Excerpted from The Beginner's Guide to Mathematica V4

By Theodore W. Gray and Jerry Glynn, published by Cambridge University Press

Will it rot my students' brains if they use Mathematica?

Jerry: I have young students who reach for their calculators to get the answer to 5×6.

My response, when I see that, is to explain that such behavior is socially unacceptable, sort of like picking your nose. Many people will see this and think the student must be brain damaged. It's a social problem, not a mathematical one.

Theo: I agree that the problem lies with the other people more than with the students. The most profound engine of civilization is the inability of a larger and larger fraction of the population to do the basic things needed to survive. Many people fail to realize this.

Jerry: I don't understand that statement at all. It must be very significant.

Theo: In a society where everyone knows how to hunt, grow food, and make shelter, and knows these things well enough to survive, no one has time for much of anything else--even for perfecting one or the other of these basic skills. In early tribal societies, some people were undoubtedly better at one thing than another, to the point where they would probably have had a hard time outside the group. The best arrow makers probably weren't very good at weaving shoes, and would have had a lot of blisters without some help from the shoe weavers.

Few people would argue that people who are bad at weaving shoes are somehow inadequate, but it's surprising how strongly people feel this way about "modern" skills such as the ability to add well.

Technology's greatest contribution is to permit people to be incompetent at a larger and larger range of things. Only by embracing such incompetence is the human race able to progress.

Irate bystander: So, you're saying civilization progresses by having technology relieve everyone of the need to learn skills, so we can choose to be incompetent idiots barely able to feed ourselves. Is this really progress? Isn't the result of too much technology an aimless, pathetic populace just moving from one senseless pastime to another? What else could explain professional wrestling and the $180 sneaker!

Theo: I didn't say incompetence is a sufficient condition for progress. I do say it is a necessary one. If you want to move beyond endless drudgery, you have to have technology (or slaves, servants, or a spouse) to free you from the otherwise all-consuming task of survival. Technology is the least-objectionable way of generating free time, in my opinion. Of course, some people will use their free time more responsibly than others.

Jerry: People are very attached to the value of their skills. They believe that the skills of their generation should be preserved, with new skills added on.

Theo: Such an attitude represents a tremendous degree of disrespect of our forepersons. It was really, really hard to be a cave person. The skills needed to live comfortably in, say, northern Europe in 20,000 BCE were extremely complex. They required then and would require now the full range of human intelligence.

To think that a modern human should be able to do everything that previous generations have been able to do (hunt, speak Latin, do square roots by hand, etc.), and also have any time left over to learn anything new (microbiology, email, calculus), is basically insulting to all those previous generations, since it implies that they under-employed their intelligence. It is also quite false.

Jerry: I think it matters that students spend their time thinking and learning. People seem happiest if they are good at something. But I agree it doesn't matter whether they learn all the same things their parents learned. Not learning Latin is a problem only if you need to speak to Latin people on a regular basis, or if people will make fun of you on the playground. Not learning to add is a problem only if you have to add regularly, or if people will make fun of you for using a calculator to do

Theo: Well, you probably do think people should learn to add. Adding is not that hard, and it's a fairly practical skill in the day-to-day world.

Jerry: In the old days (before television), being able to add up a long column of numbers without making any mistakes was a valuable skill. People would pay you a living wage to do nothing but add numbers well. Not today.

Theo: Today, it's nice to be able to add small numbers, and larger numbers in a pinch, but the specific mental tricks and habits needed to get the right answer consistently when adding lots of numbers are just not helpful. Not being able to do this does not represent a failure of the intellect, any more than not knowing which fields in your neighborhood have the best rabbit hunting: both were, at one time, failings that would get you laughed at.

Jerry: But, you'd agree that being able to estimate the sum of a column of numbers is valuable. I would spend more time learning to do that well than working to reduce my error rate in doing exact sums.

Theo: And yet, in schools you find worksheets with 100 addition problems that are supposed to be done correctly, with points taken off for errors. What a waste of time.

Irate bystander: Oh, now I get it. You're one of those romantic educational know-nothings who think it's not necessary to learn anything in particular, as long as you learn "critical thinking skills" and have good self esteem. Yuck.

