|
Mathematics, Spreadsheets and Thinking
A Computers in Education Chapter
Introduction
New tools and technologies have historically brought about dramatic
changes in our mathematical understanding and practices. This chapter on
mathematics, spreadsheets and thinking shows the dynamic nature of mathematical
knowledge over time and into the future. Common spreadsheet applications
and other computer applications will be used to assist exploring these
changing understandings and the changing instruction in math across a wide
range of ages and math topics.
Cool mathematical activities (those that are quick, easy, age and math
level relevant, and fun) are available to help introduce math on a computer.
For example, note the spreadsheet survey activity on the right for second
graders. This is not a replacement for physical manipulatives such stacking
wooden block surveys, or seat work such as pencil filled squares on graph
paper. It must follow them. The digital perspective provided by the spreadsheet
parallels and reinforces the general reading goals of second grade in taking
one more step into thinking with the abstractions of words, numbers and
images. |
Second Grade Spreadsheet Survey
|
For those ready for this level of abstraction, spreadsheets provided
an interactive environment with numerous opportunities for further manipulation.
Such manipulation is an important early step in preparation for higher
levels of scientific visualization. For examples, changes to the raw data
are immediately reflected in the graph. Further, the spreadsheet's
numerous options for graphing provide immediate access to a wide range
of perspectives on a given set of data, including charts based on bar,
line, pie, different three dimensional models and more. Spreadsheet applications
are an important vehicle for math studies, in part because of their ready
availability, ease of use, power and low cost. But there is so much more
that a computer provides for math studies.
Mathematics and scientific curriculum continue to evolve to keep up
with the latest understandings that computer-enhanced research extracts
from the world. It has not always been the case though that rulers
and governments wanted the facts known and the correct observation supported
in policy and teaching. There are many strong relationships between math,
science and social studies topics.
Credit: NASA
|
For example, there was a time when an Italian
experimenter by the name of Galileo confirmed Copernicus's perception that
the sun and planets did not go around the planet Earth and that further,
the planets, including the Earth, go around the sun. In 1633 Galileo was
sentenced to be imprisoned for life for his discussion of this fact. Galileo's
understanding came from an observation that others had not seen because
they did not have or had not experienced the technology of his telescope.
Even when they did attempt to duplicate his work, they ignored this knowledge
to keep favor with the popular view and the political power of the day.
His life is an amazing story of science, social intrigue and personal bravery.
Of course, over time, the correct observation about the science and mathematics
of planetary orbits became accepted and routinely taught to each new generation. |
|
Equally amazing discoveries continue today. What would Galileo think
of today's computer controlled Hubble space telescope? Click the picture
on the left to see more than 8,000 images from and of the Hubble Space
Telescope. Galileo is also the name of an important scientific mission
of discovery and satellite that NASA directed around the planet Jupiter.
Click the picture to the right to see more than 8,500 pictures of Galileo,
of the man, the tools and the space mission. Mathematical knowledge was
essential in the development of both of these cutting edge projects. |
|
Just as Galileo magnified what he could see with the telescope and changed
the world, today's thinkers magnified their mathematical thinking with
calculators, computer programming and spreadsheets and changed the world,
and continue to change our world. Whole new mathematical perceptions and
disciplines emerged. A part of the lesson to be learned here, and for you
to teach others, is that mathematics is important, powerful, growing, and
exciting. Computer technology makes this math more accessible than ever.
