Visualizing Elementary and Middle School Mathematics Methods

546 Pages·2011·53.806 MB·English

by  Joan Cohen Jones

Learn Methods for Teaching Mathematics Through
Visual Pedagogy
Visuals help illustrate mathematical concepts and procedures
for teaching mathematics to children. Through visuals with
guided instruction, you learn to organize and prioritize
information, select and use appropriate representations, and
integrate visuals with other pedagogical tools.
Mathematical patterns are abundant in the natural world. Here the seedhead
of a sunfl ower demonstrates the Fibonacci sequence, the chambered nautilus
illustrates the Golden Ratio, and the beehive is constructed from regular
hexagons.
Connect Mathematics to Our Everyday Lives
Children will be motivated to learn mathematics more
successfully if they understand how it is a part of their
lives inside and outside of school. Throughout this text,
mathematics in familiar contexts is illustrated in chapter
openers, discussion of children’s literature, lesson plans,
examples, and activities. Infuse your lessons with these
examples to motivate student interest and notice the
difference in how students respond.
Classifi cation is an important process linked to the acquisition of counting
skills and is also a part of our everyday lives, as illustrated by these students
sorting recyclables.
Teach Mathematics as a Social Activity
Mathematics concepts are addressed in the text through
collaborative activities as well as techniques that encourage
communication and discourse. Mathematics is foremost
a social activity that involves working with others to solve
problems and generate new ideas. Vignettes and research
projects from real classrooms appear throughout Visualizing
Elementary and Middle School Mathematics with questions
on how to apply the results of these situations in the
These kindergarteners are learning how to ask statistical questions and
classroom.
collect and interpret data in a collaborative setting.
There are four principles to consider when
Make Mathematics Accessible to All Populations
differentiating instruction for English-language
Incorporate how diverse cultures have used and contributed
learners:
to mathematics, how these contributions can be integrated 1. Comprehensible input
into the mathematics curriculum, and how mathematics can 2. Contextualized instruction
3. A safe learning environment
be made accessible to all populations. Use real-world and
4. Meaningful learning activities
cultural perspectives of mathematics to teach the strong
connection between mathematics, culture, and learning.
Finger counting has been used by many
cultures around the world. Children
who learn fi nger-counting techniques
can enhance their number sense.
(From Count on Your Fingers African
Style, written by Claudia Zaslavsky,
illustrated by Wangechi Mutu)
V I S U A L I Z I N G
ELEMENTARY AND MIDDLE SCHOOL
MATHEMATICS METHODS
JOAN COHEN JONES, PhD
Eastern Michigan University
Visualizing Elementary & Middle School Mathematics Methods offers future teachers the opportunity
to learn about teaching mathematics with real-life examples, multicultural perspectives, and powerful
visuals. This dynamic approach enables students to set aside their previous beliefs about mathematics
and to learn concepts and pedagogy from a new perspective.
For example, using a real-life visual like a lighthouse can help teach math in a meaningful way. Many
lighthouses, like the one pictured above (an interior and an exterior photo) and on the front cover,
were built with spiral staircases because they take up less floor space than traditional staircases. In
addition to being used for decorative and architectural purposes, spiral curves have been studied by
mathematicians since the time of the ancient Greeks. They appear in many forms—including the shell
of a snail, the structure of a chambered nautilus, and the shape of a whirlpool—a reminder that math
is everywhere.
Credits
VICE PRESIDENT AND EXECUTIVE PUBLISHER Jay O’Callaghan
EXECUTIVE EDITOR Christopher Johnson
ACQUISITIONS EDITOR Robert Johnston
DIRECTOR OF DEVELOPMENT Barbara Heaney
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PHOTO RESEARCHER Teri Stratford
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ISBN 13: 978-0470-450314
Printed in the United States of America
10 9 8 7 6 5 4 3 2 1
Preface
How Is Wiley Visualizing Different?