Theo: No, and let me make this very clear. No one can learn to think without having something to think about. If you try to teach someone how to think in the abstract, you are not going to get anywhere. If you try to make education "easy", by removing the content, you are cheating your students out of the most important thing you have to offer: the chance to do something hard. Only by mastering a difficult body of knowledge can a child develop into a confident, thinking adult. The point is, it doesn't necessarily have to be the same difficult body of knowledge that the child's parents learned.

And while we're on the topic of romantic educational know-nothings, let me just say that if you think you can improve your students' self esteem by letting them "succeed" at various insipid educational games, you are kidding yourself. Kids are much smarter than that. There is nothing more demoralizing to most children than being put through an educational program they know they can't fail at. Instead of teaching them self esteem, it teaches them that you expect so little of them that you have contrived special extra-stupid lessons for their benefit. Don't think for a minute they don't know what's going on.

If you start a lesson off by telling the students "This is going to be easy", you are simultaneously telling them "We had to make this easy because we don't think you're capable of doing anything hard". And when the lesson is over, the only sense of accomplishment they can feel is that they did something easy. So what?

Learning is hard work. If you are not working hard, you are not learning. Period. Kids love hard work, as long as they see where it's going and why. Instead of killing that energy by giving them something easy, we should foster it by giving them something really hard. We should tell them it's hard. We should give them the chance to do something meaningful.

Jerry: Readers should be aware that Theo is the father of one three year old and a couple-odd babies, while I am the father of four adults. It is well known that people at the beginning of the child rearing process have much stronger opinions than those who have completed at least two children. However, in this case I have to agree with Theo.



His viewpoint is nicely supported by Joseph Mitchell's story in his wonderful book Up in the Old Hotel about a bridge-building disaster which killed a number of young members of the Mohawk nation. People believed this disaster would drive the Mohawks away, but instead it had the opposite effect. The tragedy confirmed that working on high steel is serious, dangerous work, worthy of the efforts of young Mohawk men. Mohawks have gone on to erect the steel of a high proportion of all the high-rise buildings and bridges on the North American continent since 1910.

We seem to agree in general principles but how do we put them into action. If we want our children to learn some things that are hard, how do we decide which hard things are-- or are not--worth their effort?

Theo: It's not easy, and of course it changes every day. The one sure wrong answer is: The same thing we learned as children.

Therapy can probably help, if you feel you must insist that your child learn the multiplication tables up to 12 by 12 because you had to, and you would be embarrassed if her grandma finds out. If you want her to learn it because you really think it's something that will help her in her life, fine, there's nothing wrong with that. Examine your motives. Look around at our society. Be open-minded.

Jerry: I forget, what was the point of this chapter? Have we said anything yet?

Theo: The point of this chapter is to tell people that if they think Mathematica might be harmful to the process of education, they need to reconsider their fundamental belief structure. Just like the breech-loading rifle, or the pocket calculator, modern tools such as Mathematica change (maybe a little, maybe a lot) the kinds of things that ought to be learned. Some things that used to be important are not anymore, and some new things have become important.

Jerry: How about a short list? After Mathematica :

Less Important

Guessing factors for polynomials

Knowing many tricks for integration

Being careful when copying over expressions many times

Finding roots of complex equations

Knowing how to do matrix row operations

Knowing how to avoid dropping minus signs

Memorizing specific rules for derivatives of such functions as tangent and secant

Memorizing multiple angle formulas for trig functions

More Important

Translating statements about problems in natural language into statements in mathematical or procedural language

Learning how to experiment with math

Knowing which integrals should best be done numerically

Knowing how to work backwards or to use numerical methods to check symbolic results for plausibility

Knowing how to use techniques from programming

Understanding recursion and how to use it practically

Knowing which functions are discontinuous and where they are discontinuous

Knowing how to mix math and programming

Theo: So, won't people be in trouble if they don't know how to do these traditional, now less important, things? What if they have to function without computers some day?

Jerry: Well, I guess they'll just all die.

I have trouble taking this criticism seriously. How many microprocessors are there per person in the US? Too many to count. Very few of us are more than 100 meters from a computer most of the time.

Theo: I would have to agree. If you are worried that your child will suffer by not learning to solve a polynomial by hand, I would suggest worrying more about not learning how to skin a rabbit, or how to start a stalled car. Of all the failures of education likely to get your child into trouble, manual polynomial solving is not high on the list.

