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Two math education tracks? Yes, No or maybe so?

Posted 9 years ago
8 Replies
4 Total Likes

My question is a call for comments regarding the failure of most students, in USA particularly, to learn much math. When my son started high school I went out and bought the Ti Inspire calculator thinking it would help him get into the computer world sideways. Never in six years has any class allowed him to use the calculator. Granted his interest level for school was low but none of his teachers were able to harness the calculator's potential. For the most part, they all said "I don't know how to use it". If teachers are unwilling to even try a new calculator, what are the chances they will be willing to tackle mathematica?

Now for my question. My family has a long history of math failure. Given that history I would have preferred to have the option for my son do a complete computer aided or directed CAS program, learning only how to input the correct formula format into the magic box and come up with an answer. Forget about understanding the concepts and proving said understand via manual calculations, get him to the point he could plug in the questions and come up with the correct answer. At least now he would know how to use the tools to solve problems. Why am I all wet with this idea? Nearly everyone on this site of course would be able to stay in the traditional track. I understand we learn from our failures, but at some point the system needs to provide positive alternatives to that important learning lesson. All comments appreciated.

Unsatisfied former math student

POSTED BY: paul miller
8 Replies
Posted 9 years ago

Wolfram takes 20 seconds to login or even get to the forum pages. Maybe I have a problem with my software operating system which is made by the Apple corporation.

I'm thinking too much formulaic thinking by the political leadership in the education sector ends with students hating school. Don't know the answers but they are not right or left, just who is best able to direct the learning sector?


POSTED BY: paul miller

Interesting discussion from insightful people. Conrad Wolfram's 4 steps sounds good but it does have the flavor of: We're putting you through boot camp and then you can go out to the industrial front lines and start identifying problems. High school students don't have to solve any such practical problems, they're not yet equipped for it, and most of them will never use mathematics later. Besides, most of them are bored to death with the material thrown at them.

So how can Mathematica be used to open some new avenues? First, CDF documents (especially if package code and style sheets can be included) are an absolutely tremendous tool for providing new material to students. With the free player, practically anyone can read them. I don't see any reason why CDF documents (and Mathematica documents for more serious students) won't eventually replaced the present textbook format. What kind of look and content should such documents have?

They shouldn't look like a Manipulate statement! Especially one with five or six sliders with everything moving around with very little explanation.

They should look more like a tutorial, or better yet an essay, or even a story about mathematics. And mathematicians, and topics in the history of mathematics. The more advanced documents might be similar to the style of John Stillwell in several of his educational books (Numbers and Geometry or Mathematics and Its History). The documents might show at least one or two easy proofs (The irrationality of the square root of 2, visual proofs of the Pythagorean theorem?) And maybe a case where mathematics disproves some intuitive idea.They could tell stories about mathematics and art (flutes and lyres, or perspective.) They might talk about Galileo and mathematics as the language of nature, or Wigner on the unreasonable effectiveness of mathematics. Or the belief that some people have or had that everything that exists is really just mathematics! They might concede that mathematics is difficult and that it took millennia to solve some problems and there are plenty of unsolved problems. Maybe they could be assured that they aren't expected to do a lot of practical mathematics or mental calculation. They might make it interesting and stop threatening them with standardized tests on the routine mechanics of it. Do you want to find out if someone can do some mathematics? Just talk to them, you'll find out fast enough.

The material in the documents should be leisurely, with development by multiple diagrams, calculations and dynamics so they might see how a simple line of mathematical development goes. Our technical prowess should go toward making the exposition simple, clear and elegant - perhaps not so easy.

Conrad Wolfram's 4 steps (not really original with him) highlights the problems with maths education. Understanding the problem, translating the problem into maths, doing the computation, and verifying the result in the real world is a good rubric for what most people needs maths for. Traditional education has been tied up with computation, since that was an obvious 'hard part', and one that has been revolutionized by technology. Using technology has to be done correctly, though, or people just go through a different series of "magic" steps to solve problems, with the related issues of platform (HP calculators vs TI, Mathematica vs MatLab, c vs Python, etc.). There is also the issue of what learners need to be able to do by hand. Traditionalists (e.g. "maths was painful for me, it should be painful for everyone) may see n reason to change, but I think that there are reasonable things that a student should be able to do with some facility -- standard multiplication tables, differentiating and integrating polynomials, etc. This question may best be solved by what maths is going to be used for. Engineers have different needs from economists.

