Our Jonathan Bartlett (johnnyb) is the author of *Calculus from the Ground Up,* that is getting great reviews at Amazon. But get this, mostly he hates calculus texts:

I am always amazed when I read calculus textbooks. They are the most dry and boring presentations of mathematics I have ever seen, especially if you realize that calculus offers some of the most amazing insights. Unfortunately, most mathematics texts teach only the mathematics, never the insights. I felt so frustrated by this gap that I wrote my own textbook, in which I try to teach both.

Jonathan Bartlett, “Doing the impossible: A step-by-step guide” atMind Matters News

He has a serious purpose in discussing the boredom though:

One of the topics that you should learn in calculus is how to solve impossible problems. In the first semester, I always tell students, “Here are some examples of impossible problems, and next semester we will learn to solve them.” As an example of an impossible problem, think of the calculator functions, sine and cosine. It is impossible to write these functions in terms of standard algebraic functions (i.e., polynomial functions). It is literally impossible. It is provably impossible. However, every calculus student learns a method for writing sine and cosine just in this way.

Jonathan Bartlett, “Doing the impossible: A step-by-step guide” atMind Matters News

He thinks that learning calculus can help us look past other types of insoluble problems as well.

At times, talking about mathematics, he sounds like Bill Dembski, who founded this blog in 2005.

*Also by Jonathan Bartlett on calculus:*

Walter Bradley Center Fellow discovers longstanding flaw in an aspect of elementary calculus. The flaw doesn’t lead directly to wrong answers but it does create confusion.

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I SURVIVED a course entitled “Calculus for Business Majors”. It was a required course for students expecting to get an MBA. I think half the class dropped out (or at least stopped showing up for class) by mid-terms. I think I passed with a C. Shortly after that it became clear that 20 years of experience in solving real problems trumped ANYTHING I would learn by getting my MBA.

I have any number of friends who are degreed engineers. ALL of them freely admit that they NEVER use calculus in anything they do at their (high paying) jobs. And this includes marine engineers designing (or approving the design of) ship hulls and rocket scientists designing (or approving the designs) of missiles.

All SANE human beings simply use the TABLES. Any working adult who feels the need to manually CALCULATE the value of a sine or Pi or or any of the other junk for which COMPLETE BOOKS OF LISTINGS are sitting there on a shelf are clearly NOT the kind of people you want to PAY to do engineering studies. And so, by reversing the logic, ANYONE can get the right answer to “complicated calculations” simply by knowing the general formula appropriate for the occasion and typing some numbers into Excel is worth as much to most projects as an official “engineer”. I’ve met any number of REALLY dumb engineers who simply couldn’t understand which parts of the mess needed to get fixed. The basic problem is they want to use funding for YOUR project to pay them to do something “artistic” or “inventive”. The same is true for Software Engineers: they want $100,000 so they can do something they think is cute and “creative”, and will cost $500,000 to TAKE OUT of the product before it will work in the field…

I wouldn’t go quite as far as vmahuna. I use rate of change fairly often in software for education and graphics and neurology. (Because nerves are differentiators!)

But I never use integrals or partial diff eq. In curricular terms, I use the first week of college calculus.

The example Bartlett gives in his MindMatters article is completely useless. Nobody EVER needs to figure sin or cos by converging series. If you don’t have a table or a slide rule or a calculator or a computer, you can always carefully draw a right triangle and measure the ratios!

Folks, it’s complicated. Calculus requires a conceptual leap to understand rates and accumulations of change. In turn (especially through exponentials) that pops up in ever so many cases and has all sorts of pathological pitfalls that have to be worked around from the roots of Math up. That’s why the limits approach prevailed, infinitesimals as conceived were too unreliable. And Physicists get away with murder because we can with decently behaved functions. Non-standard analysis does fix infinitesimals through the hyperreals but again there are pitfalls. So, Calculus is hard, it is fundamental, it is powerful; a recipe for trouble. I should add too many Math teachers are not familiar with the Physics, Engineering or Finance where calculus issues find rich application behind the tables and nomograms, even lurking in the marginal revolution in Economics. For that matter that is implicit in Leontief input-output tabulation and modelling of an economy using huge Matrices. Statistics (which is involved in econometrics etc) uses it. Let me mention transistor modelling and parameterisation which underlie the integrated circuit revolution, transistors being extremely nonlinear systems. And of course the world of messy functions and difficulties with integration lead to huge books that are tables of integrals and integration related transforms such as Fourier and Laplace etc. Built up across centuries. Partial differential equations are the stuff of a lot of thermodynamics and things that deal with fields. Then we run into operators and so much more. I should mention difference equations and their many similarly important uses including now digital signal processing. And we can go on. I trust that JB’s book is a genuine breakthrough as we need it. KF

