## Monday, October 01, 2007

### 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.