A B C

Solutions for Session 10, Part A

See solutions for Problems: A1 | A2 | A3 | A4 | A5 | A6| A7

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.