## Do Computers Dumb Down Math Education?

September 27, 2010 — Jon McLoone, International Business & Strategic Development

Since I just heard that the video for Conrad Wolfram’s recent TED talk “Stop teaching calculating, start teaching math” will be coming out soon, I thought I would address the single biggest fear that I hear when I talk about using computers in math education.

The objection that using computers will “dumb down” education comes with the related ideas “students have to learn to do it by hand or how will they know they have got the right answer”, “they won’t understand what is happening unless they do it themselves”, and so on.

Well, let’s examine this by looking at a typical math question that I know I had to solve at some point in my education.

“The Second World War Gustav gun had a muzzle velocity of 820m/s (SI units used throughout). Assuming no air resistance, what was its range when fired at 45°?”

If I am using *Mathematica*, then I can pretty much type the differential equations and equations for the system and get the answer.

“Aha!” say the critics. “Proof that the computer has dumbed down the subject: that didn’t require any thought at all. You never solved the ODE; you never solved the equation; the computer did it all for you.”

Well, I do agree that this example is pretty dumb, though not because the computer did the computational work. The example is dumb because the answer is completely wrong.

The gun in question had a range of 48km, not the 68km that we just calculated. It wasn’t *Mathematica*‘s fault, and doing it by hand wouldn’t help—the equations are wrong. They are not wrong as far as the current education system is concerned; they would get top marks in my old school. They are wrong in the sense that they do not reflect reality.

The dumbing down was in the question, which explicitly excluded the atmosphere, and implicitly excluded any other influencing forces or complicating factors. It’s hard to imagine any scenario where that might be true.

The reason why that typical question is so dumbed down is that, without computers, it quickly becomes too hard to solve by hand. In an educational system geared to neat little hand-solvable problems, the only solution is to look at a toy version of the problem.

One key conceptual problem shown here is that, in education, assumptions are usually instructions. Instead, we should be teaching students that assumptions (even implicit ones) are choices. Each should be considered for the impact that it might have on the validity of the solution.

To illustrate the point, let’s look at a less dumbed down version of the same problem. The biggest missing factor is drag from the air.

# Drag Effects

Drag is given by this formula:

where *pb* is the base air density, *A* is effective area, and *Cd* is the drag coefficient (a measure of how streamlined the shape is). But drag as a force is applied against the direction of movement, so we need to resolve this into *x* and *y* components, based on *x* and *y* velocities.

And now let’s put that into our system. Already this has become a tricky problem that is too hard to compute in closed form (another real-world issue that students should understand). So I will solve numerically (essentially impossible by hand):

We are missing some information, so off I go to Wikipedia to gather some data on air and the Gustav gun.

Drag coefficient depends on the shape of the shell: streamlined can be as low as 0.04, a cube is 1.05, a sphere is 0.47. I am going to cheat here and claim it is 0.28 without any analysis or citation. You should deduct some marks for this!

# Altitude Effects

There are multiple altitude effects that we can add in. We might ignore those if we are hitting a tennis ball, but this shell goes up 15km and air density falls significantly at that altitude. Here is a version of our `Drag` function that takes density as a parameter:

Now I need a model for how air density varies. This alone is a tough problem. We don’t want students perpetually trapped in solving preliminary steps from first principles, so they need to be able to use existing models.

Here is the model for air density and some key values, valid to about 11000m (I am going to assume that is close enough).

It is unreasonable to expect students to learn all such formulas—and I include simple ones such as sin(2*θ*)=2sin(*θ*)cos(*θ*) and the dozen or so long-forgotten variants that I was made to memorize. So that means an “open book” approach of looking them up as needed. But before spoon-feeding the student with the right formulas, remember that recognizing which model is appropriate for the information that they have and need, and figuring out how to fill in the parts of the model that are missing, is another important skill students should learn for the real world.

Gravity also decreases with altitude, and while I am looking things up, here is a better value for *g* at the surface:

It turns out that this effect is only about 0.3% at the top of the arc. But it is easy to add in, so I will use it.

