Free Math Software






 * “Knowledge is of two kinds. We know a subject ourselves, or we know where we can find information upon it.” (Samuel Johnson; British author and father of the English dictionary; 1709-1784.)


 * “Try to learn something about everything and everything about something.” (Thomas H. Huxley; English writer; 1825-1895.)


 * “Each problem that I solved became a rule which served afterwards to solve other problems.” (René Descartes, French philosopher, mathematician, scientist, and writer; 1596-1650.)

A Philosophy of Education
Each of has our own, personally developed, philosophy about education. The three quotes given above capture quite a bit of my philosophy. I enjoy having both breadth and depth of knowledge and skills. I enjoy being overwhelmed by the progress that is occurring in expanding human knowledge. I enjoy being both a lifelong learner and a lifelong teacher.

The third quote is particularly poignant, especially with a little modification. I believe that:


 * Each problem that we humans learn to solve becomes a building block or tool that subsequently serves all of us who have the knowledge and skills to build on this accomplishment. Computers have opened up a vast vista for representing, storing, and helping to use knowledge and skill in solving problems and accomplishing tasks.

We are living at a time of exceedingly rapid growth in the totality of human knowledge and in our abilities to access and to disseminate (share) this knowledge. A modern, forward-looking educational system "understands" and builds on the ideas of the three quotations. Such a system is rooted in past, present, and reasonable forecasts of future change. However, it is not bound by all of the aspects of education that have served us well in the past. Some of these will continue to serve us well, but others are outmoded and need to give way to ideas and processes that will better serve us in the future.

This IAE-pedia document is about computer tools for the representation and solution of math problems. You know quite a bit of math, since it is a required component of the K-12 curriculum as well as a routine part of your everyday life. For example, do you know what day it is, what time it is, your weight, where you are currently located, how old you are, and the cost of items that you frequently purchase? All of your answers involve math.

Introduction
A computer is a machine designed for the input, storage, processing, and output of information. A computer system consists of a combination of hardware (physical machinery) and software (computer programs). Typically, some software in built into the hardware in a manner that cannot be changed by the casual computer user.

For example, an inexpensive handheld, solar-powered calculator contains programs for addition, subtraction, multiplication, division, and calculating square roots. You tell such a calculator what to do by depressing keys on its keyboard. What you are doing is keying in data to be processed and the operations to be used in processing the data.

Nowadays when you acquire a computer (mainframe, laptop, tablet, or Smartphone) it contains some built-in software and provisions for easily adding software. The computer applications ("apps") that people acquire free or purchase are software (i.e., computer programs).

This document focuses on free math software. Some of this software is specifically designed for use in education, and some is designed for more general purposes of helping to solve math-related problems in many disciplines.

For example, statistics is a component of the discipline of mathematics, but a great many disciplines of study and research make use of statistics. Graphical representation of data and the results of analyzing data are routine in our world. Thus, it is not surprising that there are a great many statistical "packages" (collections of statistics programs designed to work together) available both free and from commercial sources.

My 7/14/2016 Google search of the expression free math software returned about many million results. This search produced both a list of documents and the following four related search terms that I might want to investigate:


 * free math games download
 * free mathematica download
 * free algebra software download
 * free calculator software download

I found the suggestions interesting and useful. Perhaps you have never heard about Mathematica, the second item on the list. It is a very powerful aid to solving a broad range of math problems. It is a quite expensive software package, and the website offers a free trial. In addition, the company makes Wolfram Alpha (a limited version of Mathematica) available for free use on the Web. Both are discussed later in this document.

My goal is to provide you with some help in your search for free math software to fit your specific needs. I use the term your specific needs to focus on your specific knowledge, skills, and needs in representing and solving various types of math-related problems. However, you need to be aware that a reasonable level of knowledge about the types of math-related activities you are interested in is essential to locating and using math-related software that will likely meet your needs.

Think about it this way. If you know relatively little about math notation or math functions such as square root, logarithms, sin, cos, tan, and so on, your lack of knowledge will not be solved by purchasing a scientific calculator. However, if you are taking a math or physics course that involves calculations using math notation and functions, then you will find that a scientific calculator is a valuable aid to learning the subject area and in doing the homework. (And, likely the tests will require use of such a calculator.)

