Math cat

Though still essentially an out-and-out unschooler, Fiona has been taking Introductory Spanish at the local school this semester. Nominally she's in Grade 4 and the requirement for second-language learning doesn't kick in until Grade 5 within the DL program we're part of. But she was interested in doing something like this. When we first raised the possibility last fall, our DL liaison teacher who happens to also teach Spanish and Math at the local school looked at the schedule for us and we were disappointed: Spanish was scheduled for Tuesdays and Wednesdays. Tuesdays conflicted with Fiona's gymnastics, and Wednesdays with her violin master class. So we set Spanish aside in the hope that it might work out for next year.

Then the high school schedule got shuffled, and Spanish ended up on Mondays and Fridays during Semester 2. This worked out perfectly: these are Fiona's two completely empty days. Although the course is intended for students in Grades 7 through 12 (with the bulk being Grades 7 & 8, as it's an introductory course), the teacher knows her well and was happy to welcome her. Our current principal is generally quite opposed to grade-skipping, but there is a policy that DL students can be welcomed into any class with the approval of the teacher. We had that approval, and since Spanish isn't offered to younger grades no one could really ask "why wasn't she placed with her age-mates?"

She's doing well in the course. Her test and assignment marks have been great, and she's fitting in very well, participating well in class and, more amazingly still, taking on leadership roles in group projects. She loves all the trappings of school: the binder she takes, the supplies that tuck into it, the schedule of three hours a week that she needs to be there, the deadlines for assignments, the studying for tests. It's been a fabulous introduction to school for her.

She would like to attend school a bit more next year. The obvious course to add would be math, since it's the one subject in which she has been following the school's curriculum. The wrinkle is that she's four years ahead age-wise. But Spanish has given her a chance to show that she fits in socially with those kids and can handle the organizational expectations of a high-school style course.

Yesterday she was doing some algebra at home from the end of the Grade 8 workbook. It had snowed outside. The wood stove had a nice hot burn going and the living room was cozy and warm. Fiona was sitting on the sofa with a London Fog in a mug in front of her. The cat was curled up beside her being cute and loving.

Fiona decided that math should always be done this way. "Everyone should have a math cat," she said. "This will be one of my Terms & Conditions for doing math at school next year." We envisioned the scenario:

Teacher: "Okay, everyone open your books to Unit 6.2. I would like you to do the odd-numbered practice problems on page 142 and 143. Can someone help me with this box? Thanks. You can come and get your cats now."

The teacher and a student yank the lid off large Rubbermaid container. The nineteen cats inside begin to stir.

Teacher: "One cat only per student, please.... "

Between the subjects

One of the things I love about homeschooling is the way the boundaries between subjects don't need to exist at all. I know that many schools pride themselves on making "cross-curricular connections," but those are more like threads connecting otherwise discrete areas. As homeschoolers we are free to dwell for months in the spaces between the subjects.

This fall has been a case in point for Fiona. She's been learning about ... how does one describe it when it isn't a traditional subject area? ... learning about where math, art and nature overlap.

I think perhaps the thread of discovery that led us into this playground was the stylized form of doodling called zentangles. This was clearly a form of creative precision that appealed to Fiona: the fine lines, the iterations, the patterns and conventions, the interesting figure-ground relationships.

And then, suddenly there was Vi Hart and her quirky doodling in (and about!) math class. Fiona has spent hours watching these videos over and over, experimenting with the tricks and techniques. Hexaflexagons, binary trees, Fibonacci spirals, doodling music, she's loved all of Hart's stuff.

And that brought us to October, when art class began. Last year's classes focused on the alphabet and on place: our community, the natural world we treasure, and letters, and words, and art that combined them all. But this year's series of workshops, wonder of wonders, is about geometric patterns in nature and art.

How perfect for this girl at this moment in her life!

Right from the first class drawing compasses were part of the exercises. Mandalas made with concentric circles. Lines made from circles. Equilateral triangles made from circles. Squares made from circles. Fibonacci spirals made from squares and circles.

As with most homeschoolers' classes, this one is multi-age and multi-level. The instructor is amazing at meeting the needs of barely-5-year-olds, and even some of their younger siblings, as well as challenging the 9- to 11-year-olds.

Fiona is neat to watch socially in the class. She enjoys talking to the adults -- she chats away with the art teacher and the homeschool liaison teacher genuinely and comfortably as if they are good friends, and they afford her the same respect and warmth. She and the 11-year-old are at similar academic levels and are friends from way back so they banter back and forth about science, spelling, vocabulary, music, books and math. She enjoys the company of one lovely girl who is almost exactly her age; they often hang out after class for an afternoon. But her favourite classmate to spend time with is a 6-year-old. And it's not a doting babysitter-like arrangement. She doesn't play with this kid because she thinks she's cute or because she enjoys being a leader who is bigger and older and more capable. She plays with her simply because they're really good friends. She really really likes J. as a person.

