“You cannot learn just by reading. You cannot learn just by listening to lectures. You cannot learn just by watching. We all must add the action of your own mind in order to learn something.” —George Polya
Early in my career, I taught Advanced Placement (AP) statistics at the high school level. I assigned a project in which students were required to survey 100 classmates and apply the statistical analyses that we had been learning in class.
Imagine my surprise when two students displayed their first slide and said, “We studied the relationship between a student’s GPA and the number of hours of sleep they get.” Hmmm…I thought. This sound like a great study. The students seemed to embrace the assignment.
“Our line of best fit shows that the more sleep you get at night, the higher your GPA is,” they continued. “But our next slide is more accurate. It shows the same data without the liars.”
I was too stunned to comment. Here the AP exam was only a few weeks away. I blinked and swallowed and asked, “Excuse me, why did you take out the kids that you thought weren’t telling the truth?” My students replied, “You told us to take out the liars!”
“The outliers?” I asked, realizing that although my students had completed several textbook examples about removing outlying data, they could not apply their learning to this novel task. They had not transferred their book learning to this project. I realized I needed to provide not only engaging but also systematic tasks to help students generalize their learning in order to demonstrate mastery.
What Is Generalization, and How Does It Occur?
Generalization is the noticing of patterns and relationships in mathematical context. Students have generalized when they can apply skills and concepts across mathematical and perceptual variations and apply the generalization to novel problems.
Take for instance the following first-grade problem:
9 + 4 = 3 + ?
Are the authors expecting first graders to subtract 3 from both sides of the equation? Are we asking first graders to complete an algebraic manipulation? The answer is no. We are asking first graders to generalize their knowledge of making 10s to a number sentence. They have already encountered that 8 + 5 is the same as 8 + (2 + 3) or (8 + 2) + 3 in order to make 10. In fact, they can tell you that 8 + 5 is the same as 10 + 3. When students generalize, they can extend their knowledge of number relationships to a new problem or format. If a student has mastered the making 10 strategy, knowing 8 + 5 leads to knowing 9 + 4.
Through carefully crafted lessons that use the focus cycle of Engage, Learn, and Try, students move from exploration, to variation, to generalization in a diagnostic and thoughtful way.
The Focus Cycle
In Math in Focus, engaging and systematic focus cycles help students engage with, explore, and reinforce concepts as they move toward generalization of mathematical concepts.
Part I: Engage
The focus cycle consists of an Engage task that connects to prior learning and provides an inquiry-based opportunity for students to enter a task cycle. For example, consider the following Engage task: Use fraction pieces to show your thinking. Audrey had one-third of a pie. She shared her portion equally with her brother. How much pie did they each receive? Explain your method.
By providing students with the opportunity to first build and then explain their reasoning, the Engage task connects student learning to prior experiences with dividing fractions. It helps students construct meaning through the use of physical or visual models, which ultimately lead to abstract representations.
Part II: Learn
The second part of the focus cycle emphasizes exploration of the variation of a mathematical concept. In the Learn phase, students encounter both mathematical and perceptual variations of the task. One way mathematical variation occurs is by controlling the number strings that students will encounter.
Consider the number string:
210 x 4
132 x 3
204 x 2
225 x 3
What aspect is changing each time? By consciously controlling the mathematical variation of the task, we can predict where students might have difficulty and thus help them to form generalizations.
Perceptual variation occurs in the Learn cycle when students are introduced to multiple representations of a concept or problem. Consider 2-digit by 1-digit multiplication:
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