Now I will add those into the system of equations, and I’ll also add a wind speed parameter while I’m at it (assuming constant wind speed at all altitudes, for the duration of flight).

And now we have our 48km range. It’s pretty clear that the model matters:

# How Else Can We Be Less Dumb?

I am still a long way from the “truth”. I have ignored the curvature of the Earth, the fast high-altitude jet streams and gustiness of low-level wind, the Coriolis effect, atmospheric pressure, rainfall, and humidity, and I am working only in 2D. I’ve also ignored the fact that gravity varies between the equator and the poles due to the non-spherical Earth and rotation effects. There will even be a small eddy current effect in the metal as it passes through the Earth’s magnetic field and gravitational effects from the Moon, Sun, and other planets. A “perfect” model would be a major undertaking, perhaps worthy of a PhD thesis, and probably quite useless in practice.

I had to make choices to dumb this down to fit the size of a blog post, and had an intuition about which effects would give the best improvement in answer against the effort to implement. We should also consider the measurability of the parameters. These are the kinds of choices made in real-world modeling that we fail to teach people.

More importantly, I have spent no time here on validation tests. Are my models plausible? Was my intuition about what effects were insignificant correct? Did I type in the equations correctly? It is no longer a trivial task to check that I haven’t done something wrong, for student or teacher. I highly expect someone to point out a mistake that I have made somewhere in this post, and equally that most readers will not have noticed it. Validation is a skill we must teach. Does my model fit the simple model for low altitudes? Does it behave correctly if fired straight up or straight across, or at zero velocity or other known values?

It is not good enough to just check our work by hand. This is now a real problem, and like most real problems, it is tricky and messy and needs thought.

Once we are happy with our model, we could be asking much more interesting questions than what the range is, like, “How much difference does a wind speed of ±5m/s make?”

At a quick glance, we can see that it is a little over 300m.

“If the wind varies by 2mph, and muzzle velocity and launch angles have a standard deviation of 1%, what is the probability that we can hit a target with a 1km diameter?” (According to Wikipedia, the higher velocity Paris Gun had a range of 130km, but missed the whole of Paris with half its shots in the First World War).

The challenge for teachers who embrace computers as tools is to teach students to sensibly trade off complexity against correctness, to recognize what they need to know, to find out, to calculate, and how to validate. In short, to teach them to think rather than to perform computation procedures.

Perhaps more importantly, through interesting and challenging tasks, we must give students the confidence to work with problems that don’t have neat, little, pretend answers, but that are messy, have alternative approaches, and where perhaps no one knows the answer yet—just like the problems that they will face in the real world.

## 34 Comments

Very well argued post. Thanks!

My take is that mathematics needs to be taught as a second language, since the part that students have trouble with is translating the words of the problem into equations. I agree that Mathematica helps by making it easy to solve the equations.

Excellent points.

How solid is the use of the square — the second power of v — for the drag?

I think it boils down how much detail you need in problem solving. There are always people, as you mentioned as critics, insist on hand calculation.

But I want to point out that all people rely on certain ‘system encapsulation’ in their work. For example, how many of the people, even the critics, really derive chain rule in calculus, derive Z score table in statistics, derive trigonometry identity, derive basic rules in number theory, every time they work in those areas?

I think the focus is really how much deep internalization you have gained on the mathematical knowledge and how to correctly apply the knowledge in your work.

Mathematica serves as the excellent tool to learn and internalize mathematics, in my view.

Man, you are right, the traditional stuff we teach at school stops at the dumb model… what a shame. Thanks for showing the educators how to proceed from the shoulders of giants!

Bravo! The phenomena described here of dumbing down math to accommodate the _lack_ of computational power is equally if not all the more present in the social sciences.

This is a fantastic post, Jon.

It shows how important it is to understand the white-box / black-box principles of the didactics of mathematics.

Philosophically speaking, our equations do not make the world.

