This post was first posted on my personal blog as Visualizing Math. Thanks to David Wees (who has retweeted it) and David Loitz for the push to put it here as well.
I love math. Am I an expert at it? No. Do I make mistakes as I teach it? Probably–but I work hard not to, unless I am doing so deliberately for kids to figure something out. Here’s how I got to be a math loving female….
My family played both card games and board games as I grew up. Every year at Christmas, we spent the afternoon setting up and playing all the new board games Santa had brought. We had shelves and a cabinet that was full–Candyland, Parcheesi, Monopoly, Scrabble….games at various levels, for the 6 siblings (and friends) whose ages ranged over 17 years. I spent most Sunday afternoons playing Scrabble with my Mom–with a dictionary between us, not following the time rules, but instead challenging ourselves to find the very best word we could. Our games took hours–because we’d scour the dictionary, looking for that word that gave the most points and used the most letter tiles. When my grandmother came each summer to spend two weeks with us, the card game Canasta took over our evenings–and those of us too young to be in the four or six playing hung around and apprenticed ourselves to one of the players so we could learn how to play, hoping we’d get to play the next game. I was amazed at how my Dad could shuffle so many cards at once (the game calls for 4 or 6 decks, depending on how many are playing),and I also got good, as I got older and got to play, at explaining my strategy to a younger sib watching while not giving it away to my opponents. For us, games weren’t about competition–I can’t even remember who usually won the Scrabble games–they were about learning. We learned by watching “experts” and having strategies explained to us in the moment, when it mattered.
So when I hit Algebra 1 in high school, I did okay. I had a teacher who was very linear and well organized, so I learned how to do those expressions with variables. It made sense to me, as I saw it as a puzzle. I still enjoy all kinds of logic puzzles and figuring out variables. That was NOT my experience, however, with Geometry. I saw that as formulas and rules…and those of you who know me well know I am NOT a rule follower. I can still remember struggling with Geometric proofs in 10th grade and after having gotten several not-so-good grades, my mother sitting down with me and asking what the problem was. I told her I couldn’t remember the rules and the formulas…and I specifically recall her response: “Paula, Geometry is fun–you look at the figures and work the puzzles. It’s just a different kind of logic puzzle–but the fun of it is that you can see it.”
She worked me through several of my homework problems, with us following through the logical steps and proofs together and I remember feeling challenged, relieved and happy all at the same time when she left me to do the rest alone. I knew I could work puzzles–I’d been doing that all my life at home. And, I found I enjoyed the challenge of solving geometric questions and writing out the proofs. Being able to look at the figures, though, made all the difference in the world for me–I couldn’t remember those rules when given words, but when shown two similar triangles or asked to name an angle or side measurement when given pictures, I was in hog heaven–I had what I needed.
So, when Willy Kjellstrom and I, in our UVA/ACPS partnership, decided to work on a unit around art and math, for both of us it was truly about visual spatialization-helping kids to get beyond the words in a textbook to visualize shapes and their various rotations, reflections and transformations in their minds. We spent hours and hours working on our plans and getting the materials together. We spent hours revising and revamping them–and we created two wikis, as the original planning wiki got huge and pretty confusing as we added more and more material. But what we came out with was incredibly awesome. We basically implemented it in January, (having been working on planning it since early November) and our results from pre-test to post-test were statistically significant. Not only did the kids learn the math skills we had included (the geometry standards from our state list), but they also increased significantly in their spatial visualization skills and their confidence.
We taped an ending conversation where Mr.K, as the kids call him, held up a shape and asked them to close their eyes and visualize it to determine the surface area. It looked like this:
Then, we asked the kids to share their strategies.
” I saw that one side had 3 faces and doubled that to count the back, too , then counted around the other edges.”
“I saw that there were two on the bottom row and counted those and then counted the top cube in my mind.”
There were variations on this theme–picturing it and counting around, while adding the bigger face of three cube faces.
But the one that surprised me the most was the kid who said, “I knew there were three cubes that each have 6 faces, so I multiplied 3×6 to get 18. Then, I knew that the figure had two places where cubes joined together, so I multiplied 2×2 for those joints and took away 4 to get 14 cubic inches.”
Several nods accompanied that explanation, so it was obvious this was not the only kid who had jumped to a mathematical shortcut. Then, Mr. K asked them to find the surface area again with another cuboid. Again, he made sure everyone had seen it, then hid it so they had to visualize it.
We got similar answers, but many of them had adopted a better strategy than just counting. So I asked, “How many of you did it one way and then checked yourself by using someone else’s strategy? I asked that because I myself had done that. Over half the group had done it two ways in about the same amount of time it took then to figure out the first surface area we asked for. That was pretty cool!
So, as I reflect on the work we did (which you can find at our student wiki, Artful Engineering), I can’t help but think of the power of learning logic at an early age, the strength of looking at games for learning (not necessarily for winning/losing), and the benefits of visualizing math in a variety of ways. I remember India, when we asked how the kids felt differently at the end of this unit from the beginning, immediately saying-“Smart! I feel really smart because at the beginning I held up this cuboid and couldn’t see it from all angles in my mind and now I can. I can turn it and rotate it and flip it in my mind.”
Beyond fulfilling my responsibility to teach the state mandated standards, I hope Willy and I have helped along the strengthening of some more math loving females, while helping them all build visualization skills and flex their logic muscles.
Thanks to @smeech who tweeted recently, “Imagery in math would have been huge for me when I was a kid … Thanks Dan for pushing it so much. http://t.co/GkhNV6lW” and got me reflecting on this work today, when I had time to write about my thinking!