An Introduction to MED 4185:

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An Introduction to MED 4185 Curriculum
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Course Description
This course examines the ways in which high
school students acquire mathematical knowledge,
considers the particular mathematical knowledge
they should have at each grade level (as
articulated by the Principles and Standards of
School Mathematics), and applies this
understanding to the design of secondary
mathematics curricula. The course may also
include one or more of the following
discussion of how mathematics curricula have
evolved over time in response to developments in
cognitive science and other factors classroom
barriers to mathematical proficiency in the form
of gender, cultural, and socio-economic biases
and a comparison of American secondary
mathematics curricula with those of countries
that perform well on international tests of
student achievement.
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Course Rationale
We have a problem with mathematics education in
the United States. Too many students graduate
from high school lacking the mathematical and
quantitative skills necessary for success in
higher education, the workforce, and daily
decision making. This happens despite national
awareness of the problem and years of attempts to
correct it. In this course we consider the
extent of the problem, and discuss possible
causes and solutions. Although what and how we
teach forms only part of a solution, it is the
aspect of mathematics education over which we
have the most control. Our best hope for
improving mathematics education comes from way we
organize and present our subject. In this course
we focus on organization (curriculum), while MED
4145 focuses on presentation (pedagogy). Knowledg
e of how students learn mathematics forms the
basis for effective curriculum and pedagogy. In
this course we examine the ways in which people
acquire mathematical knowledge. We then consider
the particular mathematical knowledge high school
students should obtain at each grade level, as
well as the principles that guide its sound
organization and presentation. Finally we use
our understanding of how students learn
mathematics and the knowledge they should acquire
to evaluate, improve, and develop mathematics
curricula.
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Studying successful school mathematics programs
provides a source of inspiration for the design
of curriculum and pedagogy. Recent international
studies of school mathematics achievement reveal
surprising differences in mathematics curriculum
and pedagogy across countries. In this course we
consider these differences and the directions
they suggest for improvement. Ineffective
mathematics teaching and curricula contribute to
the problem of mathematics education. Some
students encounter additional barriers to
mathematical proficiency in the form of gender,
cultural, and socio-economic biases in the
classroom. Such biases arise from assumptions
about students innate mathematical talent (or
lack thereof). This course examines the
pervasive myth of ability and the effect of
teacher attitudes on student learning.
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The Problem

• Too many students graduate from high school
  lacking the mathematical and quantitative skills
  necessary for success in higher education, the
  workforce, and daily decision making.
• Before Its Too Late contains a description of
  the problem, the rationale solving it, and a
  proposed solution.
• This video shows the press conference at which
  this report was released.

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• The need for people capable of quantitative
  reasoning is not limited to the workplace.
  Mathematics and Democracy The Case for
  Quantitative Literacy describes several facets of
  daily life that require quantitative reasoning
  skills for their successful navigation. Areas
  include citizenship, culture, personal finance,
  and personal health.
• Innumeracy documents the consequences of an
  inability reason fluently about the basic
  concepts of number and chance. Effects include
  belief in pseudoscience, impossible demands for
  risk-free guarantees in medicine, anxieties over
  personal safety, and so on.

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Evidence of the Problem

• National Center for Education Statistics (NCES)
  Trends in International Math and Science Study
  (TIMSS)
• National Assessment of Educational Progress
  (NAEP) The Nations Report Card

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TIMSS

• What is TIMSS?
• How can it be used?
• How was the study conducted?
• How would you summarize the 2003 results?

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The following slides highlight several
TIMSS 1999 results.
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You can copy TIMSS tables from the .pdf file into
PowerPoint with the camera tool.
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The Nations Report Card

• What is The Nations Report Card?
• How is it used?
• How was the study conducted?
• How would you summarize the 2005 results?

