During this spring break I had the blessing of spending time with my wife’s family in Maryland. Her sister and brother-in-law joined us along with their two precious children, our niece, two years old and our nephew, 5 months old. We spent a good amount of time playing with them because we live so far away and do not get to see them often.
Learning from Children
I was especially intrigued by the memory and learning of the 2 year old (the 5 month old smiles, but that’s about it as far as “interesting” goes). It reminded me of some of the blog posts I’ve read about teachers interacting with their own children, and just how incredibly telling that can be of human behavior, no matter the age.
I wanted to describe one interaction I had with my niece, Caroline, which involved putting together a puzzle. Typically Caroline has her go-to set of games, books, and puzzles, and she will usually flying through puzzle after puzzle, going on to the next one as soon as she has completed the previous one. One day, her father found a few other puzzles in another cabinet that Caroline had never seen before, and I found out later that Caroline, when interacting with new puzzles, will do them over and over again because they’re so fun. 
This puzzle was different than the rest because the previous 5 puzzles she had done that day each had a blank background under the puzzle. (See below). I believe this is typical of puzzles and may have been the only type of puzzle that Caroline had done in her life up to that point.
The new puzzle had an image under the board which was covered by the pieces as you did the puzzle. The interesting thing was how the puzzle pieces did not exactly match the image they covered, but were similar in most cases. For example, you had a cow walking out of a barn on a puzzle piece, and it fit on top of a whole herd of cows which were penned up in the barn. Yeah, I know, a really neat puzzle. (See pictures below)
The puzzle was on the upper end of the scale of complexity for little Caroline, though not the hardest she had done that day. The first time through the puzzle, I decided to help her out by asking her about the image on the puzzle piece and the image on the background. She not only correctly identified (almost) every animal, but she also quickly recognized that the pieces on top matched the backgrounds and would help in identifying the location of the puzzle piece. Her spatial understanding surprised me when it came to getting the pieces to fit, however it was lacking when determining whether the animal was up-side-down or not, and so, after a few rotations, would fit the piece in correctly with the others.
My Niece’s Interaction
What interested me was what she did the second time around. Being very excited about this new puzzle, of course she wanted to do it again (8 or 9 times, I soon found out), and so I got to watch her try to do the same puzzle again, having mixed up the pieces. I thought that we had established the fact that “Pictures on top (nearly) match pictures underneath”, but for some reason she started the puzzle again as if she had no clue about the connection! It was like she was doing a puzzle where the underside was blank–she was hardly looking at the images on the puzzles and certainly wasn’t looking at the background to match them in any way other than the shape of the piece.
At first I was amused until I realized that this probably the way many of my students interact with mathematics.
In class, I always try to teach the “why” of an algorithm before I’ll allow students to use the algorithm (whether it is multiplying or dividing fractions, adding exponents when multiplying, or using logarithms). My reasoning, and I believe that most
math teachers would agree, “When you understand the reason why something works, then you’ll not only know when you can and cannot apply it, but you’ll be more prepared when you encounter something you’ve never seen before.” However, my niece was less interested in the “why” (making the visual connection that a cow would logically be walking out of the cow herd part of the barn) than the “what” or “how” (the physical act of connecting the puzzle pieces).
Just Keep Swimmin’ 
So I began pointing out the image on the piece and asking Caroline what the animal was, and she had complete success identifying them (very easy for her), as well as making the mental connection (again) of “Oh hey, the similar images go on top of each other” (also a very easy task for her), but I found myself having to prompt her for the images for each new puzzle piece. I would even switch it up and ask to ID the background first, and then ask “okay, where’s that puzzle piece?”, and other times I would ID the piece first and then ask “okay, where’s do we see the _____ on the puzzle board?”, and she still required me to prompt for that question.
Here’s an example of me just asking her and her making the connection (this is the 5th or 6th time consecutively)
Here is the place where if we were in math class, I would pause the whole class and we would briefly answer the question “What is the easiest thing to think about first when searching for the puzzle pieces?” or “What question should we ask ourselves as we start our search?” Unfortunately, my niece being only 2, I did not think such a discussion would be productive or interesting to her. Besides, she was in it for the fun. 
A few interesting points about the situation:
- One of the puzzle pieces was only a roof, and so it did not at all resemble the animals that were “underneath” it. However, my niece was able to identify its location and orientation just as easily as the other pieces using
- Would a child who is new to puzzles operate the same way? What made my niece “stubborn” when it came to identifying and pairing the images–was it the fact that usually most puzzles don’t have backgrounds to help?
One of the most fun things was seeing my cousin make the mental connection of the images, even on the 8th or 9th time, and realizing “that’s exactly where the piece must go” even before she put the piece in place. That feels deeply metaphorical for the math classroom on so many levels. Here’s just one level: students knowing that they can solve a complex math problem, as well as how to solve it, even before they do solve it, and being correct about that. Of course, my niece followed up by putting the puzzle piece into the puzzle, whereas often students want to ask “Why do I have to do it when I already know I can?”
Here’s a brief video of her realizing where the Ram goes 🙂
Did you hear the “Oh!”? I think that’s why I got into teaching.
 Why is it that a 2-year old will find something new and difficult engaging when 16-year olds hate new and difficult things? If I was really cynical, I could phrase the question “What is wrong with our education system that a 2-year old will find something new…”
 Was this just an April Fools, or was it for real?
 If students are finding math activities fun, are we, as teachers, responsible for pulling them aside and helping them to think meta-cognitively, even if they are learning math without the meta-cognition?