Understanding Math Rather Than Merely Learning It

There’s a line from the original Star Trek where Khan says, “Improve a mechanical device and you may double productivity, but improve man and you gain a thousandfold.” Joan Horvath and Rich Cameron have the same idea about improving education, particularly autodidacticism or self-learning. They share what they’ve learned about acquiring an intuitive understanding of difficult math at the Hackaday Superconference and you can watch the newly published video below.

The start of this was the pair’s collaboration on a book about 3D printing science projects. Joan has a traditional education from MIT and Rich is a self-taught guy. This gave them a unique perspective from both sides of the street. They started looking at calculus — a subject that scares a lot of people but is really integral (no pun intended) to a lot of serious science and engineering.

You probably know that Newton and Leibniz struck on the fundamentals of calculus about the same time. The original papers, however, were decidedly different. Newton’s approach was more physical and less mathematical. Leibniz used formal logic and algebra. Although both share credit, the Leibniz notation won out and is what we use today.

Calculus Casualties

Unsurprisingly, a quarter of calculus students at Tier 1 universities get a D or lower. Quite a few of those who fail will leave science and engineering to either drop out or move to a less technical discipline. We always heard the joke that calculus was the biggest pre-business course on campus.

Joan Horvath and Rich Cameron entered the Hackaday prize with a plan to use 3D printed models to help teach concepts for calculus. The idea is to produce 3D printed objects that show the intuitive understanding of basics like the fundamental theorem of calculus. It’s like graphing out an algebra equation for better understanding, only this moves to three-dimensions and provides a tangible foothold for your brain to understand abstract concepts of traditionally difficult subjects.

Even if you don’t want to 3D print the models, they are in OpenSCAD, so you can experiment with them virtually in that environment. This approach has been so successful, that there’s an upcoming book that will expand on the topics at greater length.

Math Intuition

I’ve never been a big fan of learning math for math’s sake and a fair number of math classes I’ve had are organized that way. No one ever comes to your office and asks what’s X if 4X+13=2. They will come in and ask something like, how do I change this resistor to make the output of this power supply go from 5V to 5.2V? Algebra is part of the toolkit you can use to answer questions like that. Calculus is like the algebra of things that change and can answer a wider range of questions.

Integrals to measure volume under curves

I’ve often thought that in modern times, teaching the intuition of math might be more important than the mechanics. You can always ask the computer to do the real work and get the right answer. The real skill is in formulating the correct question and frequently that requires a combination of algebra, calculus, and differential equations. Can you get by without it? Sometimes. Maybe even much of the time. But there’s always going to be certain cases where you really need advanced math tools. [Joan] and [Rich] can help.

Expand Your Horizons

Many things you just “know” by rote are actually applications of calculus. If you are taking the RMS value of a sine wave by dividing the peak value by the square root of two, you are using a result from calculus. If you have something other than a sine wave, you are going to need to do the math. If you know the time constant of an RC circuit, that’s a result of calculus, too. If you need to know the 63% point, you have your answer. But how about the time it takes the capacitor to charge to 10% or 80%? Calculus.

Hackaday has made attempts to demystify calculus in the past. I also love how modern computer graphics can make intuition easier to develop. Honestly, one of the best traditional calculus books I know of is Thompson’s Calculus Made Easy from 1914 (well, the second edition, anyway). His approach isn’t that different from [Joan] and [Rich] as he states:

Being myself a remarkably stupid fellow, I have had to unteach myself [overly difficult ways of doing calculus] and now beg to present to my fellow fools the parts that are not hard.

Although it is over a century old, calculus doesn’t really change much and you might enjoy reading it — along with the Hacker Calculus — if you plan to chase mathematical enlightenment.

44 thoughts on “Understanding Math Rather Than Merely Learning It

    1. Memory and association skills of the stored memories in an individuals brain, or maybe backed up somewhere else in the bodies nervous system… or maybe tissues or elsewhere for moments if the long term memory association resource time event moment/dynamic isn’t co-operating/functioning, to recall a specific identified logical phenomena, process, operation, act, event, etc. based on input information (question regarding comprehension… let’s say event) to process accurately, or at least precisely enough to be easily directed to the bulls eye (I think the precision is more physicist vs mathematician maybe), for the output questioned to be understood?