The only absolute is the real trajectory of the “bullet” and everything we can do is inversion. But ironically, we ned models to solve the inverse problem, by kind of parameter identification or iterations.

And we might begin to understand that the results can become worse if the models are too simple or too complex.

No way to do that without computer mathematics. Consequently Mathematica that is the only representation of the language of Mathematica and its most important operational semantics.

.. the language of mathematics … (sorry!)

What I think you’re proposing here is somehow more of a top-down approach. And while I totally agree with it, I must point out that students must also learn what differential equations are – and this is somewhere around the tip of the iceberg – and how they are to be solved before actually using software to solve them; and that’s where the simplified problem helps.

Both approaches are critical: the student should know exactly what DSolve does before using it. Then a bit later he can attack the more complex problems using the help of a computer.

hmm. this is now more of a ‘physics’ problem – not a Mathematics problem. at least for those of use who prefer to think of mathematics as ‘pure mathematics’ of Algebra and Calculus.

I’m a theoretical physicist and in my field Mathematica is absolutely vital for solving very complicated problems. However, I do not agree with your arguments. For a student to learn mathematics, they need to start of with simple problems and when they can truly understand them, they can move on to more complicated problems. At the point where their understanding of these problems takes them to the point that they need to use computer software to solve them, all well and good, however I believe that one needs to get familiar with the simpler systems first.

Dirac said that he truly understood an equation when he could look at it and predict the properties of the solutions without actually solving it. I believe that this is much harder to do if you study equations using a computer program. Perhaps for some people it’s possible but when you rely on the computer to solve the equations for you you lose contact with what’s really going on.

So, my belief is that by all means use mathematica to solve those questions which we cannot solve by hand, but to teach mathematics with it (when solving simple problems) i think that you lose more than you gain.

@ spkyked & Jonathan – I didn’t try to address the question of how one first understands the concepts before trying to use them, in this piece. I do believe that that is often, but not always, done by hand computation. But even if hand computation on trivial examples is the best way to understand the concept, my feeling is that you want to move on to computer computation as soon as you “get it”, so that you can spend time “getting” other concepts. Right now, the assumption is often that you should stick with hand computation as long as humanly possible.

Regression was a good example for me. I did my first least squares fit by calculating the table of error-squares by hand. Fine, I “got” it. I have never done it by hand since. And yet with integration, I also got the concept quite quickly from simple examples, but was made to spend many hours practicing ever harder examples. The difference? Only that computers are were assumed for statistics, but not for calculus.

For some ideas though, I think, computers lead from the start by giving hands-on real-time feedback. http://demonstrations.wolfram.com has some fine submissions for doing just that .

Dirac was right for his time, but he died while the human ability to compute was still the limiting factor. Today we need to prepare students for problems that Dirac would not have fully understood!

Excellent, excellent post. I’ll be showing this to my daughter who is learning high school physics now.

Thanks for the reply. I agree that once you “get” it, this allows you to move onto harder problems, and in general more interesting ones, but those problems are often not maths problems but physics problems (or analogous).

I wasn’t sure about your comment about the initial answer being dumb. As physicists we have to spend a lot of time working out what information we can cut out. You say “the answer is completely wrong” but in fact the answer to the question was correct, it stated in the question that you could neglect air resistance. Had it not stated that in the question, then the answer would indeed have been wrong.

Somebody following your example would learn some physics, but not a great deal of mathematics. For someone at the level where the question was appropriate, what they were being taught was about solving simple equations and my problem with the piece was that, for someone with the sort of knowledge for which the question was targeted would gain more from going through the simple example by hand than the complex example in detail (though they would learn more physics your way).

I absolutely agree that computers can boost the rate of learning mathematics. I spent a lot of my undergrad years plotting things on computers for precisely the reasons you state.

I think maybe it was simply the example that you gave which made me feel that doing this by hand would have been better, and that the initial answer to the question was perfectly correct.

