimic L5’s odd and even discussions!

Money Matters: J10 is keen on doing Su-Do-Kus and there is always one in the local paper. If you send it in you might win a prize:-) After sending a few away, I had him work out how much it would cost in a year to send one each week – including postage and envelope. He’s saving his money now!!!!

Statistical Challenge
(WHAT’S THAT ABOUT? I obviously put it in as a reminder to write about something and I don’t have a clue what it was!)

Cooking Disaster
This happened quite some time ago, but I’ve just remembered it now. It shows the importance of being able to read numbers correctly. LOL
J (9 at the time perhaps?) thought there were supposed to be 11 ½ tablespoons of yeast in the bread – imagine if he’d put that much instead of the one and a half!
He has also discovered recently why you put a pinch of chilli powder in curry and not a couple of tablespoons! (along with the other spices)


November 2006
A couple of months back I told J12 and J10 I was getting them a maths text book (they seemed “ready”). J10 threw his arms around me and said, “Oh thanks mum that will be so cool” (definitely seemed ready!)
A week into it J10 had decided he “didn’t like maths”. WHAT? The only thing he had said that about before was cleaning the toilet!!!! And he LOVES maths. Well, in real life he does anyway. They both loved the fact that through doing the mental drills they halved their time for one hundred sums in under a week. That was motivating! But as for the rest of it…even though he was getting at least 98% each day, he was not enjoying it much at all.
The difference between the two Js was remarkable – J10 would write the bare minimum each day. J12 wrote out all the answers, all her workings and the problems too. Plus she started a glossary in the back of her book of terms she wanted to remember.
I slowly reached the conclusion that while J10 is CAPABLE, he is not yet interested in utilising a workbook approach and as I believe it is more important to foster his interest in numbers, than stifle it with repetitive bookwork, we changed tack. Instead of doing pages and pages of sums and then doing a test….we did the test first and then used the book to learn what they didn’t yet know, and we skipped the rest. All this with a view to dropping it entirely for J10 if his interest level doesn’t increase – certainly what we’ve been doing up to now has served us well, and J12 went for an extra year before she started the book so it won’t hurt him to wait.


T2 (and a half) is just starting to get one-to-one correspondence beyond two items. It’s so exciting!


February 2007
We’ve started up some formal stuff after the holiday break. J12 naturally wanted to continue with the maths text - and she requested to be allowed to do every single sum on every single page so that she would be assured of learning it all properly! Request granted!
J11 was given permission to NOT use the text book, but he elected to do it anyway! He, however, wants to use it differently – he quite likes the idea of just doing the bits he doesn’t already have a handle on.
Having said that, he spent about an hour doing it today. I encouraged him not to overdo it and get sick of it, but he was keen to complete the bit he was working on.

I started using Family Math with T2, M4, L6, K8, K9 – we did two of the little activities. One was just putting blocks in circles drawn on a piece of paper: one-to-one correspondence. That was dead simple, even for T2! The other activity embraced the concept of odd and even. The older ones already knew about it, but M4 didn’t - yet he picked it up immediately. He looked so pleased with himself! I liked the fact that I was sitting down WITH them doing something. It only took less than five minutes, but it was a positive time. Although I don’t actually believe the kids need the book or special activities to learn maths, planning to use the book or some of the other manipulatives or games we have for a few minutes each day helps structure what I do together with the children.

June 2007
Ha that last entry is so funny. We did one more activity the next day and That Was It!!!!! But Grandpa has been spied studying a counting book with T3.
L6 has been very time-focussed. In fact in hospital last week he saw a clock which said 17:50. He asked how you could know the time from that and I suggested he’d need to take 12 off. I was about to give a more precise explanation and tell him all about 24 hour clocks when I noticed he had an engrossed expression on his face so I left it. Inside of two minutes he said to me, “So in three minutes it will be five to six” I didn’t bother saying any more!
Yesterday K8 brought me her first ever knitting pattern she had made up and written out. Cast on 20 stitches. Knit 40 rows. Cast off. She explained to me that when she was trying it out, she had cast on 20 stitches, knit 20 rows and cast off, but it was only half as long as it was wide. So she was really hoping if she knit 40 rows it would be square. Tonight both K8 and K9 sat clickety-clacking with needles and they churned out two squares!