2018.9.13
Song Birds and Pedagogy
It’s a memory indelibly imprinted in my mind. My fifth-grade teacher, Mrs. Reeves, was entertaining several of us after school until the school bus came. That day in class we had learned how to add fractions such as 1/5 + 2/5. She cut a pie into 5 pieces, took one piece and then two more, and we had 3/5 of the pie. Easy.
Now she asked, “What is 1/3 + 1/2?” I drew half a pie and a third of a pie and put them together, but that didn’t help. The denominators (bottoms of the fractions) were different, so the earlier method didn’t work. I was stumped.
After letting us struggle a while, she gave a hint: Cut the 1/2 and the 1/3 into 6ths.
Then I got it! 1/2 is the same amount as 3/6, and 1/3 is the same as 2/6, so we add 2/6 + 3/6 to get the answer: 5/6. What a feeling of satisfaction. The next day when she taught that lesson, I proudly displayed my superior intellect to the class.
The next year my science teacher asked: Why isn’t it self-sustaining for a burner to heat water, steam from the water to spin a rotor, and the rotor create friction to heat the burner? He let us think about it overnight before giving us the hint: Not ALL of the heat from the burner heats the water. Not ALL of the steam hits the blades. My hand shot up to give the answer: Energy is lost at each step in the process.
Those truths of math and nature have stayed with me my entire life. But more importantly, I still use their method of teaching. Shamelessly taking the credit, I describe it to my students as Pennings Pedagogy Theory.
Drawing a circle on the board, I explain to my students, “This is your brain. When I teach you something that you haven’t yet pondered, it’s like a stream of water [which I then draw above the circle] running over the top of the circle. A little seeps into the brain, but most runs off the sides.”
However, I continue to explain, if students become invested in the problem either via a teacher’s challenge or from their own inherent interest (the advantage of teaching sex education), then a divot is formed in the top of the circle – their brain - so the material collects and sinks in.
So that is how I teach. As I explain new material, the lecture is pitted with silent periods where I ask students to solve the next step. Either they figure it out, confirming that they fully understand, or else they get stumped, thereby forming a mental divot. Then they are (hopefully) all ears and get a satisfying “ah ha!” sensation when I give the solution.
This summer I taught a graduate course in Urban Education in Mathematics at Davenport University. The K-12 teachers and I together explored the challenges of teaching students who may come from homes where they have not been exposed to the rich learning environment that others have experienced. These students have not formed as many mental divots. For example, as a child I learned about area and volume by playing with wooden blocks. A rectangle of 4 by 7 requires 28 blocks (the area), and if I stacked them three high, then the volume is 4x7x3=84. It’s also clear by re-orientating that 4x7x3=7x3x4, so the order of multiplication doesn’t change the answer. Children self-discover these important mathematical concepts just by playing with block towers. Their minds are now primed – divots formed - to formalize the concepts in school.
It’s just as fun as we grow older. When I was advisor for the Hope College Outdoor Adventure Club, I kept vanloads of college students entertained for hours asking them math and logic puzzles. Just like Mrs. Reeves, I posed the questions and their natural curiosity and competitive spirit did the rest.
The method was applied to me this summer when I camped and fished in the BWCA (Boundary Water Canoe Area) with my college friend of 40 years. Tom knew each bird we encountered – either by appearance or song – and kept asking me if I knew. I have enjoyed bird songs all my life, but never had the curiosity to determine the name of the bird. Now, thanks to Tom’s enthusiasm and prodding, I have started forming and filling my own mental divots and consequently my life has just become richer.