Anyway, I absolutely agree with the main point in this article and will continue to promote the use of computers in education…in the appropriate places ;-)

Just from another topic. I am over 60, and I needed to learn Descriptive Geometry – about the projection of 3D objects into 2D – by hand. And then Geometric Modeling by computers became common to designers, … Simulation of many geometrical scenes fueled imagination of complex geometric objects with all their intersection curves … (even Klein’s bottle)

Descriptive geometry disappeared?

Computer mathematics is as different from mathematics, as axiomatic mathematics is different from algorithmic mathematics?

Wow, I am impressed you got the exact same result that Wikipedia quotes as the maximum range for that gun. Oh wait, was there some reverse engineering involved for getting that drag coefficent of .28? :-)

The education industry needs to change. It needs to move away from finding problems to computations that can be done by hand and move onto teaching students how to make discoveries that can be made easily by using Mathematica. But first, educators need to be educated. How about sponsoring a “lesson plan day” for the educators?

Seems to me the issue isn’t about computers dumbing down maths education, it is more about maths pedagogy making mathematical thinking opaque for bright minds. The formalist turn of the 20th century has been mistranslated into education as mastering symbolic manipulations. Mathematical thinking is not typological (ie based on category distinction as in natural languages) but is What Lemke calls topological (based on extensions of adverbial and adjectival comparatives into realms of quantification). My experience is that the fundamental concepts of mathematics are based on a visual and kinesthetic rhetoric. Mathematical thinking isn’t about solving individual problems (in respect of a pedagogy) but about the interplay of the discrete, the continuous, and the random as the fundamental problematic of the mathematical project. The deep tools that inform thinking mathematically are the semantics of “1″ (the thing, the common measure, the point”) and the verb and noun forms of addition and multiplication (operator and function). Without a sense of the deep questions it is hard for you minds to appreciate the impact of mathematical thinking (take the relationship between number and space that happens with multiplication as a model of Euclidean space, and the development of alternative metrics. I think it is an issue of rethinking math pedagogy so that computers provide the experimental basis for developing rhetorical thinking and connect it with modern syntax – but for learners the primary experience has to be the semantics of how the world is a mathematical problem. Secondary maths education is essentially about the story of how the counting “one” becomes the “one” of scales (this conceptual move is essential to understanding both negative numbers and geometric sequences, and the development of the calculus). It is unfortunate that most of the demonstrations and classroom applications of mathematica have nothing to contribute to the issue of understanding the basic semantics of mathematical thought. Mathematica is a brilliant laboratory and it could be used to reproduce the thinking behind the great experiments of the past(eg from aliquot parts to rational numbers). When students get to the end of a secondary education in mathematics and can’t articulate the distinction between a number and the numeration system in which it is expressed or the pragmatics of its application (eg discrete vs continuous context) . Jon, I agree with your reference to Conrad’s contention, the issue is how de we get the thinking about pedagogy that is needed. I wonder if anyone would be interested in doing the phenomenological analysis of the relationship between a visual rhetoric that informs arithmetic and how arithmetic statements become the visual rhetoric behind algebra and these two get formalized in a syntax. If kids can learn to read and write mathematics then they have a chance of thinking about the world mathematically.

I strongly support your view. We need a mixed world of concept, virtual reality, game, and real world play to teach these things in an enchantment. Mathematica has a role to play in the second/third of these.

Hi Jon, Another great post, and a message that I think gets lost in a lot of traditional physics teaching. You’ll be happy to know that we bash our students at Manchester on this topic.

Sirs, to solve this or any problem in math or the sciences, I would explicitly define the problem to be solved.

Make a written list of ALL the factors that contribute (with percentages) to the problem’s solution. After deciding the needed percentage of the answer, use only those items that would lead to that answer. If this first solution satisfies the problem statement, you’ve completed the assignment. If not, calculate items as needed to provide an acceptable answer.

The point is to teach and learn how to DEFINE a problem before you rush an attempt to solve it.

Sincerely,

Sheldon Rabin

I don’t agree. Mathematics is dumbed down in schools so the concept is understood. It is important to learn the concept without which the student – when he starts using computers – will not know what to input and what the output means.