The real problem is that maths has been taught almost entirely divorced from reality. People do not realize that a problem they have in the real world has any relationship to maths at all, or that they are 'doing maths' without realizing it. The insurmountable barrier to making maths relevant is not that the computation bit is (often) presented in a mindless, dull manner. The barrier is step one of Conrad's four steps.

If there is any rational for "two tracks", it is to distinguish between people who are to become professional mathematicians, and those (a vastly larger number) who need to use maths for their work -- physics, biology, finance, engineering, etc. As an example, a lot of people need to be able to solve and use differential equations in their work. For these people, being able to recognize that their problem is amenable to diff Eq, being able to set up the problem in maths, using Mathematica to solve the problem is sufficient. For a mathematician interested in differential equations (for their own sake), then being able to create the algorithms Mathematica uses may be necessary. There are other examples from pure maths that are of use (always a dangerous word) primarily to mathematicians.

Maths education is just the most obvious example of the breakdown of the model for education in general. We as a society do not need the factory model of education anymore. Preparing children for jobs on the assembly line or doing routine office work is obsolete. Technology can help with this by letting students select the mode of learning that suits them best, usually with a lot of guidance and coaching from teachers. This approach has been shown to work, and work effectively, but it is expensive and there are a lot of political and social opposition. In the US, there are a lot of people who do not want students to learn to be critical thinkers because it undermines parental and religious authority. (See recent activity in Texas about standards.) While the teaching of evolution gets a lot of press, the issue involves maths to a surprising degree. The one advantage that drill-and-kill methods have is that they are easy to test for, while learning to think critically is really hard to assess using multiple-guess tests.

The good news is that people can take the initiative on their own. They can reject the notion that there is a special "maths gene" and if you don't have it, you will be hopeless at high school or college maths. They can reject the notion that being good at maths means being able to do sums or long division quickly -- you will always lose to a computer if this is the standard of achievement. You can start with what interests you, and by pursuing those areas, you will learn maths in context. For some people, coding is a good entry point, but not everyone will be able to get hooked into maths in that way. It is entirely likely that for a large percentage of the population, a child's natural curiosity will have been killed to the extent that they are not really interested in anything to the extent that they are willing to keep working if the work becomes hard.

This is not a new problem. It was a problem when I was in school a half century ago. About 5% of students became interested in science and maths through a variety of experiences, and managed to pursue this interest in spite of the educational system.

Stephen Wolfram demoed a tool for creating interactive courseware. I hope to see that in the wild soon. I think there could be a cottage industry of people who a passionate about a particular subject preparing material so that students could learn on their own. Quality would vary, but cream rises.

Dear Paul,

As a math teacher from The Netherlands I might mention that scientific calculators are obligatory in secondary education, but this doesn't quite solve the problems that we have here, which sound rather similar as yours, but now with pupils punching buttons without quite knowing why. I am afraid that your solution of "typing in a CAS" (for you "Computer Aided System" rather than commonly "Computer Algebra System") will not work much. Check up on Wolfram Alpha, and use it to your benefit, but you will note that it still requires some understanding of how to ask questions. Conrad Wolfram is quite busy also with the Wolfram Demonstrations project but it seems that it will take a longer while before this affects education. Overall I advise that national parliaments look into their nation's math education problem.

A key problem is that mathematicians are trained for abstract thought while education is an empirical issue. The training of teachers cannot uncondition what has been conditioned before. Professor Wu laments that academic mathematicians neglect the horrors of "Textbook School Mathematics" but more attention would not necessarily make things better, see There is also truth in the statement by David Keith: "physics classes where physics professors considered math to be too important to be left to the math teachers".

