James E. White received his Ph.D. from Yale University in 1972, as a student of William Massey, and has devoted the better part of his professional life to teaching mathematics, beginning at University of California, San Diego in 1972, and oscillating between teaching positions at Carleton College, Bates College, Kenyon College, and research positions at Jet Propulsion Labs, Institute for Academic Technology, and finally, his own consulting company, Bluejay Lispware, which operates the New Mathwright Library and Café on the web. He has held visiting positions at Harvey Mudd College, University of North Carolina, Chapel Hill, Naval Postgraduate School and Stetson University.
Early in his career, he became interested in the epistemological basis for the pedagogical styles that were in general use, and that interest led him to the Piagetian "constructivist" interpretation of the psychology of mathematics learning that is couched in his Genetic Epistemology. This has much in common with ideas expressed by the mathematicians: Poincare, Hadamard, and Hurewicz, and he found it to be consistent with his personal views.
On the genetic view, the learner
actively constructs her knowledge by challenging old ideas, and assimilating
new ones into increasingly stable "structures". Since it is difficult to translate this
abstraction to pedagogical practice within the context of curricula that are
organized around different (more behavioral) premises, he began to look for ways to structure lower
level mathematics courses around the
constructive activity of the learners.
It quickly became apparent that, for a strategy like this to work, the
students would need heuristic tools
that would help them form coherent pictures of the mathematical processes and
constructions under discussion, and that would allow them quickly to test their
understanding by showing them the answers to the questions they asked.
These considerations led him, in 1982, to ponder a role for computers as heuristic devices that can support student construction of their mathematical knowledge. Following some ideas of Seymour Papert (who was also strongly influenced by Piagetian epistemology) he first developed tools for students, but soon realized that the heuristic tools that students would use ought to be created by their teachers. These teachers are a good representative sample of "the best students," and, given the right tools, can create learning environments for their students that are both relevant and accessible to them.
With an invitation from UNC Chapel Hill to develop and extend these ideas, he became the architect of MathKit and of IBM’s ToolKit for Interactive Mathematics. MathKit is a system of object-oriented tools, together with an informal and powerful mathematics scripting language, that supports the creation of interactive mathematics texts. ToolKit for Interactive Mathematics is a commercially available program published by IBM, that uses MathKit. He did this work in constant communication with teacher/authors through his role as co-chair of the Interactive Mathematics Text Project, a project sponsored by the Mathematics Association of America and funded by IBM corporation and NSF. Since that time, James has worked with teachers to develop interactive mathematical texts through a number of projects:
- IBM Corporation and Mathematics Association of America, co-Principal Investigator (with Jerry Porter), $2,500,000. (1992-1994) Interactive Mathematics Text Project. This project was funded principally by IBM and was sponsored by the Mathematics Association of America. It established 6 computer laboratory sites around the country at which teachers experimented with the creation and use of interactive mathematics texts.
- National Science Foundation (ILI
Leadership Grant) (1992)
Co-Principal Investigator (with Jerry Porter) Support for 1992 Workshops for the Interactive Mathematics Text Project.
- National Science Foundation (ILI Leadership Grant) (1995 - 97) Co-Principal Investigator, with Ladnor Geissinger, Library for Interactive Mathematical Studies
- National Science Foundation (CCLI-EMD Grant) (2000 - 04) Principal Investigator, Project WELCOME, a project that is also sponsored by MAA and that provides tools and instruction to teachers at HBCUs to develop interactive web microworlds. The 43 Microworlds developed by these teachers is now part of MathDL.
James has written several computer languages, all built on his LISP interpreter:
- The Lisper LISP Interpreter, published in 1984 by Bluejay Lispware
·
CAL: Mathematics Teaching and Learning Environment published in 1986 by Bluejay Lispware
·
ToolKit for
Interactive Mathematics, published
1994 by IBM EduQuest, (was Program Author and System Architect)
- MathKit A System of Mathematical Authoring Tools for Windows programs such as ToolBook, ToolKit for Interactive Mathematics, Visual Basic. Published in 1994 by the Institute for Academic Technology
- Mathwright published 1994 by Bluejay Lispware (A
Mathematics Authoring/Reading environment for interactive texts)
- MindScapes published 1997 by Bluejay Lispware, (A 3D graphics OpenGL based language built on LISP)
- Mathwright32 published 2001, by Bluejay Lispware (A Java version of Mathwright that supports mathematical exploration on the web)
As the Director of the New Mathwright Library and Café on the web (http://www.mathwright.com) James continues to work with many Mathwright Authors to promote and support the view that teachers (not programmers) can create dynamic and expressive mathematics learning environments for their students.
