Mathematics – Auxiliary Memory

Studying Euclid’s Elements in Retirement

by James Wallace Harris, 8/2/26

Euclid is sometimes called the Father of Geometry because of his book The Elements. A literal translation of the word geometry could be “earth measurement.” Even Aristotle knew that the Egyptians and Babylonians used geometry to survey land and construct buildings. Euclid doesn’t use numbers or make measurements in The Elements, and his geometry does not match what is taught in Geometry classes today.

What Euclid teaches is an abstract construction, yet it drives a pylon into reality. What Euclid really teaches is how to think clearly and precisely, something humans seldom do. There is a story about Abraham Lincoln that claims he didn’t become a good lawyer until he mastered the first six books of The Elements. (If you understand the foundation of what Euclid teaches, you will follow that link. If you don’t follow the link, you will after you have studied Euclid.)

The reason I’m studying Euclid’s Elements in retirement is that it’s exercise for my aging brain. Crossword puzzles, Wordle, and Sudoku are decent stretching exercises for my flabby neurons, but The Elements is Nautilus machines. I’m only as far as Proposition 12 in Book 1, which has left my brain in a puddle on the floor.

It’s not that the problems in The Elements are hard; they mostly require persistence. Numbers and measurements aren’t involved. Just logic, reason, a straightedge, compass, and pencil. Even a tiny amount of exercise can make you feel weak and lightheaded if you’re out of shape.

At 74, the challenge is whether I can learn anything new at all. Especially since I can’t seem to retain new words and concepts. I can still recall old stuff, but keeping new stuff is difficult. Maybe studying Euclid will help with that.

Before I retired in 2013, I thought I’d work on a degree in computer science as a hobby once I retired. I’ve taken many computer courses at the undergraduate level while earning an English degree. Since I spent my working life with computers, I always wanted a computer degree to validate myself. However, when I investigated the requirements for a B.S. or M.S. in computer science, I learned that I needed 24 hours of math as a prerequisite.

I only had College Algebra, Calculus I, and Statistics from the 1970s. When I looked at textbooks in those courses, I realized I had forgotten everything. I’d need to start over from scratch. Not only that, but when I tried Khan Academy as a refresher, I learned I had forgotten the fundamentals of math from elementary school. I had to start over in the 2nd grade. I eventually worked my way up to 5th grade in Khan Academy, but my brain ran out of gas. I dropped the idea of going back to college as a retirement hobby.

Twelve years later, my mind is even more mushy. I do Wordle and the Mini Crossword at The New York Times every day. Even though I often have trouble remembering words, I’m pretty good at those puzzles. I gave up on learning math over a dozen years ago. I regret that. I should have stuck it out.

Recently, I read about the history of Euclid’s Elements. Supposedly, it’s only second to The Bible in the number of editions over the last two thousand years. Its main claim to fame is that it teaches more than basic geometry – it pushes its students to think logically.

The geometry in Elements is very basic, not like the geometry they teach in school today. All 13 books of the Elements use the same 5 Postulates and 5 Common Notions. From that foundation, 48 propositions (problems) are set up in Book 1. Each proposition is proved step by step with diagrams based on the 23 definitions. 131 total definitions cover 465 propositions in all 13 books. Most students only do the first 6 books.

For centuries, an educated person in the Western world would study Latin, Greek, and Euclid’s Elements. Before the scientific method, most knowledge was opinion. Euclid’s Elements set up the idea that knowledge should be built on a foundation. A foundation that is consistent across cultures. Every proposition has to be proved with definitions, postulates, and common notions. Look at this video; the visual demonstration will probably say more than I can in words.

The simplicity and limited domain of the Elements appealed to me. I may not be able to pass a university math course, but hopefully, I can work my way through the first book of the Elements.