      Can probably trim that down and word more simply.

      Humorous, is I can’t remember the Calculus and Diff. Eq. very well since I wasn’t as solid on really… I think algebra as I want to be… however… I do recall the comment:

      “Being myself a remarkably stupid fellow, I have had to unteach myself [overly difficult ways of doing calculus] and now beg to present to my fellow fools the parts that are not hard.”

      Immediately reminding be of Calc. 3 and Diff. Eq. being way more easier to solve problems than was done in Calc 1 & 2.

      Reminds me of geometry really… I don’t use the formulas enough and I trained myself to always know the reference resource and go to that BOOK to make sure my memory wasn’t wrong. I think that is why I didn’t learn the formulas as well so wasn’t as confident going into engineering as I wanted to be versus no lack of confidence going into science. I’m sure there are other causations that I think have to do with the remote sensing and transmission stations on me (on the radar) making me not able to focus intentionally on critical thinking quantitatively and learning all the mathematical and physics theories and formulas. I almost get the feeling just because how I look/looked.

  1. Calculus is in many ways a very visual-spatial subject, and I suspect that many people who have difficulty with it have issues with visualizing shapes in the “mind’s eye”. To that extent, models could be very, very useful.

    1. On the contrary. Math is conceptually easy. Trying to use it is hard, because you have to remember all these rules of syntax and symbol manipulation that is too abstract to remember.

      Knowing and understanding that an integral is the area under a curve is useless, when you can’t actually solve a simple equation using integrals, because even the symbols and notation are just so much nonsense. This is the same problem as with the “New math” that was all the rage a couple decades ago, where instead of arithmetic and algebra, you first taught children set theory before they knew what a plus sign means.

      1. Also, while set theory is useful and again conceptually simple to learn, in order to use it for teaching and have students solve problems, they again had to introduce another universe of weird symbols, syntax and abstraction, that took all the effort to memorize and master – so in essence you had to learn how to -do- math twice and on both occasions the majority of the effort went on learning the “language” rather than the “message”.

  2. Well, Newton used tortuous geometric proofs for his calculus in the Principia because he felt in order to reach his readers, he needed to base it on solid traditional understanding. His personal methods were very sophisticated. He solved the famous Brachistachrone problem in a single night with his calculus of variations. Liebnitz had requested the time limit on the problem be extended to 18 months.

    “They started looking at calculus — a subject that scares a lot of people but is really integral (no pun intended) to a lot of serious science and engineering.” I would add that it is integral to virtually everything. The public today is expected to have opinions on lots of phenomena that involve growth, whether in temperature, or depletion of a resource. If they can’t see when a problem is being described in the form of a first degree differential equation and that all those equations have the same solution – exponential growth or decay and asymptotic limits, they are easily duped by simplistic talking points. Handling ideas about growth and decay rates may be the most important area of numeracy today.

    But I am also a fan of math for the sake of math. I had a fantastic physics professor who answered every question in his office by grabbing some scratch paper and deriving an answer from first principles. You are not going to be able to do that with intuitive or practical math. If you know some math, read Felix Klein’s books on geometry and algebra from an advanced standpoint. They were written for teachers to get them in line with expectation at the college level.

  3. “You can always ask the computer to do the real work and get the right answer.”
    This turns out not to be the case, and is the core reason for many problems.. Even if you have asked the correct question – which you are unlikely to do if you don’t understand what is going on…

    The (one of many) problems is that if you really don’t understand what is going on, you will have no idea if the answer is sensible.. You could have inputed the wrong data,, asked the wrong question, or just completely stuffed up, and you just wouldn’t know…

    1. IME as somebody who does a lot of math on a computer, but didn’t take the classes, if you know you need to ask the computer you should already understand enough about the problem to tell if the answer is sensible.