Web Publications (Interactive Mathematical Microworlds and WorkBooks)
1) Mathwright2000 workbook: fraction marathon
Description: This playbook may be used to practice arithmetic of fractions. it presents a series of five "tasks": addition of fractions,subtraction of fractions,changing "improper" fractions to "mixed numbers", multiplication and division of mixed numbers.
2) Mathwright2000 workbook: memily game
Description: This playbook is a memory game. Sharpen your memory by playing against the computer! You may play at different levels if it gets to be too easy to win.
3) Mathwright2000 workbook: mirror game
Description: This workbook explores the symmetry of a game often found in roadside restaurants. We call it the "mirror game". There are 14 red pegs arranged in the playing board. After that, the moves will be to "jump" one peg over another and to land in a hole.
4) Mathwright32 microworld: mathwright logo playground
Description: This microworld contains a dialect of classic logo. In this dialect, you may create as many turtles as you like, and move them around simultaneously with your programs. This programming environment lets readers write, save and restore logo programs.
5) Mathwright32 microworld: tactic: 3d tictactoe
Description: You are probably familiar with the traditional game of tic-tac-toe. Tactic expands that game to the third dimension, on a 4x4x4 board, and it illustrates one of the most basic strategies of game theory.
6) Mathwright2000 workbook: bezier curve designer
Description: Use this workbook to design your own curves. Click between 2 and 15 points on the screen and the bezier spline approximating those points is drawn for you. Also, you receive the function definition of that spline as a parametric curve.
7) Mathwright2000 workbook: equation assistant
Description: this workbook is designed to assist the beginner in learning how to solve linear equations. these equations may contain any number of variables, but of course, the reader must specify one of them to be solved in terms of the others. fractions are written
8) Mathwright2000 workbook: golden ratio and the fibonacci sequence
Description: This workbook uses logo style graphics to explore the relations between the golden ratio and the fibonacci sequence. The Greek view of ratio is illustrated. In this view, a ratio of segments is the shape of a rectangle whose sides are those segments.
9) Mathwright2000 workbook: mastermind
Description: Mastermind is a game of deductive logic. Learn this intriguing game while playing it. The workbook teaches you the rules of the game, then allows you to try out your deductive skills as it creates codes for you to guess. Next, watch it guess your codes!
10) Mathwright2000 workbook: pool
Description: This workbook simulates elastic collisions in the game of pool. Readers may play pool, or may do experiments in elastic collisions. For example, they may explore the world of "frictionless" pool.
11) Mathwright2000 workbook: precalculus course (complete set)
Description: This sequence, developed in the spirit of reform, is a set of 9 laboratory workbooks based on the text elementary mathematical models, written by dan kalman.
12) Mathwright2000 workbook: precalculus set: arithmetic growth
Description: Here we retrace the steps in which galileo learned the law of "freely falling bodies". When a body falls from rest under the influence of gravity alone it falls "faster and faster". It took a very long time for uniform acceleration to be discovered.
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workbook: precalculus set: difference equations
Author(s): james white and dan kalman
Description: This lab focuses on sequences. A sequence is like an indexed variable except that it does not necessarily terminate. The infinite nature of sequences shows that they are abstract models rather than data collected in experiment.
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workbook: precalculus set: geometric growth
Author(s): james white and dan kalman
Description: In this lab we introduce geometric growth sequences. They differ in only one essential way from arithmetic growth sequences, and are, in a sense, as simple as the latter. They will lead directly to the exponential (and logarithmic) functions.
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workbook: precalculus set: graphical methods
Author(s): james white and dan kalman
Description: The main purpose of this laboratory is to learn to work with indexed variables (measured quantities). We begin with a few scatter plots that describe certain data sets. The second page of the lab is an exploration that will help you practice graphing.