I started with the free web version, and then bought a 99-cent Kindle edition of The Elements of Euclid for the Use of Schools and Colleges (Illustrated) so I could read on my phone. I also bought Delphi Collected Works of Euclid (Illustrated) because it had a long history of this famous book. It was only $2.99 for the Kindle edition, and I liked the scholarly history of how Euclid wrote the Elements and how people used it over the centuries.

I’ve been able to work through the early Propositions. I even bought a compass to draw them out myself. I’m not sure I’d even call this mathematics. It’s really just logic. The work is tedious, though. With each step, you’re supposed to cite the Definition, Postulate, or Common Notion you used to justify that step. You have to think hard as you follow along.

Here’s my work for Proposition 1. Most books describe the steps in words, and then show the final diagram. I find that confusing.

Proposition 1: To construct an equilateral triangle on a given finite straight line.

Look at how David E. Joyce works the problem at Clark University. It’s very wordy, but it also explains some of Euclid’s faults. But even Euclid’s description is very confusing, at least to me. It’s only clear when you follow the problem logically, step by step.

We start with a line segment with the endpoints being A and B. Ignore point C for now. Many teachers start by drawing line segment AB and immediately put a point C above it. I think that’s wrong.

Draw:

Then draw a circle with A at the center, with AB as the radius.

Then create a circle with B as the center, using AB as the radius.

Now, it’s time for point C. Where the circles intersect, designate points. Remember the definitions. They are abstractions, but we have to always build on them.

Definition 1: A point is that which has no part.

Definition 2: A line is a breadthless length.

Definition 3: The ends of a line are points.

Definition 15: A circle is a plane figure contained by one line such that all the straight lines falling upon it from one point among those lying within the figure equal one another.

Definition 16: And the point is called the center of the circle.

Postulate 3: To describe a circle with any center and radius.

Euclid doesn’t state that when a line crosses another line, the intersection is a point, but we assume it. He should have included it as a definition.

Because the two circles intersect, we have defined two additional points, which we can label C and D; however, D is not needed for this proof.

Now we can draw lines AC and BC.

Definition 20: Of trilateral figures, an equilateral triangle is that which has its three sides equal, an isosceles triangle that which has two of its sides alone equal, and a scalene triangle that which has its three sides unequal.

Postulate 1: To draw a straight line from any point to any point.

Common Notion 1: Things which equal the same thing also equal one another.

Common Notion 4: Things which coincide with one another equal one another.

Because AB is the radius of both circles, and AC and BC are also radii of those same circles, all sides of the triangle are equal.

Thus we have proved:

Proposition 1: To construct an equilateral triangle on a given finite straight line.

In modern geometry, we would have just drawn an equilateral triangle knowing that all three angles equal 60 degrees. And we could have measured the length of each side with a ruler. The point of Euclid is to create geometric objects with basic geometric building blocks. No numbers are involved.

The rest of The Elements involve creating ever-evolving complicated 2D and 3D geometric shapes. Some of which prove other mathematical problems, like Proposition 47. It’s an alternate method of proving the Pythagorean theorem.

But even after watching a video multiple times, it’s still hard to do it myself. Without practicing with drawing lines and circles, it’s hard to see how the proofs really work, even when they are shown to you.

JWH

The Mathematics of Buying Science Fiction Anthologies

by James Wallace Harris, Friday, August 17, 2018

Like the famous vehicle routing problem or the four-color map theorem, I’m proposing the science fiction anthology problem.

We’ve just published The Classics of Science Fiction Short Stories that identified 275 short stories, novelettes, and novellas that are the most remembered in the genre. We gathered stories from 290 retrospective and annual best-of-the-year anthologies, several polls and lists, finalists from three awards (Hugo, Nebula, Sturgeon), three recommended reading lists and put them into a database. We produced what we call The Classics of Science Fiction Short Stories list consisting of the stories that were on at least 5 of those sources. The Stories by Rank (with Citations) report starts with “Bloodchild” by Octavia Butler. She got the most citations – 16. We’ve also created several other interesting reports from the data – see the site menu.

