In recent years researchers have used an interdisciplinary approach to seek to better understand how the brain works. Much of this research has direct implications for the classroom. For example, we know that students remember what they were thinking about at the time when they first learn a new concept. If this thinking is shallow, then their learning will also be shallow.8 For instance, are students thinking about the straws and gumdrops used to create different shapes, or are they focusing on the goal of the activity: the idea that triangles are necessarily rigid?
To make use of these findings, a teacher should anticipate what students may be thinking about during a lesson, be careful to link the activity to the mathematical concepts that it is designed to illustrate, and direct student thinking to the mathematical concepts at hand. The teacher should change the activity if students spend too much time thinking about material that is unrelated to the instructional goal of the lesson.
Cognitive science has also shown that the mind tends to remember in very concrete forms. If we want students to have a deeper knowledge of a mathematical concept so that they can apply the knowledge in new situations, we must offer students numerous opportunities to engage with related knowledge, facts, and examples of the concept. Teachers can guide students as they move through several stages in the process of developing deep, flexible knowledge. Teachers should revisit the same concept repeatedly over an extended period, in order to encourage thinking about the similarities and differences in the examples.
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