Facilitating Communication about Measurement, Exponents, and Scientific Notation

As students move from the middle grades to high school, they move from primarily modeling linear phenomena, which have constant rates of change, to studying non-linear phenomena, which have variable rates of changes. Exponential functions are an important class of functions with non-constant rates of change and provide students an opportunity to revisit, reflect and refine their previous experiences involving numbers (e.g., properties of exponentials, scientific notation), issues of measurement (e.g., problems of scale), and geometry and spatial sense (e.g., visualization of exponential growth) (see the 9-12 Patterns, Functions, & Algebra Standard). Students need ample opportunities to organize and consolidate their thinking about exponentials.

All students need extensive experience listening to, reading about,writing about, speaking about, reflecting on, and demonstrating mathematical ideas in order to develop the ability to express mathematical ideas coherently. Students also need to reflect on the thinking of others to broaden their understanding about their own strategies and alternative strategies for solving a problem. Active student participation in learning through small- and large-group discussions provides multiple opportunities for questioning, listening, and summarizing.

The teacher plays an important role in facilitating and fostering communication so that powerful mathematical understanding is developed (see the 9-12 Communication Standard). Not only do teachers need to encourage their students to communicate mathematically, they also need to model good communication skills. The ability to pose questions that elicit, extend, and challenge students' thinking is an essential part of creating a classroom environment where intellectual risks, sense making, and deep understanding are expected.

The class shown in the following video clips is engaged in solving problems which require the use of very large numbers, prompting consideration of the value of scientific notation. The teacher relates the notions of scientific notation and exponents to the way Alice changes size in the story of Alice in Wonderland.

Considering Large Numbers

Prior to this video clip, students have been exploring exponentials in the context of the size changes of Alice in the story Alice in Wonderland. Students have been asked to translate the experiences of Alice into mathematical symbols. The investigation leads to discussions over the differences between a constant growth rate and a constant growth factor. Being engaged in solving problems that require the use of very large numbers prompts students to consider of the value of scientific notation to communicate the answer clearly.

In the first video clip below, students discuss their solutions to the problem, "How many inches are in a light year?" As you watch the video clip the first time, think about the following focus questions.

Students need time individually to formulate their thinking. Following this time with small group discussion maximizes the number of students who have an opportunity to verbalize their thinking (as seen in the video clip above). In addition to discussing solution methods in small groups, students need opportunities to communicate their thinking to larger audiences. Presenting problems and solutions to the class provides students with such an opportunity. When a small group of students prepares a presentation, group members have the opportunity to listen to each other's ideas and evaluate their own solution methods. In the following video clip, a group of students shares their solution method to the problem "About how many atoms are there in a kilogram of carbon?"

Reflection, Writing, and Communication

There are many different ways to encourage students to think and talk about mathematics; informal writing activities can help students reflect on their understanding of mathematical concepts and encourage them to make connections among topics. In the following video clip, the teacher uses Alice in Wonderland as an analogy, describing how Alice changed size as she ate cake and asking students to consider the relationship between scientific notation and Alice's growth.

Before beginning a writing assignment, students should know who their audience will be - teachers, other students, parents, community members, and personal writing for only oneself are a few possibilities. When students share their writing in small groups, they have the opportunity to verbalize their ideas and receive feedback from other students. The next video clip shows the students as they share their writing in small groups.

In the next clip, students share their ideas with the whole class. This whole class dialogue provides an opportunity to come to a consensus as a class and to formalize the mathematics.

Conclusion, Reflection, and Discussion

• How do the key ideas generated by these students compare to goals that you might have for a lesson on scientific notation?
• Compare the connections that are being made in small group discussion early on in the lesson to the connections that are made during the small group discussion near the end. Why are both opportunities necessary in the formation of connections?

References

Video clips generously provided by WGBH Boston. All clips were taken from "Alice to the Moon ," part of Teaching Math: A Video Library, 9-12. Funded and distributed by the Annenberg/CPB Math and Science Project, P.O. Box 2345, S. Burlington, VT 05407-2345, 1-800-LEARNER.

Fendel, D., Resek, D., Alper, L., & Fraser, S. (1998). Interactive Mathematics Program, Year 2. Pages 379-434. Berkeley, CA: Key Curriculum Press.

The National Council of Teachers of Mathematics is a public voice of mathematics education, providing vision, leadership, and professional development to support teachers in ensuring mathematics learning of the highest quality for all students. The views expressed or implied, unless otherwise noted, should not be interpreted as official positions of the Council.