What is Mathematics?
Major Unifying Themes in
Mind and Body Tools
Science of Teaching &
We have a 5,000 year of history of development in the
curriculum, instruction, and assessment of our formal math
education system. During this time, a huge amount of new
mathematical discovers occurred, and many of them gradually
made their way into the K-12 curriculum. But, most of the
math curriculum content at the K-12 level is is math that
was developed hundreds of years ago.
Thus, as we look toward the future of K-12 math
education, some of the things we will likely see
- While there will continue to be changes
in the K-12 content, these will be based almost
entirely on math that was developed hundreds of yeas ago.
The math content changes will mainly be increases and
decreases on the emphasis places on various topics. There
will be a continuing emphasis on understanding of
concepts and on problem solving.
- The major changes in math curriculum content will
come about because of the steadily increasing
capabilities and availability of ICT and the direct
relevance of ICT in
math curriculum content. More specifically,
Computational Mathematics is of growing importance.
[Eventually this document will contain a major
section on Computational Mathematics. There is a strong
parallel between Computational Mathematics and
Computational Science. Computational Science is now one
of the three main ways of knowing and doing
- We will see increased use of
ICT-based math applications in non-math disciplines,
because ICT is an interdisciplinary tool that empowers
users of math,
- Significant progress will occur in instructional
methods (pedagogy). Our steadily increasing knowledge
of Brain Science, Learning Theory, and Teaching Theory
will all contribute to this progress.
- There will be continuing controversy on
assessment. Math assessment has a long history of
occurring in an environment in which students are allowed
to use "traditional" math tools such as paper, pencil,
and perhaps straight edge and compass. The past 20 years
have see significant progress toward allowing students to
use calculators while being assessed. The future will
gradually bring an emphasis on hands on ICT while being
assessed, and portfolio assessment.
- We will see continuing improvement in Computer-Assisted
Learning (which, of course, includes the build-in
"Help" features of all modern software tools. This will
facilitate "Just in time." learning and review.
- Distance Learning
and Computer-Assisted Learning will become more
closely intertwined and will become important components
of the instructional delivery system.
Each of these is affected by the continued rapid change
of ICT capabilities and availability. (Such projections are
a standard component of discussions about the
future of ITC in education. A brief discussion of this
topic is given at the end of this page.)
Each of these "predictions" is discussed in more detail
in the sections that follow.
We do not have a Federal Government-mandated national
curriculum in mathematics. However, the work of the NCTM,
major book publishers, and testing agencies has led to a
relatively uniform curriculum across the country. The
curriculum content that we have is based on input from
thousands of math education leaders and other people over a
long period of time. Over the years, there has been a steady
- Move certain secondary school topics to lower grades.
(For example, we are currently into an "Algebra for all."
movement, with algebraic concepts and then formal algebra
being introduced at earlier grades.
- Integrate some math topics of growing applicability
and importance into the general curriculum. Probability
provides a good example.
The NCTM curriculum content recommendations have a strong
emphasis on students understanding of concepts, problem
posing, and problem solving. There is a decreasing emphasis
on rote memorization and on by-hand computation.
Activity 1: Many students get through certain math
topics by rote memorization and by developing skill at
plugging numbers in formulas and accurately carrying out the
needed computations. One sees examples of this as students
encounter topics such fractions, probability, and algebra.
Working individually and then sharing in small groups,
select a math topic that you know well. Think carefully
about rote memorization of algorithms or formulas and
developing skill in their use, versus understanding of the
underlying concepts. What relationships can you see between
the concepts and the computational processes? (For example,
what does a student learn about the concepts of division by
developing skill in paper and pencil long division of multi
digit numbers? For another example, probably you have
memorized the rule: To divide a fraction by a fraction,
"invert and multiply." Think about the concept of division
versus this computational algorithm. Also, think about
whether you can explain why this algorithm works.)
Activity 2: "Algebra for all," along with an
increased emphasis on standards and accountability, is a
controversial movement in math education. Working in small
groups, discuss possible roles of Brain Science and ICT in
Roles of ICT in
Math Curriculum Content
By the late 1970s, handheld calculators had become
inexpensive enough so that many people begin to recommend
their inclusion in the K-12 curriculum. In 1979, the
National Council of Supervisors of Mathematics made such a
recommendation. In 1980, the National Council of Teachers of
Mathematics made this recommendation.