Jerry: Readers who think the topic of this chapter is unimportant are mistaken. Computer use in our schools today can only be described as pathetically limited. Attitudes we have discussed here are a serious factor. Maybe ten years ago one could argue that computers did not belong in the classroom. They were too expensive, good software was not available, and they were too difficult to operate. But today, a math classroom without a computer is a joke. Yet the number of math teachers who use computers regularly with their students is very small. In same cases, teachers have computers in the classroom, but by the time they are finished teaching all the old lessons, there is no time left over for using the computers. Try to tell them to cut out some of the old topics, and war breaks out. In that conflict, parents and teachers are often on the same side, outnumbering the voices for change. Misuse of computers is easy and a problem but no use is presently a bigger problem.

Will it rot my students' brains if they use other educational software?

Theo: Yes.

Jerry: But wait, didn't we just write a whole chapter about how great it was for students to forget everything their parents ever learned and use Mathematica instead?

Theo: Well, yes, but we were talking about one particular piece of software being used by relatively advanced students to replace manual computation, a very narrow set of skills.

It is interesting to note that many teachers and parents feel that using calculators to avoid learning multiplication tables, or using Mathematica to avoid learning integration, is an educational failure. Yet they don't seem to have a problem with using software that retards development of far more fundamental aspects of the student's humanity, such as creativity, enthusiasm, love of learning, and empathy.

Jerry: What? Have you gone completely off the deep end?

Theo: Yes, in this chapter we will be firmly off the deep end.

The sad fact is that 95% of all "educational software" and 95% of all educational use of computers is, for lack of a better word, crap. I'm not claiming that Mathematica is the only good educational software in the world. But you have to look far and wide to find the few others.

Jerry: Why do you say it's mostly crap? And if you're right, how did educational software get to be so bad?

Theo: It's bad because thousands of earnest and dedicated software engineers are working hard to make it that way. Bringing to bear all their skills in commercial software design, the best programmers in the industry have created an impressive body of absolutely worthless educational software. Worse than worthless, much of it is downright scary in its effects.

Jerry: How about some specific examples. So far this is sounding like a pretty categorical attack on the competition by someone with a vested interest in one particular product.

Theo: Let's start with something I'm sure most people would agree has little educational value, the video game. We'll discuss later how video games are related to educational software.

To understand the effects of video games, one needs to go back to debriefings conducted by the U.S. Army after WWII. Interviewing soldiers returning from battle, researchers discovered a very disturbing fact. A significant number of soldiers had been face to face with an enemy soldier, rifle in hand, enemy in their sights, gun not jammed, and had not fired . Something deep in their being, some sort of innate humanity, had prevented them from actually pulling the trigger.

Needless to say, this was very disturbing to the military. They began a research effort to figure out what to do about this problem. They discovered that in the heat of battle, under the incredible physical and psychological stress of being faced with another human being you were supposed to kill, the higher mental functions were largely absent. Under such conditions, the mind reverts to much simpler modes of operation, to deeply wired, almost instinctive behaviors. In other words, no amount of target practice at bullseye targets and classroom lectures about how you're supposed to kill the enemy had much effect when it counted.

Over the following decades and wars, the Army learned that the way to get soldiers to reliably pull the trigger was to use very basic, repetitive operant conditioning, along the lines of standard behaviorist theory. Now, behaviorism provides a very poor model for how humans act in everyday life, but it turns out to be a pretty good model for how humans act when they are under stress and have to act quickly, and are responding primarily to fear. Under stress, fearful people do what they have been conditioned to do.

The Army's solution was to replace dry target practice with realistic training grounds, complete with pop-up targets, loud noises, smoke, stress, the works. The goal was to condition the soldiers: If it moves, shoot it now, don't think about it. Repetition, repetition, repetition: Target pops up, you shoot. Target pops up, you shoot. Do that often enough, and, research shows, next time you see something pop up, you are more likely to shoot it, even if it's a real human in a real battle. This is not just a theory, it is documented by exit interviews from soldiers in later wars: The Army got what it wanted. (What armies do, and how that is similar to video games, is forcefully presented in the book On Killing by David Grossman, a former military officer (Little Brown, 1995)).

Now, what does this have to do with video games? The answer should be obvious by now to anyone who's ever seen one. The whole point is, if it moves, shoot it. Again and again and again.

