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?
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.
Answer questions (a), (b), and (c) above.
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?
Ms. Weiss gave each group of students a set of chips. What were some advantages and disadvantages of using this manipulative?
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?
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.
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.
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.
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.
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.
Session 1 What Is a Number System?
Understand the nature of the real number system, the elements and operations that make up the system, and some of the rules that govern the operations. Examine a finite number system that follows some (but not all) of the same rules, and then compare this system to the real number system. Use a number line to classify the numbers we use, and examine how the numbers and operations relate to one another.
Session 2 Number Sets, Infinity, and Zero
Continue examining the number line and the relationships among sets of numbers that make up the real number system. Explore which operations and properties hold true for each of the sets. Consider the magnitude of these infinite sets and discover that infinity comes in more than one size. Examine place value and the significance of zero in a place value system.
Session 3 Place Value
Look at place value systems based on numbers other than 10. Examine the base two numbers and learn uses for base two numbers in computers. Explore exponents and relate them to logarithms. Examine the use of scientific notation to represent numbers with very large or very small magnitude. Interpret whole numbers, common fractions, and decimals in base four.
Session 4 Meanings and Models for Operations
Examine the operations of addition, subtraction, multiplication, and division and their relationships to whole numbers. Work with area models for multiplication and division. Explore the use of two-color chips to model operations with positive and negative numbers.
Session 5 Divisibility Tests and Factors
Explore number theory topics. Analyze Alpha math problems and discuss how they help with the conceptual understanding of operations. Examine various divisibility tests to see how and why they work. Begin examining factors and multiples.
Session 6 Number Theory
Examine visual methods for finding least common multiples and greatest common factors, including Venn diagram models and area models. Explore prime numbers. Learn to locate prime numbers on a number grid and to determine whether very large numbers are prime.
Session 7 Fractions and Decimals
Extend your understanding of fractions and decimals. Examine terminating and non-terminating decimals. Explore ways to predict the number of decimal places in a terminating decimal and the period of a non-terminating decimal. Examine which fractions terminate and which repeat as decimals, and why all rational numbers must fall into one of these categories. Explore methods to convert decimals to fractions and vice versa. Use benchmarks and intuitive methods to order fractions.
Session 8 Rational Numbers and Proportional Reasoning
Begin examining rational numbers. Explore a model for computations with fractions. Analyze proportional reasoning and the difference between absolute and relative thinking. Explore ways to represent proportional relationships and the resulting operations with ratios. Examine how ratios can represent either part-part or part-whole comparisons, depending on how you define the unit, and explore how this affects their behavior in computations.
Session 9 Fractions, Percents, and Ratios
Continue exploring rational numbers, working with an area model for multiplication and division with fractions, and examining operations with decimals. Explore percents and the relationships among representations using fractions, decimals, and percents. Examine benchmarks for understanding percents, especially percents less than 10 and greater than 100. Consider ways to use an elastic model, an area model, and other models to discuss percents. Explore some ratios that occur in nature.
Session 10 Classroom Case Studies, K-2
Watch this program in the 10th session for K-2 teachers. Explore how the concepts developed in this course can be applied through case studies of K-2 teachers (former course participants) who have adapted their new knowledge to their classrooms.
Session 11 Classroom Case Studies, 3-5
Watch this program in the 10th session for grade 3-5 teachers. Explore how the concepts developed in this course can be applied through case studies of grade 3-5 teachers (former course participants) who have adapted their new knowledge to their classrooms.