Now, more than 20 years later, we can look at what has
transpired. From a hardware point of view, there has been
considerable progress. A 5-function with memory handheld,
solar cell powered calculator retails for $5 or less.
Multifunction scientific calculators retail for under $15.
Graphing and equation-solving calculators are available for
under $100. Classroom sets of such calculators are available
in most schools. And, of course, many students own such
calculators or have ready access to them at home.
One can find a number of examples of calculator-related
changes in curriculum content, instructional processes, and
assessment. Here are four general examples:
- Certain curriculum topics have greatly changes. The
by-hand calculation of square roots, and the use of math
tables with interpolation to computer trig functions,
have nearly disappeared from the curriculum. Computation
using slide rules and using logarithms has nearly
disappeared from the curriculum. The use of graphing
calculators has led to significant changes both studying
functions and solving equations.
- The use of calculators is allowed in certain types of
national and state assessments.
- Most math textbooks at the K-8 level make mention of
calculators. Some include a significant emphasis, while
others downplay the topic.
- Many elementary teachers, especially in the primary
grades, feel that student use of calculators is
inappropriate. A commonly heard statement is, "I think it
is okay for students to use calculators after they have
mastered the computational algorithms [of multi digit
addition, subtraction, multiplication, and
What is important about the types of changes listed above
is that the calculator is being used as a tool to do
mathematics. There is a growing realization that memorizing
algorithms and formulas, ad developing speed and accuracy in
their use, has been over emphasized . The calculator makes
it possible to place less emphasis on these topics, and
shift some of this time to learning concepts, problem
posing, and problem solving.
One might think that the same statements would hold for
use of computers in K-12 math education. For the most part,
however, this has not been the case. The computer has not
been viewed as a tool to help represent and solve math
problems. Rather, it has been viewed as a tool for
instruction. The attitude of many teachers is expressed by
the statement, "The computer is a tireless drill master,
helping students master the basics of
We are making steady progress toward providing every
student with a computer to routinely use as they study and
do math. Thus, it is becoming more and more feasible to
provide students with good ICT as they study math, as they
apply their math knowledge throughout the curriculum and
their everyday lives, and as they are assessed in math. Here
are a few important things to think about as this
- A computer is an information storage device (and, or
course, can provide access to the Global Library that we
call the Web.) Thus, we need to think in terms of math
curriculum, instruction, and assessment in an environment
in which the student has access to books and materials
studied in the past and currently being studied, as well
as a mathematical dictionary, a mathematical
encyclopedia, lots of math reference books, and so on.
This is a hugely changed environment from the current
situation in which many teachers are loth to even allow
students access to their current textbook while taking a
test, and most students lack ready access to an adequate
math reference library while they are studying and using
- The human-machine interface in a calculator is very
poor relatively to what can be provided by a computer
with a full sized keyboard and a decent sized screen.
Moreover, the "Help" features in a computer can provide
detailed, step by step, just in time instruction on use
of software tools. Just in time learning and review of
topics is a powerful aid to learning and using math.
- We now have available very powerful software that can
solve a huge range of the types of problems that we
currently teach students to solve by hand. (Indeed, some
versions of this software are available on calculators.
But, better user interface that computer can provide
makes such tools much more user friendly on a computer.)
This situation helps to make clear the issue of concepts
versus processes. Computers can carry out mathematical
processes. If such use of computers becomes routine in
curriculum instruction, and assessment, then there will
be much more time available to help students learn
concepts, problem posing, and problem solving.
Activity 1: Working in groups of three, assign
each person a different one of the three ideas listed above.
Each person is to think about their topic, and then develop
their own opinions of the opportunities and threats that the
idea presents for the student and the teacher. Then each
person is to do a brief presentation to the other members of
the group, and facilitate a discussion on the topic.
Activity 2: Suppose that all students had ready
access to ICT, math software applications, math library, and
math Computer-Assisted Learning materials on a routine
basis. In small groups, discuss how this would change math
curriculum, instruction, assessment, and the general overall
job of being a math teacher.
Use of Math in Non-math
We all routinely make use of math. For example, think
about the measurement of time (including day of the week,
month of the year, the year), quantity (money, your salary,
your age, your weight), location (how to go some place and
get back home), and so on. We routinely make use of maps,
graphical representations of data, and time schedules. Math
has such general usefulness that it is one of the basics of
education and is a routine part of the K-12 curriculum.