Jerry: Well yes, but it's aliens and other fantasy figures they are shooting at, not people. Does it really carry over?

Theo: Yes it does. (And by the way, it's not all aliens; many video games have photo-realistic people complete with recognizable human faces, and blood splatters. You stick your handgun in their face at point blank range and pull the trigger.) Intellectually, no one would confuse a video game with real life, but we're not talking about an intellectual situation. We're talking about a scared kid with a Saturday night special in his hand seeing a member of a rival gang move his hand around in his coat. That kid is thinking at an operant conditioning level; what matters more than anything else is how many times in the past he has pulled the trigger. In reality or in a video game, it doesn't make that much difference. For a good discussion of the current consensus opinion on the effects of violence on TV and in movies and video games, see the book Mayhem by Sissela Bok (Perseus Books, 1998).

If you think watching violent TV is bad, video games are much, much worse. TV is a passive medium, requiring no participation from the viewer. A kid watching a murder on TV may not be benefiting much from it, and maybe he's learning a certain degree of callousness, but at least he is not being conditioned to pull the trigger. He is just watching. In the video game, he is the murderer, he pulls the trigger, he participates in the violence.

When soldiers are trained to kill, it is with a certain amount of context, with an effort to teach honor, duty, self restraint, and the difference between civilian and military life. When a 12-year-old kid comes out of his bedroom after spending three hours actively participating in the killing of people, what context has he had? what debriefing does he get?

I should mention that most of this chapter was written before the recent outbreak of kids shooting their classmates at school. To what degree video games contributed to those incidence is of course subject to endless debate, but the public does seem to have been alerted to the topic. There are even reports of, for example, Disney resorts removing violent video games that include humans as targets. A fine move, but obviously mainly symbolic. As long as parents pay to have killing arcades installed in their kid's bedrooms, the harm will be done.

Jerry: I am alarmed. (Reader: I've read Grossman's book, and Theo is right, we should be alarmed.)

But what do violent video games have to do with educational software?

Theo: Video games are at one end of a continuum. Educational software doesn't usually have the same kind of violence, and it tries to be "positive". But the fact is that it's written by the same people, and more importantly, it's often judged by the same criteria, as video games and other types of entertainment software.

When software engineers design educational software, there are a few things they take for granted, and reviewers of educational software appear to agree with them:

Software should be "easy to use". That means you shouldn't have to read a manual before starting to use it; you should be able to click on just about anything on the screen and have it work; you should be able to figure out how to use the program just by flipping through the menus and trying things that seem relevant. In short, you should be able to use the software by trial and error.

Users should never feel at a loss for what to do next. The software should lead them seamlessly from one stage to the next, with no moments of confusion or uncertainty.

Software should engage and keep its users' attention with colorful graphics, sounds, animations, whatever it takes. Software without these elements is judged to be boring and of inferior quality to the better grades of multi-media educational software.

Good performance should be rewarded with a treat, such as a clever animation or a game that can be played for a while.

Obvious, right? Any software that lacks these characteristics is routinely trashed in reviews. It is called "hard to use", "boring", or "unmotivating". We are told repeatedly that kids prefer flashy graphics software.

Well, kids prefer TV over homework, late movies instead of sleep, and chocolate coated sugar bombs for breakfast. Have we all forgotten what education is supposed to be about? Shouldn't education be about preparing children to lead successful, fulfilling lives? Here are a few self-evident facts about children and learning:

Children need to know certain facts, but acquiring them is not the main point of learning, especially not in the earlier grades. Mainly children need to have effective habits of mind and an ability to think analytically. They also need to be self-motivated, because in real life there isn't always someone there to provide external motivation (unless they join the army).

The hardest things to teach are the skill of solving problems with incomplete information, the skill of figuring out which problems need solving in the first place, and the skill of finding and bringing together the resources needed to solve a problem.

Children are primed to want to learn. They start out valuing learning and accomplishment above anything else in the world. If you see a child uninterested in learning, it is overwhelmingly likely that the child was made that way by something in the child's world: Children do not start out that way. (Of course there are always exceptions, but they are just that: exceptions.)

Let's go through our list of features of "good quality software" and see how each feature affects these learning goals. (Many of these points are made very effectively in the fine book Failure to Connect by Jane M. Healy, Simon & Schuster 1998.)