      And if you’re not sure, you only have to be paying attention enough to notice that, and then you can do the math using a few different possible inputs to see if it varies as expected. You have to know something about what is going on, or you wouldn’t even know you wanted to calculate something.

    2. You have to take that in context. I said right before that that perhaps we should change how we teach math to focus more on the understanding. It really isn’t necessary that I know 80 patterns for integrating different kinds of formulas anymore. It is more important that I understand what question I need to ask. Obviously that could go too far, people do need to know basic math. And I would hope that I understand the area under a curve, for example, should be about about x. But it may not be as valuable as it once was to drill integral tables into students. Ditto for numerical methods for computing integrals and solving DE. Just like the average trig student probably couldn’t calculate transcendentals from scratch they would just look it up in a table or now use a calculator, the average calculus practitioner really doesn’t need to know how to do a Simpson’s approximation or any of the other exotic numericals. Simpson might actually be a good one to teach because it sort of exposes what’s really happening. So maybe Runge–Kutta is a better example.

      1. I agree that bulk drilling specific integral forms into students isn’t necessarily a good idea, but disagree about the numerical methods and also that I still think it is good if people can (maybe with a bit of memory assistance) still be able derive things from scratch many years later. And as soon as you said they don’t really need to know how to do a Simpson’s approximation I though ‘yes they do!’ – which you yourself went on to say :-).

        Generally, I think many numerical techniques teach quite a lot of the fundamentals of what you are doing, and understanding what you are doing I think is still key. And they are useful in their own right…

      2. The reason for understanding 80 patterns isn’t to solve those particular problems, but to recognize when a new problem has one of them as a basis. Much like chess players understand hundreds of openings and hundreds more mid-game situations. It lets memory take care of these cases rather than starting from scratch on previously explored territory. Moreover, integration is generally reversible, so it also allows seeing when a differentiation will yield a formula that can be better handled.

        For an example of this there’s a case where a certain equation is given and the task is to find the roots. There are actually 6 roots, but Mathematica can only find 4 because it doesn’t recognize two others. Humans are still superior in pattern recognition outside of limited questions of “does this can have a label.”

  4. I’ll second the recommendation of Thompson’s Calculus Made Easy. I got through an engineering degree by essentially rote learning when it came to calculus. Years later I discovered Thompson’s book and it was like a light bulb went off. I curse whoever it was at my school who decided calculus be taught as though it’s a complete abstraction without a relationship to the physical world.

    1. This is really amazing, I was completely entranced as I read through it, seems like it buttons up a lot of issues I have found too, and so succinctly! Why don’t you publish it?

      1. Well, a basic description of the New Calculus has been published free online here:

        https://drive.google.com/file/d/1CIul68phzuOe6JZwsCuBuXUR8X-AkgEO/view

        I had tried many years ago to publish my main work “What you had to know in mathematics but your ignorant educators could no tell you”. The above link is just a small part of this unpublished work.

        Given that I continue to be persecuted by the Church of Academia, it is unlikely that my book will ever be published. On at least 6 occasions they have shut down my online sites and caused the IP hosting company to terminate my account. I am called a crank, but in all truth they are the cranks and I am possibly the greatest mathematician who ever lived on planet earth.

        I debunk many mainstream ideas in my videos. Why don’t you subscribe and pass on the news?

        https://www.youtube.com/channel/UClBbBVLs3M-d3dNgU4Vop_A/videos

    2. From the “Final Remarks” (“New Calculus”) on (the last) page 134–

      “…it’s unlikely that I will write any additional book because…I will not share the most priceless knowledge I have discovered with scoundrels in mainstream academia who have libeled me and called me a crank…They are jealous, incompetent, arrogant and stupid.”

      “I am the great John Gabriel, discoverer of the first and only rigorous formulation of calculus in human history…”

      Mr. Wolfram, meet Mr. Gabriel.

