JHGFDSA
SOLUTION MANUAL
Elementary and Middle School Mathematics
Chapter 1-22
CHAPTER 1
TEACHING MATHEMATICS IN THE 21st CENTURY MAIN
IDEAS/LEARNER OUTCOMES
TU
The main purpose of this chapter is to introduce students to the changing world and ongoing revolution in school
mathematics and their role in the continuation of this transformation. The four chapters that follow this one will
create the foundational ideas for effective teaching in a problem-solving environment. Those chapters are to be seen
in the context of the revolution that has been going on since 1989.
For more than twenty-five years, mathematics has constantly undergone change. There are several significant factors
in this transformation. Over the years, the pressures influencing school mathematics have become much more
complex. The TIMSS data has caused much of the concern with the popular press pointing out that most
TO
industrialized countries significantly outperform U.S. students in mathematics and science. The National
Assessment of Educational Progress (NAEP) offers ongoing indications of what American students are learning.
These data suggests that while we continue to show improvement, we are not near where we want to be. Both
TIMSS and NAEP data are referred to throughout the book.
This chapter explains briefly how the movement toward shared standards began with the Curriculum and
Evaluation Standards for School Mathematics, followed by the Professional Standards for Teaching Mathematics
and the Assessment Standards for School Mathematics. Considerable attention is given to the content of Principles
R
and Standards for School Mathematics, Curriculum Focal Points and the Common Core State Standards. It may be
useful to use a time line to show your students the progression. The new Association of Mathematics Teacher
Educators (AMTE) Standards for Preparing Teachers of Mathematics (AMTE, 2017) along with NCTM publication
Principles to Actions (Appendix B) provide a focus on mathematical teaching practices. Perhaps a poster or bulletin
G
board with these teaching practices would be a powerful reminder to refer to throughout the semester
Pressures on teachers from state testing programs, the requirements of NCLB and the 2010 Common Core State
Standards set of focused mathematics content standards and practices adopted by 43 of the 50 states (at the date of
this publication) have a significant influence on what is happening in mathematics classrooms. You should
U
encourage students to examine Appendix A of the main text, which contains the Standards for Mathematical
Practice from the Common Core State Standards and the overarching domains. Although it is difficult to make
general statements concerning these influences across states and provinces, teacher candidates should recognize the
importance of discussing their own state/provincial standards in light of the vision of NCTM.
R
You will not be able to adequately help your teacher candidates understand the standards documents in any one class
or by having them read any chapter in this book. The task at this early juncture in the course is to heighten students’
awareness of the standards and then to consistently refer to them as you travel through your course. It is important to
U
help teachers know that Common Core State Standards are not a curriculum but a statement of what is important in
school mathematics. Most likely your own state or province has standards with which prospective and practicing
teachers must be familiar. You may want to have students contrast the NCTM Standards and Curriculum Focal
Points with the Common Core State Standards or those at the state/province level where you are located.
Other Factors to Consider about School Mathematics
The power of the individual teacher cannot be ignored. The last section in Chapter 1, An Invitation to Learn and
Grow points out that the responsibility for excellent teaching and for significant learning by all students falls on the
teacher. We added a section on how to create a whole school agreement with a cohesive mathematics message. How
prospective and practicing teachers use this text and your course instruction to enhance their content knowledge
develops their ability to persevere and demonstrates their positive attitude and passion for teaching mathematics.
jhgfdsa
, JHGFDSA
Their readiness for accepting change and their reflective disposition will play an important role in how their pre-K–8
students grow as mathematical thinkers.
SUGGESTED ACTIVITIES
Here are a few things that can be done to get students involved in the ideas just discussed.
Introduce Principles and Standards
Discuss the goals for students or read aloud the first paragraph of Chapter 1 of Principles and Actions (see Appendix
B) How is this similar or different from what they experienced as elementary or middle school students? Is this what
they are seeing in their field experiences? How possible do they believe this vision is for their classroom?