Sense of Wonder - A Century of Science Fiction edited by Leigh Ronald Grossman

Here’s the mathematical challenge that appeals to me. What’s the minimum number of anthologies to buy to get the most of the 275 stories on the list? I’ve already read 63 of these stories in 2018, and it’s been extremely rewarding. I believe anyone who reads a short story or novelette a day and takes two days to read a novella, could finish the list in one year. Sadly, there’s no 5-volume set of the classic science fiction short stories that collect them all.

Even more depressing, most of the anthologies we used are out of print. Anthologies don’t stay in print long. And some of the anthologies we used are textbooks priced far higher than the casual reader is willing to pay. I have spent the past year buying many of the anthologies on our Citation Bibliography list as I could afford, but my collection is far from complete.

My willingness to buy shelves of old SF anthologies to get all these stories isn’t typical. Thus, the mathematical problem I propose of finding the fewest anthologies that give readers the most stories from the 275.

We can’t claim these stories are the very best short works of science fiction. Neither did we pick them. They aren’t our personal favorites. We used math to identify the most remembered stories, which should be more valuable than mere opinion. By promoting the list, we are reinforcing the memory of these stories (maybe at the detriment of better stories). I could easily create “My 100 Favorite Stories Not on the List.” Those stories would be even harder to find. If you look at our Citation Sources Ranked report, you’ll see how many stories each citation source identified.

And let me be perfectly clear, not all these remembered stories are still worth reading today, at least to my taste. Time is cruel to science fiction. Chernobyl, Three Mile Island, and Fukushima has ruined “Nerves” by Lester del Rey. Even though we have a methodology for revealing the most remember science fiction stories, I’m not sure all of them are worth remembering. But I do believe the stories that got the most citations are.

I want to promote the reading of short science fiction. Most fans don’t like short stories or buying anthologies. They need to try short science fiction to see what they are missing. Maybe it will change their minds. So part of this mathematical problem is also recommending the most recommended of the 275 stories, especially the first 100.

I believe the single most useful anthology that’s in print is Sense of Wonder: A Century of Science Fiction edited by Leigh Ronald Grossman. It contains 133 stories, of which 50 are on our 275 Classics of Science Fiction Short Stories list. However, it’s $40 for a Kindle edition. (Granted, it is a huge book that’s probably best read on a Kindle.) And I sure wish it was available on audio because I love listening to short science fiction! If you eyeball our Stories by Rank (With Citations) list, you’ll see that many of the top stories are collected in this anthology. Still, $40 for an ebook book will scare most buyers off. Sense of Wonder is priced as a textbook, so it also contains essays about science fiction putting each story into context. That does add extra value.

The next volumes that are story-list full and in print, are the three volumes of The Science Fiction Hall of Fame. They contain just 48 stories, of which 39 were on our list. This year I listened to all three on Audible. But that’s a commitment of 3 credits to get 39 stories. Buying printed copies of Volume 1, Volume 2A and Volume 2B would be almost $57, so the $40 Grossman book seems less expensive now. I immensely enjoyed hearing these old stories and got them for around $30 by buying credits in bulk. Using full price credits would be $45. But 39 out of 48 stories is a very high hit rate.

Now if you’re willing to buy used, the three anthologies edited by Robert Silverberg and Martin Greenberg give you 188 stories to read, of which 53 were on our list. These books are:

If you’re lucky, you could find these books for a few bucks at a library book sale, or all three on AbeBooks.com for maybe $15-30.

The Wesleyan Anthology of Science Fiction is in print for $29. It will get you 49 stories to read, but only 34 of which are on our list. You can get it even cheaper used.

The best bargain is The Big Book of Science Fiction edited by Jeff and Ann VanderMeer. For $17.00, you get 107 stories, but only 25 are on our list. It’s also available as an ebook for $18, making it much more convenient to read all those stories. The paperback is like an old-style phonebook. Even though it only has 25 of our stories, I think it’s a fantastic collection.

