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Foundational
Information
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Goals in Math Education
Math is a very large, vertically structured
field. There are many possible goals in Math
Education, and various stakeholders hold strong
(and, often differing) opinions on what these goals
should be.
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Mathematics and Math Education are large and complex
fields. Math Education changes over time. This Website
explores two current powerful change agents: Brain Science
and ICT. This section of the Website provides information on
several important areas that we need to understand as we
explore possible changes in our Math Education system.
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Weaknesses in Our Current Math
Education System
Much of the weakness in our current Math Education system
is historical in nature and can be discerned by carefully
thinking about the following diagram. It is a simplified
4-step model of using mathematics to solve a problem.
Standard estimates are that about 80-percent of Math
Education at the K-12 level is focused on part 2 of the
diagram.
Historically, Math Education systems focused on helping
students to learn to carry out a number of different types
of "step 2" using some combination of mental and written
knowledge and skills. It takes a typical students hundreds
of hours of study and practice to develop a reasonable level
of speed and accuracy in performing addition, subtraction,
multiplication, and division on integers, decimal fractions,
and fractions. Even this amount of instructional time and
practice -- spread out over years of schooling -- tends to
produce modest results. Speed and accuracy decline
relatively rapidly without continued practice of the
skills.
During the past 5,000 years there has been a steady
increasing body of knowledge in mathematics, science, and
engineering. The industrial age and our more recent
information age has lead to a steady increase in the use of
"higher" math in many different disciplines and on the job.
Our Education System has moved steadily toward the idea that
the basic computational numeracy described above is
insufficient. Students also need to know basic algebra,
geometry, statistics, probability, and other higher math
topics.
As these topics began to be introduced into the general
curriculum, a gap developed between the math that students
were learning in school and the math that most people used
in their everyday lives. More and more, Math Education
focused on learning math topics in a self-contained
environment where what was being learned had little
immediate use in the lives of the students and little use in
the lives of their parents.
A pattern of Math Education curriculum developed in which
one of the main reasons for learning the material in a
particular course was to be prepared to take the next
course. Students developed little skill at transferring
their math knowledge and skills into non-math disciplines or
into problems that they encountered outside of school. Only
a modest number of adults maintain the math knowledge and
skills that they initially developed while studying algebra,
geometry, and other topics beyond basic arithmetic.
That brings us up to current times. Many high schools
require students to take three years of math (during their
four years of high school) in order to graduate. There is
considerable pressure to have all students take an algebra
course. The nature of the instruction and the learning in
many of these math courses follows the "80-percent on step
2" that has been noted above. Students are now learning the
underlying concepts, or how to make use of the math in other
courses or outside of a formal school setting.
As a final note in this subsection, the 4-step diagram
represents only part of the field of math. For example, it
does not include math as a human endeavor with its long and
rich history.
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Here, and perhaps elsewhere in this Website, we want to
emphasize that computers can carry out step 2 both
accurately and rapidly. Thus, our current Math Education
system is spending about 80-percent of its time teaching
students to compete with machines in a domain where the
machine is far far superior to even the best of humans.
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Note to self: The February 1999 issue of the Phi Delta
Kappan is a theme issue focussing on The Mathematical
Miseducation of America's Youth. It is an excellent
resource.
Phi Delta Kappan. Select articles are available online.
Accessed 8/15/01: http://www.pdkintl.org/kappan/karticle.htm.
See specifically the following three articles from the
February 1999 issue:
The Mathematical Miseducation of America's
Youth: Ignoring Research and Scientific Study in
Education, by Michael T. Battista
Parrot Math, by Thomas C. O'Brien
Technology, Children, and the Power of the Heart, by
Matthew M. Maurer and George Davidson
Reference
Glazer, Evan (Date?). Technology Enhanced Learning
Environments that are Conducive to Critical Thinking in
Mathematics: Implications for Research about Critical
Thinking on the World Wide Web [Online]. Accessed
1/14/02: http://www.arches.uga.edu/~eglazer/EDIT6400.html.