TU
Point out the structure of the CCSS standards document. The standards define what students should understand and
be able to do. The Clusters summarize groups of related standards. The Domains are larger groups of related
standards (PPT Slide 1-5). The standards from different clusters and domains may sometimes be closely related,
because mathematics is a connected subject. It is critical that your methods students also read the six principles
listed on (PPT Slide 1-8 as well as the five process standards (PPT Slide 1-6). Although these are included in the
text, it would be much better if students read the full discussion of these in the actual document. With a student
membership to NCTM, your teacher candidates can access a Web version of the document or NCTM graciously
allows guests to sign up for 120 days of free access which should cover the time frame of most courses.
TO
The Five Content Standards
Have students examine at least one of the five content standards or the domains from the Common Core State
Standards (downloadable at www.corestandards.org). You might want to spend some time discussing how these
standards are alike or different from the content in your state/province standards. Most importantly for pre service
teachers, you want them to be aware of these documents to get an idea of what content is appropriate at each grade
level.
Another good discussion to have surrounds the issue of basic skills. What is meant when people say “basic skills” in
mathematics? Do the standards documents support students having basic skills? (Absolutely!) A discussion
R
concerning the pressures of high stakes testing and how this pressure influences what is taught in the classroom is
most useful at this early point in the semester. If there are sample-released test items in your state/province that are
available, compare those to the standards documents to see whether there are similarities and differences.
G
Introduce Professional Journals and Publications
Pass out back issues of Teaching Children Mathematics and/or Mathematics Teaching in the Middle School. Point
out some of the regular features of each journal that may be of interest. Encourage or require your students to
become e-members of NCTM. This membership allows full-time students to select an electronic subscription to one
U
of the journals—Teaching Children Mathematics, Mathematics Teaching in the Middle School or Mathematics
Teacher (for high school)—and enable them to download 25 journal articles from a non-subscribed journal. They
can also attend regional NCTM conferences for FREE!
Also show students the NCTM Essential Understandings series including the new Essential Understandings in
Practice, which provide details of how to teach the essential understandings effectively, Navigations books, and
R
other valuable NCTM publications that they should consider for their personal professional libraries. Several of the
publications, such as the Navigations books, each include a CD-ROM with other articles and several good applets
that teachers will find quite valuable.
U
POSSIBLE ASSIGNMENTS
1. Select one of the five process standards from Principles and Actions (see Appendix B) or one of the Standards for
Mathematical Practice from the Common Core State Standards (see Appendix A) and read it carefully. Discuss how
this standard reflects a vision of mathematics that is different from the one you experienced as a K-12 student.
Decide on two ideas that you think are most important for you as a future teacher to take from this standard. Create a
“to learn” list of the things you most need to learn in order to implement that standard as a teacher of mathematics.
Use this “to learn” list approach as a first step toward developing a personal professional growth plan.
jhgfdsa
, JHGFDSA
2. Read Chapters 1 and 2 (Vision and Principles) in the Principles and Standards (pp. 3-26) or pages 3-8 in the
Common Core State Standards (downloadable at www.corestandards.org). After reading these pages, write a 3- to
5-page reflective paper entitled, “Personal Reflections on the NCTM Standards (or the Common Core State
Standards).” In this reflective paper, describe what the NCTM Standards or Common Core State Standards means
to you relative to your own personal mathematics education, how you personally perceive the importance of
mathematics in schools, and the implications for you as a teacher. You can and should add whatever other thoughts
you have about this most important document. There is no “correct” thing to say in this paper. It is important that
you read and reflect seriously on these documents that have a major impact on school mathematics in the United
States and Canada (explore the provincial standards for this assignment as well).
TU
3. (An alternative to the second assignment.) Read about the five process standards in Chapter 3 of Principles and
Standards (pp. 30-70) and/or the Common Core State Standards at a specified grade band. (Repeat the reflective
writing assignment from above.)
4. Select a grade level in the Curriculum Focal Points or the Common Core State Standards and compare the
suggested focal points or standards and related connections to a textbook in your local schools (examine the Table of
Contents) or to your state curriculum guidelines at that grade if you are not in a Common Core State. How are they
inclusive of the same mathematics content? What are the differences?