Between Sense of Wonder and The Big Book of Science Fiction, many of the top stories are collected. Unfortunately, you also get duplicates. That’s another factor in solving the science fiction anthology problem – how to keep duplicate stories to a minimum.

By now, you’re probably sensing the mathematical headaches this problem generates. How to calculate the minimum number of anthologies to buy that cover the most stories. If you factor in costs, it becomes even wormier.

I haven’t figured that out how to solve this problem. It’s very tricky. I’m open to suggestions. Just buying three of any of these volumes I’ve mentioned so far, only gets you just over a 100 of the 275 stories. It might take buying 10 books to get to 200, and 30 to get to 250. I wonder if there’s a mathematical progression involved?

The minimum number of citations to get on the list was 5. But some of the 275 stories might have only come from polls, awards, and recommendation lists. And it’s possible that several stories came from 5 different books that don’t overlap with any other missing story.

I’m not sure if the answer isn’t 290, the total number of anthologies we used.

JWH

More Sense of Wonder Than Science Fiction

by James Wallace Harris, Monday, February 20, 2017

For the first two-thirds of my life sense of wonder mostly came from science fiction, but in the last third science is supplying more wonder. I have theories as to why. First, aging is making me more fascinated with reality. Second, I’ve lived long enough to feel the real world is science fictional. For example, my science fiction book club is reading Little Fuzzy by H. Beam Piper, a 1962 novel about the discovery of cute creatures on a distant planet that might be sapient. As a kid in the 1960s, that was an exciting idea. But in 2017 we know animals are far more intelligent than we thought and in ways far more exciting than an old science fiction novel. Learning how and why has a great sense of wonder.

The dimensions of sapient behavior have become far more fantastic than fiction, including old stories about robots. For example, The Door Into Summer by Robert A. Heinlein, published in 1957, and first read by me in 1964. Heinlein’s character Dan Davis built household robots – which dazzled me back then. But today I could build my own robot with a Raspberry Pi kit, producing a completely different kind of sense of wonder. I could also download open source machine learning toolkits. This era of Makers and DIY produces a different kind of wonder. Science fiction is great, but I believe I would now give a kid a subscription to Make Magazine before telling her to read science fiction.

More and more when I watch a great documentary I want to know the details about how things are actually done. I don’t want to just be an observer. Last night I watched a wonderful episode of NOVA on PBS that has more sense of wonder than any science fiction novel I can remember reading in a very long time.

It was about origami.

Origami?

Yes, origami. You know, paper cranes…

It was titled “The Origami Revolution” – about how the art of folding paper has inspired scientists, mathematicians, and engineers. The producers completely blew me away. Origami is a fascinating craft, even an art form, but not one I ever paid much attention to. The program began by reporting the latest developments in the art, which go way beyond making simple paper cranes. Using a single sheet of paper, it’s possible to make very elaborate 3D shapes by just folding paper (and without cutting).

Cranes are simple, requiring about thirty folds. Modern advanced origami art like above requires hundreds of folds involving very complex geometry. This is where the excitement started for me – because they brought in mathematics. The program introduced Erik Demaine, showing him working on a 60-page mathematical algorithm with Tomohiro Tachi for computerized origami folding. Can you imagine the mathematics of creating the above work of origami? I can’t, but I wish I could. Tachi has developed a software program Origamizer that the two of them hope will eventually be able to create any 3D figure from a 2D piece of paper. Their theorem should prove it’s possible.

“The Origami Revolution” then goes on to survey wide-ranging work in biology, genetics, chemistry, physics, astronomy that have been influenced by what we’re learning from folding. This has been happening for decades, so I feel a little left behind. The program generated a tremendous sense of wonder in me, probably because this new research offers so much far-out potential, including building robots and spacecraft, and even claiming that dark matter theoretically reveals folded shapes in the structure of the universe.