Famous case: https://en.wikipedia.org/wiki/Gradshteyn_and_Ryzhik at Amazon https://www.amazon.com/Table-Integrals-Products-Daniel-Zwillinger/dp/0123849330/ref=sr_1_1?keywords=integral+tables&qid=1566264424&s=gateway&sr=8-1 also see digital library of functions https://dlmf.nist.gov/ with Wolfram’s math world http://mathworld.wolfram.com/ Don’t overlook Mathematica: https://www.wolfram.com/mathematica/

Polistra –

Thanks for taking the time to read the article! Unfortunately, it seems that you missed the point entirely, or even got it directly backwards.

The whole point of the article is that the technical knowledge that we learn in calculus *isn’t* very useful on its own. What is useful (and rarely taught) are the *reasoning* skills that go along with them. In this case, learning how to examine underlying (and unstated) assumptions which can bridge the gap between the impossible and the possible.

Similarly, I tell students that, even having a technical job, nothing in my world has ever relied on successfully doing prime factorization. However, I continually use the *skills* learned in prime factorization almost every day. Prime factorization teaches you how to look at something, and break it up into its constituent components so it can reorganized for something else. *That* is a skill that I use literally every day. We do it with numbers in mathematics because it is concrete enough that we can tell someone for sure when they do it wrong.

Math is valid only when it describes reality. Unfortunately, you can do math without ever considering what reality it describes, and many/most mathematicians never learn the connection. That is why they are, as a rule, lousy teachers. When I took calculus I asked “What is a curl?” The teacher sure could calculate one, but he could not explain what it meant. He had to ask a physics professor to explain it to him.

SAZ, Curl is of course an operator which was given its definition due to utility in physical contexts such as electromagnetism (as well as grad and div of course). You point out one of the key gaps between the physical and mathematical sciences. Where physics and math did not go separate ways until sometime in the 1800’s. Was Gauss a mathematician or physicist? He was both. KF

vmahuna –

You’re giving your age away. 🙂 Nowadays sane humans use computers, not tables.

Right, but knowing what the right formula is is important, as is knowing if the results make sense. One reason I’d want engineers to understand maths is so they understand what they’re doing, and don’t simply treat it as a black box. Just like I’d want a car mechanic to understand how an engine worked before they tried to fix it.

So Bob doesn’t understand that the tables are part of the computer program? Really, Bob?

Uh, no, ET. Your calculator does not have tables in it. If you press sine 42°, it does not look up the value in a table. It computes the value using the formulas JohnnyB mentions in his articles, using enough terms to get the level of accuracy for that particular calculator. A small hand-held computer might only have to use four or five terms, and a super computer 10 terms, but they all use a small number of the infinite number of terms in the relevant series. They do not use tables.

When teaching this, I would have students do a few examples “longhand”, finding the value of about four terms directly (x, -x^3/3!, etc.), adding them up, and then comparing the result to the calculator.

By the way, the calculator also uses trig relationships to simplify the process by translating the results to one involving a small angle close to zero, as the closer to zero the value of x is, the quicker the result converges.

For instance, for sine 100°, the calculator would first convert to sine 100° = sine 80°, and then to cos 10°.

Hey look – Jeffrey Shallit is trying to win the award for “pedant of the month”!

https://freethoughtblogs.com/recursivity/2019/08/20/why-cant-creationists-do-mathematics/

http://recursed.blogspot.com/

Note to folks at home – when trying to teach students how to think more broadly, the most important thing is to maintain pedantic distinctions at all times!

Why is hazel conflating a calculator with a computer? And that the tables would be the formulas used?

I didn’t enjoy Shallit’s snarky article (including the fact that this has absolutely nothing to do with creationism???), but I had some similar reactions to how Johnny described the situation.

Johnny writes,

Polynomials, by definition (not by assumption), have a finite number of terms, so it is true that you can’t write sine and cosine as polynomials. You can write sine and cosine as power series, which have an infinite number of terms. Note that each term in the power series for sine and cosine is a monomial, so technically a power series is made of “polynomial terms”, but the power series itself is not a polynomial.