  5. Everything I know about calculus started with “The Mechanical Universe”, the 80’s PBS documentary series, and then its very clear textbook.

    Physics and calculus go hand-in-hand. High school physics is usually taught with plain algebra, but Newtonian physics REALLY only makes sense with calculus, and calculus is really too abstract without, say, the example of motion to show you the way: from x-location, to speed (v, or dx/dt), to acceleration (a, or dv/dt, or d^2v / dt^2). Derivatives in one direction, integrals in the other.

    I know these are fighting words, but I think the blue first textbook of “The Mechanical Universe: Introduction to Mechanics and Heat” is more valuable then the famous Feynman lectures. Yes, Feynman is the great explainer, but this is the nitty gritty: everything from basic motion to steam engines with plenty of problems in one book. With enough detail to restart civilization if need be.

    But if you like videos, there is the TV series. Oh, and of course I should mention the 3Blue1Brown guy has a whole series of highly detailed calculus videos.

  6. The problems with people and maths start way before calculus, the root problem starts on day one at school. Infant and primary school maths teachers do not understand maths and just repeat what they have learned. This passes the flaws on to another generation.

    The single biggest problem is the form that children first see when a maths problem is written down:
    6 + 2 =
    That is mathematically incorrect. If from day one it was written as:
    6 + 2 = X, what number does X represent?

    Then children are more able to understand what is being written because X is tangible and nothing isn’t. Learning that really basic substitution from day one allows the step to algebra with no drama and that step is when most people throw their hand in on the subject.

    1. – Perfect – I love this – and along with it, it seems they misconstrue what ‘=’ is from early on to make it worse. It’s almost used more as a ‘?’ of sorts more than an ‘=’. Using equal and a variable from the beginning from day 1 sounds like a great idea… Heck, you could do it with a black box with an ‘X’ on it and marbles underneath, learning the ‘=’ means the same thing is on both sides ‘physically’, but you need to figure out what is in the box to make both sides the same. Start out very basic, but I’d bet my 5 year old could pretty quickly catch on to even going up to solving something like a 2+5 = 3 + X with a marble/box concept… take 3 marbles from each side of the ‘=’, and one side tells you how many are hiding in the box… anyway, rant over, but I love the direction of your comment.

    2. Actually, most children do not have problems with 6+2 = X, rather they have problems understanding what is 2, 6 and +.

      You can’t understand number (and hence mathematics or calculus) until you know what is a number:

      https://www.linkedin.com/pulse/how-we-got-numbers-john-gabriel-1

      A number describes the measure of a magnitude/size.

      The operations of arithmetic are all systematically derived from the most primitive which is difference/minus/subtraction.

      https://www.youtube.com/watch?v=B-CFhdGeJnE

      The chapter called How we got numbers in my free ebook explains in great detail:

      https://drive.google.com/file/d/1CIul68phzuOe6JZwsCuBuXUR8X-AkgEO/view

  7. I disagree. Many problems require solutions that are simply beyond any intuitive and threedimensional understanding. They can only be computed through abstract equations. Newton was a champion at abstract thought and that’s how he came up with calculus.

    Crunching the numbers and reaching an unintuitive solution that works is the pinnacle of engineering. And this happens in real life too. Orbital mechanics are full of unintuitive that just work if you crunch the math. The most classic example is the orbital brake to overtake which is a critical maneuver.

  8. I find it suprising how many people start to understand calculus when you use the Acceleration, velocity, time example. The concepts of calculus are not exceptionally hard from the basics, understanding that you are either looking for the slope of the curve (derivative) or the area under the curve (integral) is an easy enough concept to teach once you have the right examples to use. Even once you get to the more complicated things it is important to continue to use real world tangible examples that people can relate to. Understanding is just a matter of how to relate a certain subject to another that the person already understands.

    1. Actually, you can’t understand calculus until you understand the basics such as number and arithmetic mean. The standard definite integral is a product of two arithmetic means. My free ebook explains:

      https://drive.google.com/file/d/1CIul68phzuOe6JZwsCuBuXUR8X-AkgEO/view

      Number is the most important.
      Arithmetic mean is the second most important.

