## Learning Math: Number and Operations

# Classroom Case Studies, 6-8 Part A: Observing a Case Study (25 minutes)

To begin the exploration of what topics in number and operations look like in a classroom at your grade level, watch a video segment of a teacher who took the *Number and Operations* course and then adapted the mathematics to her own teaching situation. When viewing the video, keep the following questions in mind:

Note 2

**a.**What fundamental ideas (content) about number and operations is the teacher trying to teach?

**b.**What mathematical processes does the teacher expect students to demonstrate? How does this lesson help students achieve reasonable estimations and fluent computations?

**c.**How do students demonstrate their knowledge of the intended content? What does the teacher do to elicit student thinking?

**Video Segment **

In this video segment, Ms. Miles applies the mathematics she learned in the *Number and Operations* course to her own teaching situation by leading students through the process of converting from base ten to base five. The students then work in groups to convert base ten numbers to base five.

You can find this segment on the session video approximately 12 minutes and 2 seconds after the Annenberg Media logo.

### Problem A1

Answer questions (a), (b), and (c) above.

### Solution A1

**a.**Ms. Miles is working with her students on understanding place value and interpreting numbers in different bases. She uses manipulatives and later symbolic expressions and algorithms to help explain the idea that each place in a number stands for a particular power of the base number (i.e., in base five: 5^{0}, 5^{1}, 5^{2}, etc.). In this video segment, students convert base ten numbers to base five numbers using a different process — by decomposing the number into powers of 5. They also begin to develop an algorithm.

**b.**Students need to demonstrate that they can compute fluently and have an understanding of the order of operations and exponents. They need to be able to understand algorithms and translate visual information (e.g., manipulatives) to more abstract forms, such as symbolic representations.

**c.**The teacher is asking probing questions that help her assess students’ understanding and the level of their thinking as well as challenge the students into new understandings. The teacher is also utilizing class discussion, using students’ answers to build or scaffold the concepts she is trying to convey and to bring additional clarity.

### Problem A2

At what point(s) in the lesson are the students learning new content?

### Solution A2

Students are learning a new base. The new content is the interpretation of place value with a base other than ten.

### Problem A3

How do students transfer their knowledge of base ten to base five? What is the evidence?

### Solution A3

In the lesson, the teacher is utilizing the students’ prior knowledge and familiarity with base ten in order to help them interpret numbers in a new base. She uses both manipulatives and mental math to help them substitute powers of 10 for powers of 5 and make the necessary connections. Sometimes, as shown in this video segment, it is hard for students to separate their thinking about the two bases; an example is Britney’s answer about base five, which indicated that she was still thinking in terms of base ten. As they work on the problems themselves, students are clearly developing fluency in converting numbers from base ten to base five.

### Problem A4

What was the benefit of having students use base five, rather than another base, to examine the concept of bases?

### Solution A4

The teacher chose base five for her students because, since they are familiar with powers of 5, it would be relatively easy for them to work with them. Thus, they can use mental math to convert numbers back and forth more easily, allowing them to focus on making observations about place value rather than getting bogged down in cumbersome calculations. However, working with other bases would be beneficial as well.

Additionally, using a different base — for example, base five — is helpful because it makes the underlying concepts of place value more apparent. Students are then able to transfer this knowledge to numbers in other bases — in particular, base ten. In base ten, the same concepts can be harder to notice because of students’ familiarity with this base, which may result in their overlooking the particular meanings of a place-value system.

### Problem A5

Discuss the role of manipulatives in Ms. Miles’s lesson. How did they help deepen the students’ knowledge of the content area?

### Solution A5

Ms. Miles first uses manipulatives to help students understand the process. She is also preparing them to do the written work after they’ve learned with manipulatives.

### Problem A6

Ms. Miles’s lesson was based on Session 3 of this course. Discuss the ways in which her lesson was similar to and different from the one in this course. What adaptations did she make, and why?

### Solution A6

Ms. Miles adapted the lesson to suit her students’ grade level and mathematical abilities. Since this was the first time her class was dealing with different bases, she added manipulatives to help her students gain visual understanding, which, initially, was an important element. She then guided them toward working on a more abstract level, using the symbolic expressions as well as algorithms (later in the lesson).

Notice that she did not use the analogy of packaging that was used in the course. This technique may also be helpful in enhancing students’ understanding of this type of content.