Art class is outdoors sometimes too. We walk to and from the lakeshore, or along the creek, or along the streets of the quiet neighbourhoods near the school. The kids enjoy the less structured time, the physical outlet of getting themselves somewhere on their own two feet and the chance to socialize. Along the way they're learning to see the patterns in nature, collecting specimens and samples, building ephemeral art.

I love ephemeral art. We have a friend living nearby who makes a living at sand and snow sculpture. I admit to at first thinking it was kind of gimmicky, a sort of festival competition endeavour designed to pull in tourists and make a spectacle. But I see it differently now that I've watched his work evolve, and seen Andy Goldsworthy's work, and grown older and more comfortable with impermanence myself. For kids, I think ephemeral art lowers the stakes: there's no need to strive for perfection if you know it will wither or blow away. Just enjoy the process, and appreciate your work for as long as it lasts, and that's all. What wonderful therapy for perfectionists!

I was mystified a year or so ago when Fiona announced that architecture is something she could see herself doing as a career. Mystified because she had never expressed much interest in building toys like Lego or Knex. Mystified because she had never been rapt over cranes or framing or cement trucks as I assumed budding little architects would be. But she loves the design aspect of architecture, especially spare, clean contemporary architecture. She doesn't see many clean modern lines in our little hewn-log corner of the world, but she loves images of the sexy towers in Dubai, and of achingly spare glass-and-rice-paper rooms in Kyoto apartments.

Further, she has fallen in love with contemporary design. An IKEA catalogue was her gateway drug. Now she's browsing design blogs and awards. Who knows where it will all go. At this point I'm avoiding labelling and categorizing her interests as much as I can. Although a certain amount of labelling is helpful to her in finding resources and talking about her interests, I think it is wonderful the way she is comfortable inhabiting the overlapping space shared between various subjects.

Fall bits

Where has October gone? Where has my blogging mojo gone? My additional teaching load, and all the juggling of travel and activities by Noah and Sophie, is taking a toll.

Fall was amazing until the rain finally caught up with us. The leaves were a good two weeks later to start turning, and then when they did it was in a flash of gold glory. Still it was sunny. And (relatively) warm. 

Now it's been raining for a couple of weeks. And cold, and grey, and dark. The wood stove is working all the time and we're inside a lot more. 

Fiona has been doing a lot of neat art and math. The topo map project (previously mentioned) but also a lot of Vi-Hart-inspired doodling, zentangle style art, and circle-geometry stuff. The art classes that were offered last year to homeschoolers are underway again, and this year's focus is mathematics and patterns in art, which is just a perfect fit for Fiona's current enthusiasms. An art class where the vocabulary frequently includes words like radius, concentric, tetrahedron, equilateral, rotation and intersection? Right up her alley! They're currently working on making mandalas, using various paint techniques, geometry with compasses and straight-edges, and carved print blocks. 

A last few tidbits, clockwise as above. 

We've been doing some organized running clinics. Fiona attended the first one and enjoyed it. I'm going to continue doing some regular Sunday clinics in a neighbouring village. I am trying to work up interest in an ongoing local running group, so I've put some time and energy into hiring a clinician, setting up an email loop, and doing publicity.

My knitting bug has hit again, with the stoking of the wood stove. The only local yarn shop (90 minutes away) closed last spring, so I shall have to find a good on-line source for yarn. It's a shame not to be able to see colours and touch hanks in person though. 

My networking with other runners and cyclists has made me keen to get to know even more of the local wilderness. Last weekend I stumbled on a fabulous trail that I had never heard about. A well-kept secret, I suppose. It's low-elevation, easily accessible, rolling in profile, quite and stunningly beautiful ... yet very close by. How could I not have known? I've biked it and run it and loved it both ways. It's made me hungry to truly and thoroughly get to know the backcountry around here. In the last couple of years I've got a good handle on the trails between my area and town. But that's just a wee portion of what's out there. I want to get back into backcountry hiking and camping too.

Fiona dressed up for Hallowe'en as No Face from the Studio Ghibli anime movie Spirited Away. An easy and warm costume for a typically cold and wet Hallowe'en.

The Law and Government classes are drawing to a close. These fun theatrical classes about the evolution and structure of our constitutional monarchical government system have continued sporadically throughout the summer and into the fall. Fiona has continued her role as Head of the Executive branch of government, but she now has a new role as Miss Matter, a witness for the defence in the case of against Mr. and Mrs. Brown, who have refused the installation in their living room of a video surveillance camera according to Section 10 of the Kingdom Security Act designed to protect against the rumoured terrorist plans of Evil Lord Voldesnort. The mock trial is scheduled for the Nelson Courthouse in a couple of weeks.