Software should be "easy to use".

Life is not "easy to use". Children quickly learn that the best and most effective way to handle typical educational software is to click as fast as you can on anything you can find on the screen until something works. It's really not worthwhile to read the bothersome text; understanding the material just slows you down. Since the software is "high quality", it will always do something, help you out, not be judgmental; pretty soon you'll hit the right icon and get to the next level.

Extensive use of this sort of software supports and develops shallow thinking habits. It discourages careful analytic thought. It is dysfunctional. Real life is not a multiple choice test. In real life there are points off for making wrong choices.

The user should never feel at a loss for what to do next.

In life, it's rarely clear what to do next. Uncertainty is one of the defining characteristics of life; one of the most important things a good education can give a child is the ability to sift through seemly limitless options and make wise choices. It's usually not even clear when a choice is there to be made, let alone what the alternatives are.

Extensive use of sanitized, candyland software develops a weak, passive mode of thinking. Children reared with software that never lets them get lost often seem lost and helpless when faced with the confusing mess of the real world. Jane M. Healy gives many illustrations of this effect in her book Failure to Connect.

Now, it's possible that children have always been helpless and confused in the real world, but this sort of software isn't helping things. Sure, some kids will come out fine, and yes, some kids would never have come out fine. But what about the ones in between who would have developed into thoughtful, capable adults but for their exposure to this sort of software? How many is too many?

Software should engage and keep the user's attention.

Good performance should be rewarded with a treat.

These two are without a doubt the most damaging attributes of most educational software. It is absolutely tragic that, by rewarding learning with stupid tricks and games, such software devalues the learning itself. Children who would gladly learn for the sake of learning are, quickly and devestatingly, turned into children who demand rewards for learning.

Now, I know a lot of people will say, "Oh, kids today need to be entertained or they will lose interest!" Well, sorry, but that's no excuse for pandering. Kids in your neighborhood may be zoned-out video addicts, and maybe some of them are beyond help, but do not delude yourself into thinking you are helping the situation by giving in and giving them what they want. You are the adult here, and it is your responsibility to work with them to recapture their innate love of learning. It may not be easy, and you will probably fail with some or most of them (and you certainly will fail with many of those whose parents aren't helping). But you have to try: It is part of being an adult, just like not giving your kids candy every time they ask for it, and not giving them edutainment when they need education.

Jerry: Well, I can see you feel strongly about this. It must be nice to be so sure of things.

I think there's quite a bit of software out there that is valid and valuable. Perhaps we can help people see the difference. I assume that your attack on "easy to use" software doesn't mean you think good software should be intentionally hard to use?

Theo: I think that learning is intrinsically hard. Educational software that students just get through without working hard has not succeeded in making learning easy, it has simply replaced learning with entertainment.

Imagine taking your kids to a fun movie, having a good time, eating ice cream. No harm in that. But should you tell them that they are having an important educational experience? Should you tell them that they have accomplished something by sitting through the whole movie? What a horrible devaluation of accomplishment that would be! What's the point of actually working hard to reach success, if your parents and teachers will praise you for joking around and eating ice cream?

That's exactly what most educational software is: A stupid, insipid movie that keeps your kids entertained for an hour or two while the teacher plans a bake sale to replace the music budget that was spent on computers. Kids don't learn anything, but worse they are being taught that it's not necessary to work hard to achieve success.

Software should not be unnecessarily hard to use, but neither should it shy away from or disguise the inherent richness of the subject matter. It should be open-ended, deep, and capable of doing senseless things if asked.

Jerry: Why should software be able to do senseless things? What's the point in that?

Theo: Let's take a specific case. Give me an example of a program you like.

Jerry: Geometry programs like Geo1, Geometer's Sketchpad, Cabri, and Cyclone let students make constructions with lines and circles, then discover relationships among the elements. Students of mine have acquired a deeper understanding of geometry, trigonometry, and algebra from programs like these than they did from traditional textbook approaches alone. Contrary to your description of mindless multiple choice quiz programs, these programs have let my students build anything they like, then manipulate it freely.

Theo: Exactly! If students decide to build a completely useless geometrical construction, the program won't stop them. It lets them discover for themselves that their construction is uninteresting. This is very important: By allowing freedom to go off in the wrong direction, the software is giving students the opportunity to learn.