Quoting the Abstract from this draft paper:
Mathematics education reform has emphasized a
need for increased experience with technology and
critical thinking skills in order to better prepare
students for a modern society that is dependent on access
to and use of information. However, the complexity and
diversity of these skills leaves some uncertainty about
their successful implementation in the mathematics
classroom. The goal of this paper is to reveal a model
that promotes critical thinking in a technology-enhanced
mathematics classroom. Moreover, this paper will also
examine critical thinking opportunities that are unique
to technology-enhanced environments by examining
discrepancies from critical thinking in non-technology
enhanced environments.
A domain specific definition of critical thinking in
mathematics will be formed in order to identify research
and literature that supports this model. A review of
literature suggests that environmental factors, such as
the nature of a critical thinking task, the students'
learning responsibilities, and the teacher's role to
foster learning, influence and enhance critical thinking
opportunities. The nature of each of these elements in
technology and non-technology-enhanced environments will
be elaborated through the model. The paper will conclude
with implications for research about critical thinking
using resources from the World Wide Web.
This Evan Glazer article includes a lengthy discussion of
what is meant by critical thinking in general, and in
mathematics. Suppose one decides that one of the goals of
mathematics education is to develop critical thinking. Then,
what does progress in Brain Science, ICT, and SoTL do to
help us here?
Six General Principles
Specified by the NCTM
The National Council of Teachers of Mathematics (NCTM) it
the leading professional society for Pre K-12 Mathematics
Education in the United States. Much of the contents of this
section are direct quotes from NCTM [Online]
Accessed 7/28/01: http://www.nctm.org/standards/overview.htm.
NCTM's Math Standards are based on six general
principles:
- The Equity Principle
Excellence in mathematics education requires
equity--high expectations and strong support for all
students.
- The Curriculum Principle
A curriculum is more than a collection of activities:
it must be coherent, focused on important mathematics,
and well articulated across the grades.
- The Teaching Principle
Effective mathematics teaching requires understanding
what students know and need to learn and then challenging
and supporting them to learn it well.
- The Learning Principle
Students must learn mathematics with understanding,
actively building new knowledge from experience and prior
knowledge.
- The Assessment Principle
Assessment should support the learning of important
mathematics and furnish useful information to both
teachers and students.
- The Technology Principle
Technology is essential in teaching and learning
mathematics; it influences the mathematics that is taught
and enhances students' learning.
Comments on the Six General
Principles
The Six General Principles are carefully thought out,
powerful statements that NCTM feels should underlie
mathematics in curriculum, instruction, and assessment. Here
are a few observations that relate some of these principles
to this workshop and Website.
- Note that #3 is supportive of higher-order knowledge
and skills -- learning for understanding, rather than
rote learning.
- Note that #4 is strongly supportive of
Constructivism.
- Note that #6 is supportive of a careful examination
of roles of ICT in all aspects of mathematics curriculum,
instruction, and assessment.
When people read a list such as the Six General
Principles, many tend to accept what is listed as being
complete and comprehensive. It does not immediately occur to
them that the list may have major flaws, and that the list
is a compromise developed by a large number of people
working together over a long period of time.
For example look more carefully at #6. The statement "it
[technology] influences the mathematics that is
taught" does not provide us with much help in deciding what
to add to the curriculum or what to drop from the
curriculum. For example, should we drop paper and pencil
long division of multi digit numbers from the curriculum
because of calculators and computers? How about paper and
pencil calculation of square roots?
Or, look more carefully at #4. Constructivism is an
important learning theory. But, there are other learnng
theories that might also be important to math education.
Situated Learning is a good example. And, teaching for
transfer and learning for transfer (that is, taking into
consideratoin what we know about transfer of learning) would
seem to be an important principle.
Activities
Activity: Working in small groups or individually,
select one of the Six General Principles. Analyze it from
varying points of view such as:
- Is it consistent with and supported by your current
understanding of Brain Science?
- Is it consistent with and supported by your current
understanding of ICT?
- Is it consistent with and supported by your personal
understanding and philosophy of mathematics and
mathematics education?
Share your analysis. Does your analysis suggest changes
that you might want to consider making in your teaching? If
"yes," then suggest a specific change that you intend to
try.