My suggestion is an overall re-engineering of mathematics education. We need educational engineers who combine theory and practice. Some hospitals are linked to medical schools who train MDs, and we need schools linked to educational schools who train teachers. Re-engineering mathematics education means that also the symbolics in standard math can change. The general discussion is in my book "Elegance with Substance". My own effort at implementation has led to three books, "A logic of exceptions" on logic, "Conquest of the Plane" on" on 2D analytic geometry and calculus, and "Voting theory for democracy" on said subject. These implementation books have been written in Mathematica with available PDFs, see my website The books / courses have not been tested in class since the official curriculum is different.

I am hesitant about the MOOC by Keith Devlin. I haven't looked at it, but saw some video's from Devlin on other math subjects, and found those rather traditional and too woolly. He doesn't re-engineer math but tries to make the old un-educational ways more fun. My comments on those are here:

Hope that this helps out,


Posted 9 years ago

Hi Paul,

I think that two tracks are appropriate, but it is important not to separate them too soon. I remember my mother telling me about my 5th grade teacher who told her, "David will have to find something to do in life that does not involve mathematics." It turns out my final choice was physics, with a degree in math as well. I suspect I was tired of doing page after page of long division.

In my life, I experienced what was for the time a typical math education: various forms of "number theory" and arithmetic, until about the 9th grade, and then algebra, algebra 2, geometry, college algebra, analytic geometry, trig and a year of calculus all crammed together. And all taught by "math teachers."

Eventually, at college level, I wound up in physics classes where physics professors considered math to be too important to be left to the math teachers.

I think the most fundamental problem with early math education is that, with math taught as a stand-alone topic, for many it lacks motivation. That is not to say that students should not be expected to take on some learning as merely required, but where a connection to interesting activities can be made, it is well to do so. Who wants to spend a year sawing boards in half -- it's so much more interesting if you're building a soap box derby race car. And it matters to you if the cuts are square.

So, early on I would look for every opportunity to connect math with applications. Theory and mechanics are still important -- learning the algorithm for long division teaches you the meaning behind a base system representation for a number. But sending a 5th grader home with 10 pages of long division problems is excessive. Once the point is made, a calculator works very well. I would connect math with other academic work closely, all the way through high school, coordinating the "pure math" classes with other academics to the extent that new math learning was introduced as needed in non-math classes, where it has a purpose, and reinforced in a synergistic manner in the math classes.

This would remain one-track into early high school, and not separated until the student knew how to think quantitatively. Not just by making a budget, balancing a checkbook (if anyone still does that), calculating take-home pay, or making change -- but by solving the "dreaded story problems." Learning math at this level is more than learning math; it is learning how to think.

At this point, separating tracks makes sense to me. Being a confirmed nerd, I do think that a brief intro to calculus is beneficial to those who will never use it, merely because it introduces the concept there are things which depend on the rate of change of other things, as all of the laws of nature, or the accumulation of other things as the inverse. But I see no need to expect those who will not pursue quantitative work in science and technology to master calculus or differential equations. (Or perhaps even college algebra, trig, . . . ) But using algebra to solve story problems is a great learning experience.

There is however the the cultural need for discipline and learning in the early years. To a great extent, many of the activities we force on the young are those which they would not choose for themselves, but are none the less essential to a good life. We humans have a long childhood because we have a rich and complicated social structure. There are animal species wherein the young still learn by instinct-driven play, but we have evolved so much more culturally that physically that we must make our children learn things the genetic programming does not require of them. (And refrain from doing quite a bit which it does.) So I think this will remain an issue, and there is only so much we can do to sweeten the pills which we must force down their throats at an early age.

Kind regards,


POSTED BY: David Keith
Posted 9 years ago

Thanks David and George for your thoughtful replies. My son is 20 years old so my control is less than when he was a young lad in elementary school. Looking back on those years I believe I missed the opportunity to push more learning and less sports.

I'm really wondering if something like I suggest could help salvage math study for other students? Could students learn to do the problems using technology as opposed to learning to manually do calculations? It seems we need a major course correction to avoided the huge fallout we see now in math study. The "I'm not smart"enough factor would be taken out of the equation if all students had the same technological assistance. More like learning levels in a video game than working through 1000's of problems on their own. This is merely one suggestion to change the system and I'm curious what folks think about it. My buddy Roger Schank, a math major himself, has many ideas I seem unable to disagree with.


