## Wednesday, September 28, 2016

### Homeschool Math: The Decanomial Square and Its Extensions

Just yesterday, I told a friend that if I'd had preschool to do all over again, I'd have skipped Montessori and saved the money.

When I said that, I was thinking of 1) how expensive Montessori was, and 2) how I'd primarily thought of myself as using it for childcare and social experiences, and so I'd perhaps have been better served with a a cheaper alternative.

It didn't take much reflecting on my statement, however, to realize that I was wrong. I'm glad that we didn't continue with Montessori, of course, because homeschooling has been vastly superior to any available alternatives, but Montessori, I think, has deeply affected how my children view learning, and how I mentor and mediate their learning.

Montessori is where I developed my firm philosophy of--other than screentime limits--never disturbing a busy child. Children's focused concentration is precious and not something to be lightly interrupted. That focused concentration, whether directed at "work" or play or nothing at all, as far as you can tell, is children thinking deeply and hard and they'll later be able to apply that focused concentration to their other work and studies.

Montessori is where I first saw work plans, and now I couldn't imagine our homeschool without them. For the kids, a work plan is expectation management, empowerment in how they're going to conduct their day, and a good model of clarity and organization that leads to a solid work ethic.

Montessori is where I became firmly convinced of the vital importance of hands-on manipulatives, especially for math. We think of math as cerebral, but it's also visceral and and intuitive and sensorial; you may know how to do long division on paper, for instance, but when you physically do long division, you understand how it works for real.

Much of our own math manipulatives, then, are Montessori-style, even if I create them myself and alter them from the specified Montessori format. The decanomial square, for instance, is a physical model of the multiplication table. In Montessori, kids compile a slightly abstract paper model (Montessori relies on the reinforcement of the color coding of numerical values, which kids have long internalized by this time, so that they can get away with having a less concrete representation) that they then will recreate with the bead manipulatives that they use. Kids can do some really sophisticated extension work with this material, including bringing out some of their old preschool manipulatives (the pink tower and the brown stair, in particular) to make clear the relationship between all of these different ways of representing numbers.

We use Cuisenaire rods instead of the Montessori bead materials, and we sure as heck don't have a school's worth of them to build ourselves a decanomial square with, so I didn't want to use a decanomial square that relied on color-coding to impart much of its crucial information. Instead, I wanted to make it gridded with centimeters throughout, and make the relationship of each piece to the number that it represents (as well as its area and perimeter and the length of each side) clear that way.

Thank goodness that my husband is a graphic designer!

Matt designed the decanomial square that we used (we need to figure out how to best format it for a home printer, and then my goal is to make it available for sale if there are any other Montessori-obsessed homeschoolers out there), and although he color-coded it to our Cuisenaire rods, I printed it onto cardstock in shades of grey, on account of I'm too lazy to replace the color ink cartridges on my printer. Syd then helped me assemble the pieces into the complete square and then cut them out.

In a Montessori setting, you would model the assembly of the decanomial square in an organized way, but I'm mean, so I gave it to the kids as a puzzle, with no other clue than that the finished piece would be a perfect square. I made them figure it out completely without my assistance, and wouldn't you know it, but they eventually (after some griping, and then some settling down to get to work) did present me with a perfectly assembled decanomial square!

That process was one complete math enrichment work for one school week, but last week we played some more with the decanomial square, extending our understanding of what it can offer. First, I had the children build the square again, which they did this time with minimal fuss (other than losing the 2x2 square, sigh, which we actually really needed for this lesson. Oh, well... I do intend to replace this particular model with the colorful one as soon as I can get Matt to replace the ink cartridges for my lazy ass, so then this entire greyscale one will be just spare parts).

Then, I demonstrated one of my favorite equations in all of math: the Pythagorean theorem. Will has encountered this before, and Syd has worked with squares before, so it was a good lesson for them both. I set up the 3x3 square and the 4x4 square at right angles to each other, and told them that these were two sides of a right triangle, sides a and b. They needed to find the square whose side made the perfect hypotenuse, or side c.

And they did!

A squared plus b squared equals c squared! If you worked hand-in-hand with a good graphic design program, it wouldn't be hard to print out physical models on the spot of the hypotenuse of various other right triangles. You could print out a square with sides exactly 5.3851640713 cm long, for instance, to go with your 2x2 square and 5x5 square.

Hmmm, maybe that's another set of models that Matt should design for me?

Most of our work, however, involved using the pieces of the decanomial square to build equations (pre-algebra for the win!). We set out one piece of the square, then covered it completely by puzzling together other pieces--

--then wrote an equation to represent that model:

For instance, one equation might read:  5 squared (ugh, I wish I could find the superscript hotkey without looking it up! Soooo lazy!) = 4 squared + 4 + 5. (I didn't mention it, but do you notice that this is also the Pythagorean theorem? So cool!) Another might read: 3 squared = 3 + (2x3).

That was the extent of this particular extension lesson, but there are so many more things to do with the decanomial square, and we'll be revisiting it often throughout algebra and geometry and possibly into trigonometry. I'm currently on the lookout, for instance, for a cheap version of the Montessori pink tower (you *can* DIY it, but with a to-do list as long as mine...) so that we can have physical models of the cubes represented in the decanomial square.

When I've finally gotten my hands on one, I'll be VERY curious to see if the kids remember it from their own Montessori preschool days. Between the two of them, I wouldn't be surprised if they'd built it a hundred times over the years that they were there.