Here’s a 2008 TED Talks by Robert Lang which give more details than the episode of NOVA, including some examples that are more impressive than shown in the TV show. Follow the link in his name to his website for even more information.

Understanding how modeling 3D structures from a 2D source teaches us about nature, because once the mathematics of folding were revealed scientists began seeing folding in nature, including plants, insects, and even the cosmos. From there it goes into applied engineered structures.

(This isn’t folding per se, but I think it’s related. See SmartFlower Solar.)

If you watch “The Origami Revolution” count all the far out bits of technology. You’ll realize that many of them were never discussed in science fiction. When I was young, I thought science fiction explored ahead of science, but after all these decades I’ve learned something different. Science fiction trails science. This show could inspire countless science fiction stories. Even while watching the TV show I imagined other folks seeing it and thinking up science fiction stories as they watched. They will magnify the demonstrated concepts, extrapolate, speculate, imagine, and come up with possible future scenarios to dramatize. I’m sure they will create far-out tales.

But I think getting older is making me both more patient and less patient. I’m becoming impatient with fiction. It’s easier to skim over the drama, and just zero in on the current science. Now that I’m retired, I have more time to fool around with tech toys. I spend less time reading about imaginary futures, and more time trying to figure the details of now.

You can also watch the full episode of “The Origami Revolution” on YouTube.

JWH

Could You Pass 4th Grade Math?

By James Wallace Harris, Tuesday, January 12, 2016

One of my great regrets in life is not trying harder in school when I was young, especially at studying math and science. I did get through Calculus I in college with a B, but I laid out a year and when I returned to take Calculus II, I was lost. I always studied just enough to pass the tests, but never enough to gain a deep understanding. It was complete laziness on my part.

Now that I’m retired, and I sense my mind in decline, I’ve wondered if I could learn in my final third of life what I didn’t in my first third. It’s that age-old question: Can you teach an old dog new tricks? Would it be possible for me to relearn math and then finish Calculus II? I’ve been meaning to get started on this project for two years, but like my younger self, I put it off to play instead. I don’t know why, but about a week ago I did get started, studying math with a workbook and the Khan Academy.

My first impulse was to begin again with Algebra, but I thought I better refresh myself with Arithmetic, and tried some 4th grade math. It’s a good thing I did, because I’ve discovered I’ve forgotten how to do advanced subtraction and division problems. Decades of using a calculator has ruined my basic math skills and I discovered I was completely flummoxed by that whole carry the number thing.

What’s really amazing is how fantastic the Khan Academy is at teaching. At least the new version, with interactive assessments. Ever since personal computers came out in the late 1970s, I thought they should be fantastic teaching tools. And I assumed the best subject computers to tutor would be math. But every time I looked at math teaching programs I was disappointed. The Khan Academy programs have come up with a rather straight forward method that I’m actually finding addictive. They have drills that automatically assess my answers. Each session covers six problems. I work out the problem on paper, and put in the answer on the computer. If it’s right, I get the next problem, if it’s wrong, I’m forced to keep trying. I can ask for hints, or I can watch instructional videos.

My ego pushes me to get all six problems right in a row. I hate seeing the big X that reminds me I failed. Early on I learned that I’m careless about reading the screen properly, or transferring the problem to the paper, or the answer to the screen. But I quickly began to double check my work. Then I learned that I make casual math mistakes. I used to know my times tables cold, but evidently I’ve got some bugs in my brain. So I do everything twice or thrice. Finally, and this was most enlightening, is I’ve completely forgotten how to do some basic math skills. Which makes me glad I started with arithmetic.

This challenge is demoralizing in a way. I used to believe that with effort I could relearn all my old math and finish Calculus II, but now, I’m not so sure. It’s certainly going to take a lot of time, and hard work. What I’m actually feeling are the limitations of my mind. I’m hoping those limitations are like exercising the body, and that with daily workouts will build my math stamina. I already physically exercise three times a day, and I know my body will never do what it did in my twenties or even forties again. I might be fooling myself that I can mentally turn back the clock, but for some reason I do have hope. I believe my brain is plastic enough to still learn. I’ll learn just how adaptable my 64 year old brain is this year when I get into algebra.