4-8th
Lessons on Fractals
|
For example, another set of new perceptions in math emerged in the
1950's and 1960's about the nature of mathematics. In the following years,
the scholars in discipline after discipline found that nonlinear dynamics,
also referred to as fractals and chaos theory, applied to their work and
revolutionized the thinking in their fields. In this case it was the technology
of the calculator and the computer with their ability to quickly carry
out an enormous number of repetitive and linked calculations that enabled
these discoveries. (Click the picture of the Mandelbrot Set fractal
on the left to see more than 30,000 other pictures of fractals. Click the
"K-8 Lessons..." text to learn how to teach about fractals.) Among other
benefits, fractals have given us a more realistic mathematical model for
the formations of nature. As is well known, powerful theory eventually
creates powerful practical results. Among other things, this knowledge
has in turn been used to solve practical engineering problems. |
Like the telescope and much later the microscope, calculators
and computers were used as complexity-scopes for mathematics. These digital
tools enabled easy and regular observation of what was previously all but
invisible to the mind. For example, in modeling basic global patterns of
air, universal mathematical principles emerged. Using the computer technology
of the 1960's, Lorenz found similar repeating patterns, but never precisely
repeated patterns. He further discovered that long range prediction was
impossible as such systems diverge exponentially with time. A "gold rush"
of discovery of such systems then occurred rapidly across the disciplines.
Departments of Nonlinear Studies emerged. Forty years later this fundamental
mathematical discovery is being applied
to psychology and other life sciences. What makes this knowledge of
complexity so accessible to K-12 school curriculum is that very simple
elements and very simple steps are its building blocks. What will it take
for such ideas in enter the mathematical canon in the K-12 curriculum?
The image of the Mandelbrot set above, the Lorenz chaos butterfly on
the right and the intellectual breakthroughs they represented were not
possible nor visible until those discoverers became digitally literate
and used this digital knowledge for further exploration. These observations
are still transforming and redirecting our thinking about many culturally
important issues that go far beyond mathematics. The technologies which
enabled the development of this new "nonlinear math" can now be routinely
found in all public school grade levels. However, the facts and the intuition
that emerged from the "nonlinear math" are very hard to find when you examine
the K-12 mathematics curriculum. The new nonlinear perceptions increasingly
dominate cutting-edge scientific, social science and business research.
The old linear perceptions dominate academic or school curriculum. As in
Galileo's day, in time a new school view will emerge. Click the image
to the right for further images and organizations studying nonlinear dynamics.
Click the text links for credit and lesson material. |
Credit by
Schneider |
Chaos
in the Classroom
The "Lorenz
Butterfly" Lessons
|
This chapter's spreadsheet tutorial will require the creation of both spreadsheets
that use standard or classic math and one that uses nonlinear math. As
with Galileo, one would expect that the new perspectives will lead to new
K-12 curriculum in the years ahead. Exciting times are ahead for those
working to translate this new knowledge into college and pre-college curriculum.
In the meantime, teachers must integrate the technologies and math standards
of today with the awareness that fundamental perceptions about mathematics
are still undergoing revision. The knowledge about the planets held by
Galileo's critics was simply wrong. This is not the case of the linear
perceptions of today's math. Fractions are still accurate ways to describe
divisions and the multiplication table is still essential. The need is
not to throw out current mathematical knowledge, but to adjust its balance
and add important new ideas to it.
Knowledge of Linear Math
Credit: Pacific
NorthWest Labs |
The ten standards of school math represent deep understandings of well
defined and well supported concepts. Though as will be seen, many of these
topical areas can be expanded to use information technology (IT) to also
address the nonlinear and other issues. However, the case for the newer
perspectives of the information age is not yet won. Fortunately, a close
reading of the standards for NCTM would indicate an organization that is
open to new ideas and new frames of reference. See the left margin
links to "The Technology Principle" and the "6 NCTM Principles." |
Spreadsheets represent one of the more common software applications
for addressing these NCTM principles. See also the "Spreadsheet & Graphing
competencies for teachers" and the left margin link to "Spreadsheet Tutorial
activities and assignments" for a deeper examination of this type of computer
program. Computer programming with Logo geometry extends mathematics into
the primary years using the unique programming power of computers.
Several important concepts emerge from our current understandings of
effective teaching in mathematics. First, spreadsheets, calculators and
other math processors need to be presented in the larger context of mathematical
effort. Further:
"Teachers need to understand the big ideas of mathematics and be able
to represent mathematics as a coherent and connected enterprise." Principles
for School Mathematics, NCTM.