Math Watershed, the 4th Edition

The math watershed occurs in our family when the kids finish the Singapore Primary Math program. The 1st, 2nd and 3rd editions of the math watershed occurred around age 10 for my three elder kids respectively. That was too young for a traditional academic high school program, especially after the friendly, unintimidating, low-repetition approach of the Singapore Primary consumable workbooks. We made a lot of stabs at finding the right next step and I really didn't feel like I found the right approach with any of them. There was a lot of fallow time, a lot of treading water, or starting out one direction and then back-tracking. Eventually they all sidled into Canadian high school math courses just fine, but there was 3-4 years of aimlessness before they developed the maturity to tackle them.

When Fiona finished Singapore Primary Math at age 8, I felt it was even more important to find a suitable next step, since she was even younger than her siblings, and very keen on continuing her formal math work. So neither of us was really willing to wait through 5 or 6 years of aimlessness.

So far we've been thrilled with Challenge Math by Ed Zaccarro. At the top of the cover it says "Math is often taught as all scales and no music. This book contains the music!" It's a very apt description. It's intended to provide enrichment for children in Grades 4 through 9. Fiona fits smack in the middle of that range in terms of her math level (~Grade 7-ish) and the book is perfect for her. It has a friendly layout with occasional yet undistracting cartoon characters offering insights and fun quips. It isn't overly dense. The problem sets are appropriately varied and sometimes humorous. (eg. A snail-year correlates with a light-year, being the distance a snail can travel in a year.) There are "Einstein problems" for extra challenge, which Fiona seems to be managing just fine. And the concepts and problem-solving approaches are explicitly taught with clarity. Yet it has a more textbook-ish mature format than the Singapore Primary workbooks, and it requires more in the way of creative synthesis of skills and ideas.

I have a feeling that this book (and perhaps one or two of the others from Zaccarro's series) will provide the perfect segué into one of the Singapore secondary programs for Fiona. We'll likely give New Syllabus Math a go, one of the newer secondary programs, which has consumable workbooks and a less dense presentation than NEM or NMC. I expect Fiona will spend a year or so expanding her ability to apply K-7 math skills to deeper and more complex problem-solving through Zaccarro's books and then she'll jump into high school materials very well-prepared.

Market math

Fiona watches Sophie adeptly add the cost of three items in her head, receive a $20 bill, work out the required change and count out a $5 bill, two toonies and three quarters to make change quickly and accurately.

"Sophie's not very good at making change," she whispers to me quietly.

"What do you mean?" I ask, stupidly assuming that Fiona was unable to follow the complexity of what Sophie has just done and thinks she made a mistake.

"She should have just asked that lady if she had an extra quarter and given her a ten back. Much simpler."

Four for four

It was a proud mom day full of accomplishments.

Fiona and Sophie spent their third day on downhill skis with the school's field trip program. This time I went along. They're skiing fabulously! Sophie got moved up to the intermediate group after only one day and is carving nice parallel turns. Fiona is zipping all over the place fearlessly yet under control. Both skied part of a black (expert) run and a couple of different intermediate runs, loved zipping through the trees and tried out some deep powder. It was a great day.

Noah had been procrastinating on writing his Math 9 midterm for a couple of months. The course is mostly review, but he easily gets anxious and unsure of himself, and having never written a test that actually counts for anything, he was not keen to dive in. I finally pushed him to do it. He heard from our liaison teacher today -- he scored 96% with just a couple of little "brain fart" errors in the arithmetic: he nailed all the conceptual material.

And Erin, who couldn't think of anything better to do today, decided to go to school to do a bit of work mopping up some of the peripheral grad requirements. How fortunate. It turned out that her English 12 exam, which she was sure was tomorrow, was actually today! She wrote it, it was marked, and she scored 100%. On an English exam. How is that even possible?

The halfiest

Fiona is getting really handy in the kitchen these days. Today she baked up a batch of chocolate chip hazelnut bars all by herself from a new recipe. She got help with the mucky job of spreading the dough in the pan, but otherwise the entire recipe was assembled by her.

Oh, and she likes me to cut the pound of butter when the recipe calls for half. She doesn't have good luck cutting the brick in even halves. So I am happy to do the cut for her.

The first time I did this job for her I said "Okay, I did my best. I don't think it's perfect, but you just pick the halfiest part for the recipe."

She didn't miss a beat. She laughed. "That's impossible," she cackled.

Math humour. I love that she gets it.

Early Math Nostalgia

A week or so Fiona started Singapore Primary Math Grade 5 work and I realized that our "early math" days are gone forever in this family. These days the kids are busy with obtuse angles, polynomials and  repeating decimals. Gone are the days of "different ways to make 7" and "constructing and deconstructing tens." Today I pulled together our K-3 math materials in preparation for adding them to the library of local home-learning resources that is being amassed this year courtesy of the local public school. It was a bittersweet moment for me. I have loved watching my kids' early mathematical thinking grow through great leaps and fallow plateaus as they discover, integrate and apply all those basic concepts. But at the same time I am proud and amazed at how far they've all come.