TO
SUGGESTED ASSESSMENT QUESTIONS
This is a chapter that is designed to make students aware that significant changes in mathematics are occurring and
to introduce them to NCTM and the Standards, Common Core State Standards, and Curriculum Focal Points
documents. This is stage-setting information. The course is about teaching mathematics and how your students need
to take personal responsibility in their growth and development as a teacher of mathematics. It is important for your
students to think about their overall “to learn” list as a way to set their goals for the semester ahead.
R
G
U
R
U
jhgfdsa
, JHGFDSA
CHAPTER 2
EXPLORING WHAT IT MEANS TO KNOW AND DO MATHEMATICS
MAIN IDEAS/LEARNER OUTCOMES
This may be the most challenging chapter in the text, but its contents are at the very heart of teaching
developmentally—teaching from the perspective of a child who must develop his or her own ideas and
understanding. In the spirit of a constructivist approach, the chapter could be titled the “what and how” of teaching
mathematics. The most difficult task with teacher candidates is helping them re-conceptualize the nature of
TU
mathematics. We have come to believe that an appreciation for what it means to “do mathematics” is critical for
teachers to actually engage in standards-based instruction. The activities in the beginning of this chapter are to
provide teacher candidates with the opportunity to (1) make connections within the mathematics and (2) engage in
productive struggle. After providing these experiences, the chapter provides a discussion of what it means to be
mathematically proficient and how students learn mathematics. This includes very basic coverage of learning theory
(constructivist learning theory and sociocultural learning theory), followed by an important discussion of what this
theory means for mathematics instruction. Understanding how people learn is critical to the rationale of why using a
standards-based approach to mathematics instruction. Third, the chapter focuses on knowledge—using the work of
TO
Skemp (R. Skemp, 1978, “Relational Understanding and Instrumental Understanding,” Arithmetic Teacher, 26(3),
9–15) and the five strands of mathematical proficiencies discussed in Adding It Up (PPT Slide 2-8), the point is
made that knowing mathematics is much more than memorizing procedures or naming shapes.
Doing Mathematics
As Magdalene Lampert M. Lampert, 2001, Teaching Problems and the Problems of Teaching, New Haven, CT:
Yale University Press) points out, most people believe (incorrectly) that mathematics is associated with certainty;
that knowing mathematics means being able to get the right answer—quickly. Teacher candidates must come to
R
believe that mathematics does not have to be explained by an expert, but can emerge from carefully planned
instruction. The validity of a mathematical idea is found in the logic and order of the mathematics itself—not from
an answer key or a teacher. Ask teacher candidates to explore the tasks in this section and to think about different
approaches to each (PPT Slides 2-5, 2-6, and 2-7). This can be done as homework or during class. These tasks
provide referents for the learning theory and knowledge discussions later in the chapter.
G
Learning Theory
In both constructivist and sociocultural theory, it is proposed that learners use their prior knowledge and experiences
to build new knowledge. The more new ideas are connected to prior knowledge, the better the chance that these new
U
ideas will be understood and remembered. The “blue dot metaphor” depicted in Figure 2.8 helps teacher candidates
make sense of the basic tenets of constructivist learning theory. As the semester continues, they are likely to talk
about their own learning and teaching in terms connecting more “blue dots” themselves or of finding out what “blue
dots” children have. Teachers tend to confuse constructivism with a way of teaching. From a constructivist learning
point, children will learn by connecting to prior knowledge whether they are hearing a lecture or exploring with
R
manipulatives. The chapter makes this very explicit, but it is worth discussing in class.
Knowledge and Understanding
If knowledge is the possession of an idea, then understanding is a measure of how well this idea is integrated with or
U
connected to other existing ideas in the learner’s cognitive framework. Richard Skemp introduced the terminology
of instrumental and relational understanding. If understanding is placed on a continuum (PPT Slide 2-8), at one end
is instrumental understanding, knowledge with little or no connections to other ideas. At the other end is relational
understanding—ideas that can be related to many others. (Note that instrumental understanding is not the same thing
as procedural knowledge—we want students to know procedures and to have them well connected to their
conceptual understanding.)