Math is useful across all of the disciplines that
students study in K-12 education. Thus, curriculum designers
face a significant transfer of learning problem. How can
math curriculum, instruction, and assessment be designed to
facilitate students being able to use their math throughout
their current and future curriculum, jobs, adult lives,
hobbies, and so on? Most math educators agree that we do not
do very well in teaching math for such transfer of teaching,
and there is considerable room for improvement in this
ICT is an interdisciplinary tool. This means that many
different disciplines find it valuable to incorporate
instruction about and use of ICT. Gradually, teachers in
various disciplines are role modeling effective use of ICT
in teaching, learning, and using the disciplines they
This opens up opportunities for increasing the use of
(ICT-enabled) math within the various disciplines. For
example, consider a spreadsheet. The study and use of
spreadsheets is a perfectly appropriate topic in math
instruction at a variety of grade levels. But, a spreadsheet
is a valuable tool in all of the sciences and social
sciences, as well as in business courses. Thus, a math
teacher who is teaching spreadsheet will want to make a
determined effort to have students develop and use
applications in a variety of disciplines. And, of course,
similar statements hold for the use of the data graphing
features that are built into modern spreadsheet
Another example is provided by statistical computations,
mean, standard deviation, and such correlation., From a math
teacher point of view, one measure of success would be
seeing students taking the initiative to use various
math-oriented ICT tools in non-math disciplines -- and even
helping their teachers in these disciplines to learn how to
make such use of ICT.
Activity: Select a math topic that has useful
applications in non-math courses, and in which ICT is a
valuable aid. Think about how you can help your students to
learn the ICT aspects of this math topic in a manner that
will facilitate students making this transfer of learning to
other disciplines. Share in small groups.
Here are a few things that we know:
- The human brain has some innate abilities in counting
and spatial relationships. Formal education can greatly
enhance the usability and power of such innate abilities.
A number of important ideas about this are covered in the
- Gardner, Howard (1991) The Unschooled Mind: How
Children Think and How Schools Should Teach. Basic
Perhaps the main point that Gardner makes is that
people can learn a lot without formal education. However,
the contemporary standards in many disciplines can only
be reached by concerted and extensive study and practice.
Thus, our educational system should develop efficient and
effective instructional methods to help students gain
contemporary levels of expertise within the various
disciplines covered in school.
- Each student brings their own knowledge, skills,
attitudes, and other "baggage" to any particular learning
task. New knowledge and skills are "constructed" by
building on previous knowledge and skills. There are huge
individual differences among students. Individualization
of instruction makes a significant difference in the
speed and quality of learning that occurs. (One way of
looking at this is to compare the results of conventional
instruction in classes of 20 to 30 students, with the
results of individual tutoring by skilled tutors.
Students learn significantly faster and better with
- Learning can be substantially improved by actively
engaging students in applying what they are learning in
problem posing and problem solving environments,
metacognition, project-based learning activities, and
other ways that increase student engagement.
- It is important for teachers to role model being
active, engaged learners and users of what they are
teaching. For example, students need to see science
teachers who do and use science in their everyday lives,
writing teachers who write, art teachers who are artists,
and so on.
- Cooperative learning and cooperative problem solving
are effective instructional and learning
Note to self: This section needs more work.
Eventually we are looking for quite specific recommendations
for things that a teacher can do to enhance the
effectiveness of the instructional process. Here are a few
- Teachers are being encouraged to make use of some
form of Presentation Graphics (such as Microsoft's
PowerPoint, or the Slide Show in AppleWorks) in their
everyday teaching. I am not aware of solid research that
indicates that this leads to students learning faster and
- The research supporting Project-Based Learning and
Problem-Based Learning is fairly strong. (Note that each
is called PBL.)
- The Low-Road, High-Road model of Transfer of Learning
is useful in teaching for transfer.
- Constructivism is an important Learning Theory for
use in math education and many other disciplines.
- It is helpful to view mathematics as a language. It
is effective to have students do journaling, peer
tutoring, cooperative problem solving, and in other ways
communicate mathematics in written and oral form.
Activity: In small groups, discuss uses of
ICT-Assisted PBL in math education. What are strengths and
weaknesses (opportunities and threats) of this form of
Assessment in a Hands-on Environment
Computerized Adaptive Testing
General information about CAL
Specific results in math education
Change in effective class size
Learning to learn in this environment
Specific applications in math education
Gordon Moore's "Law" and
The density of transistors on a chip will double every 18
months, thus increasing the price performance of compute
power by a factor of two every 1 1/2 years.
