She also learns math without me. There are plenty of times when she happens upon something 'by accident' by which I mean: things (books, puzzles, making supplies, marbles, tape...) left around the house, acted upon, forgotten for a while, ultimately to be rediscovered a month or three later and acted upon again, but this time in new ways.
This happens a lot 'round here.
For instance, we have tangrams in almost every room of our house. There's a magnetic puzzle book set in the car and approximately six plastic sets on the game shelf, of various colors and all mixed together. I have four sets of magnet tangrams that I got at the NCTM Annual Meeting in Indianapolis this past April. They have been on the fridge in the kitchen since then with minimal interaction...until recently, and, last I looked, they were all in use in some kind of design or another.
Here's what I found my dear child doing with the set of car tangrams (inside for some reason) over the last few days.
By way of setting the scene, I should tell you that the kid generally initiates and works independently on any project she pleases during her afternoon quiet time, usually with the company of an audio book. A few days ago during this time she informed me she was "making a math book." I had observed her tracing tangrams, but was surprised to find that she was doing it in a deliberate way, and that she was also able to clearly describe her discoveries and observations to me. Here is the book, so far. I am just a scribe here -- these are her words as she dictated them to me, except where noted:
"Four of these triangles that you see here can make a square. If you pull these triangles apart you can see that they're little triangles. But you can see on this page that they make a square." [I see that she has drawn them as individual shapes, which, over the course of her illustration, merge together into her intended shape.]
"This rectangle you see is made up of a parallelogram and two triangles. Really they're just shapes, but when you put them together they make a rectangle." [It looks like she's numbered the inside angles of the individual shapes. It also looks like she is again showing the process individual shapes merging into the intended new shape.]
"You see the wheels of this bike as rhombuses but really they're squares turned so their points are facing up and down, and to the side." [She was gesturing this first, and at first she used the word 'flipped' to describe the orientation of the square wheels. I focused her on the orientation of the corners to describe how the square was turned.]
"The square and the rhombus that you see here, their edges are both the same length. The difference is a rhombus is a squished square, squished to its side. The rhombus has two larger angles and two smaller angles than the square. But the square has the same angles on each corner." [These are actually tracings of shapes from the pattern blocks set we have. The ruler markings was her idea for comparing the two shapes. I supplied some new vocabulary in the form of 'angles' and made some observations about the difference by overlapping the two shapes, in an effort to help her observe and further articulate the difference between the two.]
Writing about all of this made me think to look up more about the van Hiele levels for geometric reasoning. I need to spend more time with that to analyze how I'm interacting with future discoveries of hers. I also think I'd like to focus more specifically on flips, turns and slides. Maybe I'll be super ambitious and have us play around with flips and turns using both 2D and 3D shapes!