This past January I spent 2 weeks in Kasoa, Ghana. One of the goals of the trip was to encourage Ghanaian teachers to choose less didactic teaching methodologies and to, rather, adopt more interactive, learner-centered pedagogies. This is a similar message that I bring to teachers in the US, where I am often disheartened to find students sitting at their desks and filling out routine worksheets. Despite the subject area- this type of work is generally uninspiring and does not lead to the deep and lasting understandings that we all want for students.
My students and I arrived at the Blessed Life Academy late one morning. We were looking forward to spending time with teachers, speaking to the school proprietor, and assessing the needs of this PreK-8th grade school. We did not come there to teach. However, after observing nearly 45 minutes of a lesson delivered by a young teacher on the topic of area, we all grew antsy. My graduate assistant and I walked to the outside courtyard that sat adjacent to open-air school. “Let’s do a lesson on area out here,” I suggested. Immediately we began to think and discuss what we could do to make area come alive to the young 3rd graders inside. We began drawing a rectangle in the dirt using a nearby stick to make the outline.
We debated (for at least 5 minutes) the dimensions of the figure. We then made the decision to base our lesson on non-standard measurements and to make use of the many cinderblocks littering the courtyard.
After our planning, I asked the school proprietor whether or not it would be okay for me to conduct a lesson with the students on area. He graciously agreed.
The lesson began with me posing the question, “What is area?” It was no surprise that the students who answered repeated the exact same thing that their teacher had been saying to them all afternoon long. “Area is length times width,” said one eager student. I probed further. “What is area?” “Is it this,” I said touching a large classroom divider. “Is it this desk?” “Where is area?” The students sat quietly. One attempted to measure his desk. I proceeded to tell the students that area is all around us, and that we were going to spend a little time exploring what area was.
Students were invited outside and told to stand in a line near the rectangle carefully drawn in the dirt. My graduate assistant and I then told the story about our rectangle. “We drew this rectangle on the ground. We know that the area is six, but we do not remember which six it is.” We then invited the students to help us figure out the puzzle. In groups of 2, students were asked to collect 6 things that they felt would cover the area of the rectangle.
Students walked out into the field, collecting an assortment of rocks and stones. In a few minutes they were standing in line again each group carrying six objects.
Then, one group at a time was asked to see whether or not their six objects could cover the area of our figure. After numerous trials, a few which came fairly close to covering the area of the figure, I revealed to the students my secret. I had only told them that I forgot which six objects covered the area. In reality, I knew all along that 6 cinderblocks would fill it. One by one, teachers and graduate students carried over six cinder blocks and laid them side-by-side in the rectangle. I watched the children’s faces as the final block was laid. One girl placed her hand over her mouth in amazement as the final space was covered. It was the look that every teacher dreams of seeing- the one where a light bulb flickers on, and where something once cloudy becomes clear.
The lesson proceeded into a discussion of non-standard measurements and how one could measure using the length of their hand, or their foot, or their forearm. There were more light bulbs and a-has before we returned to the classroom. There I posed my question again, “What is area?” One girl raised her had stating, “area is everywhere. It is all around us.” A boy on a bench near the back of the room stated, “Area is what occupies space.”
As with every lesson I teach, I assess what the learning outcomes were, what improvements could be made in the lesson, and what the next steps would be as students move forward in their understanding of area. For example, I learned that children did not give attention to whether or not the stones they collected were the same size or not. That’s a misconception that I’d need to address were I to teach the lesson again. As well, I asked the majority of the questions in the lesson. I would certainly want this to change were I to teach the lesson again. I could go on. Instead, I’ll share a very salient take-away, one that my education professors and the countless theorists they forced me to read confirmed- learning is an active mental process. It happens as we closely examine our current understandings and abandon them for better ones. Teaching, then, involves the designing of learning experiences and tasks that push students to examine and subsequently abandon their misconceptions- adopting in their place stronger, more powerful understandings. This is what we attempted to do with the lesson on area. We wanted these students to engage their minds and to push themselves to make sense of area- for themselves. Developing ones own understanding is one of the most powerful gifts that education can give. When a student develops their own understanding and not merely a set of passed-down facts or procedures, they are transported from a station of redundancy and inertia to a space of creativity, action and transformation. Given the great responsibility that teachers hold, and the great hope residing in communities like Kasoa, my students and I felt humbled to engage in this small, yet transformative act of teaching area for understanding.
Dr. Joi Spencer is an Assistant Professor in Mathematics Education in SOLES and the recent recipient of the Greater San Diego Mathematics Council’s 2011 Post-Secondary Mathematics Teacher of the Year Award. Her latest article, “Views from the Black of the Math Classroom,” was recently published in Dissent Magazine.”