"The teacher is responsible for creating an intellectual environment
where serious mathematical thinking is the norm. More than just a physical
setting with desks, bulletin boards, and posters, the classroom environment
communicates subtle messages about what is valued in learning and doing
mathematics. Are students' discussion and collaboration encouraged? Are
students expected to justify their thinking? If students are to learn to
make conjectures, experiment with various approaches to solving problems,
construct mathematical arguments and respond to others' arguments, then
creating an environment that fosters these kinds of activities is essential."
Principles for School Mathematics, NCTM.
"Using a variety of strategies, teachers should monitor students' capacity
and inclination to analyze situations, frame and solve problems, and make
sense of mathematical concepts and procedures. They can use this information
to assess their students' progress and to appraise how well the mathematical
tasks, student discourse, and classroom environment are interacting to
foster students' learning. They then use these appraisals to adapt their
instruction." Principles for School Mathematics, NCTM.
"One of the most robust findings of research is that conceptual understanding
is an important component of proficiency, along with factual knowledge
and procedural facility." (Bransford, Brown, and Cocking, 1999)
NCTM Standards now used to teach Linear school math
|
Selected New & Nonlinear Ideas Awaiting Further Integration
|
Number & Operations |
basic Boolean logic operations such as if, and, or, not (K and up) |
Algebra
patterns, models, analyze change, linear functions in middle school |
iteration, recursion in Logo from primary, nonlinear models (primary
and up) |
Geometry |
turtle graphics, 3-D and virtual reality space (primary and up) |
Measurement |
infinite accuracy is needed for absolute prediction, an impossibility;
complexity comes not from number of variables but from depth of interaction,
fractal scale
|
Data Analysis and Probability |
the normal or bell curve is just one distribution probability, see
fractals |
Problem Solving
This means that the solution method is not known in advance. "Some of
the best known strategies can be found in the work of Pólya (1957).
Frequently cited strategies include using diagrams, looking for patterns,
listing all possibilities, trying special values or cases, working backward,
guessing and checking, creating an equivalent problem, and creating a simpler
problem." |
Logo programming,
computer programming creates rich playful environment for problems |
Reasoning and Proof |
(the same as above) |
Communication |
Internet systems, Web composition |
Connections, not only to math sub-categories but to social issues as
well |
Computational limitations |
Representation |
new forms of charting and graphing (GIS systems) |
Our knowledge of the linear world goes back thousands of years before
computers. Computers have added to our understanding of this linearity
in important ways. The major contribution of the computer has been to enable
us to see the nonlinear world that had been largely invisible if not actively
avoided by mathematicians (e.g., Henre Poincare) because of its radically
different perspective and inaccessibility when doing nonlinear calcuations
by hand.
NCTM and Computer Programming
The relationship between math and computer science (computer programming)
seems an odd one. Mathematics curriculum in the public schools appears
to keep Computer Science at arms length, instead of a close embrace in
spite of both practical and conceptual reasons for being otherwise. Computer
programming is one of the central skills of the information age. Nothing
happens in a computer without it being given explicit directions on how
to proceed. The need for skilled programmers with strong mathematical backgrounds
has lead to a general shortage of such people to fill such positions. Its
integration with mathematics is so complete that many college campuses
have degrees in Computer Science and Mathematics, which indicates their
heavy dependence on one another. Yet, introduction to computer programming
is not a standard or required part of the mathematics curriculum for all
students.
A typical example of the use of programming by NCTM would be the introduction
of important Geometry concepts through the use of elements of a programming
language called Logo. Click the image on the right for a "live" or interactive
example of NCTM's development of Geometry concepts at the primary level
using some basic programming language concepts. |
Credit: NCTM |
Computer programming could be seen as a logical but more flexible and
powerful extension of the calculator, which has made it into the canon
of math education. Instead, the use and reference to programming in math
is still somewhat oblique. Computer Science grew up and left the math department
to become its own discipline. The impact of this logical development on
public education is that in K-12 this has left computer programming a somewhat
homeless waif outside of the officially supported content areas. Most high
schools and some middle schools have some optional courses in computer
programming, and the subject is generally not required to any degree in
primary, elementary or middle school in spite of significant curriculum
work introducing this topic at the K-8 age level by Semour Papert and others.
