|
Showing 1 - 16 of
16 matches in All Departments
Why are there so many formulas for area and volume, and why do some
of them look alike? Why does one quadrilateral have no special name
while another has several, like square, rectangle, rhombus, and
parallelogram-and why are all these names useful? How much do you
know ... and how much do you need to know? Helping your students
develop a robust understanding of geometry requires that you
understand this mathematics deeply. But what does that mean? This
book focuses on essential knowledge for teachers about geometry. It
is organized around four big ideas, supported by multiple smaller,
interconnected ideas-essential understandings. Taking you beyond a
simple introduction to geometry, the book will broaden and deepen
your mathematical understanding of one of the most challenging
topics for students-and teachers. It will help you engage your
students, anticipate their perplexities, avoid pitfalls, and dispel
misconceptions. You will also learn to develop appropriate tasks,
techniques, and tools for assessing students' understanding of the
topic.
What is the relationship between fractions and rational numbers?
Can you explain why the product of two fractions between 0 and 1 is
less than either factor? How are rational numbers related to
irrational numbers, which your students will study in later grades?
How much do you know... and how much do you need to know? Helping
your upper elementary school students develop a robust
understanding of rational numbers requires that you understand this
mathematics deeply. But what does that mean? This book focuses on
essential knowledge for teachers about rational numbers. It is
organised around four big ideas, supported by multiple smaller,
interconnected ideas-essential understandings. Taking you beyond a
simple introduction to rational numbers, the book will broaden and
deepen your mathematical understanding of one of the most
challenging topics for students and teachers. It will help you
engage your students, anticipate their perplexities, avoid pitfalls
and dispel misconceptions. You will also learn to develop
appropriate tasks, techniques and tools for assessing students'
understanding of the topic. Focus on the ideas that you need to
understand thoroughly to teach confidently.
This volume shares and discusses significant new trends and
developments in research and practices related to various aspects
of preparing prospective secondary mathematics teachers from
2005-2015. It provides both an overview of the current
state-of-the-art and outstanding recent research reports from an
international perspective. The authors completed a thorough review
of the literature by examining major journals in the field of
mathematics education, and other journals related to teacher
education and technology. The systematic review includes four major
themes: field experiences; technologies, tools and resources;
teachers' knowledge; and teachers' professional identities. Each of
them is presented regarding theoretical perspectives,
methodologies, and major findings. Then the authors discuss what is
known in the field and what we still need to know related to the
major topics.
Why does it matter whether we state definitions carefully when we
all know what particular geometric figures look like? What does it
mean to say that a reflection is a transformation—a function? How
does the study of transformations and matrices in high school
connect with later work with vector spaces in linear algebra? How
much do you know… and how much do you need to know? Helping your
students develop a robust understanding of geometry requires that
you understand this mathematics deeply. But what does that mean?
This book focuses on essential knowledge for teachers about
geometry. It is organised around four big ideas, supported by
multiple smaller, interconnected ideas—essential understandings.
Taking you beyond a simple introduction to geometry, the book will
broaden and deepen your mathematical understanding of one of the
most challenging topics for students—and teachers. It will help
you engage your students, anticipate their perplexities, avoid
pitfalls, and dispel misconceptions. You will also learn to develop
appropriate tasks, techniques, and tools for assessing students’
understanding of the topic. Focus on the ideas that you need to
understand thoroughly to teach confidently. Move beyond the
mathematics you expect your students to learn. Students who fail to
get a solid grounding in pivotal concepts struggle in subsequent
work in mathematics and related disciplines. By bringing a deeper
understanding to your teaching, you can help students who don’t
get it the first time by presenting the mathematics in multiple
ways. The Essential Understanding Series addresses topics in school
mathematics that are critical to the mathematical development of
students but are often difficult to teach. Each book in the series
gives an overview of the topic, highlights the differences between
what teachers and students need to know, examines the big ideas and
related essential understandings, reconsiders the ideas presented
in light of connections with other mathematical ideas, and includes
questions for readers’ reflection.