Math Education Software
There are many good sources for various types of math-related and math education-related free or inexpensive software. Some are briefly discussed in the following subsections.

Concrete and Virtual Manipulatives
Math manipulatives such as an abacus, bead frame, or bean (counting) sticks have a very long history. Students use them both in learning certain aspects of math and as an aid to solving certain types of math problems.

As computers became more and more available, virtual manipulatives (computer-based manipulatives) came into common use. The manipulatives that did not require use of a computer came to be called concrete manipulatives.

A 1999 article by Douglas Clements presents some of the research on concrete manipulatives as well as early (circa 1999) thoughts on virtual manipulative. Quoting from the article:


 * Students who use manipulatives in their mathematics classes usually outperform those who do not, although the benefits may be slight. This benefit holds across grade level, ability level, and topic, given that use of a manipulative "makes sense" for that topic. Manipulative use also increases scores on retention and problem solving tests. Attitudes toward mathematics are improved when students have instruction with concrete materials provided by teachers knowledgeable about their use.


 * However, manipulatives do not guarantee success. One study showed that classes not using manipulatives outperformed classes using manipulatives on a test of transfer. In this study, all teachers emphasized learning with understanding. In contrast, students sometimes learn to use manipulatives only in a rote manner. They perform the correct steps, but have learned little more. For example, a student working on place value with beans and beansticks used the (one) bean as ten and the beanstick (with ten beans on it) as one.

Virtual manipulatives can be thought of as interactive videos. They are like concrete manipulatives, but the come with "built-in" compute power that allows a variety of operations to be automated.

Think about coins both as concrete manipulatives and as virtual manipulatives. Students can arrange concrete coins into patterns, such as squares, rectangles, and triangles. They can do coin flipping experiments of examining frequencies of heads versus tails, or lengths of runs of heads or tails. For example, see Random.org. They can practice counting money and making change.

Similar software exists for rolling dice. One of my earliest childhood memories is playing board games that involved rolling dice and moving pieces around the board. With practice, a glance at the pair of faces showing produces the sum, and a glance at the board produces the place to move one's piece to. This illustrates an important idea in math education. I was intrinsically motivated to gain these two skills, and through meaningful practice (practice while playing a game with friends) I gained the knowledge and skill.

The real "power" of virtual manipulatives can be seen in a collection of free math manipulatives available from the Math Learning Center. Click here for information about a specific math manipulative named Number Line that is available in the Math Learning Center's free collection.

Graphing Software
Graphs and charts are great because they communicate information visually. For this reason, graphs and charts are often used in newspapers, magazines, and businesses around the world.

Creating and using graphs of data are important components of a good math education. Let's take a specific example. At what grade level (or cognitive development level) can a child learn to make use of a pie chart (a circle graph)? At what grade level is a child likely to have the math knowledge and skills to create a pie chart?

Likely you came up with two different grade levels. A pie chart provides a visual representation that a relatively young child can use to determine smaller pieces from larger pieces. A quarter, third, or half of a circle is easy to "see." It is not necessary to know that there are 360 degrees in a circle, 90 degrees in a quarter circle, and so on. But creating a pie chart requires working with percentages, degrees, and a measuring instrument such as a protractor.

This raises an interesting and currently unresolved issue in math education. If a computer or calculator can perform a certain task, what do we want students to learn about understanding and making use of that capability? An inexpensive 6-function calculator has a square root key. A relatively young student can understand the meaning of square root and can use such a calculator. As a consequence, the "by hand" calculation of square roots has practically disappeared from the secondary school math curriculum.

The National Center for Educational Statistics (NCES) collects a huge range of data about education and prepares reports it makes available free. The NCES is continually faced by the challenge of how to represent its findings in an understandable and useful manner. Part of the answer is provided by the use of graphs and charts.

The NCES provides free software to create Bar, Line, Area, Pie, and XY graphs. The following figure is from the home page of their website.