In many states, including North Carolina, it is not even an element of
the computer literacy curriculum strand. Consequently, it is not difficult
for a student to arrive at high school making career decisions without
any prior exposure to its potential as valuable career path and a powerful
aid to thinking in our digitally based knowledge economy. Examples of curriculum
resources that can correct this problem are free and online and therefore
readily available to all. As later exercises in this chapter will show,
a small amount of easy to teach curriculum activity, from the primary level
on up, can readily correct this increasingly serious oversight in curriculum
policy and politics.
Should basic knowledge of computing become a more regularly part of
the mathematical curriculum?
Extending our Math Perception
Linear math is what most of us know and it is the mathematical world around
which current school curriculum is organized. The new nonlinear math perceptions
do not invalidate what we know. Instead, they opened up the realization
that the linear branches of math so heavily and ornately explored in standard
school curriculum were describing a fraction of the nature of the world
around us. Our predictable linear systems turn out to be the exception,
not the rule. Our inorganic and organic world is interactive, iterative,
recursive, and scalable in astonishing ways. Until computers, it would
have taken one or many human lifetimes to carry out the high number of
calculations needed to test and prove the nonlinear models. There is deep
irony in the fact that the computer which was invented to enable the long
term prediction of our most important human, physical and biological systems
proved through nonlinear math that such prediction was in fact fundamentally
impossible. As will soon be seen, a tool as common as a spreadsheet is
sufficient to demonstrate the classic and new world view. See the spreadsheet
tutorial link in the left margin for more on this topic.
See also the left margin link to the reading "Math and the LEAP Problem
Processing Model. " How is mathematical thinking related to the thinking
and problem solving language of other content areas?
Math, Curricula and Higher Order Thinking
These chapters show that different disciplines use different terminology
for the same problem solving process. Math is no exception. Note the link
in the left margin to Math and the LEAP Problem Processing Model. Here
terminology from math problem processing can be compared with computer
activities at different stages discussed in the last chapter. Though ideas
are often expressed in the language of words, mathematics provides another
language of expression, and spreadsheets provide a generally two dimensional
perspective of numerical thinking.
Spreadsheets and other math processors fit within the Evoke stage tools
for the organization and composition of information. Because of the many
different structures that computer technology brings to information, the
term information architecture is coming into vogue. Teachers must know,
use and create information architecture just as a building architect must
know, use and create physical structures. Interlinked web sites create
vast nonlinear structures. Word processors provide a one dimensional view
of a stream of words. Outline processors provide many different two-dimensional
and collapsible views of concepts and ideas. Spreadsheets create two dimensional
designs for text numerical data. Desktop publishing will represent different
two dimensional architectures that includes text, images and more.
Credit: California
North Coast Watershed Assessment |
Even though spreadsheets are commonplace, spreadsheets are but the
tip of the iceberg in the use of numbers, calculation and charts to foster
better understanding. Scientific visualization is an important concept
whose potential has exploded with the power of computers. Imagine
a tool that can go far beyond showing numerical relationships visually
as bar charts and exploded 3D pie charts.
Imagine a software tool which converts numerical data into a 2D or 3D
detailed real world map of your county showing significant water quality
differences. Imagine the capacity to add and subtract from this view
different layers such as a tree cover layer or a building location layer
to better understand the source of pollution. GIS (graphic information
systems) software provides this capacity and because of its free software
versions is readily available for school use. See the optional left margin
link to "ESRI's GIS systems." A free ARC Explorer GIS application can be
downloaded from there. |
There are also many other computer based tools that support scientific
visualization from multiple dimensions. Further, the very storage and distribution
of numerical data is undergoing the same radical change that text and multimedia
went through with the invention of the World Wide Web in the 1990's. The
Center for Advanced Computing is well along in developing DSTP or Data
Space Transport Protocol. "The web today provides an infrastructure
for the remote viewing of multi-media documents, but does not provide a
similar infrastructure for remotely exploring data. DataSpace is our attempt
to provide such an infrastructure. DataSpace supports a) remote data access,
analysis, and mining, b) distributed data analysis and mining, and c) event-driven,
profile-based real time scoring" (Center for Advanced Computing).
