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?

Paul

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.

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,

David

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 (http://www.youcubed.org) 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.