I am reminded of that wonderful novel, Flowers for Algernon, about a guy name Charlie, with an IQ of 68. Charlie volunteered for a medical experiment to boost his intelligence. The procedure worked, and eventually Charlie became a genius, but then the treatment wore off, and tragically Charlie returned to his low IQ existence. Getting old feels like being Charlie after the treatment starts wearing off.

Essay #997

Mathematica versus Sage

Quick version: If you want to learn math get Mathematica. If you have access to Mathematica use it. If you have the money, buy it. If you want to study mathematics, pray that your school provides it for free. It’s wonderful. If you don’t believe me watch these videos or look at the Wolfram Demonstrations Project. I believe if every K-12 kid or college student was taught math with Mathematica far more of them would becomes scientists and engineers. Unfortunately, Mathematica costs a lot of money. If you don’t have the dough, consider Sage, the open source alternative. But if there’s any way to get Mathematica, go that route. If you can’t, let me tell you about Sage in a roundabout way.

When I was a kid I wanted to be an astronomer. I even took astronomy and physics courses when I started college, but I hit a math wall – I finished Calculus I, but then stayed out several semesters. When I returned to Calculus II, my math knowledge was gone. This was partly due the distraction of girls and getting high, but I mostly blame myself for being lazy. I didn’t have whatever it took to focus and work hard. I’ve always wondered how my life would have been different if I had taken school more seriously when I was young, and applied myself.

Now forty years later I fantasize about testing my aging brain by studying math again. Could I go back and relearn math, catch up to what I had learned, and go further? It’s the old question: Can an old dog learn new tricks? My regrets about life involve two kinds. First, all the real jobs I wanted, astronomer, computer scientist, robot engineer, etc. involved mastering math. Second, my fantasy ambitions were about writing science fiction or popular science, and those involved intense verbal skills. I think I failed at both because I’m lazy or I can’t focus deeply enough. Now that I’m older, with fewer distractions in my life, I wonder if I could break through those barriers.

Kids today should have a better time of it because of technology. If young grade school kids could start out learning with Mathematica it could give them a tremendous edge. It might make the abstract and boring subject of mathematics real and alive.

One test of my old brain would be to study math again. I eventually finished college and went into computer programming, but with office applications and databases, not with computer science concepts. I’ve wondered if I could take my computer programming skills and apply them to learning math. Could programming a math problem teach me to understand how math works?

Searching the web, I looked for people who had already tried this, but what I thought of as an obvious match made in heaven doesn’t bring up many hits. Then I found “Mathematical Software and Me: A Very Personal Recollection” by William Stein. Sage is system for using dozens of mathematical programs that have evolved on Unix/Linux OS over the years and tying them together with a Web 2.0 front end and using the programming language Python as the underlying user input language. It’s a free, open source alternative to Mathematica and similar expensive commercial programs. From reading many blogs I had already decided that Python was probably the best programming language to use with learning math, so Sage intrigued me.

When I started out on this project I imagined myself finding a beginning math book, maybe just a 7th grade algebra book and seeing if I could write Python programs to do the problems. But there’s another kind of problem – math has its own language and character set of symbols. Programs like Sage and Mathematica have to create a way to enter formulas without using the traditional symbols of math. Imagine putting this formula into code:

If I just used plain Python I’d have to develop my own subroutines of conversion and I didn’t want to do that. Also, there is the problem of binary to decimal accuracy. Often computer programs will produce 3.99999999 when I need 4.000. Programs like Sage and Mathematic have already solved those problems with custom formula editors and built in subroutines that are time tested. They created programming conventions for entering mathematical formula and subprograms to show that code with standard mathematical symbols. Think of word processing for mathematicians.