See the optional left margin links to DataSpaceWeb and Lab for Advanced
Computing. A demonstration of ARCExplorer or DataSpaceWeb activity
to the class would be a nice extra credit activity for more advanced students.
This Center envisions a future in which it will be as easy to load a spreadsheet
or other numerical applications with data from various web sites as it
is to load a web browser with text and pictures for a web page. This has
the potential to provide math, science and social studies classes with
the same significant access to authentic or real-world mathematics data
as language arts classes now have for using the web to retrieve significant
quantities of current and relevant web pages of text and pictures.
How would a lesson look that used such tools?
Changing the Accent Mark to Questions
One use of these wondeful math tools is to collect information from our
observations about the world. This is a significant phase of scientific
thinking, observing and making accurate observations. Because of the Internet,
never have so many answers been available in seconds to so wide a range
of people. Is there anything left to do or teach if the answer to anything
can be found so easily? But though finding facts is now much easier, it
does take significant search skills and knowledge about information storage
to find and recall such information. But once those facts have been found,
then what?
A following critical step is to wonder, or ask questions about what
we find and about what is missing or cannot find. With the breakup of the
space shuttle Columbia on February 1, 2003, enormous quantities of
facts and data were known and available about the disaster. Unfortunately
those facts did not make simple and clear what had happened. What followed
that event was an enormous amount of questioning, questioning that involved
analysis, comparison, inference, evaluation and more among many teams of
people. As the high pace of change continues and increases, the percentage
of facts that are still useful, current and accurate continues to decline.
There remain tens of thousands of known important social and scientific
problems that are in urgent need of answers. How many more are unknown?
Given the increasingly dynamic and changing nature of life, whether dealing
with tragedy, disaster or exciting and cheerful creativity, the need for
higher order skills continues to grow.
A system for learning can be assembled from the finding and posting
of our questions to a wonder web and the use of spreadsheets for numbers
and word processors for words to compose a response. Though these steps
are important, they are insufficient. Something is missing. But how do
we know if we are asking the right questions or the best questions? The
major missing element has to do with the nature of the question itself.
The quality of an answer depends on the quality of the question. Some stage
of the learning process must ask if there is not a better question that
will lead to a better response. Further, the more interesting or more difficult
a problem, the more that many types of questions which provide many types
of perspectives are needed.
To address this problem, this chapter turns to the third major branch
of the CROP site, a branch called THINK. The THINK branch of the CROP site
addresses the many structures for higher order thinking questions that
are currently given formal recognition in our culture. In particular it
asks the reader to examine a five level structure for questioning activity
that has its roots in Bloom's taxonomy (1956).
What we produce depends on what we ask. Long before computers were
in vogue, structures for thinking and questioning were given shape and
definition. Whether using the language of math or of words, the effort
is similar. The many sophisticated math tools provided by computers readily
involve the many categories of higher order thinking. Higher order thinking
is the infrastructure or foundation for thinking among the major cultural
forms of thinking that have emerged: ethical, critical and creative. Through
such thinking the problem finds the perspective from which it can best
be framed. |
|
Take the left margin link to the NorthWest Regional Labs. Study
and memorize the NorthWest Regional Labs model for higher order
thinking skills: learn the definitions for recall, analysis, comparison,
inference, evaluation and learn the question formats or special words used
for each of these skills. As you study the computer applications of this
chapter, become more conscious of when you are using these different thinking
skills for different stages of your study and work. As you practice your
teaching, look for opportunities to "walk the rungs" up the ladder of higher
order thinking.
Can you invent questions that use math to run the range of these higher
order thinking skills? What types of questions do you hear and use in today's
classrooms? Do some content areas use more higher order skills than others?