Math Forum @ Drexel
The Math Forum @ Drexel is a large and steadily growing collection of aids to the teaching and learning of math. Their website provides links to the following:
 * Ask Dr. Math®
 * Discussion Groups
 * EnCoMPASS
 * Financial Education
 * Ignite!
 * Internet Mathematics Library
 * Math Forum Internet News
 * Math Images
 * Math Tools
 * Math Topics by Course
 * PCMI@MathForum
 * Problems of the Week
 * Teacher2Teacher (T2T®)
 * Virtual Math Teams

Computer Algebra Systems
Quoting from the Wikipedia:


 * A computer algebra system (CAS) is a software program that allows computation over mathematical expressions in a way which is similar to the traditional manual computations of mathematicians and scientists. The development of the computer algebra systems in the second half of the 20th century is part of the discipline of "computer algebra" or "symbolic computation", which has spurred work in algorithms over mathematical objects such as polynomials.


 * Computer algebra systems may be divided in two classes: the specialized ones and the general purpose ones. The specialized ones are devoted to a specific part of mathematics, such as number theory, group theory, or teaching of elementary mathematics.


 * General purpose computer algebra systems aim to be useful to a user working in any scientific field that requires manipulation of mathematical expressions. To be useful, a general purpose computer algebra system must include various features such as


 * • a user interface allowing to enter and display mathematical formulas
 * • a programming language and an interpreter (the result of a computation has commonly an unpredictible form and an unpredictible size; therefore user intervention is frequently needed)
 * • a simplifier, which is a rewrite system for simplifying mathematics formulas
 * • a memory manager, including a garbage collector, needed by the huge size of the intermediate data, which may appear during a computation
 * • an arbitrary-precision arithmetic, needed by the huge size of the integers that may occur
 * • a large library of mathematical algorithms


 * The library must cover not only the needs of the users, but also the needs of the simplifier. For example, the computation of polynomial greatest common divisors is systematically used for the simplification of expressions involving fractions.

There are a number of free CAS available on the Web.

Many people have suggested that a modern math education includes having students learn to make use of a relatively general-purpose Computer Algebra System. However, our overall math education system has made only limited movement in that direction. Perhaps the best example of such movement is the requirement that many students should learn to use graphing calculators with some CAS features in many math courses at the high school and college level.

Some calculators are powerful enough to contain a fairly comprehensive CAS. Quoting again from the Wikipedia:


 * In 1987 Hewlett-Packard introduced the first hand held calculator CAS with the HP-28 series, and it was possible, for the first time in a calculator, to arrange algebraic expressions, differentiation, limited symbolic integration, Taylor series construction and a solver for algebraic equations.


 * The Texas Instruments company in 1995 released the TI-92 calculator with an advanced CAS based on the software Derive [a CAS]. This, along with its successors (including the TI-89 series and the newer TI-Nspire CAS released in 2007) featured a reasonably capable and relatively inexpensive hand-held computer algebra system.

GeoGebra
GeoGebra is a powerful educational tool. Quoting from the website:


 * [Geogebra is a] graphing calculator for functions, geometry, algebra, calculus, statistics, and 3D math. [It is] dynamic mathematics for learning and teaching.

Here is a screen shot providing links to some of its tools:



Here is the first page of its 3D Graphics Application:



 Maxima, a Computer Algebra System
Quoting from the Maxima website http://maxima.sourceforge.net/:


 * Maxima is a system for the manipulation of symbolic and numerical expressions, including differentiation, integration, Taylor series, Laplace transforms, ordinary differential equations, systems of linear equations, polynomials, and sets, lists, vectors, matrices, and tensors. Maxima yields high precision numeric results by using exact fractions, arbitrary precision integers, and variable precision floating point numbers. Maxima can plot functions and data in two and three dimensions.


 * The Maxima source code can be compiled on many systems, including Windows, Linux, and MacOS X. The source code for all systems and precompiled binaries for Windows and Linux are available at the SourceForge file manager.


 * Maxima is a descendant of Macsyma, the legendary computer algebra system developed in the late 1960s at the Massachusetts Institute of Technology. It is the only system based on that effort still publicly available and with an active user community, thanks to its open source nature. Macsyma was revolutionary in its day, and many later systems, such as Maple and Mathematica, were inspired by it.

SAGEMATH: Open Source CAS Software
Quoting from the SageMath website: SageMath:
 * SageMath is a free open-source mathematics software system licensed under the GPL. It builds on top of many existing open-source packages: NumPy, SciPy, matplotlib, Sympy, Maxima, GAP, FLINT, R and many more. Access their combined power through a common, Python-based language or directly via interfaces or wrappers.