In the hand-me-down pile I've amassed:

Cuisenaire rods. My favourite manipulative ever. My kids weren't big on actually using manipulatives, especially Erin and Noah, but having familiarity with the cuisenaires, even if they didn't actually shuffle them around on the table to solve problems, gave them a visual-spatial reference for thinking about numbers and relationships.

A base-ten set to match the cuisenaire rods. We bought a few extra 10-rods, a dozen hundred flats and a single thousand cube. We didn't use these much, but they were invaluable at times for giving the kids a visual model of place value relationships. We augmented these last spring when introducing Fiona to decimals.

Miquon Math. The first "curriculum" I introduced the kids to. They all ended up in Singapore Primary Math by Grade 3, but I think the early time spent with Miquon was the best possible curricular foundation for them. The First Grade Diary, while I never followed anything in there as outlined, gave me a great sense of what "discovery-oriented learning" meant in the context of a cuisenaire-rod math lab. The Lab Annotations book was essential. The heart of the program is in the activities more than the actual workbooks, but we also used the workbooks.

Pattern blocks. A lovely large wooden set. For free play with shapes, patterns, symmetry, angles, pictures. Two unbreakable locker mirrors, hinged together with duct tape, allowed for nifty mandala-like reflections.

The Cuisenaire Discovery Book and cards. I made this years ago for Sophie. Fiona loved it too as a pre-Miquon playful approach to getting familiar with the cuisenaire rods and many of the concepts and relationships they illustrate. You can print and download your own via the links in this post.

A Touch 'n Tell Me depression board for multiplication facts. We just lucked into this as a freebie on an eBay purchase of used kids' clothes I made once, but it turned out to be a wonderful low-tech tool for my older kids. You push down the button of the multiplication fact you're interested in and through the translucent plastic of the depressed button you can see (or almost see, anyway ... perhaps opacity of the buttons is increasing as this unit ages into its fourth decade of use) the answer. My older kids did not memorize their facts naturally in concert with their precocious conceptual mastery, so there was a period of time when it was handy to have this as a reference tool. As they gradually learned the facts, we put little Avery dot stickers on the buttons they no longer needed to use. It was fun to watch the dots gradually take over the board as they mastered more and more.

And so there it is. Sniff, sniff. I'll never again witness a kid delightedly notice that 9x6 and 6x9 are the same thing! Or that 7+9 is the same as the in-between number doubled! Sure, I am still witness to nifty epiphanies, like discovering that the repeating digits in the decimal equivalent of 11ths correspond to the numbers from the nine timestable. But it's not quite the same. It's more abstract, and less likely to be accompanied by shrieks and giggles. Ah, passages...

The gazinta bar

Ah, the family lexicon. Ours is bizarre and extensive, including such neologisms as "threehead," "agilitous," "wobbits" and "clape." We come by it honestly. My dad referred to the white residue left on one's toothbrush as "spinge." We carry that one forward in homage.

For the most part we remember where the words came from and we're all careful to keep these words within the privacy of our weird family conversation at home. But occasionally a word becomes so much a part of our lives that one or another of us is no longer aware that it's not universally understood. Such was the case with the gazinta bar.

Fiona knows that there are two ways to think about division. You can think of it according to the fractionalization model: you are splitting a larger number into so many equal smaller pieces. Or you can think of it according to the measurement model: you are figuring out how many pieces of a particular size fit into the larger number. Twenty divided by four is a quarter of twenty (i.e. 5) or else it is the fact that five fours fit into twenty.

And when it comes to long division, we use the measurement model. We draw this symbol: it's a right parenthesis with a horizontal bar attached to the top of it, extending to the right. It's a gazinta bar, because we use it when we're figuring out how many times 9 gazinta 28.9. I coined the word back when Erin was working in Singapore 4B I think. I thought it was pretty clever. It was a reminder of which conceptual model of division we needed to use when doing larger problems or those involving remainders or decimals.

Except that I realized today, after months of working with Fiona on various forms of division, that I had never explained to her the derivation of the word, nor that it was a family neologism. She was just matter-of-fact calling it a gazinta bar, as easily as she might have called it a widget or a scuzzlewhit. When I explained that our liaison teacher wouldn't have a clue what we were talking about if we used the term, and in fact that no one in the world outside our family would have any idea, she started laughing her head off. We googled it to be sure. Indeed, no relevent hits. This term is ours and ours alone.

Except that now we've both gone and blogged about it. So now all of you know.

