Table of Contents

Unit 5: Seeing Structure and Generalizing (MP7 and MP8)

5.2.1 Summary and Reflection

In this section, you looked at various ways to obtain a general formula for a pattern, representing how students look at figures and see visual growth from different perspectives. For example, one student might see one square in the middle and the increase in the number of squares on each of the four spokes. Others might see an “X” figure and count the squares on each diagonal and subtract the one in the middle. They might even count the number of squares on one diagonal and notice one less square on the other diagonal.

Knowing that students have different perspectives is important for teachers to consider, as students need time to explore and discover how to make sense of the growing pattern.

Time to Reflect

In your Metacognitive Journal, reflect on the critical thinking and problem solving skills used to first solve the problem. Include in your reflection the responses to the questions posed in the Time2View activity on the previous page .

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Time to Extend

To learn more about the role of visualization in forming generalizations, refer to the following publications:

Rivera, F., “Changing the Face of Arithmetic: Teaching Children Algebra,”
   Teaching Children Mathematics. 12 no. 6 (2006): 306–311.

Rivera, F., Ferdinand D. & Becker, J., “Algebraic Reasoning through Patterns,”
   Mathematics Teaching in the Middle School. 15 no. 4 (2009): 212–221.

Rivera, F. & Becker, J., “Figural and Numerical Modes of Generalizing in Algebra,”
   Mathematics Teaching in the Middle School. 11 no. 4 (2005): 198–203.