Learning Math: Number and Operations
Classroom Case Studies, 3-5 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.
In the video segment, Ms. Donnell introduces students to That’s Logical! puzzles, which can be solved by using logic, spatial clues, and number theory clues. Each puzzle consists of a three-by-three grid and a set of clues to help students decide where to place the numbers 1 through 9 on the grid. When the numbers are placed correctly, all the clues are true. Read the information on clue grids and the clues below before watching the video segment.
The Clue Grids
Each clue grid consists of several cells from the puzzle grid. Each cell contains a symbol that tells you something about the digit in that cell. These clue grids may be put in the puzzle grid in any way they’ll fit without turning or flipping. Some clues are fixed — i.e., the clue grids can be fit onto the puzzle in only one way. Other clue grids can be fit onto the puzzle in a few different ways.
• The letter E stands for an even number (2, 4, 6, or 8). An E with a slash through it means the number is not even.
• A square stands for a square number (1, 4, or 9). A square with a slash through it means the number is not square.
• A triangle stands for a triangular number (1, 3, or 6). A triangle with a slash through it means the number is not triangular.
• A cube stands for a cubic number (1 or 8). A cube with a slash through it means the number is not cubic.
• The letter P stands for a prime number (2, 3, 5, or 7). A P with a slash through it means the number is not prime.
• A number stands for itself. A number with a slash through it means any number but that number.
Before watching the video segment, you might like to try this sample puzzle and its solution. When viewing the video segment, 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
In this segment, Ms. Donnell explains the That’s Logical! puzzles and has students try to solve a puzzle with two fixed clues.
You can find this segment on the session video approximately 6 minutes and 56 seconds after the Annenberg Media logo.
Answer questions (a), (b), and (c) above.
At what point(s) in the lesson are the students learning new content?
Discuss the role of manipulatives in Ms. Donnell’s lesson. How do they help deepen the students’ knowledge of the content area?
Ms. Donnell’s lesson was based on Session 6 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?
Puzzles in Part A adapted from Findell, Carol, and Greenes, Carole. That’s Logical! Series K-8: A Unique Puzzle System for Logical Thinking. © 2000 by Creative Publications, Wright Group/McGraw-Hill. The above materials may not be reproduced without the written permission of Wright Group/McGraw-Hill.
This session uses classroom case studies to examine how students in grades 3-5 think about and work with number and operations. If possible, work on this session with another teacher or a group of teachers. Using your own classroom and the classrooms of fellow teachers as case studies will allow you to make additional observations.
a. This lesson deals with basic ideas of number theory — for example, understanding specific characteristics that numbers may have, such as being prime or even. The students also need to know factors in order to be able to decide which numbers are square, as well as addition of consecutive numbers in order to find triangular numbers. All this in combination makes them think about the multiple characteristics that numbers may have as well as the relationships between those numbers.
b. The students are using spatial reasoning, number theory, and logical reasoning. They use spatial reasoning to place the clue in the grid, number theory to identify the characteristics of the clue for a particular square in the grid, and logical reasoning when clues fall on top of one another. Since the puzzles are beginner-level, students can use these processes one at a time as they solve each puzzle. In later puzzles, as the level of difficulty increases, students have to use number theory clues and spatial clues simultaneously to place the clues in the solution grid.
c. After reviewing the categories (clues) with students and modeling the activity, the teacher has students work in groups to demonstrate their knowledge and understanding of the given material. The nice thing about this activity is that students can often detect on their own when their thinking is off and the clues don’t match, as is the case in this video segment. Then the teacher’s role is to help students review and correct their work.
The students first review some terms and clues that they are relatively familiar with (except perhaps such terms as triangular or square numbers, etc.). Then, as the activity gets more challenging, they need to consider multiple characteristics of each number and compare the numbers with one another. Again, that may be new for some students. Finally, the lesson reinforces spatial reasoning, which is rarely taught in classrooms but is extremely important.
Manipulatives play a key role in this type of lesson. By manipulating and playing with the physical clues, students are able to make visual connections that help enhance their understanding. Also, the grid greatly helps students organize and keep track of their data and, as a result, solve the problem correctly. Doing the lesson strictly in abstract terms without the aid of manipulatives would pose a much greater challenge. Notice, however, that the lesson helps guide the students to gain more familiarity with abstract reasoning (for example, figuring out that if a number is prime and even, it can only be equal to 2).
Many topics in this course were based on number theory. This lesson adapts some of those ideas to a level suitable to Ms. Donnell’s class and presents them in an introductory manner. The use of manipulatives, the teacher’s modeling of the activity, and the presentation of activities that increase in challenge level are all examples of techniques that Ms. Donnell used to adapt the lesson to her classroom.
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.