Micro-manipulatives

Fiona's interest in math has re-ignited the past few months. She is definitely thriving with Singapore Math. I'm already dreading the day she finishes it and we need to find something else. It's just her style and just her pace. She loves the friendly unintimidating style of presentation, and the conceptual approach usually fits her to a tee. Not only that, but when she is muddled or confused by something, the sort of explanations that make sense to me work perfectly for her. It's so easy guiding her through math learning!

Since November she's completed a year and a half of the Singapore program. Every time she started a new semester I would look at the upcoming syllabus and think "Ah, here's where she's finally going to hit concepts that aren't going to come easily." But I've been wrong every time, so I don't even bother thinking it any more.

Recently she began the book (4B) that covers decimals in depth. She has an awesome grasp of place value, which I credit partly to the way her mind organizes things, but partly also to the use we made of cuisenaire and base-ten manipulatives back when she was four. I thought that might be a good starting point as we began exploring place value on the other side of the decimal, so we got them out again. Shown in the photo are the large orange 1000-cube (back right), three orange 100-flats (middle left), five orange 10-rods (upper middle) and seven white 1-cubes, all from our cuisenaire and base-ten sets. 

The new fun came with the miniature and near-microscopic new manipulatives we created, shown to the right of the white 1-cubes. We got a piece of cardboard of the appropriate thickness and cut it into 1-cm squares. Stacked up, ten of them were just about exactly the size of a 1-cube, so these were our tenths. Fiona was very impressed. She thought it was hilarious when I started cutting one of the tenths into ten miniature rods. And when I then cut one of the mini-rods into practically microscopic 0.001 cubes that look more like grains of sand than math manipulatives, she thought I was crazy. 

But it was fun, and it worked. She could see that, just as we could (theoretically) combine 1000-cubes to make bigger and bigger manipulatives denoting 10,000 or 1,000,000, we could (theoretically) continue to slice and dice our sand-grain 0.001's into smaller and smaller bits. And she had fun creating manipulative representations of numbers like 1425.174, and 20.32, and even 1000.001.

Once she has explored concepts concretely or symbolically, Fiona easily internalizes those ideas and rearranges them in her imagination, developing short-cuts and seeing patterns. The manipulatives will be put away now, possibly for good. But I don't think we'll bother saving the 0.001's to pass onto another homeschooling family!

Show your work

On the way to Calgary last week Noah was doing some math work. He was in a good mood and things were going well. He finds his current textbook (MathPower9) to be okay for the most part. It's rife with practice exercises, most of which we skip, but they're there in case he stalls for a bit. The math is logically presented and there's a fair bit of challenge. Much of it is review for Noah, but he's enjoying being systematic with his math work and filling his gaps.

But the textbook is made for classroom use and it's got a fair number of silly "working with a group" and "learning to solve problems together" activity sidebars. We normally ignore all the ancillary stuff.

So in the middle of one of these "learning to solve problems" pages Noah pulled out an puzzle problem to solve algebraically ... a number which, if thirteen is added to it and the result cubed and divided by three will equal half of itself times seventeen, or something like that. He formulated the algebraic equation, solved the problem and that was that.

And then he noticed the guidance preceding the actual problem, which said "formulate a strategy, test your strategy, evaluate the results, revise your strategy" and asked students to "create a flow chart to show your problem-solving process."

He quickly scrawled a flowchart in his notebook as a parody of the approach. It went like this:



which I thought showed off Noah's sense of humour and general-purpose mocking skills very nicely indeed. "I got the right answer. Isn't that enough?"

Graphing

The arrival of Sophie's graphing calculator as I mentioned has led to an interest in functions and graphing. I love scurrying up mathematical rabbit-trails with my kids. Yeah, I'm a math geek. I guess it comes of discovering a surprising aptitude for higher math after years of being considered an artsy kid.

We started with the concept of a function machine, something I gleaned from way back in Miquon Math. A function machine is a number-cruncher. You drop a number in the top, and the function machine does stuff to it, and out the bottom drops the crunched number. You can have fun trying to figure out what the function machine is doing inside. You put in 1 and out comes 2. Make a guess about what the function is. Is it adding 1? Drop in a 2 and out comes 5. Hmm, adding 1 doesn't explain that one. Drop in 3 and out comes 10. Drop in 4 and out comes 17. Eventually you realize the equation is "square the number and add one." These are great mathematical puzzles for little math geeks.

Next I had Sophie work with a function machine that doubled the number and added 3. She created a table of results. The input number went in the left-hand column, and the output number on the right.

input	output
0	3
1	5
2	7
3	9
-1	1
½	4

All we did next was change the names. Instead of "input" we called that number x. Instead of "output" we called it y. And instead of drawing a function machine that said "multiply by 2, then add 3" we drew the function y = 2x + 3.

Next we went on to Cartesian co-ordinates. Sophie was familiar with grids and co-ordinates even if she hadn't worked specifically with Cartesian graphing. But we had some fun with the secret puzzles like those shown above. I wrote out the co-ordinates and her job was to plot them and drawn lines connecting them to find the secret shape.