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workbook: precalculus set: introduction
Author(s): james white and dan kalman
Description: This laboratory introduces indexed variables. These are the basic objects that we use to represent and study data. We will also start to describe some of the conventions that we will use in these labs. This lab has tools that we will reuse in the future.
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workbook: precalculus set: linear graphs
Author(s): james white and dan kalman
Description: In this laboratory, we will study linear functions and their graphs. First, we review on this page basic graphing technique. this will give exercise in plotting points, and describing the line through a pair of points in a number of standard ways.
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workbook: precalculus set: quadratic graphs
Author(s): james white and dan kalman
Description: We study various forms of quadratic expressions, the translation from one form to another, and the shapes of quadratic graphs. We discuss various methods for solving quadratic equations: completion of the square, the quadratic formula, and factorization.
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workbook: precalculus set: quadratic growth
Author(s): james white and dan kalman
Description: Here, we study a new model of growth. We saw that arithmetic growth models are characterized by straight-line graphs. When the difference equation has the form: a(n) = a(n-1) + m*(n-1)+b for some constants m and b the rule of growth is quadratic.
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workbook: precalculus set: rational functions
Author(s): james white and dan kalman
Description: We study polynomial and rational functions and try to see the relationship between the algebraic representation of a function and its graph. We have studied linear and quadratic functions, and generalize those to the polynomial and rational functions.
21) Mathwright2000 workbook: space filling curve
Description: This workbook illustrates hilberts space-filling curve algorithm. It lets the reader construct various stages of the construction, and helps the reader convince herself that the uniform limit of the sequence of curves exists, and maps onto the square.
22) Mathwright2000 workbook: space shuttle
Description: This workbook is a 12-page interactive mathematics/science workbook that celebrates the role of mathematics in rocket science. It is designed to be used by students ages 14 and older for private self-directed study and recreation.
23) Mathwright32 interactive web playbook: the magical gravity tour
Description: Gravitation is an interactive microworld designed to be used by students for self-directed study and recreation. It has playful explorations such as a lunar lander or a space shuttle launch that also teach. Students may experiment with kepler's 3 laws.
24) Mathwright32 microworld: 3d game of life
Description: This playbook is about the game of life. In this 3-dimensional version, cells are cubes in a 21x21x21 array. In each generation, a cell survives to the next by counting neighbors. Empty cells are born if they have the right number of living neighbors.
25) Mathwright32 microworld: duality in the mirror game
Description: This workbook explores the symmetry of a game often found in roadside restaurants. We call it the "mirror game". There are 14 red pegs arranged in the playing board. After that, the moves will be to "jump" one peg over another and to land in a hole.
26) Mathwright32 microworld: mastermind
Description: Mastermind is a game of deductive logic. Learn this intriguing game while playing it. The workbook teaches you the rules of the game, then allows you to try out your deductive skills as it creates codes for you to guess. Next, watch it guess your codes!
27) Mathwright32 microworld: points and lines
Description: This microworld exercises your skills in writing and solving equations. It is designed for students who would like to understand how to write equations of lines in various forms, and to determine the distance from a point to a line.
28) Mathwright32 microworld: rocket science 101
Description: This microworld is an interactive playbook that celebrates the role of mathematics in rocket science. It is designed to be used by students ages 14 and older for self-directed study and play. The player must dock the space shuttle with a space station.
29) Mathwright32 microworld: sat math practice and tutorial 1
Description: This microworld will give you unlimited opportunity to prepare for the mathematics section of the scholastic aptitude test (sat). In its present form, it generates a new 25-question diagnostic test each time you ask for one. Each test is unique.
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workbook: calc1 set: analyzing data
Author(s): james white and samad mortabit
Description: This laboratory continues to model relations between sets of data, using an automatic procedure to obtain the line. The reader gets experience collecting, entering, and analyzing data. It has an "automatic fit" button to obtain the best linear fit.
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workbook: calc1 set: basketball stats
Author(s): james white and samad mortabit
Description: This laboratory investigates variables associated with data, such as scatter plots associating statistics with players on basketball teams. A "skill builder" page is provided to give a little practice drawing and recognizing the graphs of lines.
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workbook: calc1 set: compound interest
Author(s): james white and samad mortabit
Description: Here we explore questions around the calculation of interest. We wish to learn the use of recursive definitions for defining exponential functions; and to explore a particular type of limit behavior that leads to the definition if the euler number e.
