When I was a novice teacher, I remember getting into arguments with my colleagues about the best use of class time. Should we go over the concrete facts and basics first? Or should we introduce a concept in the context of realistic applications? If our students were struggling, should we spend a lot of time in remediation, or should we challenge them with new, possibly broader ideas?
Both arguments are absolutely important. We can’t discount the importance of accurate computation, or the mastery of simple ideas. But some students struggle so long with basic skills, that they only experience brief moments of rich exploration in class. And if a student’s entire experience is arithmetic facts and drills, then they’ll never have a real understanding of what math is.
One of the best lessons I ever used in my 8th grade Algebra class was an activity with lines of best fit. It was one of the first times that I handed a realistic, messy example to my students and let them struggle with it. In this activity we investigated how nations evolve as their infrastructure develops, average education rises and birth rates start declining. We collected data on countries like the United States, Canada, Italy, China and Nigeria and compared average education levels to average family sizes.
We analyzed the data in small groups, found appropriate scales for the axes and plotted our graphs. These were important but basic tasks and that were more interesting in this context than in rote practice. Every student, from the most struggling to the most advanced, had an opinion about where the line of best fit should be drawn. The best part was seeing how different students focused on different elements of the situation. Most students confirmed the negative correlation, and discussed the steepness of the slope. They pointed out which nations fit close to our line and which were outliers.
As the conversation became more advanced, we discussed the details of our line of best fit. What did the slope and intercepts mean? Some insightful students asked about cause and effect: whether high education led to smaller families, or smaller families allowed more time for education, or whether both of these were caused by a different variable. We used mathematical models to predict the average family size of other countries, based on levels of education. Best of all we had meaningful conversations about culture and economy in a math class, where most students had expected to talk about numbers and formulas.
A math class can’t really spend every day in big, subject-spanning investigations. Sometimes students need to develop automaticity and solidify their skill with procedures. Sometimes they need to talk explicitly about the way symbols, graphs and equations are related, outside of the context of specific applications. But I think too often, in those early years, I was afraid to burden my struggling students or my lagging classes with meaningful activities. Now I think that these are exactly the moments that give all students a chance to engage in truly worthwhile math lessons.