Another way to explain understanding is the extent to which you have the five proficiencies for a particular topic
(PPT Slide 2-10). Two of these proficiencies are conceptual and procedural knowledge, which have been well
established as distinct and mutually beneficial in learning a concept. In fact, it is worthwhile to engage teacher
jhgfdsa
SOLUTION MANUAL
Elementary and Middle School Mathematics
Chapter 1-22
CHAPTER 1
TEACHING MATHEMATICS IN THE 21st CENTURY MAIN
IDEAS/LEARNER OUTCOMES
TU
The main purpose of this chapter is to introduce students to the changing world and ongoing revolution in school
mathematics and their role in the continuation of this transformation. The four chapters that follow this one will
create the foundational ideas for effective teaching in a problem-solving environment. Those chapters are to be seen
in the context of the revolution that has been going on since 1989.
For more than twenty-five years, mathematics has constantly undergone change. There are several significant factors
in this transformation. Over the years, the pressures influencing school mathematics have become much more
complex. The TIMSS data has caused much of the concern with the popular press pointing out that most
TO
industrialized countries significantly outperform U.S. students in mathematics and science. The National
Assessment of Educational Progress (NAEP) offers ongoing indications of what American students are learning.
These data suggests that while we continue to show improvement, we are not near where we want to be. Both
TIMSS and NAEP data are referred to throughout the book.
This chapter explains briefly how the movement toward shared standards began with the Curriculum and
Evaluation Standards for School Mathematics, followed by the Professional Standards for Teaching Mathematics
and the Assessment Standards for School Mathematics. Considerable attention is given to the content of Principles
R
and Standards for School Mathematics, Curriculum Focal Points and the Common Core State Standards. It may be
useful to use a time line to show your students the progression. The new Association of Mathematics Teacher
Educators (AMTE) Standards for Preparing Teachers of Mathematics (AMTE, 2017) along with NCTM publication
Principles to Actions (Appendix B) provide a focus on mathematical teaching practices. Perhaps a poster or bulletin
G
board with these teaching practices would be a powerful reminder to refer to throughout the semester
Pressures on teachers from state testing programs, the requirements of NCLB and the 2010 Common Core State
Standards set of focused mathematics content standards and practices adopted by 43 of the 50 states (at the date of
this publication) have a significant influence on what is happening in mathematics classrooms. You should
U
encourage students to examine Appendix A of the main text, which contains the Standards for Mathematical
Practice from the Common Core State Standards and the overarching domains. Although it is difficult to make
general statements concerning these influences across states and provinces, teacher candidates should recognize the
importance of discussing their own state/provincial standards in light of the vision of NCTM.
R
You will not be able to adequately help your teacher candidates understand the standards documents in any one class
or by having them read any chapter in this book. The task at this early juncture in the course is to heighten students’
awareness of the standards and then to consistently refer to them as you travel through your course. It is important to
U
help teachers know that Common Core State Standards are not a curriculum but a statement of what is important in
school mathematics. Most likely your own state or province has standards with which prospective and practicing
teachers must be familiar. You may want to have students contrast the NCTM Standards and Curriculum Focal
Points with the Common Core State Standards or those at the state/province level where you are located.
Other Factors to Consider about School Mathematics
The power of the individual teacher cannot be ignored. The last section in Chapter 1, An Invitation to Learn and
Grow points out that the responsibility for excellent teaching and for significant learning by all students falls on the
teacher. We added a section on how to create a whole school agreement with a cohesive mathematics message. How
prospective and practicing teachers use this text and your course instruction to enhance their content knowledge
develops their ability to persevere and demonstrates their positive attitude and passion for teaching mathematics.
jhgfdsa
, JHGFDSA
Their readiness for accepting change and their reflective disposition will play an important role in how their pre-K–8
students grow as mathematical thinkers.
SUGGESTED ACTIVITIES
Here are a few things that can be done to get students involved in the ideas just discussed.
Introduce Principles and Standards
Discuss the goals for students or read aloud the first paragraph of Chapter 1 of Principles and Actions (see Appendix
B) How is this similar or different from what they experienced as elementary or middle school students? Is this what
they are seeing in their field experiences? How possible do they believe this vision is for their classroom?