      I have never met any academic who understood number and calculus, and I have corresponded with the so-called best in academia.

  9. I seem to remember reading a really good explanation of Euler’s original derivation of the Euler-Lagrange equation somewhere. It was very visually intuitive, and didn’t involve needing to integrate by parts (as in the typical textbook derivation). It just made sense. I’ve forgotten it now, and can’t remember where I saw it (Damned memory!).
    One of my maths teachers used to bring sound into Calculus, by making the noise of a car’s engine as it sped up. Every time I think of a rate of change I can hear the sound “ruuum rum rum rum rum….”.

  10. I consider myself clueless on the subject but I made it some of the way in to learning DE after some algebra and calculus… The coolest fields are abstract algebra, “engineering math”, and dynamical systems. None of them are even half covered even in the best universities.

  11. I thought you learn calculus at high school and only the advanced like (ln(x))dx at univeristy. In Sweden you learn derivate and calculus before univeristy and it is not as hard as the article describes, the basic calculus.

  12. Too bad modern education is run by academic pin heads who compartmentalize everything. I think I would have been more successful with math if I could have seen some practical applications of it as opposed to paper theory. But gone are the vocational aspects of education especially for boys.

  13. While there are many advantages to living out in the woods, one of the most common disadvantages is that satellite internet is too expensive to waste on any online videos. Without having transcripts of this being available for us poor rural dwellers, they might as well be speaking Thracian or Liki. Maybe I should visit here again once Elon has Skylink working, assuming I’m still alive whenever that happens? Meanwhile, I still ain’t got no education…

  14. Me and a friend were just talking about a related concept on facebook.
    As a primarily autodidactically educated person, I never fully appreciated
    school, particularly primary schooling.

    But I found a pertinent quote from a book on history, which emphasized
    that after the French Revolution the modus operandi of general schooling
    switched to a memory based system, where as before it did include
    development of intuition and understanding in the curriculum.

    https://books.google.com/books?id=YnIwhrdjt6QC&q=%22lesson+of+words%22#v=snippet&q=%22lesson%20of%20words%22&f=false

    I suspect that most people do believe they are being educated while
    matriculating through collegiate systems (I can mainly draw my experience
    from the US school system), I think there is a dangerous line between
    being indoctrinated into a set of probable facts and being able to
    understand the effects of such an indoctrination, in addition to being
    able to understand whatever general subject is being explored.

    Then there is Sir Ken Robinson talk (http://youtube.com/watch?v=zDZFcDGpL4U) which also talks about the modern
    educational paradigm, its historical formation, and how it benefits and
    fails the individual.

    Overall, a very different take on education..

  15. Title says it all!! I was trending for a D… Never studied so much for a class in my life. Somehow the final exam proctor lost my exam. I was honestly shocked. Majority of my freshman 2nd semester studies were dedicated to this class. I had study buddies! Anyway, I scheduled an office hour visit. Basically demanded he give me a 100 on the exam vs the 75/”average” or above average score than my previous tests. I demanded a 100. It would change my D to a C!

    I got a C in calculus. Siddique waited until the start of the following semester to file a grade change. I sat in a Business Calc class, 1st day, and new there was no way I was doing it again. Emailed a strong worded letter and the grade shortly after.

    Chyea

  16. I stumbled across Silvanus Thompson’s book the first day at my new high school and devoured it in a week. A couple of weeks later and I’m tutoring university liberal arts majors using it. It really is one of the best books on the calculus that I’ve ever met and I have gigs of maths texts to draw from as it’s one of the subject areas that people approach me for help, along with physics and engineering (of all sorts).

    He gives an intuitive explanation to limits and why it’s somewhat important as introduction. Then he goes on to explain the mechanics which is what someone really needs to know. The use, not necessarily the derivation even if it’s given in passing. So, thanks for the Gutenberg link, my hardbound ain’t leavin’ my cold dead fingers ;-).

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