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workbook: calc1 set: exploring derivatives
Author(s): james white and samad mortabit
Description: This lab aims to help you form ideas about derivatives, ask your own questions and create your own examples. First we study position and velocity portraits. Next, we explore limits of difference quotients. Finally we look at slopes of tangent lines.
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workbook: calc1 set: functions and data
Author(s): james white and samad mortabit
Description: In this lab, there are four "scatter plots". These plots report the number of aids cases in thousands by date, describe the position of a freely falling body, show the federal deficit, per year, and a plot to experiment with and modify.
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Mathwright2000
workbook: calc1 set: functions and graphs
Author(s): james white and samad mortabit
Description: We experiment with the role of the derivative in critical behavior (local extrema, inflections) and linearization which, using newton's method will yield approximate roots of equations: f(x) = 0. The reader may use built-in examples or supply her own.
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Mathwright2000
workbook: calc1 set: initial value problems
Author(s): james white and samad mortabit
Description: Here, we explore initial value problema. There are several types of initial value problems, and several ways to observe and test solutions. We begin with differentiable, 1-dimensional ivps. In the next section, we explore a 2-dimensional discrete ivp.
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Mathwright2000
workbook: calc1 set: introduction
Author(s): james white and samad mortabit
Description: The main purpose of this laboratory exercise is to learn to work with scatter plots, and to learn some of the conventions used in the calc laboratories. The pages of this lab have tools that will be reused from time to time in future labs.
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workbook: calc1 set: keplers third law
Author(s): james white and samad mortabit
Description: This lab visits a question that inspired the birth of calculus. The data are the orbits of planets and other objects in sky. We launch our own probes around the sun to generate data, and make an hypothesis about it based on a log-log plot.
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Mathwright2000
workbook: calc1 set: logarithmic analysis
Author(s): james white and samad mortabit
Description: This lab derives and checks model formulas from the observed pattern of data. We often check hypotheses about the data using semi-log or log-log plots. This often requires a "linear best fit" to model data that is transformed to logarithmic form.
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Mathwright2000
workbook: calc1 set: population growth
Author(s): james white and samad mortabit
Description: This lab starts with the derivative of a changing quantity (in this case, population size or position as they change in time), models that can make predictions about the quantity that is changing. We study mainly models of population growth.
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Mathwright2000
workbook: calc1 set: process models
Author(s): james white and samad mortabit
Description: In this lab, there are 5 scatter plots: sales of a pharmaceutical company, inclined planes, pendulum, and gravity. The latter describes galileo's famous "thought experiment" about falling bodies that led to his principle of uniform acceleration.
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Mathwright2000
workbook: calc1 set: raindrops
Author(s): james white and samad mortabit
Description: In this lab we develop a model for analyzing initial value problems focusing on the behavior of resistance-inductor-capacitor circuits, and falling raindrops. We use euler's method for numerical approximation, but also use symbolic algebraic techniques.
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Mathwright2000
workbook: calculus 1 course (complete set)
Author(s): james white and samad mortabit
Description: This sequence, designed in the spirit of reform, is a set of 13 laboratory workbooks based on project calc, a reform curriculum designed by david smith and lawrence moore.
44) Mathwright2000 workbook: command line
Description: Students may use this book to type commands to do graphics, decimal arithmetic, matrix or symbolic algebra. The online documentation provides a tutorial for the mathscript language, the over 270 mathscript commands and built-in functions.
45) Mathwright2000 workbook: derivatives and the graphs of functions
Description: This workbook experiments with some of the general applications of the derivative to the study of the behavior of functions. This includes their critical behavior, local extrema, inflections. The reader may work with built-in examples or supply her own.
46) Mathwright2000 workbook: discrete mathematics 1: set safari
Description: This is the first half of a 14-week course in discrete mathematics. The lectures cover: sets,functions sets and logic,composition of functions,set operations,permutations of sets,boolean algebra,graphs and directed sets,relations,order,cardinality.
47) Mathwright2000 workbook: discrete mathematics 2: set theory
Description: This is the second half of a 14-week course in discrete mathematics. The lectures cover: iteration and recursion,search,sets defined by propositions,critical path analysis,counting,automatic problem solving,relations and functions,graphs and logic.