The depth and breadth of a mathematics teacher's understanding of
mathematics matter most as the teacher engages in the daily work of
teaching. One of the major challenges to teachers is to be ready to
draw on the relevant mathematical ideas from different areas of the
school curriculum and from their postsecondary mathematics
experiences that can be helpful in explaining ideas to students,
making instructional decisions, creating examples, and engaging in
other aspects of their daily work. Being mathematically ready and
confident requires teachers to engage in ongoing professional
learning that helps them to connect mathematics to events like
those they live on a daily basis. The purpose of this volume is to
provide teachers, teacher educators, and other facilitators of
professional learning opportunities with examples of authentic
events and tools for discussing those events in professional
learning settings. The work shared in Facilitator's Guidebook for
Use of Mathematics Situations in Professional Learning (Guidebook)
resulted from a collaborative effort of school mathematics
supervisors and university mathematics educators. The collaborators
joined their varied experiences as teachers, coaches, supervisors,
teacher educators, and researchers to suggest ways to scaffold
activities, encourage discussion, and instigate reflection with
teacher-participants of differing mathematics backgrounds and with
varying teaching assignments. Each guide has ideas for engaging and
furthering mathematical thought across a range of facilitator and
participant mathematics backgrounds and draws on the collaborators'
uses of the Situations with in-service and prospective teachers.
The events and mathematical ideas connected to each event come from
Situations in Mathematical Understanding for Secondary Teaching: A
Framework and Classroom- Based Situations. A Situation is a
description of a classroom-related event and the mathematics
related to it. For each of six Situations, school and university
collaborators developed a facilitator's guide that presents ideas
and options for engaging teachers with the event and the
mathematical ideas. The Guidebook also contains suggestions for how
teachers and others might develop new Situations based on events
from their own classrooms as a form of professional learning. Both
teacher educators and school-based facilitators can use this volume
to structure sessions and inspire ideas for professional learning
activities that are rooted in the daily work of mathematics
teachers and students.
The depth and breadth of a mathematics teacher's understanding of
mathematics matter most as the teacher engages in the daily work of
teaching. One of the major challenges to teachers is to be ready to
draw on the relevant mathematical ideas from different areas of the
school curriculum and from their postsecondary mathematics
experiences that can be helpful in explaining ideas to students,
making instructional decisions, creating examples, and engaging in
other aspects of their daily work. Being mathematically ready and
confident requires teachers to engage in ongoing professional
learning that helps them to connect mathematics to events like
those they live on a daily basis. The purpose of this volume is to
provide teachers, teacher educators, and other facilitators of
professional learning opportunities with examples of authentic
events and tools for discussing those events in professional
learning settings. The work shared in Facilitator's Guidebook for
Use of Mathematics Situations in Professional Learning (Guidebook)
resulted from a collaborative effort of school mathematics
supervisors and university mathematics educators. The collaborators
joined their varied experiences as teachers, coaches, supervisors,
teacher educators, and researchers to suggest ways to scaffold
activities, encourage discussion, and instigate reflection with
teacher-participants of differing mathematics backgrounds and with
varying teaching assignments. Each guide has ideas for engaging and
furthering mathematical thought across a range of facilitator and
participant mathematics backgrounds and draws on the collaborators'
uses of the Situations with in-service and prospective teachers.
The events and mathematical ideas connected to each event come from
Situations in Mathematical Understanding for Secondary Teaching: A
Framework and Classroom- Based Situations. A Situation is a
description of a classroom-related event and the mathematics
related to it. For each of six Situations, school and university
collaborators developed a facilitator's guide that presents ideas
and options for engaging teachers with the event and the
mathematical ideas. The Guidebook also contains suggestions for how
teachers and others might develop new Situations based on events
from their own classrooms as a form of professional learning. Both
teacher educators and school-based facilitators can use this volume
to structure sessions and inspire ideas for professional learning
activities that are rooted in the daily work of mathematics
teachers and students.
Why do some equations have one solution, others two or even more
solutions and some no solutions? Why do we sometimes need to
""switch"" the direction of an inequality symbol in solving an
inequality? What could you say if a student described a function as
an equation? How much do you know...and how much do you need to
know? Helping your students develop a robust understanding of
expressions, equations and functions requires that you understand
this mathematics deeply. But what does that mean? This book focuses
on essential knowledge for teachers about expressions, equations
and functions. It is organised around five big ideas, supported by
multiple smaller, interconnected ideas - essential understandings.
Taking you beyond a simple introduction to expressions, equations
and functions, the book will broaden and deepen your mathematical
understanding of one of the most challenging topics for students -
and teachers. It will help you engage your students, anticipate
their perplexities, avoid pitfalls and dispel misconceptions. You
will also learn to develop appropriate tasks, techniques and tools
for assessing students' understanding of the topic. Focus on the
ideas that you need to understand thoroughly to teach confidently.
Like algebra at any level, early algebra is a way to explore,
analyse, represent and generalise mathematical ideas and
relationships. This book shows that children can and do engage in
generalising about numbers and operations as their mathematical
experiences expand. The authors identify and examine five big ideas
and associated essential understandings for developing algebraic
thinking in grades 3-5. The big ideas relate to the fundamental
properties of number and operations, the use of the equals sign to
represent equivalence, variables as efficient tools for
representing mathematical ideas, quantitative reasoning as a way to
understand mathematical relationships and functional thinking to
generalise relationships between covarying quantities. The book
examines challenges in teaching, learning and assessment and is
interspersed with questions for teachers' reflection.
