In my previous posts, I wrote about issues we had with a traditional, North American approach to teaching and learning math basics. Utilizing popular approaches and activities weren’t working, so we had to try something new.

Natural Math was a lifeline for us to explore learning math with our kid. I liked the philosophy of opening up math, not putting in artificial age-based gatekeeping, that it de-emphasized counting based activities. I loved the focus on exploration, personalization of math and fun. This article, 5 Year-Olds Can Learn Calculus was an inspiration to us.

Reflection and Symmetry

One of the first activities we tried from Natural Math was body mirroring. I would stand and my son would stand opposite. I would move my right arm to a position, he would move his left arm to match me. I would extend my left leg, and he would match with his right leg. Then it was his turn. He would waggle fingers, kick a foot, or pull a funny face for me to match. Reflecting each other could easily devolve into silly giggling and fun, if someone got it wrong or did something funny

To add more variation, I also had him stand in front of a mirror and move himself, and watch carefully what happened. We would add more variation in with movement, such as moving to the side so only part of his body, or half his body was reflected. I would ask him to think like a scientist and carefully explain what he was observing. I was cautious with feedback, and I praised whatever he came up with. I also started modelling behaviour by doing it myself, and describing what I saw using different approaches. Then he would try and add more to his observations based on what he had seen me do.

My Gen Xer brain didn’t feel like this was really math, but at least he was enjoying it. I started to research reflection in mathematics to assuage my parent guilt, since I didn’t feel like I was doing a very good job in math learning.

After a few days of reflecting fun, I started to hold up a number of fingers for him to reflect as well. He would reflect different combinations of fingers on both hands with ease. I had to be careful though, if I called attention to it with: “How many fingers am I holding up”, he would get suspicious and stop having fun. I also ordered a small foldable mirror for him to play with, and once that arrived, I would put in a Lego character or a small toy, and show him the initial reflection, then fold one side in to increase the number of reflections.

Kiddo would play with that for extended periods of time, looking at things in the mirror, shifting the angle of the reflection to increase or decrease the numbers, and adding and subtracting items to add variation. It was also interesting to see what happened to an object placed in the centre, and how the object itself would get partially reflected. Reflecting half a Lego character would lead to us exploring symmetry. If you don’t get your Lego person exactly in the centre of the foldable mirror, it can reflect in unexpected ways.

Mom is an artist, and she suggested using small colorful and geometric objects to make patterns. Tangrams are fantastic items to use with a foldable mirror. He could make mandala-like reflections using tangram shapes and folding in the mirror. This led to conversations about infinite numbers, and to a topic that piqued his interest: fractals. We found videos on youtube discussing fractals, and we started to look for repeating patterns in nature. He would spot patterns in leaves, in rocks, in certain plants growing in the neighborhood or the river valley and shout out FRACTAL!

Since this was appealing to kiddo, I decided to try to tie fractals back to our reflection work, and found instructions on making a Sierpinski triangle. You draw out a large triangle on a piece of paper, find mid-points in each side of the triangle, and then draw a new triangle within the larger one. You repeat this process as many times as you can.

The Sierpinski triangle was a bit advanced for his learning at this point, and I realized we needed to look at symmetry more. Determining the line of symmetry in an object was intuitive from mirror play, but now we needed to be more explicit. I downloaded drawing and coloring worksheets to help learn more. I started with symmetry drawing exercises where a simple object would be drawn on half the paper, but at the line of symmetry, the rest of the paper would be blank and the child would need to draw it in. They would reflect the drawing in the other axes on the blank half. He found this a bit hard, so I started looking for exercises that were on a grid. The grid helped provide a guide, but the grid also had some hidden benefits. We were sneaking more math thinking we could build on later.

Counting away from the line of symmetry and then drawing a smaller part of the overall shape within that box requires more math thinking. There is counting, but now you are thinking in two dimensions. You need to keep track of rows and columns in a grid. There is counting involved, but the underpinning mathematics brings us to a matrix. Matrices lead us to arrays, and arrays lead us to different powerful abstractions for mathematics and programming. Cool!

Kiddo enjoyed working through the symmetry worksheets. We also found that the side that the drawing was on could have an effect. It was easier if the drawing was on the right side, with the left side blank, while a drawing on the left side with the blank right side left to fill in was a bit more difficult. The further away from the line of symmetry, the harder it was to get right. To get precision he was happy with counting, and keeping track of rows and columns. If he forgot, I coached him to write down the co-ordinates on his white board.

To add variation, especially when little hands get tired of drawing and coloring, we found worksheets that used the grid to make pixel-style art. He declared them to look like retro videogames or that they looked like Minecraft, and he would color in a block, rather than drawing with precision all the time. We also used bingo daubers to color in objects, as well as manipulatives like Mathlink cubes and color counting chips. These required less fine motor effort, but provided more time for variation and play.

After a few weeks, I noticed that our math time during the days was turning into a lot of fun. Instead of dreading it, or checking out when I put the math sign up to help with transitioning, he was starting to look forward to it. There was positive energy, questions, frustration with execution rather than with concepts, and he was becoming comfortable and fearless. Discussions would lead to math in real life, and he would spot symmetry in nature, or point out things around the house. Helping out with cooking and baking in the kitchen would lead to more topics for us to research and explore together.

Things were falling into place, now we needed to build on it.