CHAPTER OVERVIEW 1
CHAPTER 2 Sets and Whole-Number Operations and Properties
Chapter Overview
Major Objectives of the Chapter
Preservice elementary teachers will be able to:
• use manipulatives and ideas of sets to develop and understand the concept of whole number;
• use discrete and continuous models to represent the variety of physical situations associated with
each whole-number operation;
• use the properties of and the relationships between the whole-number operations to develop
strategies for learning basic facts;
• compare and contrast different numeration systems and identify the important characteristics of
our base-ten Hindu-Arabic numeration system; and
• use their understanding of whole numbers and whole-number operations to solve problems.
Meeting Some Challenges
One difficulty when planning instruction for Chapter 2 is the fact that the basic ideas are so familiar to
preservice teachers. Whole numbers, place value, addition, subtraction, multiplication, and division are
topics that have been in the first chapters of their mathematics texts since first grade. Trying to view a
world without our whole number symbols, as requested in Mini-Investigation 2.1, requires some
motivation and imagination on the part of both the preservice teachers and the instructor. Frequent
references to their future influence on young children’s understanding of number and attention to the
Connection to the K-8 Classroom feature can be used effectively as sources of the motivation and
imagination needed to keep preservice elementary teachers intellectually engaged in thinking about whole
numbers.
Another challenge involves the properties of the whole-number operations. The mathematical properties
of closure, commutativity, associativity, and identities can seem very abstract and disconnected from what
preservice elementary teachers expect to teach. It is important to make connections wherever possible
between these properties and their implementation in developing operation sense and strategies for
learning basic facts.
A third challenge is encountered when asking preservice teachers to work with other numeration systems.
They feel that their experience with our commonly used base-ten Hindu-Arabic numeration system is
sufficient for teaching elementary mathematics. In order to bring some relevance to the study of other
numeration systems, it is important to have the preservice teachers focus on the specific characteristics of
the different numeration systems that provide clarification and efficiency and how these characteristics
have been brought together in our current numeration system. It is also helpful to point out to the
preservice teachers that the questions they encounter in trying to understand the other numeration systems
are very similar to the questions that their elementary students will have as they develop understanding of
our base-ten system.
2 CHAPTER 2 SETS AND WHOLE NUMBER OPERATIONS AND PROPERITES
Teaching Section 2.1 Sets and Whole Numbers
The emphasis in this section is on the formation of formal definitions for some very intuitive ideas such as
one-to-one correspondence, equivalent sets, equal to, and greater than. It is essential that the preservice
teachers begin to develop some comfort with the use of the mathematical language in these definitions,
language such as “if and only if” and “for all.” As much as possible, these formal definitions should grow
out of the preservice teachers’ language used in small-group and large-group discussions. Some key
questions to stimulate discussion are:
• How do you think the idea of number developed?
• How could we communicate whole number ideas if we didn’t have words like three or symbols
like 3?
• How could a child who doesn’t know how to count verify that two sets have the same number of
objects? That a set has more objects than another set?
Exercises 9 - 11, 17, 18, and 22 focus particularly on connections between informal language and formal
mathematical definitions.
Teaching Section 2.2 Addition and Subtraction of Whole Numbers
To promote ideas for the comparisons between addition and subtraction, some key questions to ask are:
• In what physical situations do we use addition? Subtraction?
• What kinds of questions can we answer with addition? With subtraction?
• What kinds of problems could you make up to go with this addition (subtraction) equation?
• What kind of thinking do you do to find the answer to an addition equation? A subtraction
equation?
• What kinds of models can you use to understand addition? Subtraction?
Exercises 5 - 7, 12 - 19, 28 - 30, 36, 38, and 40 build ideas of appropriate models for and relationships
between addition and subtraction. The number line models and the inverse relationship between addition
and subtraction are particularly important ideas that will be needed to explore addition and subtraction of
integers in Chapter 5.
Teaching Section 2.3 Multiplication and Division of Whole Numbers
Some key questions to ask to promote discussion in this section are:
• In what physical situations do we use multiplication? Division?
• What kinds of questions can we answer with multiplication? Division?
• What kinds of problems could you make up to go with this multiplication (division) equation?
• What kind of thinking do you do to find the answer to a multiplication equation? A division
equation?
• What kinds of models can you use to understand multiplication? Division?
Exercises 4, 5, 10, 13 - 17, 19 - 22, 29, 30, and 43 build ideas of appropriate models for and relationships
between multiplication and division. It is particularly important to emphasize the area model for
multiplication and the two questions answered by division (how many groups and how many in each
group) to prepare for the discussions in Chapter 6 on multiplication and division of rational numbers.
CHAPTER OVERVIEW 3
Teaching Section 2.4 Numeration
This section should involve preservice teachers in recognizing the need for efficient numeration systems.
The emphasis of this section is on the useful characteristics of numeration systems in general–
characteristics such as tallying, grouping of different sizes, place value, and use of a symbol for zero–
rather than the memorization of any specific numeration system. Small-group discussions and debates
provide opportunities for preservice elementary teachers to practice using mathematical language in order
to highlight the important mathematical similarities and differences among the various numeration
systems, as well as to evaluate the advantages and disadvantages of each one. Exercises 13 - 15, 18, 21,
and 24 - 26 highlight comparisons between different numeration systems.