So my impression of Johnny’s article that it made way too much of the “possible/impossible” distinction. The fact that sine and cosine can be written as power series is cool, and a major thing for students to learn. However I think the way Johnny describes the situation to his students is a bit misleading about some key terms.

And I want to make it clear that I am in general agreement with Johnny about the need to make calculus more meaningful and less mechanical to students, and I applaud him for writing a book with a different approach. I also wrote my own calculus curriculum to help make an introduction to calculus more accessible to middle-of-the-road students. I didn’t write a book, but I did pass my material on to other calc teachers who have used it with success.

ET, a calculator is a small computer. And big computers (I mentioned such in my post) use the same approach,

And a table is not a formula. Here is a table: Table of sines.

There is nothing like this in a computer: it doesn’t “look up” values that have already been calculated and stored, it calculates them as needed. Surely you understand the difference.

Oh my. Bob said computer, not calculator. And there is a difference. Tables are constructed from the formula(s).

Yes, a table’s values (such as those in the back of a book or online) are constructed from the formulas. But there are not any tables in a computer, or a calculator. As I wrote above,

Does anyone still practice interpolations and differences? (Or, does THAT tell

myage? As in look up and interpolate. I remember going into exams with the Cambridge 4-figure tables [we were allowed to write in a list of formulas in the back!] and my dad’s financial tables. Along the way I also used an old mechanical calculator. By 6th form, I also had my all time favourite trusty HP 21. Wish the batteries were better though. Dad? When he first got calculators he cross checked their results in his head, he knew how to add up three columns of digits in his head as fast as he could run up them with a finger. He was well in his 80’s when I saw he didn’t do that anymore. When he tried to teach me the algorithm it was an overload while studying for exams. I think he could have qualified for the older sense of computer, a job description. Me, I make sure I have emulator HP 48s to hand (including on cell phones), a 50 and a 12. These for sure now run on ARM processors.)In any case, oddities notwithstanding JB has some solid points.

Technically an endlessly continued power series is not a polynomial, but then place value decimal notation is a power series in disguise.

I also think degrees go to radians first to do calculations.

I look on at Mathematica with awe, especially now that the price has come down so far.

How is stuff like that going to shift teaching of Math?

I learned trig with tables in the back of the book in high school and college, and used interpolation. I also used log tables (and anti-logs) to find numerical values that involved multiplying and dividing by trig values, such as 50/tan 20°, as well as for such things as the cube root of a number. However, calculators came out right when I started teaching in the mid 70’s. They revolutionized teaching practical trig problems. For a while I would briefly have the students do a few problems with tables, but I quit doing that after not many years. I also learned how to use a slide rule when I was a kid, but never used one in school.

And yes, the calculator has to first converts degrees to radians to use the formulas, which of course adds another issue for levels of accuracy since an approximation for pi must be used.

Now Mathematica is changing things even more. For a simple example, a young teacher who replaced me lets the kids use an online tool to use the quadratic formula, which I disagree with. But time changes.

P.S. I also think that graphing calculators have been a next step in his technological revolution, and are wonderful. All sorts of things once taught involving graphs, such as synthetic division to find the roots of polynomials, are now outdated.

P.P.S. I always emphasized factoring simple quadratics to solve quadratic equations, but I had good European foreign exchange students routinely use the quadratic formula for the simplest of problems, and who didn’t factor well.

But, to disagree strongly with vmahuna in 1, it is very important to teach the concepts in a meaningful way even if technology has removed the need for a great deal of manipulative and numerical technique. To take a simple example, even if one always uses a calculator or online tool to solve quadratic equations, one needs to have a good understanding of parabolas: whether they open up or down, axis of symmetry, number of real roots, etc. Doing some longhand work when first teaching, and introducing technological shortcuts as appropriate, is good pedagogy.

By the way, if anyone wants a *super* cheap version of Mathematica, buy a Raspberry Pi. These days, you can get a Pi with 4GB of RAM, and Mathematica comes on it by default for free with Raspbian. So, for $80, you can get Mathematica *and* an new computer.

Interestingly, I used to work for the Wolfram Research guys a long time ago. Not specifically in math, mind you – I was their web guy for a couple of years way back in the day. But I did get to know a lot of the great minds over there, and sometimes I still reach out to them for help. For instance, in my paper on changing calculus notation, I asked one of my old Mathematica friends to read it, and he gave me some really good feedback. Interestingly, a lot of my work in Intelligent Design is largely inspired by Wolfram’s “A New Kind of Science,” which was more-or-less required reading when I worked there.

JB, the Pi sounds better and better. A math tool too. KF