Click above image for "Spreadsheets in Genetics" video. High speed networks can use the larger video format.	Click above image or here for more on live Astronomy classroom webcam.

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.

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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