Extending patterns

Fiona and I were waiting for Sophie's Aikido class to finish. We were hanging out in the van. She had brought along her math but after a couple of minutes decided she didn't want to do any more of that, but would like to do some "different math." On a paper I drew pictures:

square, triangle, circle, triangle, square, _____

She easily drew a triangle to continue the pattern. We used to play games like this with pattern blocks. No big deal. I decided to up the ante. I gave her:

5, 7, 9, 11, ___

which she got easily. Told me it was too easy, it was just odd numbers, and could she please have something harder. Next came:

102, 213, 324, 435, ___

and she got that one too. I was sure of stumping her with:

1/2, 1, 2, 4, ___

but she examined it for a minute, said "I can't explain it, but it's 8." I had failed to outwit her again, and she thought this was hilarious. So in a final attempt to challenge her I wrote out:

0, 1, 3, 6, 10, ___

and she looked at it for about 5 seconds and said "oh! it just steps up and up!" and laughed and told me the next one was 15. I put my face in my hands and wailed aloud about my inability to fool her. She laughed and laughed until she was giddy.

I just love it when kids can take a rudimentary understanding of something, and extend and extend their learning, moving from what they know into what they don't know with enthusiasm, pushing the envelope bigger and wider with every new challenge.

Math passages

Sophie finally got back to work and finished the final two review exercises in Singapore Primary Math. She's been mostly-done for a while, but we agreed that it was a good idea to do all the revision to kind of wrap it all up and make sure her retention is good. It was a long slog -- I think there are 7 long revision sections in the last workbook. But she did them, and did them well.

Over the past year we've enjoyed little mathematical diversions in an attempt to broaden and enrich her primary math education without moving forward too fast. But now she's truly on the cusp of secondary math and we're investigating possibilities. Teaching Textbooks is too slow and repetitive. Singapore New Math Counts is too college-like in its presentation for a 9-year-old. Life of Fred is under the exclusive ownership of Noah for the time being. We've looking into some of the Canadian school textbooks, since they seem mathematically fairly robust and don't partake of the odd American practice of giving kids nothing but algebra for a year or two at a shot, and then nothing but geometry for a year after that. Now we're looking into some other more esoteric fare. I think we'll find something fun eventually that will challenge her keen mathematical mind but not overwhelm her with dryness and density.

While I wasn't looking, Fiona finished Singapore 2B. She started level 2A last February I think, and after a bit of an early summer hiatus moved to 2B in August. She finished the first half in mid-October, was feeling the mathematical wind in her sails and ran with it through the second half. She still has a revision exercise left to do, but the content is all mastered.

A couple of weeks ago we were at our friends' place for dinner. Fiona was sporting her analogue wristwatch. Our friend asked her what time it was. (It was 6:15.) Fiona glanced at her watch. I expected her to say "3 after 6" or "6 to 3." She said "six-thirty." Well, close.

The reason we'd gone to the trouble of rehabilitating the hand-me-down little watch she was wearing was that I knew the Singapore unit on time was coming up, and figured she had half a chance at getting it this time around. And it was the very next night that she turned the page in the 2B book and encountered time. And it clicked. Just like everything else at this level has clicked. She's very much ready for the learning she's been doing and that makes it a very successful, motivating experience for her. So we'll move ahead, I guess, though I'm also going to make an effort to do some less curricular math exploration with her. My other kids have enjoyed Penrose and others of Theoni Pappas' books for children, and I think Fiona is probably ready for some of these.

Shower steam math

If Fiona's the first one up she usually comes for a cuddle in bed with me. When I get up for a shower she usually likes to follow me and hang out in the bathroom, chatting. This morning was no exception.

When I came out of the shower, I discovered she'd drawn in the condensation on the window. Not a smiley face. Not her name. Nope, with this kid it's math she doodles, as often as not.

Lower line: -2 + 6 = 4
Middle line: -4 + 10 = 6
Upper line: 1 + 6 = 7

A very small celebration

Yesterday Fiona made brownies. Today she decided to use her allotted brownie to celebrate the fact that she gets to start the Dark Green math book. We stuck a candle in it and called it breakfast, since she'd finished the last review exercise in the old book right after rolling out of bed. After getting all sugared up, she dived right into the new book. That was the best reward of all for finishing her first math book -- more math!

