## Learning Math: Number and Operations

# Classroom Case Studies, K-2 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:

See Note 2 below.

**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?

**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. Weiss applies the mathematics she learned in the

*Number and Operations*course to her own teaching situation by having her students use counting chips to solve a subtraction problem. The children solve the problem and then discuss their methodology. When they’ve finished discussing the problem, they must write a mathematical sentence to justify their answers.

You can find this segment on the session video approximately 3 minutes and 58 seconds after the Annenberg Media logo.

**Problem A1
**Answer questions (a), (b), and (c) above.

**Problem A2**

At what point(s) in the lesson seen in the video segment are the students learning new content? Are they applying what they already know?

**Problem A3**

Ms. Weiss gave each group of students a set of chips. What were some advantages and disadvantages of using this manipulative?

**Problem A4**

Ms. Weiss’s lesson was based on one from Session 4 of this course. Discuss the ways in which her lesson was similar to and different from the one in Session 4. What adaptations did she make, and why?

### Notes

**Note 2**

The purpose of the video segments is not to reflect on the teaching style of the teacher portrayed. Instead, look closely at the methods the teacher uses to bring out the ideas of number and operations while engaging her students in an activity.

### Solutions

**Problem A1
a. **The content is focused on the meanings of and models for operations — in this case, subtraction. Using different strategies and manipulatives, such as counting chips, gives students some basic insight into the nature of subtraction. They begin doing subtraction in the forms of take-away, comparison, or missing addend, even though they are not aware of it. Indirectly, the students also begin to gain knowledge of place value.

**b.**The students are using counting strategies to solve the problems. They are also able to count large numbers.

**c.**The students have determined that they need to subtract to solve the problem, and in most cases they do not analyze how they know they need to subtract. In general, students have trouble understanding that a comparison problem requires subtraction, perhaps because of the wording, which usually contains the phrase “How many more?” Some students count up from the lesser number to obtain the answer. Other students count back from the greater number. Some count out all the chips, take the given amount away, and then recount for the answer.

To elicit student thinking, Ms. Weiss asks students to justify their answers — to provide a convincing argument that their answer is correct. She insists that they write something on their paper that shows why their answer works. Sometimes this brings to light an error the students made in their computations. In these cases, students often catch the error themselves, self-correct, and then continue their justification as though this new answer were the original one.

**Problem A2**

These students clearly understand the concept of subtraction and can apply the concept to the solution to a problem. However, there is no evidence that the students are using their knowledge of place value to help them solve the problems. The manipulative does not lend itself to any shortcuts in computation. Most students count each chip separately. Very few students attempt to group the chips so that counting or subtraction is more efficient — this appears to be new content to most of the students.

**Problem A3
**By having students use this manipulative, Ms. Weiss is trying to help them see the connections between their work with the chips and their work on paper. However, using the chip model does not force any grouping. Most students frequently lost their place, double-counted, or skipped chips as they tried to solve the problem.

**Problem A4**

The lesson touches on the concepts covered in Session 4 of this course. Ms. Weiss presents different meanings of subtraction in an introductory manner, where the students are gaining different experiences and making their own observations about those meanings. Unlike Session 4, you will see that this lesson focuses on comparing different manipulatives and induces students’ initial understanding of the efficiency of counting in groups (of 10). The lesson deals only with whole numbers.