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The following slides highlight several results
from The Nations Report Card 2003.
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The TIMSS Video Study investigated intended
curricula and teaching styles (reported by
teachers), as well as the actual curricula and
teaching styles (captured on video), in a
selection of classrooms in seven countries.
Comparing high-scoring TIMSS nations (such as
Japan and China) with the United States can
suggest possible explanations for the problem
with mathematics education.
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Sources of the Problem

• Teaching
• Curricula
• Facilities
• Students
• Families
• Administrators
• Society
• and more

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When we compare typical Japanese and American
classrooms, the difference is striking. Japanese
classrooms tend to embody ideals described by
American math education researchers, whereas
American classrooms tend not to. We should
not, however, place blame for the achievement gap
solely on American teachers. They form only part
of an intricate educational system. Its
important for us to have a big picture of the
problem so we can more clearly define the role we
can play in its solution. The Learning Gap
describes this picture for us.
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Comparative Studies

• The problem of math education begins with the
  mathematical preparation of elementary school
  teachers.
• The following slides illustrate two situations in
  which elementary school teachers might find
  themselves.
• How would you respond?

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You are teaching division with fractions. To
make the topic meaningful for students, you would
like to present problems in a real-world context.
What would be a good context in which to set the
problem 1? ½ ?
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One of your students comes to class very
excited. She tells you she has figured out a
theory that you never told the class. She
explains that she has discovered that as the
perimeter of a closed figure increases, the area
also increases. She shows you this picture to
prove what she is doing
Side 4 Sides 4 and 8 Perimeter
16 Perimeter 24 Area 14 Area 32 How would
you respond to this student?
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Liping Ma posed these and similar questions to
groups of American and Chinese elementary school
teachers. She found that Chinese teachers begin
their careers with a better understanding of
elementary mathematics than their American
counterparts. Through on-the-job teacher
training, this comprehension grows into what she
calls a profound understanding of fundamental
mathematics (PUFM).
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You are teaching division with fractions. To
make the topic meaningful for students, you would
like to present problems in a real-world context.
What would be a good context in which to set the
problem 1? ½ ?
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One of your students comes to class very
excited. She tells you she has figured out a
theory that you never told the class. She
explains that she has discovered that as the
perimeter of a closed figure increases, the area
also increases.
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Chinese students consistently score significantly
higher than Americans on international tests of
mathematics achievement. This happens despite the
fact that Chinese teachers receive considerably
less mathematical education. In Knowing and
Teaching Elementary Mathematics, Ma suggests PUFM
(or lack thereof) as an explanation for this
paradox.
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Other explanations for the difference in
mathematics achievement among American and Asian
students can be found in the results of five
studies summarized in The Learning Gap.
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Explanations Based on Stereotypes

• Japanese students are innately smarter than
  American students. (False.)
• Japanese students are innately more docile than
  American students. (False.)
• Japanese students engage in endless hours of rote
  memorization and drill. (False.)
• Japanese parents stress academic skills long
  before their children begin formal schooling.
  (False.)
• American children spend more time watching TV.
  (False.)

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The Learning Gap provides explanations based on
careful studies. The sources of the problem of
mathematics education are as varied as they are
plentiful.
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Teaching
Although what and how we teach forms only part of
the problem of mathematics education, it is the
aspect of the solution over which we have the
most control. Our best hope for improving
mathematics education comes from way we organize
and present our subject. In this course we focus
on organization (curriculum), while MED 4145
focuses on presentation (pedagogy).
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Cognitive Science
Knowledge of how students learn mathematics forms
the basis for effective curriculum and pedagogy.
In this course we examine the ways in which
people acquire mathematical knowledge.
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The Psychology of Learning Mathematics

• The Formation of Mathematical Concepts
• The Idea of a Schema
• Symbols
• A New Model of Intelligence
• Relational and Instrumental Understanding

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Office Chair
My Chair
Ags Chair
Agness Chair
Terrys Chair
My Office Chair
Front view
Back view
Side view
Top view
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Furniture
Chair
Sofa
Bed
Bookcase
Chair
Office Chair
Kitchen Chair
Lounge Chair
Lawn Chair
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What does this say about teaching math? A
definition should follow an appropriate set of
examples, not precede them. How many times have
you squinted your eyes at a definition then
waited for the book or the instructor to provide
the examples? How much more natural would it
have been to read that definition after you had
worked through the examples?
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(a b)2 a2 2ab b2