(For the record I should say that I don't really believe in early academics. But this particular kid is so enthusiastic, easy-going and joyful in her pursuit of academic skills that I've had to bend to her way of doing things, and I don't really worry about it. After all, it's about her, not me.)

This week is a lot about Erin and Noah, because of their more extensive music program involvement. So it's nice to have a little moment at home to put the focus on Fiona and what she's doing. Now, where is Sophie .... ?

Math night times four

In our house homeschooling is often inseparable from life but occasionally the kids parcel off little parts of their lives to dedicate to some sort of formal learning. Currently all four kids happen to be studying math somewhat formally. Most everything else occurs with only peripheral parental involvement, but for whatever reason the kids relish my involvement in any formal math they want to do. At times like this I begin to get a sense of what the days of "school at home" style homeschooling moms are like.

Here "school at home" starts at about 9:45 p.m.. Fiona wants to do some math. I sit on the couch with her and she cracks open her current Singapore book. She's usually good for two or three lessons, though lately there's been so much practice in triple-digit subtraction that she's slowed her pace a little. She especially likes to have help with the writing component. Her little five-year-old hand just isn't ready for the smallish arrays of numbers expected of Singaporean 2nd-graders. So she gives me verbal instructions ("cross out the 8, make it a seven, put the extra hundred with the tens, beside the 3 to make 13...") and I dutifully scribe. Sometimes I purposely misinterpret her instructions and make a silly error. She loves this. She laughs a lot while doing math.

Within a few minutes the other kids have realized, because Fiona's started math, that it's getting late, and they begin staking out their claim on my time too. It's unusual to have all four interested in math during the same month or season, so I get taken aback by the cascade of requests. Sophie is there, book in hand, before Fiona has finished. For a few minutes each girl is a little irked by the interference of the other. I manage to make Fiona feel like she is done at the end of a page partway through a second exercise. She moves down the couch and lies her head on a cushion.

Sophie is wanting to be done with Singapore and move on to something different. We've taken a few diversions this year in an attempt to feed her math interest without moving her too quickly to the end of Primary Math, but she definitely feels it's time to move on now. She's working in the last book now, working with geometric formulae. She is happy doing some of her work without me involved, so once I help her get started, she moves to a corner of the couch and carries on.

Noah sidles over next. He's working steadily through Life of Fred Beginning Algebra now. Most of it's review, but he loves the presentation and it's helping increase his confidence. He's now coming up with humorous Fred-style answers to the practice problems (which we do orally mostly, with the whiteboard on our laps in case we need it). Sophie and Erin both enjoy eavesdropping on the stuff from Fred. By this time Fiona is asleep.

Now it's after 11 p.m. and Erin wants a turn. She brought home the MathPower9 textbook from the school a couple of weeks ago and is now about half way through. Like with Noah, most of this new book is review for her, but unlike Noah she's fairly confident in her ability to make sense of unfamiliar things so she's pushing herself through it quickly. She does about half an hour on factoring polynomials and fussing with exponents and roots.

Math finishes up a little after 11:30 pm. And so our readaloud starts not very long before midnight. I read two chapters, but my deal with the kids is that if I'm to read any more than that (which will take me past my preferred bedtime) they must comb my hair. This helps keep me awake, and makes me happy too.

After four chapters I finally snap the book shut, my hair thoroughly free of tangles, my scalp tingling happily, my eyes drooping. I carry Fiona to bed, say goodnight to Sophie and encourage Noah and Erin to go to bed before too long. They're on the computers; who knows when they'll hit the sack.

Math programs

At various ages and stages my kids have gravitated to formal math programs. Capriciously, for the most part. Erin and Noah didn't touch anything of the sort until after age six. Noah only dabbled until about age 10. Erin worked in bursts up until the age of 10 and then turned other directions. Fiona and Sophie, in bids to do everything their older siblings were doing, started programs around age 4 1/2. Sophie's been on and off, though on a fairly short cycle. Fiona's mostly been 'on', but has it's all still pretty new to her.

For the early years we've found Miquon (to begin) and Singapore Primary Math (added or substituted later) to be a great progression. But as the kids neared the completion of Singapore Primary Math by age 9 or 10, we haven't really found a logical next step.

Singapore's secondary programs were awfully college-like in presentation for such young pre-teens. Erin eventually got through a book and a bit of New Math Counts, after a long break, but it really didn't seem 'friendly' enough.