TU
Point out the structure of the CCSS standards document. The standards define what students should understand and
be able to do. The Clusters summarize groups of related standards. The Domains are larger groups of related
standards (PPT Slide 1-5). The standards from different clusters and domains may sometimes be closely related,
because mathematics is a connected subject. It is critical that your methods students also read the six principles
listed on (PPT Slide 1-8 as well as the five process standards (PPT Slide 1-6). Although these are included in the
text, it would be much better if students read the full discussion of these in the actual document. With a student
membership to NCTM, your teacher candidates can access a Web version of the document or NCTM graciously
allows guests to sign up for 120 days of free access which should cover the time frame of most courses.
TO
The Five Content Standards
Have students examine at least one of the five content standards or the domains from the Common Core State
Standards (downloadable at www.corestandards.org). You might want to spend some time discussing how these
standards are alike or different from the content in your state/province standards. Most importantly for pre service
teachers, you want them to be aware of these documents to get an idea of what content is appropriate at each grade
level.
Another good discussion to have surrounds the issue of basic skills. What is meant when people say “basic skills” in
mathematics? Do the standards documents support students having basic skills? (Absolutely!) A discussion
R
concerning the pressures of high stakes testing and how this pressure influences what is taught in the classroom is
most useful at this early point in the semester. If there are sample-released test items in your state/province that are
available, compare those to the standards documents to see whether there are similarities and differences.
G
Introduce Professional Journals and Publications
Pass out back issues of Teaching Children Mathematics and/or Mathematics Teaching in the Middle School. Point
out some of the regular features of each journal that may be of interest. Encourage or require your students to
become e-members of NCTM. This membership allows full-time students to select an electronic subscription to one
U
of the journals—Teaching Children Mathematics, Mathematics Teaching in the Middle School or Mathematics
Teacher (for high school)—and enable them to download 25 journal articles from a non-subscribed journal. They
can also attend regional NCTM conferences for FREE!
Also show students the NCTM Essential Understandings series including the new Essential Understandings in
Practice, which provide details of how to teach the essential understandings effectively, Navigations books, and
R
other valuable NCTM publications that they should consider for their personal professional libraries. Several of the
publications, such as the Navigations books, each include a CD-ROM with other articles and several good applets
that teachers will find quite valuable.
U
POSSIBLE ASSIGNMENTS
1. Select one of the five process standards from Principles and Actions (see Appendix B) or one of the Standards for
Mathematical Practice from the Common Core State Standards (see Appendix A) and read it carefully. Discuss how
this standard reflects a vision of mathematics that is different from the one you experienced as a K-12 student.
Decide on two ideas that you think are most important for you as a future teacher to take from this standard. Create a
“to learn” list of the things you most need to learn in order to implement that standard as a teacher of mathematics.
Use this “to learn” list approach as a first step toward developing a personal professional growth plan.
jhgfdsa
, JHGFDSA
2. Read Chapters 1 and 2 (Vision and Principles) in the Principles and Standards (pp. 3-26) or pages 3-8 in the
Common Core State Standards (downloadable at www.corestandards.org). After reading these pages, write a 3- to
5-page reflective paper entitled, “Personal Reflections on the NCTM Standards (or the Common Core State
Standards).” In this reflective paper, describe what the NCTM Standards or Common Core State Standards means
to you relative to your own personal mathematics education, how you personally perceive the importance of
mathematics in schools, and the implications for you as a teacher. You can and should add whatever other thoughts
you have about this most important document. There is no “correct” thing to say in this paper. It is important that
you read and reflect seriously on these documents that have a major impact on school mathematics in the United
States and Canada (explore the provincial standards for this assignment as well).
TU
3. (An alternative to the second assignment.) Read about the five process standards in Chapter 3 of Principles and
Standards (pp. 30-70) and/or the Common Core State Standards at a specified grade band. (Repeat the reflective
writing assignment from above.)
4. Select a grade level in the Curriculum Focal Points or the Common Core State Standards and compare the
suggested focal points or standards and related connections to a textbook in your local schools (examine the Table of
Contents) or to your state curriculum guidelines at that grade if you are not in a Common Core State. How are they
inclusive of the same mathematics content? What are the differences?
TO
SUGGESTED ASSESSMENT QUESTIONS
This is a chapter that is designed to make students aware that significant changes in mathematics are occurring and
to introduce them to NCTM and the Standards, Common Core State Standards, and Curriculum Focal Points
documents. This is stage-setting information. The course is about teaching mathematics and how your students need
to take personal responsibility in their growth and development as a teacher of mathematics. It is important for your
students to think about their overall “to learn” list as a way to set their goals for the semester ahead.