POSTED BY: paul miller

I can agree with David Park that anyone who learns maths in the US (unless you are very lucky) more or less has to do it on their own. There were some exceptional teachers at Wolfram Technology conferences who used Mathematica very effectively for high school and college, but it was clear that these are exceptions.

You might check out Jo Boaler or Keith Devlin. They have a presence on the web, and they have been working in this area for quite a while. Dr. Devlin has a MOOC: Introduction to Mathematical Thinking, which is worth taking. Any student in middle school or beyond could do the work, and any parent who wants to provide a suitable environment for learning could benefit as well. Dr. Boaler has a website ( with a lot of excellent material, and also teaches a MOOC. If you do any reading at all about maths education, you will find that it is a mine field.

You did not mention how old your children are. For young children, I think that a hands-on approach, anther than immediately going to the computer is likely to be effective. Playing with ZomeTool, Cuisenaire rods, ore even legos (in their plain varieties) can help with an intuitive understanding of what maths is about. Making the connection between maths and science is also essential at this age (K-6), but a lot of schools have neglected science in elementary school because sit is not being tested. Most of maths arose either from science or finance, and these practical activities -- making change, doing simple experiments -- probably do more to allow children to learn maths than any amount of drill and kill.

There is a lot on the web about computer based maths -- Wolfram research is involved in this. I think that learning to code is a great activity in general, and as a replacement for typical drill-and-kill activities in the classroom. Mathematica can be an excellent first language, but it can be very intimidating. I was already an adult when I learned to program in 1970, and people were pleased if you could do anything at all -- not like today. BASIC on a time-sharing system was how I (and Bill Gates and a lot of other people) learned how to program. Of course, I already had a maths degree by that time, so solving scientific problems in code was not that hard. Mathematica has some of the flavor of my early experience, in that a learner can do some really elegant things right away. The task for the teacher or parent is to see that some maths is learned along the way.

I just finished a book "How not to be Wrong" by Jordan Ellenberg. I think anyone interested in maths -- and a lot of people who think that they don't do maths -- should read this book.

If you do chose to go down this path, it is likely that you will have to contend with the system. Students will need to satisfy the requirements of the school system to get the grades they need to get into a decent school, and then do the extra work to actually learn maths.

good luck.

You are certainly to be commended, Paul, for your interest in this subject, your analysis, and your efforts on behalf of your son.

But if you expect your children to get an education in the American public school system, especially in regards to mathematics - forget it. The only ones who get educated are those who take it upon themselves.

I'm not an educator and not really qualified to answer the question: "What kind of mathematics education should a general student receive." As far as "how to input the correct formula format into the magic box and come up with an answer." there are plenty of Apps to do that and will be even more in the future and one doesn't have to understand any mathematics to use them.

But let's say instead that you and your son want to learn about mathematics because of its beauty, because of its long history in our culture, because of its bedrock position in the infrastructure of our civilization - and just for the fun of it.

Then one approach that might be useful to you is to look into the National Association of Math Circles, which has a hands-on, extra-curricular approach to teaching mathematics.

Another approach might be to explore mathematics with Mathematica, since this is a Mathematica forum. Will this really work? I think it might if the person is curious and has a sincere interest. It is much easier to talk about Mathematica as an educational tool than to actually do it. The thing is to get Mathematica and just try it. Use it to explore SIMPLE things. There are simple things that are nevertheless quite profound. Mathematics is not just calculating numbers, but seeing patterns, creating and manipulating structures, proving or at least convincing yourself of things, creating informative representations of things, seeing how one thing changes with another thing. Another trick is not to just calculate but to also write about what you're doing - which you can do with Mathematica notebooks. Don't produce just a number but a short essay or tutorial on what you're learning. Then you or your son would have something substantial to show for your efforts. You might even publish a book from it!

I should also mention that you can get plenty of help through the various Mathematica forums.

So, that's my two-cents.

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