Then it just remained to combine the "function machine" and its output tables of x and y values with the graphing we'd had so much fun with. After she'd plotted y = 2x + 3 and y = x² - 1, recognized and filled in the lines produced, we fed her calculator the same functions. She was thrilled to see the exact same lines taking shape in the display.

Today we were talking about why a knowledge of functions and graphing is useful. My example was doing a study to determine how many bath towels households have, as a function of the number of household members. You would collect data and then plot them, find the line of best fit and then describe a function that would explain the relationship between number of family members and number of towels. And allow you to make predictions based on that function. This was just silly enough for her to appreciate.

Video O4L

My mom got a new camcorder and let us borrow it this week. Fiona and I decided that for this week we would do her reporting for the SelfDesign program by way of video. There are some fairly random snippets here, just showing off a few different activities that illustrate the range of things she puts her energy into over the course of a week. We put in some of her detail-slogging on violin, because I wanted to show how much more mature she's getting in her practicing. She was really enthusiastic about demonstrating Hands-On Equations, so we put a fair bit of that in too.

Of back burners

It was gone.

Now it has returned.

The interest in math. She is barrelling ahead in her Singapore book again, easily internalizing things that I expected a child so young would really struggle with. The efficiency of her learning still surprises me. She's excited to be mastering new concepts and algorithms, to be filling in pages, starting new sections, seeing her progress through the program. But that's not what this post is about. This post is really about three or four months of Fiona not doing math.

Blogs and on-line communities present a very distorted view of homeschooling. You frequently see only the "best of". You don't see the long stretches of nothing. For three or four months Fiona's interest in math has been negligible. For more than a hundred days over the past few months I could have posted "another day of no math." But I didn't. Why not? Because I wasn't worried so I didn't feel the need to speculate about what was going on. Because it's hard to take cute photos of a kid "not doing math." Because I was just not struck by the urge to write about how she had barely touched a pencil since Hallowe'en.

But today, recognizing again how easy it is to inadvertently present a distorted image of homeschooling on-line, I am posting this cute picture but instead choosing to inform everyone of the many many recent days when Fiona did not do any math at all. The scene above is a change from what has been our recent "normal."

Why do kids go through phases like this, when they suddenly lose interest? I think there are two main reasons. The first is that they recognize that they're not as ready as they'd like to be for what's next in their learning. The second is that they've got something else they're busy learning that is taking precedence. I can see that Fiona has struggled a little with the multi-step word problems Singapore introduces at this level; the bar-diagram method of pre-algebraic problem-solving for these sorts of problems just doesn't sit well with her yet. And there's no doubt that piano is where her intellectual drive has been flowing. This time it's been easy for me to see why math shifted onto the back burner for a while.

However it's not always clear to me what the reasons are. Not at the time. Sometimes I only see it in retrospect, because during the loss of interest phase the stuff that's instead come to the front burner hasn't necessarily revealed itself yet. But I now, after years of watching children learn, know it's cooking. I've seen it enough times -- several months of aimlessness and loss of apparent interest followed by a sudden bursting forth of new abilities, motivation and productivity. I don't doubt it any more.

The math watershed

This issue is one that has been coming up lots on message boards, blogs and within the discussion community at the SelfDesign program we're part of. What do we do with young children who have quickly and efficiently mastered the primary mathematics curriculum? The obvious answer is to move ahead into the secondary mathematics curriculum. But it's not quite that simple, for two main reasons.

First, the presentation of the material, when intended for a teen audience, often becomes dry and dense all of a sudden. Our beloved Singapore Math curriculum is a case in point. Look at the difference in presentation between the end of Primary and the beginning of the Secondary NMC books:


Goodbye friendly cartoon kids. Goodbye white space. Goodbye clever self-checking 'secret code' exercises. From a book which covers two problems in a two-page spread, we move to a book with over 70 short problems on a single page. My unschooled 8- to 10-year-old kids, who think of math as a nifty brain game occasionally aided by some fun pencil-and-paper exercises, have not been ready to make the transition illustrated above. It just looked too nasty and grown up, even though they were ready for the mathematics.

The other major issue is that the vast majority of homeschool curriculum choices out there are US-based, and follow the standard US practice of discarding almost all the interesting mathematical threads for two years in order to delve deeply into one particular thread, that of algebra. Variety and the discovery of inter-relationships between different areas of mathematics has always been what has kept math interesting to my kids. To set aside probability, number theory, geometry, statistics, series, patterns, algorithmic logic, topology and all that in order to focus exclusively on algebra seems such a misguided approach, particularly for fairly young children for whom math has typically been interest-led, capricious and playful.

