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:
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?
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.
Answer questions (a), (b), and (c) above.
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: 50, 51, 52, 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.
At what point(s) in the lesson are the students learning new content?
Students are learning a new base. The new content is the interpretation of place value with a base other than ten.
How do students transfer their knowledge of base ten to base five? What is the evidence?
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.
What was the benefit of having students use base five, rather than another base, to examine the concept of bases?
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.
Discuss the role of manipulatives in Ms. Miles’s lesson. How did they help deepen the students’ knowledge of the content area?
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.
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?
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.
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.