How do composing and decomposing numbers connect with the
properties of addition? Focus on the ideas that you need to
thoroughly understand in order to teach with confidence. The
mathematical content of this book focuses on essential knowledge
for teachers about numbers and number systems. It is organised
around one big idea and supported by smaller, more specific,
interconnected ideas (essential understandings). Gaining this
understanding is essential because numbers and numeration are
building blocks for other mathematical concepts and for thinking
quantitatively about the real-world. Essential Understanding series
topics include: Number and Numeration for Grades Pre-K-2 Addition
and Subtraction for Grades Pre-K-2 Geometry for Grades Pre-K-2
Reasoning and Proof for Grades Pre-K-8 Multiplication and Division
for Grades 3-5 Rational Numbers for Grades 3-5 Algebraic Ideas and
Readiness for Grades 3-5 Geometric Shapes and Solids for Grades 3-5
Ratio, Proportion and Proportionality for Grades 6-8 Expressions
and Equations for Grades 6-8 Measurement for Grades 6-8 Data
Analysis and Statistics for Grades 6-8 Function for Grades 9-12
Geometric Relationships for Grades 9-12 Reasoning and Proof for
Grades 9-12 Statistics for Grades 9-12
How do you refute the erroneous claim that all ratios are
fractions? This book goes beyond a simple introduction to ratios,
proportions and proportional reasoning. It will help broaden and
deepen your mathematical understanding of one of the most
challenging topics for students - and teachers - to grasp. It will
help you engage your students, anticipate their perplexities, help
them avoid pitfalls and dispel misconceptions. You will also learn
to develop appropriate tasks, techniques and tools for assessing
your students’ understanding of the topic. Essential
Understanding series topics include: Number and Numeration for
Grades Pre-K-2 Addition and Subtraction for Grades Pre-K-2 Geometry
for Grades Pre-K-2 Reasoning and Proof for Grades Pre-K-8
Multiplication and Division for Grades 3-5 Rational Numbers for
Grades 3-5 Algebraic Ideas and Readiness for Grades 3-5 Geometric
Shapes and Solids for Grades 3-5 Ratio, Proportion and
Proportionality for Grades 6-8 Expressions and Equations for Grades
6-8 Measurement for Grades 6-8 Data Analysis and Statistics for
Grades 6-8 Function for Grades 9-12 Geometric Relationships for
Grades 9-12 Reasoning and Proof for Grades 9-12 Statistics for
Grades 9—12
How can you introduce terms from geometry and measurement so that
your students' vocabulary will enhance their understanding of
concepts and definitions? What can you say to clarify the thinking
of a student who claims that perimeter is always an even number?
How does knowing what changes or stays the same when shapes are
transformed help you support and extend your students'
understanding of shapes and the space that they occupy? How much do
you know ... and how much do you need to know? Helping your
students develop a robust understanding of geometry and measurement
requires that you understand fundamental statistical concepts
deeply. But what does that mean? This book focuses on essential
knowledge for mathematics teachers about geometry and measurement.
It is organized around three big ideas, supported by multiple
smaller, interconnected ideas-essential understandings. Taking you
beyond a simple introduction to geometry and measurement, the book
will broaden and deepen your understanding of one of the most
challenging topics for students-and teachers. It will help you
engage your students, anticipate their perplexities, avoid
pitfalls, and dispel misconceptions. You will also learn to develop
appropriate tasks, techniques, and tools for assessing students'
understanding of the topic. Focus on the ideas that you need to
understand thoroughly to teach confidently.