 * Mission: Creating a viable free open source alternative to Magma, Maple, Mathematica and Matlab.

Quoting again from the website, some of the features of this software package include:


 * Tools for Teaching. Create projects for your students, hand out assignments, then collect and grade them with ease.


 * Collaboration Made Easy. Edit documents with multiple team members in real time.


 * All-in-one Programming. Write, compile and run code in nearly any programming language.


 * Computational Mathematics. Use SageMath, IPython, the entire scientific Python stack, R, Julia, GAP, Octave, and much more.


 * LaTeX Editor. Write beautiful documents using LaTeX.

This latter resource is a very valuable tool for those who want to write (or, write and publish) using the full notation of mathematics.

Wolfram Alpha</Center>
The free Wolfram Alpha CAS is designed both as an aid to doing mathematics and as an aid to learning and understanding mathematics. The following diagram from Wolfram Alpha's home page provides an overview of some application areas.



The site is based on Mathematica, a commercially available CAS system software developed by Wolfram. Thus, Wolfram Alpha can do many of the things that Mathematica can do. However, Mathematica can do a great deal more—both in math and in other areas.

Statistics Software
Quoting from the Dictionary.com:


 * 1. Statistics is the science that deals with the collection, classification, analysis, and interpretation of numerical facts or data, and that, by use of mathematical theories of probability, imposes order and regularity on aggregates of more or less disparate elements.


 * 2. Statistics are the numerical facts or data themselves.

Quoting from History of statistics:


 * The History of statistics can be said to start around 1749 although, over time, there have been changes to the interpretation of the word statistics. In early times, the meaning was restricted to information about states. This was later extended to include all collections of information of all types, and later still it was extended to include the analysis and interpretation of such data. In modern terms, "statistics" means both sets of collected information, as in national accounts and temperature records, and analytical work which requires statistical inference.


 * Statistical activities are often associated with models expressed using probabilities, and require probability theory for them to be put on a firm theoretical basis: see History of probability.


 * A number of statistical concepts have had an important impact on a wide range of sciences. These include the design of experiments and approaches to statistical inference such as Bayesian inference, each of which can be considered to have their own sequence in the development of the ideas underlying modern statistics.

A wide range of statistical software packages are available. A 7/10/2016 Google search of the expression free statistics software produced tens of millions of results.

For example, see Free Statistical Software. Quoting from the home page of this website:


 * This page contains links to free software packages that you can download and install on your computer for stand-alone (offline, non-Internet) computing. They are listed below, under the following general headings:


 * * General Packages: support a wide variety of statistical analyses
 * * Subset Packages: deal with a specific area of analysis, or a limited set of tests
 * * Curve Fitting and Modeling: to handle complex, nonlinear models and systems
 * * Biostatistics and Epidemiology: especially useful in the life sciences
 * * Surveys, Testing and Measurement: especially useful in the business and social sciences
 * * Excel Spreadsheets and Add-ins: you need a recent version of Excel
 * * Programming Languages and Subroutine Libraries: customized for statistical calculations; you need to learn the appropriate syntax
 * * Scripts and Macros: for scriptable packages, like SAS, SPSS, R, etc.
 * * Miscellaneous: don't fit into any of the other categories
 * * Other Collections of Links to Free Software

Some free statistical packages are particularly oriented toward use in education. For example, see SAS® University Edition. Quoting from the website:


 * With SAS University Edition, you get SAS Studio, Base SAS, SAS/STAT, SAS/IML, SAS/ACCESS and several time series forecasting procedures from SAS/ETS. It's the same world-class analytics software used by more than 80,000 business, government and university sites around the world, including 93 of the top 100 companies on the Fortune Global 500® list. That means you'll be using the most up-to-date statistical and quantitative methods. And did we mention it's free?

Calculators
A huge number of "virtual" calculators are available on the Web. Such virtual calculators provide the Web user with a computer simulation of a calculator. Many of the types of calculators available are quite specialized. For example, the site http://www.math.wvu.edu/~ef/150/calc/jcalc.html provides an example of a generic scientific calculator:



I enjoyed playing with the "rand" (random number generator) key. It appears to generate random numbers in the range of 0 to 1, or perhaps in the range of .00000000000 to .99999999999. I was curious about how long it would take me to generate and add a list of ten random numbers. It took me less than a half minute, and my sum was 6.429752239725. I was surprised by my result; I expected an answer closer to 5.