I wish I could say that we'd found the solution. We have found a few non-solutions, and maybe a mix of partial solutions. First, the things that didn't work:

1. Singapore's secondary programs. Though Erin did eventually get through Book 1 and part of 2, these were too dry and killed her interest in math for about four years.
2. Teaching Textbooks was mathematically shallow and too slow-paced. The computer-based presentation was fairly friendly, but my computer-loving kids preferred their math not to be dressed up as computer entertainment (just like they prefer to just eat their broccoli, rather than having it sneaked into something else) and opted to use just the textbook. We all detested the algebra-only mono-diet as well as the excessive repetition and review.
3. Life of Fred was a refreshing change. The humour and quirky narrative approach grabbed Noah's attention for a while. But again it was that algebra-only mono-diet that tired him out. And after a while the narrative thread began to feel like broccoli being disguised as something else.

What I've decided so far for certain is that for young pre-teens a straight algebra course is not the best way to go. That pretty much eliminates most US-based math curricula. And that's not a bad thing, though it makes for pretty slim pickings. In terms of English-language offerings it leaves Canadian school curricula, the Singaporean selections, and a few eclectic American offerings. (There may be some British or Australian or New Zealand stuff out there, but I haven't stumbled upon anything yet that has been readily available or impressive enough to warrant ordering sight-unseen from overseas.)

Eclectic is really where we're at right now. Over the past year and a bit, as Sophie closed in on the end of the Singapore Primary Math sequence, we've done a lot of eclectic grazing. I wish we'd done more, and I intend to keep grazing with her for a while. By the time I take Fiona through this transition maybe I'll have it down!

Here's what we have used, or have on hand, or are planning to use for our eclectic grazing.

1. Calculus by and for Young People. Sophie delved into this at age 8 and it was great fun.
2. Theoni Pappas' children's books. Delightful exploration of mathematical topics through stories, explanations and demonstrations.
3. Hands-On Equations by Henry Borenson. Wish I'd had this for the older kids. Fiona is loving it.
4. Alge-Tiles manipulatives and resource binder. Great fun for factoring quadratics and exploring negative unknowns and integers.
5. The Man Who Counted by Malba Tahan. A story about a fictional mathematician full of brain-teasers woven into the story.
6. The Number Devil by Hans Enzensberger. A very fun fictional story which romps through some of the basic concepts of number theory in creative ways.
7. Art of Problem Solving introductory books. These are US-based but unique in that various topics like number theory, probability, geometry and algebra can be taught in parallel. I can't see us working systematically through the whole shebang, but we'll likely keep at least two or three of these around as resources. We just received our first AoPS book in the mail today and we'll see how it pans out as time goes on.
8. MathPower textbooks. Overall these are the best systematic texts I've seen for my kids, and I found them right under our nose at the local public school. They're Canadian and cover a variety of topics at each level, but are friendlier and more varied that their Singaporean counterparts. They're visually busy though, with some classroom-oriented garbage ("discuss with a classmate and formulate a hypothesis, then present it to your class...") but they feel less intimidating than the book above. And mathematically robust? To a degree. They are conceptual and expect a fair bit from students, especially in the sidebar challenge projects and exercises.
   

Shooting craps

Sophie was working through some number theory and probability problems from "The Art of Problem Solving" today, things way beyond her working math level, and doing pretty well at exploring them creatively and coming up with some valid approaches and correct answers. She really enjoyed the challenge. So this afternoon after she finished some algebra work, we headed off into the probability section of her current textbook to see what we could find.

In there was some basic work and a few dice problems. She had fun with that, but as usual we pushed beyond the learning there, recalling the probability dots on the "Settlers of Catan" board game we have. We decided to mathematically work out the probabilities of different rolls from two six-sided dice and then compare to the actual frequency. Once we started rolling dice Fiona got involved too and for a while I had two girls fanatically rolling dice and calling out their sums while I marked them down.

It was the sort of thing that would have flopped with the kids if I'd told them I had a lesson plan. But because it grew out of Sophie's own tangent of interest there was lots of enthusiasm. We rolled the dice over 500 times.

The graph shows our tallies over time. The lowest line is the plot of our snapshot above. Each successive line was taken from a subsequent snapshot. Eventually our tally did approximate the nicely triangular predicted pattern, but there was a mysterious lack of 9's and a surfeit of 7's. I suspect that either our dice are a bit loaded, or Fiona occasionally mis-called her sums, opting for speed over accuracy. I guess you'd call those experimental variables.

Supper was a little late as a result of all our crap-shooting.

On another tack, today represents Noah's first day back on the math horse. We had one amazing spurt of math progress, he and I, a couple of years ago, when he completed four grade levels of Singapore Math in about 6 months. But since finishing Singapore Primary Math we have found it tough to hit our stride again. We've tried a couple of different programs with middling success. When we'd hit a little bump my tendency was to give it a few days and try again. But that didn't work well. He would dread the trying again, and I would dread his disinterest and anxiety, and we'd put it off, and it would be harder than ever to find a groove when we tried again.