Wiley Visualizing differs from competing textbooks by with an abundance of class-tested hands-on activities and
uniquely combining several powerful elements: a visual full Lesson Plans based on NCTM and Common Core State
pedagogy, integrated with comprehensive text; the use Standards. In the Classroom features present images and
of authentic classroom situations and activities, actual research-based classroom practices, and Multicultural
materials from children’s literature and publications such as Perspectives in Mathematics features provide content-rich,
Mathematics Teaching Today, Teaching Children Mathematics, culturally relevant examples of mathematics and its place in the
and Mathematics Teaching in the Middle School, and the world. Each chapter presents illustrations from children’s books
integration of Teachscape videos. that contain exciting connections to mathematics content
and offers detailed teaching strategies. These authentic
1. Visual Pedagogy. Wiley Visualizing is based on decades of
situations and materials immerse the student in real-life issues
research on the use of visuals in learning (Mayer, 2005).1 Using
in mathematics education, thereby enhancing motivation,
the Cognitive Theory of Multimedia Learning, which is backed up
learning, and retention (Donovan & Bransford, 2005).2
by hundreds of empirical research studies, Wiley’s authors select
visualizations for their texts that specifi cally support students’
3. Teachscape Videos. Through a partnership with Teachscape
thinking and learning. Visuals and text are conceived and planned
professional development series, Wiley Visualizing provides
together in ways that clarify and reinforce major concepts while
a collection of online videocases featuring rich, authentic
allowing students to understand the details. This commitment to
classroom situations, teacher refl ection, and interviews. Each
distinctive and consistent visual pedagogy sets Wiley Visualizing
of the videocases is referenced within the chapters, supporting
apart from other textbooks.
the relevant content. The combination of textbook and video
2. Authentic Classroom Situations, Activities, and provides learners with multiple entry points to the content, giving
Materials. Wiley Visualizing provides the pre-service teacher them greater opportunity to explore and apply concepts.
Wiley Visualizing is designed as a natural extension of how we learn
To understand why the visualizing approach is effective, it is engaged in learning or exert positive effort to create mental
fi rst helpful to understand how we learn. models; and unproductive cognitive load, which occurs when
the brain is trying to make sense of needlessly complex content
1. Our brain processes information using two main channels: visual
or when information is not presented well. The learning
and verbal. Our working memory holds information that our
process can be impaired when the information to be processed
minds process as we learn. This “mental workbench” helps us
with decisions, problem solving, and making sense of words and exceeds the capacity of working memory. Well-designed visuals
pictures by building verbal and visual models of the information. and text with effective pedagogical guidance can reduce the
unproductive cognitive load in our working memory.
2. When the verbal and visual models of corresponding information
are integrated in working memory, we form more comprehensive,
Research shows that well-designed visuals, integrated with
lasting mental models.
comprehensive text, can improve the effi ciency with which a
3. When we link these integrated mental models to our prior learner processes information. In this regard, SEG Research, an
knowledge, which is stored in our long-term memory, we build independent research fi rm, conducted a national, multisite study
even stronger mental models. When an integrated (visual plus
evaluating the effectiveness of Wiley Visualizing. Its fi ndings
verbal) mental model is formed and stored in long-term memory,
indicate that students using Wiley Visualizing products (both print
real learning begins.
and multimedia) were more engaged in the course, exhibited
The effort our brains put forth to make sense of instructional greater retention throughout the course, and made signifi cantly
information is called cognitive load. There are two kinds of greater gains in content area knowledge and skills, as compared
cognitive load: productive cognitive load, such as when we’re to students in similar classes that did not use Wiley Visualizing.3
1Mayer, R. E. (Ed.) (2005). The Cambridge Handbook of Multimedia Learning. New York: Cambridge University Press.
2Donovan, M. S., & Bransford, J. (Eds.) (2005). How Students Learn: Science in the Classroom. The National
Academy Press. Available online at http://www.nap.edu/openbook.php?record_id=11102&page=1.
3SEG Research (2009). Improving Student-Learning with Graphically-Enhanced Textbooks: A Study of the
Effectiveness of the Wiley Visualizing Series.
Preface iii
How Are the Wiley Visualizing
Chapters Organized?
Student engagement requires more than just providing visuals, text, and interactivity—it entails
motivating students to learn. It is easy to get bored or lose focus when presented with large
amounts of information, and it is easy to lose motivation when the relevance of the information is
unclear. Wiley Visualizing organizes course content into manageable learning modules and relates
it to everyday life. It transforms learning into an interactive, stimulating, and outcomes-oriented
experience for students.
Each learning module has a clear instructional objective, one or more examples, and an opportunity
for assessment. These modules are the building blocks of Wiley Visualizing.
Each Wiley Visualizing chapter engages students from
the start
Chapter opening text and visuals introduce the subject and connect the student with the material
that follows.