48) Mathwright2000 workbook: discrete mathematics course (complete)
Description: This sequence: "Sets, Functions and Relations", and "Recursion and Logic" can together be used as text for a 14-week formal course in discrete mathematics. This course might also be used for private self-paced or independent study.
49) Mathwright2000 workbook: elastic collisions
Description: This workbook lets you set up and launch 2 hockey pucks on an ice rink. You determine their masses, their velocities and their positions by dragging the pucks to any desired position. Momentum is conserved as you see from the initial and final vectors.
50) Mathwright2000 workbook: firemans ladder
Description: A fireman stands on the bottom rung of a ladder. The base of the ladder is 1 meter from the wall and is about to slip at a constant speed away from the wall. The player chooses how fast the fireman climbs the ladder and sees the path through the air.
51) Mathwright2000 workbook: gaussian elimination
Description: This workbook uses gaussian elimination to solve systems of linear equations in up to 12 variables. The reader enters a system of equations and learns how the solution (if any) may be parametrized by the free variables, is unique, or is inconsistent.
52) Mathwright2000 workbook: geometry
Description: Geometry is a venerable pillar of modern mathematics. Do ruler and compass constructions with lines, circles, polygons, and ellipses in this environment. Save and restore your constructions and proofs, and make "movies" of them for demonstration.
53) Mathwright2000 workbook: lunar lander
Description: This book is a game of a rocket landing on the moon. It has at each instant a height and a velocity, all represented by dials. There is a throttle that the player uses to accelerate the rocket. The graphs of height and of velocity versus time are drawn.
54) Mathwright2000 workbook: reflections
Description: This workbook illustrates some facts about motions of the plane that preserve distances between points. Such motions are called "isometries" or "rigid motions.". Examples of these motions are "translations" or "rotations" about a fixed point.
55) Mathwright2000 workbook: synthetic division
Description: In this command-line workbook, students explore synthetic division of polynomials with rational coefficients. There is, among others, a command called synthetic that returns the quotient of one rational polynomial by another together with the remainder.
56) Mathwright32 microworld: stay afloat!
Description: this playbook explores archimedes' bouyancy principle and optimization problems in a story in which players build a boat by cutting and assembling planks from a board and putting them together. They may view their product in three dimensions.
57) Mathwright32 interactive web course: discrete mathematics and computational stru
Description: This is the first half of a 14-week course in discrete mathematics. The lectures cover: sets,functions sets and logic,composition of functions,set operations,permutations of sets,boolean algebra,graphs and directed sets,relations,order,cardinality.
58) Mathwright32 interactive web course: discrete mathematics and computational stru
Description: This is the second half of a 14-week course in discrete mathematics. The lectures cover: iteration and recursion,search,sets defined by propositions,critical path analysis,counting,automatic problem solving,relations and functions,graphs and logic.
59) Mathwright32 microworld: calculus in action: chapter4: section 2: the grand de
Description: In this microworld, we follow isaac newton in the deduction of the first and third laws of planetary motion around the sun of john kepler from his single hypothesis of universal gravitation.
60) Mathwright32 microworld: calculus in action: chapter1: section 1: ballistics
Description: This microworld studies the derivative as a means of approximation. But galileo used another method, which was due to archimedes, to calculate the exact distance fallen by a uniformly accelerated object. This method will be called integration.
61) Mathwright32 microworld: calculus in action: chapter1: section 2: newtons method
Description: This microworld contains an illustration of the conventional picture of newton's method. At each step, the student sees a ray trace the tangent line to the x axis, and determine a new value of x. The student defines the function and starting point.
62) Mathwright32 microworld: calculus in action: chapter1: section 3: newtons method
Description: Newton's Method leads to a sequence of numbers that may converge to a solution. This sequence may lead to stable oscillations that do not converge to a solution from almost any starting point but oscillate among several values. And it may lead to chaos.
63) Mathwright32 microworld: calculus in action: chapter2: section 1: polar coordina
Description: In this microworld, we introduce a style of description that will serve us well throughout the book. It is called polar coordinates. Any satellite moving under the influence of a central force (like Newtons law of gravity) must remain in a plane.