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Ten General Standards
Specified by the NCTM
NCTM's Standards for school mathematics describe the
mathematical understanding, knowledge, and skills that
students should acquire from prekindergarten through grade
12. Each Standard consists of two to four specific goals
that apply across all the grades. For the five Content
Standards, each goal encompasses as many as seven specific
expectations for the four grade bands considered in
Principles and Standards: prekindergarten through grade 2,
grades 3&endash;5, grades 6&endash;8, and grades
9&endash;12. For each of the five Process Standards, the
goals are described through examples that demonstrate what
the Standard should look like in a grade band and what the
teacher's role should be in achieving the Standard. Although
each of these Standards applies to all grades, the relative
emphasis on particular Standards will vary across the grade
bands.
Five Content Standards
- Number and Operations
Instructional programs from prekindergarten through
grade 12 should enable all students to--
- understand numbers, ways of representing numbers,
relationships among numbers, and number systems;
- understand meanings of operations and how they
relate to one another;
- compute fluently and make reasonable
estimates.
Number pervades all areas of mathematics. The other
four Content Standards as well as all five Process
Standards are grounded in number.
- Algebra
Instructional programs from prekindergarten through
grade 12 should enable all students to--
- understand patterns, relations, and
functions;
- represent and analyze mathematical situations and
structures using algebraic symbols;
- use mathematical models to represent and
understand quantitative relationships;
- analyze change in various contexts.
Algebra encompasses the relationships among
quantities, the use of symbols, the modeling of
phenomena, and the mathematical study of change.
- Geometry
Instructional programs from prekindergarten through
grade 12 should enable all students to--
- analyze characteristics and properties of two- and
three-dimensional geometric shapes and develop
mathematical arguments about geometric
relationships;
- specify locations and describe spatial
relationships using coordinate geometry and other
representational systems;
- apply transformations and use symmetry to analyze
mathematical situations;
- use visualization, spatial reasoning, and
geometric modeling to solve problems.
Geometry and spatial sense are fundamental components
of mathematics learning. They offer ways to interpret and
reflect on our physical environment and can serve as
tools for the study of other topics in mathematics and
science.
- Measurement
Instructional programs from prekindergarten through
grade 12 should enable all students to--
- understand measurable attributes of objects and
the units, systems, and processes of measurement;
- apply appropriate techniques, tools, and formulas
to determine measurements.
The study of measurement is crucial in the
preK&endash;12 mathematics curriculum because of its
practicality and pervasiveness in so many aspects of
everyday life. The study of measurement also provides an
opportunity for learning about other areas of
mathematics, such as number operations, geometric ideas,
statistical concepts, and notions of function.
- Data Analysis and Probability
Instructional programs from prekindergarten through
grade 12 should enable all students to--
- formulate questions that can be addressed with
data and collect, organize, and display relevant data
to answer them;
- select and use appropriate statistical methods to
analyze data; develop and evaluate inferences and
predictions that are based on data; understand and
apply basic concepts of probability.
To reason statistically--which is essential to be an
informed citizen, employee, and consumer--students need
to learn about data analysis and related aspects of
probability.
Five Process Standards
- Problem Solving
Instructional programs from prekindergarten through
grade 12 should enable all students to--
- build new mathematical knowledge through problem
solving;
- solve problems that arise in mathematics and in
other contexts;
- apply and adapt a variety of appropriate
strategies to solve problems;
- monitor and reflect on the process of mathematical
problem solving.
Problem solving is an integral part of all mathematics
learning. In everyday life and in the workplace, being
able to solve problems can lead to great advantages.
However, solving problems is not only a goal of learning
mathematics but also a major means of doing so. Problem
solving should not be an isolated part of the curriculum
but should involve all Content Standards.
- Reasoning and Proof
Instructional programs from prekindergarten through
grade 12 should enable all students to--
- recognize reasoning and proof as fundamental
aspects of mathematics;
- make and investigate mathematical
conjectures;
- develop and evaluate mathematical arguments and
proofs;
- select and use various types of reasoning and
methods of proof.
Systematic reasoning is a defining feature of
mathematics. Exploring, justifying, and using
mathematical conjectures are common to all content areas
and, with different levels of rigor, all grade levels.
Through the use of reasoning, students learn that
mathematics makes sense. Reasoning and proof must be a
consistent part of student's mathematical experiences in
prekindergarten through grade 12.
- Communication
Instructional programs from prekindergarten through
grade 12 should enable all students to--
- organize and consolidate their mathematical
thinking though communication;
- communicate their mathematical thinking coherently
and clearly to peers, teachers, and others;
- analyze and evaluate the mathematical thinking and
strategies of others;
- use the language of mathematics to express
mathematical ideas precisely.