Chapter Introductions Alongside
Chapter Outlines provide Key
striking photographs, narratives
Questions to guide students
recount intriguing classroom
experiences to evoke student interest through the chapter.
in the chapter’s central mathematics
concept.
For each chapter, the
NCTM Principles
and Standards are
highlighted for the
relevant grade-level
band, giving the
reader an overview of
the standards-based
mathematics the
chapter will present.
The Chapter Planner gives students a path
through the learning aids in the chapter.
Throughout the chapter, The Planner icon
prompts students to use the learning aids and to
set priorities as they study.
iv VISUALIZING ELEMENTARY AND MIDDLE SCHOOL MATHEMATICS METHODS
Wiley Visualizing guides students
through the chapter
The content of Wiley Visualizing gives students a variety
of approaches—visuals, words, interactions, video, and
assessments—that work together to provide students with a
guided path through the content.
Learning Objectives at the
start of each section indicate in
behavioral terms the concepts
that students are expected to
Process Diagrams provide in-depth explanation
master while reading the section.
of how to use mathematics pedagogy. Clear,
step-by-step narrative enables students to grasp
important topics with less effort.
Throughout the text, visuals provide prospective teachers
with samples of tools to use in the classroom. Several visuals
offer tools for differentiating instruction to meet the needs of
all learners.
Other visuals support
the text by providing
glimpses of students
using the materials Education InSight features are multipart visual sections
and learning the that focus on a key concept or topic in the chapter,
concepts presented in exploring it in detail or in broader context using a
the narrative. combination of visuals.
Preface v
In each chapter, the
Strategies for the
Children’s Literature
Classroom offers detailed
feature presents illustrations
suggestions of how to use
from children’s books that
children’s books to motivate
Multicultural Perspectives in
Strategies for the Classroom contain exciting connections mathematics learning.
Mathematics present content-
guide prospective teachers to to mathematics content.
rich, culturally relevant examples
analyze the material, develop
of mathematics and its place in
insights into essential concepts,
and use them in the classroom. the world.
Fully-developed Lesson Plans Prospective teachers are given an
model ways to make mathematics abundance of hands-on Activities,
culturally relevant and refl ective of which include illustrations of materials
students’ lives outside the classroom, and complete instructions. They can
while fulfi lling standards-based be used as mini-lessons for children to
mathematics objectives. practice using mathematics concepts.
vi VISUALIZING ELEMENTARY AND MIDDLE SCHOOL MATHEMATICS METHODS
In the Classroom features provide a real-life look into a
classroom and give students access to a wide range of ideas and
classroom research. Many are from the pages of Teaching Children
Mathematics.
Through a partnership with Teachscape’s professional
development series, a collection of videocases featuring
rich, authentic classroom situations supplements
the textbook’s instruction. In the textbook, Virtual
Classroom Observations highlight a videocase that
corresponds to the content in the text and provides
Teaching Tips provide applications of best
focal points for the viewer.
practices.
Concept Check questions at the end of
Tech Tools help prospective teachers learn each section allow students to test their
how to integrate technology in the classroom. comprehension of the learning objectives.
Preface vii
Student understanding is
assessed at different levels
Critical and Creative Thinking Questions challenge students to think
more broadly about chapter concepts. The level of these questions
Wiley Visualizing offers students lots of practice
ranges from simple to advanced; they encourage students to think
material in several modalities for assessing their
critically and develop an analytical understanding of the ideas discussed
understanding of each study objective.
in the chapter.
The Summary revisits each major
section, with informative images
taken from the chapter. These visuals
reinforce important concepts.
In the fi eld provides opportunities for Using Visuals calls upon students
prospective teachers to explore the to use the visuals in this textbook
concepts developed in the chapter in as a springboard for creating their
a variety of real-world situations, from own classroom materials or for
analyzing textbooks to observing and understanding the concepts of the
interviewing teachers and students. chapter.
What is happening in this picture? presents a new
uncaptioned photograph or illustration, such as children’s
work, that is relevant to a chapter topic.
Visual end-of chapter Self-Tests pose review
questions that ask students to demonstrate
their understanding of key concepts.
Think Critically questions ask the students to describe and
explain what they can observe in the image based on what
they have learned.
viii VISUALIZING ELEMENTARY AND MIDDLE SCHOOL MATHEMATICS METHODS