Eventually we decided to try Teaching Textbooks. The presentation looked really good. It's a 'friendly' program with fun word problems and an engaging style. My main reservation was that, being an American program, it stops teaching many areas of mathematics for a year or two, focusing on algebra at the expense of geometry, probability/statistics and trigonometry. But we dived into Algebra I anyway, with both Erin and Noah. Erin progressed quickly, Noah, having moved directly into it from Singapore 6B at age 10, less quickly.

But oh my, the pace was deadly. So much explanation, so much practice, so little in the way of new concepts. Noah especially tended to get bogged down by the over-explanations, worried that he didn't understand, only to discover after periods of intellectual panic that the exhaustive explanation of an entire lesson was in order to help him understand something that had long been patently obvious to him. We began alternating "book sessions" with more free-form sessions, the latter being much more enjoyable to us, following tangents and exploring things together. But our free-form nights were taking us well beyond the scope of Algebra I, which was in the long run going to make working through the book even more painful. We stopped using the book entirely two or three months ago. He just didn't seem to be getting anything out of it; everything he was learning he was learning from the other stuff we were doing, but I didn't feel I could continue to lead him forward into more advanced algebra and other realms of mathematics without any sort of framework. Erin too had ground to a halt in a slough of boredom in Teaching Textbooks.

On the recommendation of our LC at the Wondertree SelfDesign program, a woman who knows Noah's penchant for story-telling and imagination, we purchased Life of Fred. It's still an American program, so it's algebra-only for the first book, but it's refreshing anyway. Noah dived in and is thrilled. Life of Fred has him grinning, laughing, screwing up his face and rolling his eyes. It is totally his style ... narrative in style, quirky, philosophical, with random bits of weirdness, unaffected and filled with humour. So far everything is review, and it will be for a while, but he's so engaged by the humour and personality of the book, as well as by the mind-bending questions and discussions in the "Home Companion" book, that he is loving working through everything.

Erin, who has decided she would like to take Grade 10 math next fall, wants to fill in her gaps with respect to the Grade 9 curriculum this summer. So she borrowed the Grade 9 standard academic math text (MathPower) from the school and asked me to help her skim through and find the gaps to fill. In the past week we've got through about a third of the course, which is about the pace I expected. She'll easily be on track to start Math 10.

But about the text. It's pretty darn good. The mathematics is robust, way beyond the level of Teaching Textbooks or even Life of Fred, more in line with Singapore's secondary programs. There are a lot of silly tangential 'brain-buster' stuff and hokey full-colour illustrations which we both find visually distracting. But all that stuff aside, it's darn good math, far better than any of the American programs I've seen. Who'd have thought? Right here, in our own BC backyard.

Set game

I'm not actually much of a family game player myself. I make an effort, and sometimes I enjoy the conversation and the kid-watching, but it's a rare family game that I enjoy for the game itself. Set is the exception. It's a visual-spatial pattern-recognition game that taps into mathematical intuitiveness. There's no strategy. It's difficult to explain what constitutes a Set to people who are language-based learners. Often the people who are Set whizzes are people who can't for the life of them explain in words what a set is.

(My succinct but oblique verbal explanation: Every card has four attributes. A set is a group of three cards about which it cannot be said that there are precisely two cards with some particular attribute. You can check it out at the game's website.)

Anyway, tonight Fiona wanted to play a big family game in the worst way. No one else really wanted to play. As she's been 'cryish'¹ lately, I offered to play a game with her. I was delighted when she chose Set. Then I remembered that she's only five and she probably wouldn't understand what defines a set. Oh well, we gave it a whirl anyway.

When Sophie was 6 she gradually got the knack of Set by playing with the purple cards only. That was one less attribute to fuss about, and the patterns became more apparent. With Fiona I tried a different tactic. Once I found a set amongst twelve face-up cards, I'd pull two out of three of the set out and show them to Fiona. Then I'd challenge her to find the third member of that set. She did amazingly well. Before long she was eagerly seeking to find her own sets, and managed to successfully identify a couple.

¹cryish [krahy-ish] adj. Of a fragile emotional state characterized by episodes of sobbing initiated by trivial triggers, particularly when due to as-yet-undeclared developmental reorganization of psyche or intellect.

