In math class, how can we ensure students gain a comprehensive knowledge of content while allowing them to notice, wonder and grow?

When standards are imposed on the learning of content, educators face a gripping dilemma. On the one hand, to encounter a topic like mathematics with wonder and personal interest means that students must learn with independence, following their own noticings and in unexpected directions. On the other hand, the need to master a specific set of problems to pass examinations means that students must move quickly from topic to topic, or else fall behind and eventually lose all personal interest in mathematics.

Like every real dilemma, there are no easy solutions here. There is no balancing act that erases the problem. Rather, educators should try to bring the sense of wonder and exploration precisely to the aspect of their teaching that seems most cut-and-dry. They could allow students to explore the way assessments and standards govern their learning. Let me try to explain this briefly.

The most common complaint from students in math classes takes the form of a question: “Why do we have to learn this stuff?.” The problem, of course, is that adults are often not sure how to answer that question especially if they are trying to hide the fact that most standards were imposed on them from above, and without exact justification. But what if, in classes, the educators take that question as seriously as possible? What if they invite students to think about the reasons that one topic follows another?

The most dangerous thing in learning mathematics is for the student to begin to think that what he or she is learning no longer makes sense. This is exactly what happens to students who “fall behind” — who, for example, have not yet formed a strong number sense, and nonetheless are pushed to learn new content, like fractions. Math then turns into dark magic — a set of conjuring tricks you perform without really knowing how they work.

Educators could speak openly with students about examinations and standards. They can also try hard to show the students how the topics do or do not relate to each other. They can present students with alternative standards — showing them, for example, how in different times and places different types of content were valued and different standards were expected of students.

Most importantly, and with regularity, students should be allowed time to work on at least one problem that links their existing understanding with a new topic in mathematics. They should be given time to work their way through this one problem, to wonder and notice at will, in order to come to see that not only does math make sense, but that they are capable of making sense of math.

When a student has developed confidence in his or her own ability to question the content, then he or she is much less likely to feel helpless in the face of standards. This student can bring a sense of wonder and autonomy even to an exam paper.

To achieve an environment like this, however, the first step is for educators to do the hard work of joining students in asking the  “why” questions. And then educators should speak openly with students about why things are the way they are, even, and especially, if the educator does not have all the answers.

Takeaways

• To ensure students remain motivated math learners, join them in asking the “why do we have to learn this” question.
• Regularly allow students the time to work freely on at least one problem that links existing understanding with a new topic in mathematics.
• Bring a sense of wonder to those parts of the curriculum that seem most routine and imposed. Explore, as a class, the standards that are expected of students, and why those standards may have been set.