After I left my full-time high school teaching job, during my first term as a graduate student, I taught as an assistant in a Native American boarding school in Riverside, California. One sharply cold Tuesday that fall, one of the mathematics teachers I had been assisting asked if I would be interested in teaching the next day's lesson. I said, ``Okay" first, then I asked what the topic was. His response was, ``Area. That length times width equals area." He then gave me his lesson plan (slightly dog-eared from several years of use and updated to the extent that the page numbers in previous editions of the book had been scratched out and the numbers for exercises in the current text added). I asked if I could alter the lesson at all and he said, ``Sure, do whatever you want.'' I took him at his word...
I brought to class a bag of scraps of yarn varying in length from several yards to several hundred yards provided by a friend who knitted. I began the class by telling students how, as a child, my friends and I had used finger knitting to make rope out of every scrap of string, wire, floss, and yarn we could find. I told them that at age 12 or so we had lots of use for rope (like to leash Scoobie, a neighborhood stray dog: big, black and very hairy, who tended to dig around in our dirt forts in the vacant lot; to weave together to make a roof for the fort; to use as a jump rope).
To begin the working with the students, we pushed the desks back from the center of the room and I sat on the linoleum tiles to demonstrate finger knitting. Most of the students sat on the floor with me, while a few sat in nearby desks. Two students helped me distribute the small skeins and balls of yarn and we piled the extra in the middle of the room for folks to grab from if they ran out of yarn before the ten minutes of finger knitting were done. I explained, as I demonstrated, that we would be using the rope to find out about enclosed area, remarking that we were going to build on the previous day's lesson on perimeter. I also mentioned that we would finger knit until 9:15 and then make a rope from our pieces. We sat there finger knitting for about ten minutes as the instructor of record, who had declined to join us, sat at his desk in the front corner of the room, near the blackboard, with his arms crossed and a look of annoyance on his face (more on this below).
After each student had successfully tied off her/his piece of knitting, I asked students to pair up and use the strongest knot they knew how to make to tie together their two pieces, then pairs paired up and tied off, until there was one long piece of rope. Two students used the class yardstick to measure the length of the rope (after we had all given each part a good pull to make sure it was stretched out and would hold together). The class of 18 students (and myself) had created a rope about 31 feet long. At this point I asked two students, with the rest of the class watching, to tie the two ends together so that the resulting loop would be as close as possible to exactly 30 feet long.
``Okay," I said, ``Everyone inside the loop.'' For the next 10 minutes we experimented with ways of shaping ourselves into a rectangular group inside the perimeter created by the rope. Four students were made into keepers of the vertices and their movements dictated the amount of room we, as a group, had to move around. We determined the shape with the least room and the most room and marked where the keepers had been standing (with masking tape on the floor) for those shapes. When I asked students how long the sides each of the two extreme shapes were, their guesses were often dissimilar and usually included a comment that it was hard to see past all the people in the group. I suggested at that point that we all leave the rectangle and that the keepers of the vertices also step over the rope to hold the corners from the outside, because ``Maybe we need a different perspective, a different view of the problem.''
We went to the outside and the keepers held the corners on the floor, in the places we had marked. The floor tiles were one foot square (I had measured them the previous day, before I left school). I gave students a piece of paper with four questions: How many floor tiles fit in the roomiest rectangle? What is the measure of the perimeter of that rectangle? How many floor tiles fit in the most cramped rectangle? What is the measure of the perimeter of that rectangle? They rapidly answered the questions and many noted that a fast way to count the floor tiles was to count how many in a row (or column) and multiply by the number of rows (or columns). At this point we were about 30 minutes into the 45 minute class period.
I gave each student a piece of 1/4 inch grid graph paper, a piece of string between 6 inches and 18 inches long, and asked them to estimate the area in square quarter inches and the perimeter in quarter inch increments for at least two different rectangles. They didn't have to be the roomiest and smallest, they could be any two rectangles. Their goal was to state the length of the rectangle in 1/4 inch units, its width, its perimeter, and its total area in square units. When they had it figured out they were to pair up with another person and double check each others' work. Some students sat back down on the floor, some at the scattered desks. Within ten minutes this part of the lesson was complete.
In the last five minutes of the class I asked students to put the chairs back in place, sit down, and decide on any properties about length, width, area, and perimeter that we would be able to use the next day and write them down on the sheet of graph paper, along with their names. I collected their graph paper pages and assigned the ten problems in the textbook from the instructor's original lesson plan.
I asked the instructor, after the class, about his views on what we had done. He was angry and said he resented the fact that a day had been ``wasted'' and that he would have to teach the ``real lesson'' the next day. I asked him to expand on his concerns and he said he doubted any of the students would be able to do the homework, a set of eight problems that gave length and width measurements and asked for perimeter and area, and two word problems where length and area were provided and width was to be determined. I asked him to glance over what the students had written on their graph paper, but he declined, waving me impatiently away.
The students had written things like:
The peremeter [sic] is kinda like edge distance and the area is more like filling distance, how full something is of squares or tiles or people.
For perimeter you add up length (and width) twice, for area you multiply how many times the row of the width will fit in the length.
Area is the number of little square edges on one side times the number of little edges of the squares on the side next to it.
Perimeter is the length around the shape and area is the room inside the shape. You get perimeter by adding lengths and you get area by adding up little areas.
To me, the lesson was a success. What the students turned in on their sheets of graph paper indicated they had begun to see that there was some sort of non-linear or two-dimensional relationship involved in area while perimeter was linear or one-dimensional. When they turned in their homework the next day, the average on the ten problems from the text was 9 out of 10 points and several students commented on how the homework was one of the ``easiest'' they had ever had in math.