Place value in bed

Fiona joined me for a cuddle in bed this morning. I'm not sure how (it happens a lot with this kid) but talk turned to math. Recently she's been using "the stacking trick" to add multidigit numbers, stacking them vertically and adding the different place value columns. After doing this sort of math mentally or with manipulatives for a couple of months, she thinks that doing it on paper like this is a sneaky fun shortcut and has been loving it. No regrouping, though -- just straightforward addition in columns. She was first introduced to it a couple of days ago and it's at the front of her mind. She started talking about it in bed.

"What do you suppose would happen if you added up your ones sometime and got ten?" I asked.

"Whoa!" She giggled at the thought of something so interesting.

"Do you think you'd have to squeeze a ten into the answer place?" I asked. I figured I'd get a clue from her answer as to whether she was ready to go further with this.

"I'd probably just put a zero," she said. Aha, I thought. I think she's ready.

"Cool!" I replied. "I think that makes a lot of sense. But somehow you'd have to fit that ten into the answer, right? You're on the right track. What we do is to take that extra ten and combine it with the other tens in the tens column. I'll show you when we go downstairs."

And so I did, right after breakfast. With coins and a whiteboard. And she got it. And extended the concept on her own into other scenarios, eventually ditching the manipulatives and merrily inventing her own problems to solve on the whiteboard.

I've paid careful attention to Fiona's evolving concept of place value over the past year or so. Her growth in understanding has been steady and uncomplicated ... and impressive for one so young. The cuisenaire rods (with the addition of a few ten-rods and hundred-flats from a base ten set) have been extremely helpful for her. Over the past few months she's gone from using them to imagining using them to merely thinking in place-value categories while performing mental math.

This week's progress feels more like a leap, though. She's now working easily with abstract symbols rather than tangible manipulatives or pictorial symbols. And she's applying those abstract symbols to more and more complex tasks -- yet without losing the mathematical understanding of what those symbols and their manipulation mean.

I wonder if it is just a coincidence that this week she also began reading dotted rhythms and compound time easily and with accuracy in music? Sometimes you can almost hear the synapses forming in this kid's head.

The cupboard under the stairs

We rearranged a few things in the family room last week. For the first time in many years it feels like we've made some significant progress towards decluttering and simplifying our most chaotic of multi-purpose rooms.

One of the results of the rearrangement was the creation of this little nook under the ladder-like stairs up to the loft. It's really just a case of the file cabinet not quite filling the space, but once we added a lamp, a stool and a naugahyde desk surface, it became the perfect little desk for Fiona. Here she is merrily working on some math, though it could be handwriting, drawing or colouring. This kid loves seatwork.

I love the Sonlight catalogue. We have found so many great readaloud suggestions from amongst the pages of the World History Core programs. And we order things from Sonlight occasionally, since they have an inexpensive ship-to-Canada system that gets around all the common glitches. My new copy of the catalogue arrived the other day.

As I flipped through it I wondered to myself -- maybe I finally have a kid who would like a curriculum. Not that I'm about to dive in a buy Fiona some school-in-a-box program at age 5. But maybe, some day, she'll be the one of my children who will want, and thrive on, an organized curricular approach to academics. She consistently and persistently enjoys bookwork. She asks for it daily. She likes working with me. She enjoys being taught things. She enjoys reading, and being read to. She's less perfectionistic, less introverted, than any of her older siblings. She's much more willing to learn "in public", with her mom sitting nearby available to help.

Time will tell. In the meantime, she's enjoying her cupboard under the stairs as she merrily plods through Miquon Math.

Backwards 9's

This is a page of math Fiona did not too long ago. I love it for a few reasons. First of all, I'm proud of what she understands about math. I'm delighted that she enjoys putting it down on paper and has no hangups about workbooks -- they're just fun the way crossword puzzles might be for adults. But I also love this example because it puts an unschooling spin on what could be a pretty conventional schoolish program.

I mean, look at this ... here she is doing addition and subtraction easily and correctly, with an understanding of place value, but she's still writing 9's backwards. And about half of her 3's too, and occasional 6's and 4's as well, though you don't see those here. For Fiona a solid vibrant foundation in math has nothing to do with knowing how to make tidy numerals that point in the right direction.

I remember Sophie at age 7, doing multi-digit multiplication, and incorporating fantastic mental math shortcuts that demonstrated a firm mastery of place value, but still occasionally writing "13" as "31" and other such incongruous mistakes.