What’s the difference between Mathematica and Sage? For some people it’s thousands of dollars. Sage has the goal of providing a free and open source alternative to the commercial Mathematica. Since I work at a university I have access to Mathematica, and thus I’m offered a choice. It’s an odd choice too! Mathematica is gorgeous, elegant, refined and advanced. Mathematica is like being at NASA with state of the art tools. Sage is like a poor garage inventor who has to buy their own.

If I would retire from the university I would no longer have access to Mathematica. Also, if I develop something cool and wanted to share it, with Sage I could, but if I used Mathematica, I could only share notebooks with other Mathematica users. Mathematica is a black box, users don’t know how the results are calculated. With Sage you can look at the source code.

Sage seems like an obvious choice, doesn’t it. Well, there’s one huge stumbling block, you need Unix/Linux to run it – there’s no native Windows application.

Now anybody can go to the free online version of Sage called The Sage Notebook, create an account and start using it for free. A lot of people do, and that’s the problem, sometimes processing is iffy because of demand. Next in ease of use, is to get a Live boot CD with Sage installed on it. Just put it in a PC, reboot and make sure the CD is the first drive to boot – this bypasses Windows on your hard drive and boots Linux instead, leaving Windows untouched. This is a great solution so long as you don’t really get into Sage heavily.

If you happen to already use Linux or Mac OS X, you can get binaries to install on your machine, but that still leaves out all those Windows users. The way to actually run Sage in Windows is to install a virtual machine on your Windows PC. Currently the Sage docs recommend VirtualBox, but that solution seems to be on the way out, and you need to use the free VMWare Player because at the Sage mirrors all they offer is the sage-vmware distributions.

Sage constructed a VMware distribution that you can load directly and run – no installing Ubuntu and Sage in steps. The VMware distro has been pre-customized with all the Sage utilities. This works very slick. You can run Sage from within the virtual machine, or get it running as a server app, minimize the VMware window and call Sage from your Windows browser (the Sage notebook is just a Web 2.0 app.).

I’ve used all four different methods, online, LiveCD, Linux box, and Windows with VMWare. All work. Depending on how heavy duty your math processing needs are, will determine which version you want. However, you have to get used to using a program that’s running other programs under Linux, and that can be tricky. If you are a math teacher and want to use Sage with your students you’ll want to set up a Linux box that has some horsepower and then run Sage as a server app to Windows and Mac machines in your lab.

If Mathematica was free like Sage, I’d just recommend everyone use it. It’s much easier to set up and far more consistent in its use. It’s a shame that Mathematica isn’t given to every K-12 and college kid in the world. Mathematica would be a fantastic teaching platform, but it’s just so damn expensive. But if little kids were taught to use Mathematica (or Sage) when they got their first math lessons a far greater percentage of the population would think mathematically.

What William Stein offers is a free alternative to Mathematica. It requires a bit more work and knowledge to set up and use. In fact, its Unix/Linux origins will turn off most users, so I’d recommend to math teachers to set up a Sage server and just get the kids used to Sage Notebook online.

Sage doesn’t teach math. Mathematica and Sage are like the ultimate graphing math calculator, but with the notebook feature, it can record and animate math and statistics. To see the potential of Sage see “Exploring Mathematics with Sage” by P. Lutus, especially the pages that start with “Trapezoidal Storage Tanks.” This is fairly advanced math, but it illustrates what math teachers could require of their students. Set up a problem, illustrate how to break it down mathematically, and then show the formula working with Sage.

You can visit the Sage Notebook site where users have saved and posted their notebooks online for all to see. Studying these notebooks show the diverse way mathematics is applied to many problems. This is the language of math, science and engineering. I’d like to think if I had access to Sage when I was in grade school my life would have been significantly different.

Like I said, it would be best if Mathematica was given to all kids. If that isn’t practical, I would recommend trying Sage.

JWH – 8/1/10