How can you build on young children's interactions with the world
to develop their geometric thinking? What can you say to a student
who claims that a diamond isn't a square because it "stands on a
point"? How can knowing what does not change when shapes are
transformed help you extend your students' thinking about the area
of geometric figures? How much do you know ... and how much do you
need to know? Helping your students develop a robust understanding
of geometry and measurement requires that you understand this
mathematics deeply. But what does that mean? This book focuses on
essential knowledge for teachers about geometry and measurement. It
is organized around four big ideas, supported by multiple smaller,
interconnected ideas-essential understandings. Taking you beyond a
simple introduction to geometry and measurement, the book will
broaden and deepen your mathematical understanding of one of the
most challenging topics for students-and teachers. It will help you
engage your students, anticipate their perplexities, avoid
pitfalls, and dispel misconceptions. You will also learn to develop
appropriate tasks, techniques, and tools for assessing students'
understanding of the topic. About the Series: Focus on the ideas
that you need to understand thoroughly to teach confidently. Move
beyond the mathematics you expect your students to learn. Students
who fail to get a solid grounding in pivotal concepts struggle in
subsequent work in mathematics and related disciplines. By bringing
a deeper understanding to your teaching, you can help students who
don't get it the first time by presenting the mathematics in
multiple ways. The Essential Understanding Series addresses topics
in school mathematics that are critical to the mathematical
development of students but are often difficult to teach. Each book
in the series gives an overview of the topic, highlights the
differences between what teachers and students need to know,
examines the big ideas and related essential understandings,
reconsiders the ideas presented in light of connections with other
mathematical ideas, and includes questions for readers' reflection.
How does a statistical model differ from a mathematical model? What
are the differences among the sample distribution, the sampling
distribution, and the population distribution? In an experiment,
what effect does the sampling method have on the results? What are
the implications of the use of processes of random selection and
random assignment? Can a small sample yield accurate estimates of
population parameters? This book examines five big ideas and
twenty-four related essential understandings for teaching
statistics in grades 9–12. The authors distinguish mathematical
and statistical models, explore distributions as descriptions of
variability in data, focus on the fundamentals of testing
hypotheses to draw conclusions from data, highlight the importance
of the data collection method, and recognise the need to examine
bias, precision, and sampling method in evaluating statistical
estimators. Recognising that analysing data is an important part of
understanding the world, the authors discuss the growth of
students’ ideas about statistics and examine challenges to
teaching, learning, and assessment. They intersperse their
discussion with questions for teachers’ reflection.
How does working with data in statistics differ from working with
numbers in mathematics? What is variability, and how can we
describe and measure it? How can we display distributions of
quantitative or categorical data? What is a data sample, and how
can we choose one that will allow us to draw valid conclusions from
data? How much do you know? and how much do you need to know?
Helping your students develop a robust understanding of statistics
requires that you understand fundamental statistical concepts
deeply. But what does that mean? This book focuses on essential
knowledge for mathematics teachers about statistics. It is
organised around four big ideas, supported by multiple smaller,
interconnected ideas. Taking you beyond a simple introduction to
statistics, the book will broaden and deepen your understanding of
one of the most challenging topics for students and teachers. It
will help you engage your students, anticipate their perplexities,
avoid pitfalls, and dispel misconceptions. You will also learn to
develop appropriate tasks, techniques, and tools for assessing
students' understanding of the topic. Focus on the ideas that you
need to understand thoroughly to teach confidently.
What is the difference between "proof" in mathematics and "proof"
in science or a court of law? In mathematics, how does proof differ
from other types of arguments? What forms can proof take besides
the traditional two-column style? What activities constitute the
process of proving? What roles do examples play in proving? Can
examples ever prove a conjecture? Why does a single counterexample
refute a conjecture? How much do you know...and how much do you
need to know? Helping your students develop a robust understanding
of mathematical proof and proving requires that you understand this
aspect of mathematics deeply. But what does that mean? This book
focuses on essential knowledge for teachers about proof and the
process of proving. It is organised around five big ideas,
supported by multiple smaller, interconnected ideas-essential
understandings. Taking you beyond a simple introduction to proof
and the activities involved in proving, the book will broaden and
deepen your mathematical understanding of one of the most
challenging topics for students...and teachers. It will help you
engage your students, anticipate their perplexities, avoid
pitfalls, and dispel misconceptions. You will also learn to develop
appropriate tasks, techniques, and tools for assessing students'
understanding of the topic. Focus on the ideas that you need to
understand thoroughly to teach confidently.
Unpacking"" the ideas related to multiplication and division is a
critical step in developing a deeper understanding. To those
without specialised training, many of these ideas might appear to
be easy to teach. But those who teach in grades 3-5 are aware of
their subtleties and complexities. This book identifies and
examines two big ideas and related essential understandings for
teaching multiplication and division in grades 3-5. Big Idea 1
captures the notion that multiplication is usefully defined as a
scalar operation. Problem situations modelled by multiplication
have an element that represents the scalar and an element that
represents the quantity to which the scalar applies. Big Idea 2
relates to the algorithms that problem solvers have invented - some
of which have become "standard" - for multiplying and dividing. The
authors examine the ways in which counting, adding and subtracting
lead to multiplication and division, as well as the role that these
operations play in algebraic expressions and other advanced topics.
The book examines challenges in teaching, learning and assessment
and is interspersed with questions for teachers' reflection.
|
|