Note that bandwidth of connectivity is currently doubling
approximately every nine months.
Storage has been increasing at a similar rate.
In 1983 there were 74 "Super Computers" in the world.
Each had a "blinding speed" of 200 to 300 million operations
This year (2001) many millions of people throughout the
world are buying microcomputers that are several times this
fast and cost about 1/5,000 to 1/10,000 as much in 1983
My (no longer very new) laptop has approximately the same
compute power as the Super Computer of 1983
The new iBook has an educational list price of $1,299. It
is a 500 million operation/second machine. It is more
powerful than the 1983 Super Computers.
Short Term Future
Today's Super Computers have speeds in the one to six
teraflops (trillions of floating point operations per
second) Thus, they are approximately a thousand times as
fast as top of the line microcomputers.
Back in the early 1980s, a Super Computer had one or a
modest number of Central Processing Units. Now, of course,
higher overall speeds are being achieved by building a
machine that incorporates a large number of relatively
inexpensive microprocessors. There is a brief news item:
NSF Funds $53 Million Computing Grid. The
National Science Foundation will fund development of a
$53-million computing grid called the Distributed
Terascale Facility (DTF), which when it is completed in
2002 will be more than 1,000 times faster than the IBM
"Deep Blue" supercomputer famous for defeating chess
champion Garry Kasparov in 1997. DTF will be used by four
U.S. research centers: the National Center for
Supercomputing Applications (NCSA) at the University of
Illinois at Urbana-Champaign; the San Diego Supercomputer
Center at the University of California at San Diego;
Argonne National Laboratory; and the California Institute
of Technology. The system will be built by IBM, using
Intel Itanium McKinley chips and the Linux operating
system, and more than 1,000 IBM server systems will be
connected through a 40-gigabit-per-second network created
by Qwest Communications. NCSA's Daniel Reed said the grid
will transform "the way science and engineering research
is done.'' (Reuters/New York Times 9 Aug 2001)
Daily, 10 August 2001)
Deep Blue could analyze more than 200,000,000 chess board
positions per second. It had 256 processing units. Thus, the
gain in speed in the new machine is achieved by a
combination of more processors and the greater speed of the
The concept of a "grid" is very important. We are moving
toward huge numbers of people having access to both the Web
(for connectivity and the Global Library) and the grid (for
access to processing power). A slightly different way of
looking at this situation is that the Web will consist of
connectivity, huge amounts of storage, and huge amounts of
processing power. People using the Web will be able to draw
upon the amounts of storage and processing that they need --
and typically will not have any idea where the storage or
processing power is located.
15 Year Projection
Fifteen years is the time for 10 doublings of chip power
if Moore's Law continues to hold. Ten doublings is a factor
This might mean that children now in the 1st grade will
have routine access to computers that are a thousand times
as fast as today's microcomputers -- that is, a trillion
operations per second.
"Way Out" Future Ideas
Ray Kurzweil, Hans Moravec, and others indicate that
another 15 years of Moore's law holding will bring us
supercomputers with the capability of a human mind. Thirty
years will see microcomputers with this level of
The meaning of these forecasts is unclear to me. But,
they seem relevant to education.
The following book takes a quite different approach to
predicting the future:
Dertouzos, Michael (2001). The Unfinished
Revolution: Human-Centered Computers and What They Can Do
for Us. New York: HarperCollins Publishers. Quoting from
the dust jacket:
If cars were as difficult to drive as our
computers are to operate, they would never leave the
garage. Yet every day we put up with infuriating
complications and incomprehensible error messages that
spew forth from out technology: software upgrades
crash our machines, Web sites take forever to
download. e-mail overwhelms us. We spend endless time
on the phone waiting for automated assistance.
In effect, we continue to serve our machines' lowly
needs, instead of insisting that they serve us -- a
situation that will only get worse as millions of new
mobile devices arrive on the scene.
Our world doesn't have to be this way. It shouldn't
be this way.
Activity: Continued progress in ICT hardware and
software is steadily increasing the capabilities (the power,
the knowledge) of ICT systems. In small groups, discuss some
of the changes we should be making in our current math
education system to prepare students to make effective use
of current and future ICT as an aid to learning and using
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