For another calculator example, recently I was reading about the Commodore-64 microcomputer that was first produced in 1982 and sold at retail for $595. What would this be in today's dollars? My Google search of the expression inflation calculator returned a number of hits. I used the one. I just provide the year (1982), the amount ($595), and this year (2016 ), and it provides me with the answer $1,481.25. I recall that, when I was a teenager, I used to baby sit for $.75 per hour. That is roughly equal to $7 per hour in 2016 dollars.

In the example above, notice that I had to know something about inflation. I also happened to know that Web sites exist to calculate past effects of inflation. My search was poorly stated. I should have asked for a United States inflation calculator. The Google search engine I was using was "smart enough" to provide me with examples for the United States. (Hmm. I wonder if Google is so smart that, if I were asking the question in French from a location in France, it would have given me links to calculate French inflation?)

How about a metric calculator? My Google search of virtual metric calculator returned about 675,000 results. Amusingly, the second item on the list was a virtual slide rule. A slide rule can be thought of as a type of math manipulative, and both concrete and virtual versions are available.

Looking for still more? Go to Martindale's Calculator On-line Center. Quoting from the site:


 * Currently the Calculators On-Line Center contains over 22,125 Calculators and Spreadsheets, over 4,420 Courses, Lectures, Manuals, Handbooks, and 1,000's of Movies, Videos, Simulations and Animations.

Quoting from the Mathematics Center section of this website, the following topics are included:


 * Abacus, Slide Rules, Unit Conversion, Basic Math, Adding Machines, Scientific Calculators, Calculus, Algebra, Computational Algebra, Geometric Algebra, Linear Algebra, Abstract Algebra, Chaos & Catastrophe, Differential Equations, Encryption, Cryptography & Quantum Cryptography, Ergodic Theory, Geometry, Differential Geometry, Computational Geometry, Graphing/Plotting, Integrals, Statistics, Mathematical Constants, Mathematical Physics, Number Theory, Operations Research & Logistics, Quantitative Finance and Advanced Mathematical Finance, etc.

Final Remarks
Here is an addition to my personal philosophy of education. A good education helps a student to pose questions, seek answers, and understand the answers to problems and tasks of two major types:


 * 1)  Problems and tasks of personal interest.
 * 2)  Problems and tasks that others (such as one's children, students, friends, teachers, employers, and so on) might pose.

Achieving and using a good education is a lifelong endeavor. Formal schooling is helpful, but the ultimate responsibility lies with the individual learner. One of the ways that I do this for myself is to build and maintain online collections of resources I will want to use in the future and that I want to share with other people. I store this information in what I call Digital Filing Cabinets. Click here to access my Math Education Digital Filing Cabinet.

I recommend that you build a personal "My math electronic aids" collection of hardware, software, and links to online software. Each item in your collection should be something that you personally have used long enough so that it has continuing practical value to you.

Probably your collection will include a handheld calculator. Perhaps it will include a spreadsheet program. If you are a student, look for math-related software relevant to the various courses you are taking and/or are planning to teach. Remember, math is apt to be useful in representing and helping to solve some of the problems in each course you take or will teach.

If you are currently a teacher, look for software that helps to solve the problems and accomplish the tasks of a teacher in the disciplines you teach. For example, many teachers make use of grade book software—and that certainly is a math-related area! Many teachers make use of graphics art software to prepare materials useful in teaching. Remember, the items in your "My math electronic aids" collection should be items that you have used to do something personally useful to you.

Lists of Free Online Resources </Center>
Moursund, D. (2016). Fair use. IAE-pedia. Retrieved 7/17/2016 from http://iae-pedia.org/Fair_Use.

Moursund, D. (2016). Free educational videos. IAE-pedia. Retrieved 8/17/2016 2016 from http://iae-pedia.org/Free_Educational_Videos.

Moursund, D. (2016). Free IAE math education materials. IAE-pedia. Retrieved 8/17/2016 from http://iae-pedia.org/Free_IAE_Math_Education_Materials.

