

Click above image for "Spreadsheets in Genetics" video. High speed networks can use the larger video format.  Using the world's webcams for animal and plant observation provide many opportunities for measuring & counting. 
IntroductionNew 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.
Sometimes, in order to find x (see graphic on the left of student answer on an exam), a little help from a computer on the calculations and procedures is welcome relief. It is the tedium of calculations which in turn lead to many human errors that has driven that explosion of computer use for calculation. 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 computerenhanced 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. 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.
48th Lessons on Fractals Click the picture of the Mandelbrot Set fractal on the left to see thousands of other pictures of fractals or the term Mandelbrot Set itself to see images from this specific kind of fractal, the parent of all fractals. Click the "K8 Lessons..." text to learn how to teach about fractals. What are fractals? New powerful mathematical ideas are constantly being discovered. For example, another set of new perceptions in math emerged in the 1950's and 1960's about the nature of mathematics and predictability. In the following years, the scholars in discipline after discipline found that the concept of nonlinear dynamics, which includes fractals and chaos theory, applied to their work and revolutionized the thinking in their fields. This work is still being translated into concepts that those with less mathematical background can understand. The National Center for Education Statistics for Kids describes fractals in this way. "A fractal is a rough or fragmented geometric shape that can be subdivided in parts, each of which is (at least approximately) a reducedsize copy of the whole. Fractals are generally selfsimilar and independent of scale. Chaos theory describes the complex and unpredictable motion or dynamics of systems that are sensitive to their initial conditions. Chaotic systems are mathematically deterministicthat is, they follow precise laws, but their irregular behavior can appear to be random to the casual observer."
That part of the explanation about fractals may work for mathematicians but what about the rest of us? It helps to start with the familiar and move to the known but less familiar. We can all use rectangles, triangles, circles and squares to draw houses and objects on a piece of paper and we see those shapes in manmade product design and buildings everywhere. This is kindergarten knowledge today, ideas that Euclid explored in ancient Greece with more mathematical language in 300 B.C. Now turn to the plants outside your school window. Now use those same basic shapes to build an accurate maple leaf, the branch structure of a dogwood tree or the bark of a Elm tree. This Euclidian geometry does not work at all for such tasks. It does not explain the underlying mechanics of how a brainless plant can create some mechanism stored genetically that precisely and repeatedly produces the complex and beautiful patterns of a rose, maple tree or fern leaf, that is most of what we see and the structure of the very things that sustain the life that is all around us. The Euclidian geometry that dominates the knowledge of geometry in our curriculum explains but a tiny fraction of the real world. Fractal geometry offers powerful and visually compelling explanation for the rest. Think of it as geometry built with fractions. Why is the rest of the geometric picture not in our curriculum? Like in Galileo's day, there is more to the story that is not yet being told. Can it really be that hard if a plant can do it? This chapter can just crack the door to the mystery, the solution and the ongoing research. Additional opportunities will have to be sought out to unravel this more thoroughly. Part of the point here is to realize just how much there is to help keep a sense of wonderment and vitality to our thinking.
Now, what about that "chaos theory" part of the explanation above. Again, the translation may work better for mathematicians. It might be better to start with the weather. How far ahead of time are you willing to plan an outdoor picnic and really count on the reliability of the weather forecast and not make backup plans? Since we have massive weather data collecting systems, and we fully understand the underlying physics of the chemical and physical reactions of the elements of weather and we have massive computers to do the math, why are not even long range forecasts quite reliable? What math explains how tiny changes can build on each other so quickly that the best forecasts give us cause for concern in planning that picnic? Nonlinear math and its special subdivision of chaos theory can explain emerging disorder. How do such dynamic yet seemingly random events in nature also lead to the formation of living things and social systems? Once again nonlinear math gives us a way to mathematically model such solitonlike development (e.g., emerging order). Since computers make this easy to model, where is knowledge of this math in our curriculum?
Why did it take so long to discover this? The mathematical formulas and procedures were not complex but repeating those simple formulas dozens, hundreds or thousands of times until novel patterns emerged was so time consuming it was never done. It was the technology of the calculator and the computer with their ability to quickly carry out an enormous number of repetitive, simple and linked calculations that enabled these discoveries. Your encounter with May's Equation in a spreadsheet in this chapter will provide a more concrete example. Among other things, this knowledge has in turn been used to solve practical engineering problems. As is well known, powerful theory eventually creates powerful practical results.
This chapter's spreadsheet tutorial will require the creation of both spreadsheets that use standard or classic math. There is also one in the tutorial that uses nonlinear math. As with Galileo, one would expect that the new perspectives will lead to new K12 curriculum in the years ahead. Exciting times are ahead for those working to translate this new knowledge into college and precollege 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.
Like the telescope and much later the microscope, calculators and computers were used as complexityscopes 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 K12 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 K12 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 K12 mathematics curriculum. The new nonlinear perceptions increasingly dominate cuttingedge 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 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. A further example is the way programming with Logo geometry concepts can extend programming and mathematical thinking 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, 3D 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 problemsReasoning and Proof (the same as above) Communication Internet systems, Web composition Connections, not only to math subcategories 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 calculations 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 led at times 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: NCTMComputer 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 K12 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 K8 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 exercises in later chapters of this textbook 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 twodimensional 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.
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.
Credit: ArcExplorer: Census Population
& Elevation DataScientific 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 left margin link to "ESRI's GIS systems." A free ArcVoyager GIS application can be downloaded from there. Links to the also free ArcExplorer can be found from the links on this page but the free instructional resources for ArcVoyager are better.
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. Instead of HTTP (hypertext transport protocol) that helps manage web page organization and distribution, DSTP or Data Space Transfer Protocol has been developed for sharing numerical information from a number of databases, files and other data structures distributed among multiple servers using a key that can find all the related data of an object across all data (WhatIs.com). "The web today provides an infrastructure for the remote viewing of multimedia documents, but it took longer to build 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) eventdriven, profilebased real time scoring" (Center for Advanced Computing). See the optional left margin links to DataSpaceWeb and Lab for Advanced Computing. A demonstration of special features of ArcVoyager, ArcExplorer or DataSpaceWeb activity to the class would be a nice extra credit activity for more advanced students. This Center for Advanced Computing 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 realworld 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 " Compare different models of higher order thinking skills (HOTS)". Study and memorize the Bloom's revised model for higher order thinking skills. Learn the definitions for remember, understand, apply, analyze, evaluate and create. 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?
Once Around the Spiral  Organization of the Online Textbook
How does this chapter fit into the thinking and ideas of the earlier 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 reencountered 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 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 "problemsolving 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 noncomputer interactions stimulate thought and create opportunity for intellectual teamwork are addressed in chapter two. Interaction through the use of many media 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: digital literacy; language arts; and mathematics (current chapter). With the addition of thinking skills in this chapter, readers have made it at least once around the spiral in all five strands.
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 noncomputer 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 and a four stage model for problem solving (LEAP). 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.
Optional Activities
Annenberg Media: Workshop 7. Algebra: It Begins in Kindergarten (90 min. video)
This workshop traces the fundamental concepts of algebra that students can develop through the grades. See its Content Guide.Annenberg Media: Workshop 8. The Future of Mathematics: Ferns and Galaxies (90 min. video)
The advent of new technologies allows for amazing mathematics that could not exist without computers. This workshop program looks to future directions for mathematics in the 21st Century. See its Content Guide.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