R
G
U
R
U
jhgfdsa
, JHGFDSA
CHAPTER 2
EXPLORING WHAT IT MEANS TO KNOW AND DO MATHEMATICS
MAIN IDEAS/LEARNER OUTCOMES
This may be the most challenging chapter in the text, but its contents are at the very heart of teaching
developmentally—teaching from the perspective of a child who must develop his or her own ideas and
understanding. In the spirit of a constructivist approach, the chapter could be titled the “what and how” of teaching
mathematics. The most difficult task with teacher candidates is helping them re-conceptualize the nature of
TU
mathematics. We have come to believe that an appreciation for what it means to “do mathematics” is critical for
teachers to actually engage in standards-based instruction. The activities in the beginning of this chapter are to
provide teacher candidates with the opportunity to (1) make connections within the mathematics and (2) engage in
productive struggle. After providing these experiences, the chapter provides a discussion of what it means to be
mathematically proficient and how students learn mathematics. This includes very basic coverage of learning theory
(constructivist learning theory and sociocultural learning theory), followed by an important discussion of what this
theory means for mathematics instruction. Understanding how people learn is critical to the rationale of why using a
standards-based approach to mathematics instruction. Third, the chapter focuses on knowledge—using the work of
TO
Skemp (R. Skemp, 1978, “Relational Understanding and Instrumental Understanding,” Arithmetic Teacher, 26(3),
9–15) and the five strands of mathematical proficiencies discussed in Adding It Up (PPT Slide 2-8), the point is
made that knowing mathematics is much more than memorizing procedures or naming shapes.
Doing Mathematics
As Magdalene Lampert M. Lampert, 2001, Teaching Problems and the Problems of Teaching, New Haven, CT:
Yale University Press) points out, most people believe (incorrectly) that mathematics is associated with certainty;
that knowing mathematics means being able to get the right answer—quickly. Teacher candidates must come to
R
believe that mathematics does not have to be explained by an expert, but can emerge from carefully planned
instruction. The validity of a mathematical idea is found in the logic and order of the mathematics itself—not from
an answer key or a teacher. Ask teacher candidates to explore the tasks in this section and to think about different
approaches to each (PPT Slides 2-5, 2-6, and 2-7). This can be done as homework or during class. These tasks
provide referents for the learning theory and knowledge discussions later in the chapter.
G
Learning Theory
In both constructivist and sociocultural theory, it is proposed that learners use their prior knowledge and experiences
to build new knowledge. The more new ideas are connected to prior knowledge, the better the chance that these new
U
ideas will be understood and remembered. The “blue dot metaphor” depicted in Figure 2.8 helps teacher candidates
make sense of the basic tenets of constructivist learning theory. As the semester continues, they are likely to talk
about their own learning and teaching in terms connecting more “blue dots” themselves or of finding out what “blue
dots” children have. Teachers tend to confuse constructivism with a way of teaching. From a constructivist learning
point, children will learn by connecting to prior knowledge whether they are hearing a lecture or exploring with
R
manipulatives. The chapter makes this very explicit, but it is worth discussing in class.
Knowledge and Understanding
If knowledge is the possession of an idea, then understanding is a measure of how well this idea is integrated with or
U
connected to other existing ideas in the learner’s cognitive framework. Richard Skemp introduced the terminology
of instrumental and relational understanding. If understanding is placed on a continuum (PPT Slide 2-8), at one end
is instrumental understanding, knowledge with little or no connections to other ideas. At the other end is relational
understanding—ideas that can be related to many others. (Note that instrumental understanding is not the same thing
as procedural knowledge—we want students to know procedures and to have them well connected to their
conceptual understanding.)
Another way to explain understanding is the extent to which you have the five proficiencies for a particular topic
(PPT Slide 2-10). Two of these proficiencies are conceptual and procedural knowledge, which have been well
established as distinct and mutually beneficial in learning a concept. In fact, it is worthwhile to engage teacher
jhgfdsa