@selvarajan: I think that’s the exact point of this article! The concept is more important than the computation. Sure you must know what is multiplication and what is power of x and the conecept of functions as values dependent on variables, but you don’t necessarily need to know how to reduce and expand polynomials, nor go through hours of derivating and integrating to get to real world values. It is FAR more important to teach kids how to “speak” the langauge of math, than how to evaluate numerically highly repeatable algorithms. Given a real world problem, can you convert it into the series of parameters and relationships that will then best land themselves to variables and functions to answer real world problems? Children must be taught to write any correlation as a mathematical structure, and to read any mathematical structure as a meaningful context. the computation inbetween is a waste of time. My wife decided to go to college after many years, and she is now preparring for COMPASS test. Mind you, this is after graduating a decent GED school 15-some years ago. She’s having to re-learn basic algebra. a line equation of y=ax+b confuses her, and having to plot it scares her even further! She wants a formula given and then a set of multiple choice answers to pick from because she was taught to compute, but never connected it with the ability to convert real world problems to the language of math nor to interpret results as real world meaning. “Solution of a system” does not convert in her mind to “a point where lines cross”. She’s far from the only example. Professional statisticians at a pharma company I am familiar with closely, rely heavily on means and medians as characteristics for phenomena without seeing the real world diversity in the events and subjects involved. This is not only incomplete and limiting thinking, but often plain dangerous. We raise COMPUTErs, while we should let machines do the computation, and groom thinkers and interpreters.

@Sheldon Rabin: Amen! But I would also argue that the point is to visualize and understand the problem and how all aspects impact it, so that you can make sense of the complex periphery (which often complicates problems into the realm of unsolvable, or trivializes problems which ought to have more complex solutions in real world, as illustrated above). Sure, you can solve the problem in the way you outlined, but can you then explain what your solution means? Why should I care? That question is the real world, not the number in the final line. Don’t tell me that the median life expectancy of a given cancer diagnosis is 8 months, just because you know the formula for median, or even if you’re clever enough to explain moments. Show me the skewness, the shape of the distribution, the length of the long tail, the reasons for people to fall into one or the other end of the spectrum… Why should I care about your math in the real world? THAT is what we fail to teach our children about math, and THAT is what I find so beautiful about it and wake up to do every day.

@Adrian: I think you just proved the point of the article! :-) It made you ask a complex question, and I bet it even got you thinking on how this system, once set up to consider a large number of relatively fixed real world entities, could be used to estimate drag and to optimize gun and ammo designs… Makes computation irrelevant, and math very real and fun and usable, and your friend, not foe.

GREAT blog. True math understanding happens when we aren’t dragged down by the insignificant details. Without computing power, many of modern day mathematics/statistics would not have come to be.

No. Way. Computers are useful to solve given problem. Because it is an mindless machine.. It can calculate thousands or millions time… Whereas the human may get tired. But that does not mean that “Use computer / Calculator every possible time.

USE YOUR BRIAN… This is the place where all your ideas come from. So, if we have not practiced by our own brains, it is really make the person to “dump”. Because he never “Experienced”, he only have witnessed.

Specially the students should try & practice the math & calculations by their own hands using their own brain.

I see two things mixed, still I agree with the while post. One thing is to learn mathematics which I think is quite simple in general and another is to use mathematics to model and solve problems, either simplified or hypothetical versions of reality or closer to reality propositions. I see teach-and-learn of mathematics more like a method of thinking and reflecting in an abstract way and that is the reason for being a simple task, under my view. On the other hand applying mathematics from physics to economics passing through psychology and others, is where maybe using a dumb down version of problems, just for the sake of drawing out a number to later give a score on an exam or homework, is not the best approach to provide significant learning to students is more like to avoid the opportunity to think and reflect on reality and complexity of it, is like to trying to distort reality into something very abstract and over-simpified version that can be solved in a sheet of paper with just a pencil in less than 60 minutes.