Fortunately my kids don't read curriculum-writers' scope and sequence. They don't know that multi-digit multiplication shouldn't be taught until they've sorted out whether twenty-four is written as 24 or 42. They don't know that work with parentheses and multiplication should wait until after you have learned to print a numeral 9 correctly.

I love these little incongruities. They're like the little stories I can sometimes read outside, the ones made of a bicycle left near a copse of trees, a discarded jacket, a pile of special stones, a stick in the ground. And one sandal. How odd. Where is the other one? What happened here?

These little trails of clues are evidence of my kids charting their own unique course, led by interest, veering off in unexpected directions, finding unusual ways of looking at things and moving where they decide they should. The backwards 9's are evidence that my kids are going at things their own way, according to their agenda.

Toilet paper algebra

Fiona is working through the Miquon Blue book and was thrilled to hear that the pages which assign letters to numbers and manipulate them in code were exercising her pre-algebraic skills. She has older siblings whom she knows are doing algebra, and she thinks algebra must be pretty cool. I explained to her that algebra was a process of trying to figure out what a mystery number was in balanced equations. I said I was pretty sure that she'd be able to do some simple algebra if she wanted to. She was keen, so I got some wooden checkers, the pan balance and a piece of toilet paper labelled 'x'. We had great fun hiding unknown numbers of checkers under the toilet paper, setting up a balanced 'equation', and using our math skills to solve the mystery.

My other kids weren't intrigued by algebra this young, but Fiona seems to be -- likely because she's drawn magnetically to whatever the other kids are doing. Her eyes really lit up with this play. If she keeps asking and expressing an interest, she might be a candidate for Borenson's Hands-On Equations which I once looked over and really liked. We have an obscure manipulative-based program intended for high schoolers (Alge-Tiles) which I've used a bit with the older kids, but it gets into exponents and negative numbers right near the beginning. Borenson's program would be much more at Fiona's level and I've always wished for an excuse to purchase it.

Khet

We seem to have a fondness for family board games played in relative darkness with small sources of light part the the strategic gameplay. Khet seems destined to be another favourite. This was our Christmas Eve game this year.

Basic Khet is played with four types of gamepieces. Almost every piece shares the same two basic moves. Where they differ is in their reflective capacity. Djeds reflect the light at 90-degree angles from both sides, pyramids do the same from only one side. Obelisks block light. The pharoah is the "king" of Khet, naturally, and the aim of the game is to protect him from being hit with the laser beam that originates in the corner of the board and bounces off the various pyramids and djeds. The expansion set adds two "Eye of Horus" beamsplitters. These split the laser beam, transmitting one part of it straight through and relecting another portion just like a djed.

This game taps into visual-spatial, logic and strategic skills in a huge way. My kids all seem to have a lot of these things, so, combined with the dark and the lasers, the game seems likely to get plenty of play, both according to the rules and outside-the-box-wise.

A pretty good math-er

Erin had a friend, another young unschooled teen, staying over last night after our community orchestra performance. In the late evening we were all sitting around the fire reading, chatting, knitting, whatever. Fiona decided to do some math. She's almost done the Miquon Red book now and was working through a page of combined multiplication and division. There were a lot of answers that were 12 in the first half of the page. She noticed how many there were and counted them up. "Wow, that's a lot of 12's. Ten 12's."

The general conversation around the fire went on for a moment. Erin's friend J. was talking about something or other.

"That's ten 2's more than a hundred," a little voice piped up. J. stopped talking and looked at Fiona.

"Ten 2's is twenty," continued the little voice. "So, a hundred plus twenty, that's a hundred and twenty."

J's eyes bugged out. "Wow," she said, "I could do that easily, but I'd still have to think about it. And I don't know anyone who could have done that when they were four."

"Well, I'm a pretty good math-er," explained Fiona.

Structured formal instruction

The other day I got thinking again about some ideas I'd picked up reading the Gordon Institute for Musical Learning website. Gordon talks about different types of instruction and how they're appropriate for children at different ages. I think the distinctions he makes get at something very fundamental. Here's how I interpret his ideas.

First comes the stage of readiness for informal unstructured learning. The adult presents things (or "creates a rich environment") in an unstructured fashion and expects no particular response from the child. This is the closest to unschooling, and it is all that most children are ready for prior to the age of 3 or 4, regardless of their IQ or academic level or whatever. You might liken this to a fallow field that is scattered with wildflower seed and then carefully watered and protected from wind. Amazing things may very well grow, but it's pretty serendipitous.