64) Mathwright32 microworld: calculus in action: chapter2: section 2: conservation o
Description: In this microworld, we discover another conserved quantity for satellite motion. That is the energy of the satellite. For that, we must make Newtons assumption about the actual form of the gravitational law.
65) Mathwright32 microworld: calculus in action: chapter3: section 1: plane curvatur
Description: In this microworld, we study plane curvature. We take up the notion of invariance under parameterization. We study arc length parameterization, the gauss map, and the fundamental theorem of calculus.
66) Mathwright32 microworld: calculus in action: chapter3: section 2: logarithmic sp
Description: In this microworld, we study the equiangular spiral, and develop the general techniques of differential equations that are needed to analyze it. We develop first eulers number and natural exponentials and logarithms and continue to apply eulers method.
67) Mathwright32 microworld: calculus in action: chapter4: section 1: geometry of pl
Description: In this microworld, we will show that planetary orbits are actually plane curves, and we establish the existence of the first conserved quantity, the angular momentum to establish keplers second law.
68) Mathwright32 microworld: calculus in action: precalculus introduction
Description: This precalculus introduction to calculus in action covers the following topics. free fall, ballistic trajectories, inclined plane, energy conservation, momentum conservation, pendulum, escape velocity, brachistochrone, symbolic calculator.
69) Mathwright32 microworld: derivatives and graphs of functions
Description: This microworld experiments with some general applications of the derivative to the study of the behavior of functions. This includes their critical behavior, local extrema, inflections. The reader may work with built-in examples or supply her own.
70) Mathwright32 microworld: hifi: personal household finance manager
Description: HiFi is a fully functional object-oriented, LISP-based Expert System that can make managing and planning your household finances easy and fun.
71) Mathwright32 microworld: implicit surfaces
Description: Implicit Surfaces is a utility that students may use to draw the solution of an equation: f(x,y,z) = 0 in 3 dimensions. They may draw any pair of surfaces simultaneously and view them solid or wireframe.
72) Mathwright32 microworld: mathscript author toolkit
Description: This microworld has a command-line that teachers can provide for their students. Students may use the book to type commands to do graphics, decimal arithmetic, matrix or symbolic algebra, for example. It provides a tutorial for the mathscript language.
73) Mathwright32 microworld: rational matrix calculator
Description: This microworld calculates exact rational matrices. All operations are exact rational operations,unless the reader enters decimals in the matrices. The reader has a control panel to create and edit new matrices, and a command line for matrix expressions.
74) Mathwright32 microworld: spherical logo
Description: Logo has long been a favorite way of introducing students to computational geometry and logic. Well, Terra, the turtle in this microworld lives in a sphere. Her geometry is Riemannian (spherical) geometry and she has a great many things, to show you.
75) Mathwright32 microworld: surfaces of revolution
Description: This microworld discusses surfaces of revolution. Each topic is developed through an interactive exploration that allows the reader to experiment with the ideas and to visualize its consequences.
76) Mathwright32 microworld: symbolic calculator
Description: Students may use this microworld to type commands to do graphics, decimal arithmetic, matrix or symbolic algebra. The online documentation provides a tutorial for the mathscript language, the over 270 mathscript commands and built-in functions.
77) Mathwright32 microworld: synthetic division
Description: In this command-line workbook, students explore synthetic division of polynomials with rational coefficients. There is, among others, a command called synthetic that returns the quotient of one rational polynomial by another together with the remainder.
78) Mathwright2000 workbook: 3d graphics
Description: On the first page is a simulation that allows readers to define a plane then to view the way the plane intersects a cone. Next, the reader may view the two-dimensional graph of the same intersection. In this way, the idea of conic section is reinforced.
79) Mathwright2000 workbook: bernoulli slider
Description: This simulation explores the problem of john bernoulli to discover the curve connecting two points for which the time of descent of a sliding bead moving from higher to lower is minimized. Experiment with curves you create and compare with the solution.
80) Mathwright2000 workbook: boats
Description: This playbook explores archimedes' bouyancy principle and optimization in a story in which players build a boat by cutting and assembling planks from a board and putting them together. The problem is to build a boat that will carry two across the river.
81) Mathwright2000 workbook: expert system
Description: This workbook explores symbolic algebra in mathwright using unification pattern matching, an artificial intelligence technique. It introduces factoring polynomials or integers, and has a page on which it explains step-by-step how it solves an equation.