Moursund, D. (2016). Free math education videos. IAE-pedia. Retrieved 7/17/2016 from http://iae-pedia.org/Free_Math_Education_Videos.

Moursund, D. (2016). Free math software. IAE-pedia. Retrieved 8/17/2016 from http://iae-pedia.org/Free_Math_Software.

Moursund, D. (2016). Free open content libraries. IAE-pedia. Retrieved 6/27/2016 from http://iae-pedia.org/Free_Open_Content_Libraries.

Moursund, D. (2016). Free open source and open content educational materials. IAE-pedia. Retrieved 8/17/2016 from http://iae-pedia.org/Free_Open_Source_and_Open_Content_Educational_Materials.

Moursund, D. (2016). Free open source online databases. IAE-pedia. Retrieved 8/17/2016 from http://iae-pedia.org/Free_Open_Source_Online_Databases.

Moursund, D. (2016). Free open source software packages. IAE-pedia. Retrieved 8/17/2016 from http://iae-pedia.org/Free_Open_Source_Software_Packages.

Moursund, D. (2016). Free science education software. IAE-pedia. Retrieved 8/17/2016 from http://iae-pedia.org/Free_Science_Education_Software.

Moursund, D. (2016). Free science education videos. IAE-pedia. Retrieved 8/17/2016 from http://iae-pedia.org/Free_Science_Education_Videos.

Popular Free Online IAE Books </Center>

 * Moursund, D. (February, 2016). Math Methods for Preservice Teachers. Eugene, OR: Information Age Education. PDF file: http://i-a-e.org/downloads/free-ebooks-by-dave-moursund/283-math-methods-for-preservice-elementary-teacher-1/file.html. Microsoft Word file: http://i-a-e.org/downloads/free-ebooks-by-dave-moursund/282-math-methods-for-preservice-elementary-teacher/file.html. HTML file: http://iae-pedia.org/Math_Methods_for_Preservice_Elementary_Teachers.


 * Moursund, D. (August, 2015). Brain Science for Educators and Parents. PDF file: http://i-a-e.org/downloads/free-ebooks-by-dave-moursund/271-brain-science-for-educators-and-parents-1/file.html. Microsoft Word file: http://i-a-e.org/downloads/free-ebooks-by-dave-moursund/270-brain-science-for-educators-and-parents/file.html. HTML file: http://iae-pedia.org/Brain_Science.


 * Moursund, D. (2/28/2015). Technology and Problem Solving in PreK-12 Education for Adult Life, Careers, and Further Education. PDF file: http://i-a-e.org/downloads/free-ebooks-by-dave-moursund/267-technology-and-problem-solving-in-prek-12-education-1.html. Microsoft Word file: http://i-a-e.org/downloads/free-ebooks-by-dave-moursund/266-technology-and-problem-solving-in-prek-12-education.html. HTML file: http://iae-pedia.org/Technology_and_Problem_Solving.


 * Sylwester, R., and Moursund, D., eds. (2012). Creating an Appropriate 21st Century Education. Eugene, OR: Information Age Education. PDF file: http://i-a-e.org/downloads/doc_download/243-creating-an-appropriate-21st-century-education.html. Microsoft Word file: http://i-a-e.org/downloads/doc_download/242-creating-an-appropriate-21st-century-education.html.


 * Moursund, D., and Albrecht, R. (2011). Using Math Games and Word Problems to Increase Math Maturity. Eugene, OR: Information Age Education. PDF file: http://i-a-e.org/downloads/doc_download/211-using-math-games-and-word-problems-to-increase-the-math-maturity-of-k-8-students.html. Microsoft Word file: http://i-a-e.org/downloads/doc_download/210-using-math-games-and-word-problems-to-increase-the-math-maturity-of-k-8-students.html.

All Free Online IAE Books </Center>

 * Free Books by Dave Moursund.
 * Free Books by Bob Albrecht.
 * Free Books Co-authored by Bob Sylwester & Dave Moursund.

Moursund's Collections of Quotations </Center>

 * Math Education Quotations.


 * Quotations Collected by David Moursund.

Digital Filing Cabinets </Center>

 * Digital Filing Cabinet: Overview.


 * Digital Filing Cabinet: Math Education.

Author
The original version of this page was created by David Moursund and edited by Ann Lathrop.