As students are asked to communicate about the
mathematics they are studying--to justify their reasoning
to a classmate or to formulate a question about something
that is puzzling--they gain insights into their thinking.
In order to communicate their thinking to others,
students naturally reflect on their learning and organize
and consolidate their thinking about
mathematics.
- Connections
Instructional programs from prekindergarten through
grade 12 should enable all students to--
- recognize and use connections among mathematical
ideas;
- understand how mathematical ideas interconnect and
build on one another to produce a coherent whole;
- recognize and apply mathematics in contexts
outside of mathematics.
Mathematics is an integrated field of study, even
though it is often partitioned into separate topics.
Students from prekindergarten through grade 12 should see
and experience the rich interplay among mathematical
topics, between mathematics and other subjects, and
between mathematics and their own interests. Viewing
mathematics as a whole also helps students learn that
mathematics is not a set of isolated skills and arbitrary
rules.
- Representation
Instructional programs from prekindergarten through
grade 12 should enable all students to--
- create and use representations to organize,
record, and communicate mathematical ideas;
- select, apply, and translate among mathematical
representations to solve problems;
- use representations to model and interpret
physical, social, and mathematical phenomena.
Representations are necessary to students'
understanding of mathematical concepts and relationships.
Representations allow students to communicate
mathematical approaches, arguments, and understanding to
themselves and to others. They allow students to
recognize connections among related concepts and apply
mathematics to realistic problems.
Comments on the Ten General
Standards
Remember that formal mathematics is a 5,000 year old
discipline. It is a cumulative, vertically structured
discipline. Thus, the work of Pythagorus and Euclid, more
than 2,000 years ago, is still important in our current
mathematics curriculum. Currently there are thousands of
mathematics researchers throughout the world who are
building new results on the old, seeking out new
mathematical ideas, and continuing to build the accumulated
knowledge of the field.
The Ten General Standards can be viewed as ten strands
that weave together. They cover topics that the NCTM
considers to be important in our current society. Knowledge
and skills in the areas covered is useful in many different
aspects of learning, play, work, and being a responsible
adult in our society.
As with the Six General Principles, we should recognize
that the Ten General Standards were developed by a large
number of people working over a long period of time. Many
compromises had to be made. And, it may well be that
continued rapid changes in ICT and Brain Science are not
adequately reflected in the Standards.
For example, consider the "compute fluently" part of the
third bullet in Standard #1. What should the specific goals
be? Should we try to "train" students to have a substantial
level of speed and accuracy in carrying out multi digit
computations with integers, decimals, and fractions? It
takes a great deal of effort over a number of years for a
student do meet current paper and pencil computational
standards. One might ask whether this is an appropriate use
of a student's time -- especially given the fact that most
students are unable to develop and maintain a high level of
speed and accuracy. (A passing score on a computation test
might be 70% or 80%. In multi digit computations, completing
one computation per minute might be considered a high rate
of speed. This is not a very useful level of speed and
accuracy. We expect a computer to carry out billions of such
computations in a few seconds. with no errors.)
Activities
Activity: Working in small groups or individually,
select one of the Ten General standards. Analyze it from
varying points of view such as:
- Is it consistent with and supported by your current
understanding of Brain Science?
- Is it consistent with and supported by your current
understanding of ICT?
- Is it consistent with and supported by your personal
understanding and philosophy of mathematics and
mathematics education?
Share your analysis. Does your analysis suggest changes
that you might want to consider making in your teaching? If
"yes," then suggest a specific change that you intend to
try.
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A Mathematics
Expertise Scale Approach
The material given in this section is original, not
quoted from the NCTM Standards.
For any domain of human endeavor, one can think about
levels of performance along an Expertise Scale.
When a learner first begins informal and formal
instruction in a given domain, the learner is a novice. With
training, education, and experience over a period of time,
the learner moves up the scale. The scale point "Useful
Level of Competence" is not carefully defined. It varies
with the learner. And, since "useful" may also be defined by
people other than the learner, the definition may well
change over time and be dependent on the society that is
making the definition. At the current time, a medical doctor
with knowledge and skills that were the standards of medical
school in 1900 would not currently be considered to have a
Useful Level of Competence in medicine.