We now think that the real ticket to math success with this boy is the same as it is with viola -- do a fair bit of it every day so that it gets easy, because when it's easy, it's fun. Noah easily gets discouraged and panicky if stuff feels at all unfamiliar, and that makes it hard, and then the fun is gone. So we're on a quest for math momentum now, trying to prime the pump that results in self-reinforcing success and enjoyment. Today went well. All review, but with a fresh outlook and some interesting brain teasers and some solid independent self-checking work that left him feeling confident and independent. Onwards!

Cuisenaire Discovery Book

Someone recently brought to my attention (thank you Christine!) that when I changed ISPs and retired my old website last winter my Cuisenaire Activity and Exploration Book disappeared from cyberspace. So I've uploaded it again.

It's a booklet I made back when Sophie was 4 and begging for a math book of her own like her older siblings. Though she was very precocious in terms of her mathematical ability, I didn't want to start anything formal with her at that age, even anything as hands-on and discovery-oriented as Miquon. So I decided to create a little booklet for her that would include some math games that would get her familiar with Cuisenaire rods and with some mathematical concepts. I was really trying to slow her down, to distract her away from starting Miquon. As I recall she used the booklet fanatically with me for about 3 weeks until she felt like she'd got an understanding of most of what was in it. And then she asked plaintively "now can I have a real math book?"

I caved in and bought her the Miquon Orange book and she did fine with it. Better than fine. She was far more ready for it than her siblings had been because of the pre-Miquon work we'd done. Much of what was in the Orange book was just a matter of formalizing things she'd already mastered. So while it wasn't successful in terms of slowing down her progress through formal curriculum, I think she really enjoyed it and probably gained some foundational tools and concepts through it. Fiona and I used the booklet from time to time when she was 3 and 4. Several of the games are enjoyable and suitable for much older people and Sophie, then age 8 and almost finished Singapore Primary Math, joined us in playing them again.

If you're interested you can download and print your own copy of the package from the pdf files. You'll need both the booklet and, for many of the games, the cards. If you have a colour printer, you may get reasonable colour reproduction from the colour version of the cards. If you only have a b&w printer, or if you don't find the colours in the colour version match up well enough with your Cuisenaire rods, you can instead print out the B&W cards and colour them with pencil crayons.

Christmas Games

Several new games and activities showed up under the tree this year, all of which are very enticing. So far the one that's got the most play so far is Colorku. I'm a Sudoku fan from way back and love my pencil & numbers, but this game has tactile, social and aesthetic appeal to recommend it.

The other hit in game-land was Quixo, another nice wooden game with simple rules and great depth in the game-play.

More hands-on equations

We don't do much that looks like "school" in this family, but every once in a while there's something that comes up that is so quintessentially academic and visual that I just have to take pictures. Hands-On Equations is certainly one of those pursuits. I can't seem to resist snapping photos of Fiona at work with this program. These are the photos I should put in a Christmas newsletter for skeptical relatives who could never understand unschooling.

Borenson describes Hands-On Equations as "Piagetian learning," in that children are learning by doing, without direct teaching of principles and theory, but absorbing the concepts all the same. It is pretty neat stuff. Fiona has easily learned the simple basics of what are considered "legal moves" and the whole thing plays out like a game for her.

The program is divided into three levels, with Level 2 introducing negative x and Level 3 introducing negative integers. We're working through Level 1 fairly systematically to ensure that the rules of play are well understood and are already almost done. We've had a couple of short sessions and a couple of longer ones. It is working beautifully for Fiona. She now often "sees" her way to the solution a couple of steps before the end. For instance she'll see that 2x + 4 = 10 means x=3 just by looking at the initial set-up of the problem with the manipulatives. I'm amazed.

We're almost ready to start Lesson 6. At this stage the expectation is that students will start working without the manipulatives, instead drawing their symbols pictorially on the page. I'm not sure if we'll do this step or not. I'm not sure a 5-year-old is as ready for this as a 10-year-old. Plus she loves the game pieces and that's much of the allure of the program. Although she writes pretty well for a 5-year-old, I also think that sketching out the problem visually would be a lot of pencil-work for her and might detract from the fun of the program. I'm not in a hurry to get her solving algebra on paper, so we might just skip this expectation.

	Fiona has set up her equation and done the preliminary simplification. She started with 4x + 1 - x = x + 13 and has simplified it to 2x + 1 = 13 It's time for the next move.
	She removes 1 from both sides, leaving 2x = 12 That's easy.
	x = 6! It's always fun to find the solution!

Algebra for breakfast

Oh happiness! Fiona's long-awaited Hands-On Equations kit finally arrived. Breakfast today is therefore cheerios and algebra.

Shown here: x + 8 = 3x.

And cheerios.