82) Mathwright2000 workbook: gravitation
Description: Gravitation is an interactive workbook designed to be used by students for self-directed study and recreation. It has playful explorations such as a lunar lander or a space shuttle launch that also teach. Students may experiment with kepler's 3 laws.
83) Mathwright2000 workbook: implicit functions and tangent pencils
Description: An equation of the form f(x,y) = 0 often defines a curve in the plane. We say that the curve is defined implicitly but we may not be able to solve for y in terms of x. Implicit differentiation often allows us to solve for dy/dx as function of x and y.
84) Mathwright2000 workbook: iteration and recursion
Description: In this workbook, students define recursive sequences. The points are plotted graphically and printed. Once the sequence is defined, another may be defined from it. E.g. the student might take ratios of successive terms in the fibonacci sequence.
85) Mathwright2000 workbook: logic programming
Description: Given a database of facts, together with certain rules of inference that you create, it is possible to discover many new facts that are "provable" from these. Mathwright implements prolog to let you explore the world of automatic theorem proving.
86) Mathwright2000 workbook: newtons method and chaos
Description: This workbook studies a strange behavior of the newton's method iteration. However, it may not converge. What is worse, the iteration may enter into a stable periodic orbit, or it might actually become "chaotic".
87) Mathwright2000 workbook: taylor polynomials and their graphs
Description: This workbook studies taylor-maclaurin series approximations, for which one needs only the value of the function and of its derivatives at a single specified point. These polynomials lead to infinite series representations of the function.
88) Mathwright32 interactive web article: special relativity and conic sections
Description: An interesting property of conic sections leads to the focus-locus description. We will explore in this microworld a link between the focus locus definition and the plane slicing cone definition, based on the hyperbolic geometry of special relativity.
89) Mathwright32 microworld: 3d graphics
Description: This microworld illustrates various uses for 3d graphics. On the first page is a simulation on conic sections in three dimensions. Next is a differential equation solver for ordinary differential equations in R3 starting with the lorenz equations.
90) Mathwright32 microworld: color portraits of complex mappings
Description: This microworld explores complex analysis in a colorful way. analytic maps also have some deep connections with geometry, but are somewhat more difficult to visualize than real maps. In this book we use colors to represent the properties of these maps.
91) Mathwright32 microworld: implicit functions and level sets
Description: An equation of the form f(x,y) = 0 often defines a curve in the plane. We say that the curve is defined implicitly but we may not be able to solve for y in terms of x. Implicit differentiation often allows us to solve for dy/dx as function of x and y.
92) Mathwright32 microworld: taylor polynomials and their graphs
Description: This microworld studies taylor-maclaurin series approximations, for which one needs only the value of the function and of its derivatives at a single specified point. These polynomials lead to infinite series representations of the function.
93) Mathwright2000 workbook: cubic equations
Description: This workbook develops an approach to the study of cardano's method for solving cubic equations that discloses certain new symmetries and points the way to generalization to higher degree equations. Those generalizations are to the quartic case.
94) Mathwright2000 workbook: herons formula
Description: This workbook explores heron's formula for the area of a triangle in terms of its sides. The formula may be understood by asking which quadrilateral with given side lengths has the largest area. Here the reader to varies the shape of the quadrilaterals.
95) Mathwright2000 workbook: odds and integrals
Description: This workbook explores the following question. Given n random numbers chosen from the unit interval [0,1], what is the expected value of the kth shortest segment so determined? We translate this problem from probability to geometry.
96) Mathwright32 interactive web book: cardano
Description: This microworld develops an approach to the study of cardano's method for solving cubic equations that discloses certain new symmetries and points the way to generalization to higher degree equations. Those generalizations are to the quartic case.
97) Mathwright32 interactive web book: herons formula
Description: This microworld explores heron's formula for the area of a triangle in terms of its sides. The formula may be understood by asking which quadrilateral with given side lengths has the largest area. Here the reader varies the shape of the quadrilaterals.
98) Mathwright32 interactive web book: odds and integrals
Description: This microworld explores the following question. Given n random numbers chosen from the unit interval [0,1], what is the expected value of the kth shortest segment so determined? We translate this problem from probability to geometry.