Earlier on this web page we noted that the Ten NCTM
Standards can be considered as interwoven strands. Let's
look at this idea in a simpler setting. Here are three major
types of domains that might form the basis for setting goals
in mathematics education:
- Mathematics as a human endeavor. Mathematics has a
very long history. Mathematics has beauty. Mathematics is
an important aspect of aspect of past and current
cultures. Mathematics is "the queen of the
sciences."
- Mathematics as as an interdisciplinary language and
tool. Mathematics can be used to help represent,
communicate about, and solve problems in many different
disciplines. Many jobs and other aspects of responsible
adult life in our society require some mathematical
knowledge and skills.
- Mathematics as a discipline. The formal study of and
research in mathematics is at least 5,000 years old. It
is a deep and wide discipline with a huge amount of
accumulated knowledge.
Think about a math educator's Expertise Scale for each of
these three domains.
A person who teaches mathematics has some level of
expertise along each of these three expertise-strand scales.
Presumable the combination of these levels of expertise is
adequate to meet contemporary standards for being a teacher
of mathematics. Unfortunately, for many math teachers, this
is not the case. Moreover, a math teacher is faced by
difficulties such as:
- Contemporary standards increase with time (consider
the medical doctor of 1900) and with the changing needs
of a society.
- A person's knowledge and skills within a domain tend
to decrease over time if they are not frequently used.
(Do you remember all of the details of the math you
learned in high school and college?)
- ICT and Brain Science are rapidly expanding domains
of knowledge and skills that are important in each of the
three mathematics domains that are listed above.
Activities
Activity 1: Working alone, decide on a point on
each of the three scales that you feel would most
appropriately represent the level of expertise that a math
teacher doing your math teacher work should have at the
current time. Describe each of these three points in a
manner that communicates well to you and can be used to
communicate to another person. Then, place yourself onto
each of the three scales. Analyze your strengths and
weaknesses relative to the standards that you have
defined.
Activity 2: Next, add two more expertise domains
to the diagram. One is labeled "ICT and Mathematics
Education" and the other is named "Brain Science and
Mathematics Education." Repeat Activity 1 for these two new
scales. Share your insights in a small group discussion.
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Other Topics That Might be
Added to This Page
- Incremental and continuous change versus disruptive
(large jump) change. Clayton Christensen's work in the
business field.
- Discussion of varying points of view both as to what
constituters mathematics at the K-12 levels, and what it
means to improve math education at these levels.
- Totality of accumulated math data,
information, knowledge, & wisdom is huge and
growing.
- Societal needs in this area have changed and are
changing.
- There are diverse opinions of goals, how to reach
them, & how to measure progress.
- Learning to learn. To what extent do we help our
students learn how to learn mathematics and to be
independent, self-sufficient learners, lifelong learners
of mathematics? This seems like a particularly important
topic.
The human mind (at both a conscious and unconscious
level) is designed to learn and is a lifelong learner.
(It perceives and processes threats and opportunities,
and learns from doing so.) Thus, when we talk about being
a lifelong learner, we need to specify more carefully
what we really want to have happen. We are defining some
external things (such as the steadily accumulating
totality of data, information, knowledge, and wisdom) and
indicating that a person should actively, consciously
engaged in learning certain parts of this. Hmm. We then
need to consider:
- New dimensions, such as distance learning,
computer-assisted learning, intelligent
computer-assisted instruction, learner-centered
software, and brain theory.
- "Just in time" learning.
- Continual learning (a routine, everyday part of
one's job and life).
- Modeling and Simulation. I assume that this topic is
one of the major themes in mathematics education.
Certainly ICT plays a significant role in it.
- ICT-Assisted Problem Solving.
- One of the most useful strategies in problem
solving is breaking big problems into smaller, more
manageable sub problems.
- Increasingly, IT is a tool that can solve these
sub problems -- thus, greatly increasing the
problem-solving capabilities of computer users. (This
ties in with Effective Procedures.)
- Trial and error -- or exhaustive search.
- Library research (for example, using the Global;
Library that we call the Web).
- Graphing, motion graphics, and other aids to
visualization
- ICT as Content of Various Parts of Mathematics.
Examples include spreadsheet, geographic information
systems, computer-aided design, and mathematics systems
such as Mathematica and Maple.
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