Confirmation of my math education dreams came in an unexpected place last weekend.
I sat cross legged in the doorway of my niece’s dance studio, thinking that she had grown up so much since my last visit home. Six years old, and dressed in pink leggings and leotard, she tried to coordinate her legs while somersaulting in time to “Who Let the Dogs Out.”
She finished her dance lesson and we walked hand in hand(always looking both ways) to eat lunch with the rest of the family. I gave her a purple sparkly necklace which she adored and decided was going to be added to her special box that was reserved for her favorite possessions. We began to attempt to replicate a flamingo on the kiddie place-mat when she noticed the legs of the flamingo looked like the letter four.
Four! My math educator brain returned from the vacation it had been taking to the land of necklaces, dance rehersal and sparkling jewelery, and I asked her about what she was doing in math class.
P-nut(my nieces nickname) launched into an animated demonstration of adding and subtracting integers.
I can’t remember a time when I felt interested in adding and subtracting integers-now these are essential skills in my quest to model the world with higher mathematics. However, p-nut was quite excited about adding the new numbers I gave her over and over again. She was also happy to talk about five groups of four cookies and draw related pictures. I noticed that if I asked her to add 43 + 12 she would count on from 43. While marking with her fingers as she went using a method that helped her keep track of how far she’d counted that I didn’t quite understand she said 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54.
Hoping to lead her to a new way of thinking about adding I asked wrote down in pink colored pencil 40 + 3 = 43 and asked her if this was true. The question was interesting to her but non-trivial as we had to discuss the equals sign which she had seen before but wasn’t quite sure of. Next I asked her if 43 +12 could be written as
40 + 3 +10 + 2 and again she pondered it, wasn’t sure and eventually agreed that they were the same. At first these two addition problems looked quite different and it required some real thinking to sort out what was happening, in addition to confirmation counting. Then we discussed the commutative property in kid-friendly language. 40 + 10 + 3 + 2 is how I thought about adding this all, and I think she finally agreed my way made sense, but resorted to counting on her fingers on the next problem.
We kept talking about the meaning of each digit, breaking the numbers a part, and after awhile of this discussion she turned to me and said “I see now. 40 + 30 is seventy because 4 plus 3 is ten! And 40 plus 40 is 80 because 4 + 4 is 8.” She seemed just as excited about figuring this out as she had been about dance class. She wouldn’t have to count on her fingers as much once she has internalized this new knowledge. I hadn’t been trying to teach her that concept consciously, though I had been emphasizing the meaning of the ten’s place and continued to do so. I said “Yes! Four tens plus four tens is 80 tens. We call 8 tens eighty.” I was so excited that she’d figured out a new and important idea as we discussed adding strategies together. P-nut’s excitement reminded me of times I’d solved a graduate problem successfully and I realized that no matter how easy a problem is on a relative scale, if it is a doable challenge for the solver, it can be fun. I used to think that teaching elementary school math would be boring because the math is so simple for me, but as I started listening to the reasoning of my niece, I realized how many ways there are to think about adding.
Next she started asking some amazing questions.
“I’m thinking of two numbers that add up to 22. How many numbers combinations can I have?” Or “I have 29 and subtract a number to get 23. What is that number?” I thought to myself that she was developing the beginnings of algebraic thought that would later help her understand unknowns in equations. There were so many possibilities for exploration in the questions she posed to me-I couldn’t believe my niece was coming up with questions worthy of being in an elementary math book.
It was amazing that the child who asked these questions was still unsure about such fundamental mathematical ideas. It was as if her capacity to ask and understand more difficult questions was not at all impaired by the newness of addition and place value.
After seeing so many college students who are afraid of math and try to cope with their fears by memorizing everything, it was wonderful to see that the natural state of a child is to be curious about how numbers fit together. Perhaps the school system will damper this enthusiasm later, but for a moment in p-nut’s life she was just as interested in talking about addition strategies, unknowns, representing multiplication visually, and the commutative property as she was in drawing a flamingo or playing with the toys at the restaurant. Moments like these help me know that I’m on the right path when I suggest that mathematics education can be more than memorizing and that there can be critical thinking at all levels.