Jerry: Maybe I'm starting to see what you mean. As the saying goes, you learn from your mistakes. Software that prevents you from making mistakes limits what you can learn.

Theo: Exactly. I'm not saying that educational software should encourage mistakes, or have outright flaws, such as buttons that are confusingly labeled and crash the program. But I am saying that programs should be rich enough to allow both right and wrong paths to be followed, and followed in a more than superficial way.

Software can be divided into page-turning vs. simulation-based. Page-turning software, which is very common, allows the student to following only certain pre-determined paths through screens that have been laid out in advance by the authors. Breaking out onto more creative paths is impossible, because there is nothing to break out into: It's a closed box.

Simulation software in contrast has a set of rules and algorithms (the laws of physics; sociological models; geometrical relationships; etc.). It is able to apply these rules to a fairly open-ended set of inputs. Simulation software is inherently more difficult to write; not surprisingly there is little of it out there. In fact, good simulations are so difficult to create that one could almost name all the examples that have ever been done. Page-turning software is incredibly easy to make, and there are countless thousands of titles available, virtually all of very low quality.

Interestingly, while violent video games may be evil, they are largely simulation-based. No page-turning software could hold anyone's attention for the hundreds of hours that video games capture our children. In a good video game, you have a huge world to roam about in, complete with back alleys, multiple levels, and great detail in all the parts. The only problem is that you have to keep killing people to see the next back alley. Oh well, if people get in your way, killing them is OK, right?

Jerry: OK, OK, let's not get started on that again. Now, by that definition, Mathematica is simulation software, as is most of the educational software I like. But this kind of software is much harder on the teachers. Students are constantly going off on tangents and asking questions about what they've done.

At some schools, teachers (and maybe administrators) would rather use software that the students can use for an hour without needing a teacher. From their point of view, what's the point of spending your school's entire music, PE, and after-school programs budget on a shiny new computer lab, if you just have to hire new teachers to baby-sit the students? For that kind of money, shouldn't it be "automatic"?

Finding and training teachers able to work with students using untamed, open-ended software is very hard. Potential teachers may be computer-phobic, or may not want to be put in the position of answering questions instead of lecturing. It's much harder to answer students' open-ended questions than it is to spill out a prepared lecture. They may not always know the answers to all the questions; from time to time, a student might actually end up knowing more than the teacher does about something.

Theo: I think we can all agree that it would be sad to see classroom teaching replaced by computer lab brain damage, just because school administrations don't know what they are doing. It would be far better not to bother with the computers in the first place, if they're not going to be used in ways that help people learn.

Skeptical Bystander: It's all well and good to support the discovery method of learning, but we all know that if you just throw students in a lab with open-ended software and wait for them to discover the great laws of the universe, most of them won't. Sometimes it's necessary to tell them to read one page, then the next page, then the next. And what about learning multiplication tables or the reverse declination of irregular French pronouns? Isn't there a place for drill-and-practice, page turning software?

Theo: Yes, there is a place for well-designed drill and practice software. If you have a group of highly motivated, mature students who wish to learn a defined body of knowledge (say, The Knowledge--all the streets and cul-de-sacs of London--so they can pass the London taxi driver exam), they can do this very effectively using page-turning, quiz-giving, flash-card-simulating software.

Drill and practice software is used quite effectively in industry, in the military, and by students of medicine, law, and other fields that require a lot of memorization. It's not flashy, and it doesn't have any video games at the end as a reward (the learning itself is the reward, as it should be).

But this software has almost nothing in common with the "educational software" we've been ranting about. It differs in these ways:

Good drill and practice software has no externalized reward: Learning is the reward. As discussed above, providing a video game or animation at the end is exactly like rewarding the eating of vegetables with ice cream. It devalues the vegetables and increases the attractiveness of the ice cream. It doesn't work at the dinner table, and it doesn't work in teaching either.

Good drill and practice software spends the vast majority of its time on teaching, practice, and evaluation. Bad educational software spends far too much time on the (harmful) reward. Jane Healy's book cites examples of software in which the students spend more time playing the reward game than they do in the learning portion of the program.

Life, and classroom time, is simply too valuable to waste this way.

Converted by Mathematica May 30, 2002