Next comes the stage of readiness for informal structured learning. Here the adult presents things in a structured fashion -- sequentially or in a way that is contrived to hopefully produce certain types of learning. But again, no particular response is expected from the child. Some examples of this type of learning might be offering to play math games, presenting opportunities for playful literacy learning, or demonstrating for a child as he tries to learn to tie his shoes. Some children will be ready for this type of learning by age 3 or 4, others not until 5 or 6. You might liken this to a fallow plot in a garden into which the adult has planted specific varieties of flowers in carefully designed rows or beds. The important thing is that no particular response is expected from the child. All you can do is create opportunities. At most you give guidance -- a metaphorical gentle touch at the elbow. Whether the child learns or performs or participates is entirely up to him and as the adult you need to be okay with that. I think that the beauty of the Suzuki method of music education, when properly applied, is that it capitalizes on this stage of readiness.

The final stage is that of readiness for structured formal learning. Children will be ready for this between ages 8 and 12. Learning is presented in a structured fashion and particular responses are expected from the child. One might liken this to a commercial market garden, where specific crops are planted, with thinning and pruning and fertilizing taking place in order to maximize yield.

Now, Gordon believes structured formal instruction ought to start at age five. I think that's far too young for most kids, especially boys. But even more fundamentally I disagree with Gorden in the "should." He believes that chronological / developmental readiness for a particular instructional approach obligates its use. I believe that autonomous motivation is also required. The child should be requesting (whether with words or actions) a shift into formal learning before that shift should take place.

The difference between informal and formal structured learning is chiefly in the expectations of the instructor. The instruction is given in a structured format in either case, but in the latter case certain responses are expected. It's all in the adult's attitude and expectations!

We're back in a phase of fairly enthusiastic math bookwork in our family (that's Erin in the photo above enthusiastically doing extra algebra). Sometimes I sit back and think "How can this be? How can I have children who are perfectionistic, private, highly autonomous learners with strong aversions to anything that smells of school-like expectations, and yet who actually like sitting down with their mom to do math bookwork?" I wonder if the secret is that I am not really attached to outcome. What we do at the kitchen table looks a lot like what school-at-homers do, but my quirky, anti-schoolwork, oppositional kids actually enjoy math bookwork because I don't expect it of them. If they decided not to do any (as most of them have, for long periods lasting up to two or three years at a stretch) they know that's okay.

Music instruction has been structured in our family from a young age, but I think that in some sense there's a suspension of specific expectations there as well. I believe they have the ability to do very well, and that ultimately, if they want, they'll be very fine musicians. But I don't expect mastery of anything in particular on any particular timetable. I only expect that if they want lessons, they will make a reasonable effort to use the teaching they've been given at their previous lessons, because to me that's a matter of respect. But specific mastery, specific types of practicing, specific amounts of work, no, I provide guidance but if they aren't interested in following it, that's okay.

Odds have middles

Generally speaking, we've got a good thing going on with math at our house. We don't have a ton of manipulatives and educational games (cuisenaire rods, base-ten add-ons, and pattern blocks, mostly) but we have kids who, at least through the K-6 levels, enjoy discovering and implementing things about numbers, operations and relationships between them. I don't quite know how we ended up with this lovely math tradition, but we did.

Photo left: the face Fiona gets when she suddenly thinks her way to the solution to an equation using an "easy path," some mental math trick that helps her solve it easily. In this case, 2 x 15 became two 5's plus two 10's, and that was easy.

Fiona's curiosity about math continues to be boundless. I swear, this kid thinks about numbers non-stop. Probably in her sleep. On our trip home from Calgary she awoke from a nap and said "I just figured something out. Nine is half of eighteen."

A couple of nights ago Sophie was doing some multi-step word problems using ratios. She'd sorted out all the conceptual stuff and was down to the final arithmetical step. As often happens once the fun of formulating the equations is done, the arithmetic was less interesting. She needed to find a fifth of ninety, but her attention was wandering. "Use the division the other way around," I prompted her. "Try to figure out how many fives in ninety. You know how many fives are in a hundred."

"That'd be twenty," piped up a little voice beside me. Not Sophie. A littler voice. It was Fiona, delighted to be "helping" Sophie with her math.