The Ongoing Unit Plan
The development of classroom curriculum that integrates ideas encountered
in this chapter continues. See the left margin link on updating "unit plan
sections III, VII, IX."
Once Around the Spiral - Organization of the Online Textbook
How does this chapter fit into the thinking and ideas of the earlier three
chapters? An educational theorist by the name of Bruner articulated the
idea of a spiral curriculum. The first time around the spiral, a set of
interlinked and basic ideas are presented briefly and simply in a circle
of strands or threads. As a learner completes the circle and returns
to the first basic idea, the thread is then continued at a higher level
with each strand that is re-encountered in this continuous cycle around
major ideas. Each time around the spiral the fundamental ideas of each
strand are addressed in greater detail and sophistication. With the new
educational concepts for this chapter, readers will have completed an introduction
to the basic ideas of the spiral for this course.
|
The strands of the spiral form the conceptual framework around which
this course is woven. The first strand is information technology
(IT), also called computer literacy in the North Carolina Standard
Course of Study. The IT strand far has so far introduced: an introduction
to a wide range of ways to think about how computer operating systems overlap
with other types of non-computer operating systems that we use daily (chapter
one); the language and skills of the Web and the Internet including the
first steps in the development of your web site (chapter two); and the
dynamic and powerful nature of writing (chapter three) when centered both
in the use of an outline processor which is carefully integrated with the
full range of word processing tools and in the use of a prioritized strategy
for gathering information interactively with web browsers and other Internet
tools. This strand is extended in this fourth chapter to include spreadsheets
and other related developments. The second strand is problem
processing, which so far has introduced: the basic idea a web site
for problem processing called CROP (Communities Resolving Our Problems)
(chapter one); new tools for the publish stage of problem solving through
web pages and sites (chapter two); and a four stage model for problem solving
(LEAP). |
The LEAP model generalizes across all academic content areas that includes
the idea of the information pyramid at the look stage of seeking information
and the power of computer composition tools at the evoke stage (chapter
three). It is based on an understanding that "problem-solving ability is
the cognitive passport of the future" (Martinez, 1998) and the concept
that "questions are the seeds of solutions" (Houghton, 2000). The third
strand, introduced in this chapter four, includes models of different thinking
skills. The fourth strand, introduced in the first chapter,
evolves around the concept of interaction. Activities in the first
and second chapters led us through exercises that introduced paper based
methods for problem sharing and responding, the Wonder Web with its SUP
and FAQ organization. Further, many ways in which computer and non-computer
interactions stimulate thought and create opportunity for intellectual
teamwork are addressed in chapter three. Interaction is the foundation
condition from which educational leadership can and must emerge. The fifth
strand adds curriculum goals and development across the many content
areas taught in our schools, including: computer literacy (chapters one
and two); language arts (chapter three); and mathematics (chapter four).
With the addition of thinking skills in this chapter, readers have made
at least once around the spiral in all five strands.
Continue This Chapter
Digital technology has changed and continues to change mathematics and
our capacity for teaching it. But you must do more than read about
it. It is time to experience it more directly.
As with prior chapters, turn now to the left margin of this page, and
follow the steps (or links) down the column to complete this chapter.
Bibliography
-
Bransford, John D., Ann L. Brown, and Rodney R. Cocking, eds. How People
Learn: Brain, Mind, Experience, and School. Washington, D.C.: National
Academy Press, 1999.
-
Principals and Standards of School Mathematics (1998) NCTM. http://standards.nctm.org/document/index.htm
-
Houghton, R.S. (1989). A
Critical Analysis of Educational Theory.
-
Houghton, R.S. Nonlinear
Educational Theory, a web site.
-
An overview of Seymour Papert's work in
computer and mathematics integration.
-
Pascal's
Triangle.
-
NCTM Standard Geometry: Developing
Geometry Concepts Using Computer Programming Environments
-
NCTM Illuminations
-
NCTM Tools
and Games by Grade Level
Page author: Houghton
Updated: 6.20.2003 |
|
|