Here’s an essay I wrote some years ago for my school’s website. It’ll give you an idea about my approach to math. More recently, I wrote a whole book describing many of the discovery lessons I used in my teaching. For a preview of that, click here: Higher Math in the Lower Grades.
Why do I love teaching math to young students?
Well, first of all, I love teaching these classes because of my students! They are full of excitement, wonder and curiosity. They’re very willing to work hard, and they love learning. Each year, beginning in September, I put a lot of effort into building a positive classroom atmosphere of mutual support and helpful collaboration. And my young students do make this happen. I love watching them work together, and I rejoice each time one says to another, “I respectfully disagree.”
I also need to thank the teachers who’ve worked with my math students in earlier grades, both at PFS and at other schools. My students almost always come to me with good math skills and a positive attitude towards math. I am also grateful to the parents of my students, who have prepared them to be active learners and helpful class citizens, and who support their learning in many ways.
Finally, I need to say a big thank you to my boss, Jane Fremon, who, since the founding of our school, has encouraged me to teach math my way. And “my way,” I shamefacedly admit, is to make class more fun for me. (I hope this benefits the students too!) Naturally we cover much the same topics any third or fourth grade class will cover. But I do look for ways to teach the standard skills and practice the standard drill inside discovery lessons. A few examples follow.
Subtraction with borrowing!
Most young people need some repetition and drill on this skill. I have learned to use Kaprekar’s Constant (named after the Indian mathematician D. R. Kaprekar) as a great way to practice borrowing. We start with any four digit number where not all four digits are the same. We arrange the digits to make the biggest possible number and the smallest possible number, and then we subtract the latter from the former. Then we iterate, using that answer to begin the process again. We all start first with a “seed” I select, so we can all help each other. I tell the students that if they work carefully and check their work with each other, after a few iterations they’ll make an amazing discovery. And they do! After the first discovery, and a few sentences describing it, students are allowed to pick their own seed and see what happens with that starting number. We generally continue this work over two or three days, and it’s always an engaging and enjoyable lesson. And oh, thanks to the way it’s set up, every subtraction has borrowings!
One of the ways we practice our multiplication facts is with the whole-class game Buzz. We go around the room and students simply say counting numbers in order, but they substitute “buzz” for any multiples of the times table we’re practicing. (For example, “1, 2, buzz, 4, 5, buzz, 7, 8, buzz…”) Great drill in and of itself, and a good opportunity to reinforce clear speaking and careful listening. But then come my questions. “Did the buzz move? Or did the same kids say buzz as we kept going around?” This leads to interesting conjectures, and eventually the idea of factors. After some experience with the game, I show them how we can model our buzz games by drawing star polygons. (With a class of 12 students we can model the class by 12 equally spaced points on a circle, and see how the buzz travels around.) This usually leads to some initial confusion and then bursts of creativity. Children often turn their star polygons into colorful art works, and many delight in exploring “funny” situations. Like the ones times table. Or the six times table with twelve points. Or the thirteen times table with twelve points! And of course we get to practice using our rulers and protractors…
The Invention of Chess!
Young children seem to love doubling, and they love large numbers. There is an ancient story about the invention of chess, in which the inventor asks for a reward of rice, one grain for the first square, two for the second, four for the next, and so on, doubling each time. This lesson gives third graders practice adding a number to itself, or multiplying a number by two (and seeing that these processes are equivalent). There are lots of opportunities for discovery here: for example the repeating pattern in the ones column of the powers of two, or the fact that the sum of the powers of two up to any power of two (for example, 16) is always one less than the next power of two (1 + 2 + 4 + 8 + 16 = 31 = 32 – 1). And there are also lots of opportunities to learn about estimating. (What could we fill with 1023 grains of rice?) It’s also an opportunity to keep a neat table with three columns:
square grains total so far
1 1 1
2 2 3
3 4 7
4 8 15
A Table of Factors!
This is yet another way to practice multiplication facts, but it offers much much more. As the students work I ask them to list the factors of each counting number in a chart, along with how many factors the number has, and then a place for comments. We learn that one is a unit, and we learn to label the other numbers as prime or composite. We also learn that six is called a perfect number because 1 + 2 + 3 = 6. I let them know that the next perfect number is smaller than 30 and ask them to search for it. I also ask them to keep track of square numbers. This project goes on for several days, with lots of team-work, checking and correcting. And students do make discoveries. (And I need to say my students are wonderful at writing down a discovery and never calling it out, thus allowing their classmates the chance to make the same discovery.) By the end of the lesson we’ve all learned (I hope!) that square numbers, and only square numbers, have an odd number of factors, and that 28 is the second perfect number (1 + 2 + 4 + 7 + 14 = 28). Students groan when I tell them the next perfect number is bigger than 400. But then I tell them there’s a shortcut. We observe that 6 = 2 x 3 and 28 = 4 x 7. And I tell them that only numbers with this form can be even perfect numbers. (No one knows if there are any odd perfect numbers.) And amazingly we’ve already run into these numbers with the Invention of Chess story! (See the chart above.) The next candidate is 120 = 8 x 15. I let them know that 120 isn’t perfect because its factors add up to way more than 120. But the following one works! 496 = 16 x 31 is perfect! Hey guys, I ask, what makes 8 x 15 different? Look!
2 x 3 is perfect! 4 x 7 is perfect! 8 x 15 is not perfect! 16 x 31 is perfect!
We brainstorm for a while. And almost every year a little third grader raises a hand and says, often timidly, “Is it because 15 is composite and all the others are prime?” Gotta love it!