Fortunately Sophie found it funny.

And then this morning, fresh out of bed and with her math mind churning, Fiona explained to me that odd numbers are the ones "with middles." I asked her to explain.

"Well, like five," she said. "If you count five things, the 'three' one is in the middle. Same for seven, nine, eleven, thirteen, fifteen, seventeen, nineteen, twenty-one and twenty-three. All odds have middles."

This observation wows me more than any of the other clever math problems she's solved. Why? Because she's observed a property that some numbers have but not others, described it to herself, developed a hypothesis about what is common to all the numbers that have it, tested the hypothesis in her head and correctly generalized and described the pattern.

All while still wearing her favourite long-outgrown size 2T snowflake pyjamas.

Academic precocity

Back in June I wrote about Fiona's desire to start a formal math program. Shortly after that post I caved in and ordered her a new copy of Miquon Math. I'd used Miquon with my older kids, to a greater or lesser extent, and figured its playful, casual, "guided discovery" approach based on manipulatives would be a good fit for Fiona. Sophie had started Miquon at about the same age, and though she'd moved much more slowly through it than her older siblings had when they'd started at age 5.75 or 6, she'd enjoyed it and ended up with strong math skills.

So I didn't fret too much over Fiona's equally early start. The summer was upon us and we were all too busy with musical things to put much energy into math. So at first I thought she'd just putter through the first (Orange) book over the course of a year or so. But about a month ago, Fiona ramped up her math interest and fairly rollicked through the entire Orange Book. She's now well into the Red (second) Book and I'm beginning to wonder if she'll ever slow down. In the past couple of weeks she's made some big leaps in her understanding of place value and new concepts are falling into place quickly. Miquon is supposed to be a "conceptually advanced 1st through 3rd grade program" in that it introduces all four operations in the first year and progresses to things like algebraic-style problems, Pascal's triangle and cartesian co-ordinates over the next couple of years. Common wisdom cautions parents about starting their kids prior to the age of six, and suggests that they expect to move very slowly with five-year-olds. I've never been one to follow common wisdom, but I do try to take it into account as I find the right path for me and my children.

I thought I was past these worries, having dealt with various flavours of academic precocity
with my other kids over the years. Yet once again I find myself worrying about burnout and fundamental gaps and frustration. Should she really be moving into this material at her age? Isn't it too much too soon?

Trust, I remind myself. Trust the child. She wouldn't be eagerly devouring if it wasn't doing positive things for her. She knows what's right.

Two and a half

What kind of a four-year-old misunderstands the following question in a math workbook:

1/2 of ___ = 5

and answers

2 1/2

without missing a beat? Sometimes her mistakes are the smartest things!

Snowflake fractal

Here's a fractal Sophie and I worked on together, guided by the "Calculus by and for Young People" program we've been dabbling in. If you click to go to the full-size image, you'll see that we actually began the insane task of drawing in the 6th iteration in the lower left corner.

Sophie hypothesized that the series that defined both perimeter and area of the fractal through successive iterations would diverge. In fact, we were able to work out that only the perimeter diverged, while the area converges.

You can see that I have been having fun with the "scan" function on my new copier.

I'm thinking of a number

The big kids all played this at some stage too. Fiona wanted to give it a try today while she and I were taking an extra trip to Nelson. It goes like this:

"I'm thinking of a number that's bigger than 5 and smaller than 10."

"Is it 7?"

"Too small."

"Is it 8?"

"Yep! You got it!"

The variations are endless and when the parent is doing the "thinking of a number" the game can be used to stretch a child's understanding and interest into new areas. I've been known to try to "trick" my kids with fractions, negative numbers and irrational numbers. When played as a "multiple guess" game it's a great one for helping kids develop rational strategies based on probability. For instance, if the number is known to be between 50 and 100, 75 is a good guess, while 51 is not. It often takes kids practice to get the hang of that sort of problem-solving.

Today we played some "multiple guess" rounds, but I also tried pushing towards the limits of Fiona's conceptual understanding to see where those limits are. Among the questions she got:

"I'm thinking of a number that's how old Daddy will be in two years."

"I'm thinking of a number that's half Sophie's age."

"I'm thinking of a number that's two after 12 on a clock face."

"I'm thinking of a number that's one less than 0."

"I'm thinking of a number that's how many days there are in two weeks."