Congratulations for the post.

I would disagree because it seems to me that the problem that have been stated by some above, and with which I agree, is not solved by using mathematica. The problem is that students tend to head straight for rote memorization instead of understanding the concepts. Why does it work, and can you express not just the problem from words to mathematical symbols and variables, but do the inverse, and write out an explaination of a formula with words, graphs, and pictures. Just because mathematica can help us solve real-world problems filled with excessive complexity does not change the fact that people need to use formula’s and whether someone memorizes the formula or uses an open-book approach, they still will not necessarily understand it. They will just regurgitate the information they have been fed, or read out of the book. Mathematica does not help this problem at all, instead it just makes it harder to get a grasp of understanding as the problems are more complex. I feel that classical techniques should be used, but maybe have a focus on computer-aided-mathematics as a module near the end that would take the understood concepts and expand on them.

I would like to know whether to store the formulas of chemistry and physics is “important”… because i have an admission test and i’ve never been able to memorize the formulas of this school subjects!

Excellent post!

We always have to be judicious. Different people learn at different rates and (unfortunately) at different times from those scheduled for the classroom. Here is where the computer comes in.

Learners are not all adults: babies start with a relatively empty ‘hard drive’ in mind. The learning must be short enough to teach the unit. After we ‘get’ one concept, we can then go to the next, from pure projectile mechanics to projectile + fluid mechanics, etc.

All the while we motivate by alerting the student that the ultimate goal is to describe reality. Let them choose at what point they want to give up in the particular research, because “Vita brevis ist, ars longa.”

They may then devote their energies to solving real world problems they are interested in, having successfully acquired the mathematical building blocks.

I think your example is interesting, but not (generally) a secondary school mathematics concept. Let’s look a more cogent example for high school teachers.

Suppose your objective is to have students learn how the coefficients of a polynomial affect the shape of the polynomial when it is graphed, so that they see that, for example, a negative coefficient in the highest power of the graph makes a huge impact on the final shape. Your students have already mastered graphing, and have demonstrated they understand what graphing is, so you aren’t trying to give them more practice graphing.

If the students use a program which can quickly and accurately graph many polynomials for them, they can use the program to experiment with different values of the variables and see many more examples of what is going on, thus helping to reinforce the fundamental notion you are trying to teach, that the coefficients of a polynomial affect it’s shape in different ways depending on which power of the polynomial to which they are associated. The computer in this case does not dumb down the concept what-so-ever, but it allows many more examples to be explored than would be possible with the traditional pencil and paper method.

You can then, as a teacher, spend time walking around and clarifying misconceptions students have as they explain their solutions to you and give them feedback. Since students are spending more time working on learning the intended concept, they get more directed feedback in that area from you.

@ David Wees

The relevance to the secondary school curriculum is something I would like to change, but widening the school curriculum. Freed from the time-drain of practising hand computation that, in later life, will be done on the computer, we could do more interesting math with the time.

I contribute to a project trying to do just this:

http://computerbasedmath.org

I’m an avid programmer and enthusiastic supporter of Mathematica and CDF, and I think computer-based math will, in the long run, revolutionize and greatly enhance math education, particularly for those in technical fields. My only concern centers upon a core aspect of mathematics for which we’ve yet to see great success from the CBM initiative: proofs. Consider Euclid’s classic and elegant proof that there are infinitely many prime integers: He posits a finite number, n, of such primes, then constructs a new, larger prime by multiplying those primes and adding 1. This proof-by-contradiction method is clearly a style of thinking student—all students—should have at their disposal. It is unclear how CBM would help develop such cognitive skills. Surely writing simple code to generate new such primes misses the core notion of PROOF. Perhaps some types of proof structures will have to be taught separately, particularly when calculation isn’t essential, and in such cases CBM simply doesn’t help (but it might help indirectly by freeing class time for such instruction).

Perhaps with effort from mathematicians, the use of existential qualifiers and new interfaces may enhance the education of proof methods. I hope so!