Between 6 & 8: Learning Math by Ear, Part Two

My daughter may be headed to school for the first time in over two years, but she's apparently still learning at home, too.

A big maker, doer and thinker (but, interestingly, not a big builder) my child has historically loved making maps (math!), drawing diagrams (math!) including sewing patterns, and generally thinking math 'out loud' in a visual way. The first image below is from when she was six (a map to Grandma's house the next state over) and the second two are from when she was seven (a map from our house to our 'second classroom' at the local co-op via the college campus, and a sequential diagram of how to turn a whole piece of cloth into a dress).

She's eight now and just look what she did this weekend!

She had been trying to build a fort on the couch but it kept falling down. "Oh well, I guess I'll make a plan for a new fort," she sighed.  

At that point I left to do other things. When I came back much later there was essentially a large cube in the middle of our living room made out of the PVC piping and connectors that had been lying around unused for the last half year.

I took a closer look at her plans. "Why are there two squares?"

"One is the bottom and the other is the top."

"And how did you get those numbers?"

"I measured the pipe."  (Apparently with an 18" metal ruler)

Here's what I noticed:

Neither drawing is quite to scale (inch to foot) but what stands out to me are her very clear markings and groupings of twelve inches in each segment as well as the fact that each edge is 12, 12 and 6.  And, it is also fascinating that she was measuring in 12" segments even though her ruler was 18" long.

Also, she drew the image of the top of the structure with the ruler (square on right) which, for some reason, I find important as it is indicates a shift toward precision that I have not seen before.

This is so exciting to see because it is the first time she's modeled/planned something before she's made it -- a level of abstraction that is brand new. She also obviously wanted to accurately portray how long each edge really was as well as the fact that there was a top and a bottom to the real space she was creating.

As I put together this post I realized I was getting a bird's eye view of the mathematical development of one particular child over a two-year period, specifically the progression of mapping and modeling between the ages of six and eight.  I suppose it's because I was there for every bit of it that I find the view so compelling. Also, none of this happened in a vacuum; despite my child-led, seemingly hands-off approach I have been quite intentional about the structure and tone of her learning environment, especially in terms of having a chance to play with and talk about mathematical ideas in multiple ways. For that reason, I humbly submit this post as another example of what it means to learn math by ear. 

A Vision of Precision, Revised

Every day this week we've been playing with math dice. Enthusiastically.

I'm not going to name the company because not only do I not review or endorse any product on this blog for money or power (not that they asked) but it is also quite easy to go out to your local games shop and get your own set of two 12-sided and three 6-sided dice. (The rules are also pretty easy to figure out: multiply or add the two numbers on the 12-sided dice and then roll the six-sided dice and try to find a way to make the target number using as many operations as you know.)

Did I mention the enthusiasm?

My newly eight-year-old is enthusiastic about many things but has always been a little standoffish with her affinity for math, probably because, I think, she perceives it as my 'thing'. So, it's been nice to be able to truly enjoy a math game together.  (It's been a while -- we were heavy into UNO a couple years back which was super fun.)  It's clear my kid is on her way to a happy relationship with operations, but there's something even more interesting developing...

I had always thought my girl was not what I would call 'systematic' or 'precise'.  I know for sure she is prone to intuitive leaps of connection or understanding and lots of messy tinkering, none of it looking either precise or systematic to my eyes.

As I've been drafting and revising this post I've realized that maybe she has been those things, I just haven't been able to see it.  And, as we've been playing the dice game I've watched her systematically running through different combinations of the six-sided dice (by moving the dice physically to different positions) and  reasoning to herself out loud as she thinks through the different ways to use the hand she's rolled.

I guess I always thought that precision in mathematical problem solving looked, well, neat and orderly and on paper.

Anyhow, I am not (too) ashamed to admit that I was wrong. I think she's been precise and systematic in her own way for a while now. In retrospect, I realize I've heard this kind of  'talking herself through' a series of moves or ideas before. Systematically. In math and in many other contexts. For years. In a messy, verbal, highly enthusiastic way.

Okay, so I'm a slow learner I guess, but pretty open minded all the same. I think it's worth considering that there must be a difference in the way children and adults go about their reasoning. Or, at the very least, that I have a deeply ingrained image of 'what it looks like to do math'. I'm going to keep thinking about all this. If you have any observations or resources to share on this subject, I'd be tickled pink.

In the end, I'm super impressed that not only is she beating the pants off me but she has also created her own strategy for combining operations to reach a target number.  And it's all her.  The only thing I did was bring out the dice.