• A student with an instrumental understanding of
  this identity would justify it by citing the FOIL
  method.
• A student with a relational understanding might
  justify it using the illustration below.

a
b
a
a2
ab
b2
b
ab
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Relational vs. Instrumental Understanding PROS

• Instrumental
• Within its own context it is easier to understand
  (invert and multiply).
• The rewards are more immediate (a page of correct
  answers).
• The correct answer comes more quickly.

• Relational
• More adaptable to new tasks.
• Easier to remember.
• Can be a goal in itself (the teacher doesnt need
  to provide external rewards).
• Organic in quality (act as agent of their its
  growth).

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Relational vs. Instrumental Understanding CONS
Relational ?

• Instrumental
• Easier to forget or remember incorrectly.
• Harder to apply a procedure learned in one
  context to another.
• Needs to be reviewed frequently.
• Leads to a perception of mathematics as a set of
  meaningless rules for the manipulation of
  symbols. The rules bear little relationship to
  each other or to real life.
• Results in frustration and anxiety for a large
  proportion of students.
• Creates innumeracy.

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Other Classics

• Cognitive Science and Mathematics Education
• Mathematical Problem Solving
• Learning Mathematics The Cognitive Science
  Approach to Mathematics

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We use our understanding of how students learn
mathematics to evaluate, improve, and develop
mathematics curricula and pedagogy. In this
course we focus on curricula, but our
understanding of cognitive science should inform
pedagogical decisions as well.
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In this course we use the Principles and
Standards for School Mathematics developed by
NCTM as a tool in evaluating mathematics
curricula.
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National Council of Teachers of Mathematics (NCTM)

• Provides vision, leadership, professional
  development, resources, and funding for teachers
  and their supporters.
• World's largest mathematics education
  organization, with nearly 100,000 members.

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Principles and Standards

• Comprehensive and coherent set of goals for
  improving mathematics teaching and learning
• Resource to use in examining and improving the
  quality of mathematics programs
• Guide for the development of curriculum
  frameworks, assessments, and instructional
  materials
• Tool to stimulate ideas and ongoing conversations
  about how best to help students

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Principles
Describe particular features of high-quality
mathematics programs and provide guidance for
educational decisions.

• Teaching
• Assessment
• Technology

• Equity
• Curriculum
• Learning

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Standards

• Descriptions of the mathematical understanding,
  knowledge, and skills that students should
  acquire from kindergarten through grade 12
• Content Standards explicitly describe the content
  students should learn.
• Process Standards highlight ways of acquiring and
  using content knowledge.

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Content Standards

• Number and Operations
• Measurement
• Algebra
• Data Analysis and Probability
• Geometry

Process Standards

• Problem Solving
• Connections
• Reasoning and Proof
• Representation
• Communication

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Failure of Reform

• The contributions made by curriculum and pedagogy
  to the problem of math education have been known
  for decades.
• Both aspects of teaching have been regularly
  overhauled since the 1950s.
• The consistent failure of these reforms have lead
  to the Math Wars and critiques such as Why
  Johnny Cant Add.

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Equity

• Successful mathematics programs embody
  equity---the principle that all students can and
  have the right to learn meaningful mathematics
  with understanding.
• As a result of gender, cultural, and
  socio-economic biases found within mathematics
  classrooms, this right is denied to many
  students.
• Americans tend to believe that mathematical
  ability is innate in some people and not in
  others. Many students sink to the low
  expectations of their teachers.

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The Math Gene

• The math gene is an innate ability for
  mathematical thought.
• Language and math are both made possible by the
  same feature of the human brain.
• This feature is the ability to form concepts by
  the process of abstraction, name concepts, and
  consciously manipulate them.
• Anyone who can use language can do mathematics.