"I'm thinking of a number that's how many days there are in two weeks if you don't count the Saturdays."

"I'm thinking of a number that's double 8."

"I'm thinking of a number that's one less than two tens."

Among those she didn't get:

"I'm thinking of a two-digit number that's less than 11." (I don't think she really understands the difference between digits and numbers.)

"I'm thinking of a number that's halfway between 3 and 7."

Fiona is positively driven in the area of math lately. She's pouring just as much passion into this as she did into violin last spring and into painting through last winter. She's now almost three-quarters of the way through the Miquon Orange book, and this isn't half the math she does -- she's always peppering people with questions and announcing numerical relationships she's been working out. I thought Sophie was wild about math at this age, but she wasn't half as obsessed as Fiona.

Sophie's Calculus

A few weeks ago I ordered a copy of "Calculus by and For Young People". I had hoped to get it in book form, but it turned out to be the CD version of it that I got -- the product description said it was a book, so it was misleading. It also isn't the main book, it's the worksheets book. That's actually the bigger, thicker book (300-plus pages, rather than less than 200 for the text), and it's the one with the activities and exercises all nicely laid out, so I guess if I'd known that and had decided to order only one of the two components, I would have chosen this part. But I think I want the text too. I don't want to pay shipping if I don't have to, so I've got the author looking into the possibility of selling downloadable copies. It's only about a 15-20 MB pdf. We'll see. Anyway, it only seems to be available in electronic format, so my printer is getting a bit of a workout with the worksheets and instructions, and I'm looking forward to getting the textbook at some point.

Sophie (8) did first unit with me, with Noah gravitating in towards the end, his interest piqued. This is a section exploring fraction series like...

3/5 + (3/5)^2 + (3/5)^3 + (3/5)^4 + .... (3/5)^n

(with the ^ indicating an exponent). When I thumbed through to the end of the section I thought "there's no way my kids will get this!" but when we actually worked through it a step at a time, they did. It starts out with a colouring exercise ... colouring in half of an 8x8 grid, then colouring half of what's left, then half of what's left now, and so on, until you see that almost every last speck, but not quite, of the original area, is coloured in. Then we plotted the cumulative sum at each step on a graph and so how it got closer and closer to 1.

Then we repeated the exercise with (1/3) and (1/4). In our case we used our calculator to compute the cumulative sums, and I certainly wouldn't have wanted to do otherwise. The program is clearly not about arithmetic (i.e. computation), it's about mathematics (i.e. mathematical concepts) and if one tried to work the computation out with kids of this age, or any age, really, it would get very very onerous. So I think liberal use of a good calculator is essential. It allows you to quickly see the patterns arising. At a minimum I'd suggest a 2-line calculator that handles fractions as fractions and y^x exponents. We have the TI-34II and find it works well. It's not programmable, which might be nice, but for $25 rather than $175 we'll deal with the (minimal) limitations.

There are some computational and notational skills that need to be taught unless the children have already learned those skills. For instance, the concept of exponents, the short-cuts for computing, say y^2 multiplied by y^4, use of parenthesis, order of operations, negative numbers and the factoring out common factors in equations. While the program says that kids from 7 on up can use it, not all of these little bits and pieces are actually taught in the program. The parent would need to be able to fill in any gaps. So, for instance, I had to teach Sophie how to recognize that ...

9/5 + 9/25 + 9/125 + 9/625 ...

could be re-written as

9 (1/5 + 1/25 + 1/125 + 1/625 ....)

and how recognizing that factoring pattern was crucial to solving the series (which, in case you're interested, is equal to 2 1/4).

Sophie is really really loving the conceptual challenges she's finding in this program, as well as the introduction of enticing advanced arithmetical skills like working with negative exponents and so on.

So I would say that so far our experience has been very positive. But I'm not sure 7 is really a realistic age guideline for the program. Sophie is 8-almost-9, and doing Grade 6 math, has a math-savvy parent to help her fill in the gaps and a keen attitude -- and she's doing fine with it, though it's stretching her a lot. I think ages 11 and up would probably be a more realistic guideline ... and the parent would still need to help fill in a few gaps if the child hadn't done, say, Grade 9-10 level math already. We're looking forward to exploring the program further in the future. It's a really enjoyable diversion that I hope will give my math-smart kid some opportunity for lateral growth in her mathematical understanding, rather than just continuing to plow forwards through a sequential curriculum.