Mathematics, as it is taught in schools, sometimes falls short in its mission to educate the pupils. This is the view of [Joan Horvath] and [Rich Cameron], particularly with respect to the teaching of calculus, which they feel has become a purely algebraic discipline that leaves many students in the cold when it comes to understanding the concepts behind it.

Their Hacker Calculus project aims to address this, by returning to [Isaac Newton]’s 1687 seminal work on the matter, *Philosophiae Naturalis Principia Mathematica.* They were struck by how much the *Principia* was a work of geometry rather than algebra, and they are seeking to return to [Newton]’s principles in a bid to make the subject more accessible to students left behind when it comes to derivatives and integrals. They intend to refine the geometric approach to create a series of practical items to explain the concepts, both through 3D printed items and through electronics.

We can see that this is an approach that has considerable merit, given that most Hackaday readers will have at some time or other sat through a maths lesson and come away wondering what on earth the teacher was talking about and having been baffled by further attempts to explain it through impenetrable maths-speak. If you were the kid who “got” calculus when the relationship between speed and acceleration – another thing we have [Newton] to thank for describing – was explained in your physics lessons, then you will probably understand.

The pair have some Hackaday Prize history, you may remember them from such previous entries as their 3D prints for the visually impaired project from last year.

meh, your basic calculus for non-majors class always uses geometric examples — calculating the slope of a line, or the area under a curve. newton’s specific approach is impenetrably awful for education. at best, i think this project will recreate the status quo.

anyways, the algebraic result that gives you a closed form, for example, of the effective resistance of a series capacitor given a 60Hz AC voltage is really the meat of it for most of us…understanding the calculus that goes into deriving it is not so important to us, and at any rate needs to be done algebraically.

Algebraically and with complex numbers.

I don’t even remember which math was from what module.

I think calculus was about polynomials, derivatives, integrals, quadratics and things like that and algebra was about how to manipulate the equation by transposing, translating or cancelling parts of it, things like cos^2*sin

I have a prototype robotic device that I haven’t been able to do the kinematics for because my math is too rusty now. I was attempting to do the math on an 8-bit uC which means integer math and I haven’t been able to even analyze the polar math (I call it Rotesian). It’s a folded (half inverted) arc delta. Kinda like three scorpions facing each other.

It was put aside in frustration. I am tempted to redesign it *just* to make the math simpler.

It’s here – https://hackaday.io/project/18186-arc-delta-3d-pcb-drill-and-other-failures

Don’t give up! I love it!

LOL, “Austraaaaaaaalai” :)

Spoken like a fair dinkum ozy.

Well. The Principia uses geometric proofs because Newton wanted to make the proofs as concrete and fundamental as possible. (And maybe to show he had a more basic understanding than Hooke or Liebniz. There was a lot of animosity in the air. Were the geometric proofs created after the methods were developed in other ways?) Newton’s geometry proofs are complicated and rather strained. I can’t recommend them for anything but the history.

Later, mathematicians began to prove geometrical theorems with algebra. But I agree somewhat that starting off with epsilons and deltas and the Fundamental Theorem of Calculus can leave people lost in a sea of algebra. In most physics curricula, it is a pretty natural move from measuring differences on a spark tape or video of a falling ball, to finding limits and areas.

In the math department, the emphasis is much different. I hope these tools work out really well.

It’s also worth pointing out that the Fundamental Theorem of Algebra serves to generalize geometry to algebra. Geometry is the original math.

The book is in Latin. I don’t know

many people who learn Calculus

in Latin.

Necesse est ut ultra vos sunt.

Google won’t translate that to English that makes any sense. Try this :-)

Domum tuam debes deserere saepius

Lots of open source and free translated versions (links in the project.)

Really calculus didn’t sink in for me until I began to work with HLSL and GLSL shaders doing advanced things that make AAA games look so pretty. Especially after developing a tool to do live compilation and a live preview of the most recent successful compilation. When you begin to see live feedback to changes you make when you have those, “what if it was this way” thoughts. You begin to develop a far deeper understanding than any numbers on paper (or a screen) could hope to convey.

To be clear I was using calculus to process both geometry and color. It was in processing color in particular that it sunk in hardcore. Geometry only gets you so far, the subtle changes to hues of color and the stark contrast when your equations break down under special cases are where you really see the guts of the subject spilled out before you.

I relate with this experience entirely.

It wasn’t until late highschool that I really got a firm understanding of Algebra and Geometry, because that’s when I started taking physics and electronics. Having an application which did not follow the standard assembly line approach that mathematics courses followed helped me greatly.

It wasn’t until I started working with interpolation and quaternions in embedded systems that I understood the benefits and uses of calculus. At the time I was mostly focusing on dead reckoning for light weight airborne GPS transceivers. At most they were under ten grams if I remember, I think target weight was about 5.

Anyway, I totally agree that having a dynamic application aids in learning to a far higher degree than a static application

I’d love to hear more details about how calculus informed your shaders.

seconded!

The problem with the schools now is the students have to use the exact method the school teaches otherwise even the correct answer is marked wrong.

it’s always depended on the teacher. i’ve had teachers who would cut my grade to try to compensate for their own insecurities, and i’ve had teachers give me partial credit for getting the wrong answer with an amusing technique.

Teachers are telling us their hands are tied. Once a school system adopts the ‘common core’ they have to follow the process exactly and can’t deviate.

Hi folks- thanks for the stories. My experience (both as an engineering student long ago and teaching college math) is that learning calculation without intuition rarely leads anywhere. Getting good intuition, though, usually makes the calculation possible (if not easy.) That’s where we are trying to go with this project… we imagine that our main audience is the person who never took calculus (and never will, most likely) but who would like that intuition.

My Topology and Complex Analysis professor who studied in Europe has indicated that the longstanding continental/Russian/Chinese educational approach is to introduce sequences and series early on, as early as high school, which makes subsequent contemplation of continuity, metric spaces, topology, and analysis more intuitive.

He noted that this contrasted this with the more Newtonian approach taken in Anglophone countries. His personal preference was for the continental approach.

See also:

http://historyofmathematics.org/wp-content/uploads/2013/09/2005-Collingwood.pdf

Yes, the math has evolved over 350 years, and if you are going to be calculating stuff you will want to do it algebraically. However, the audience for our project is not the practicing engineer or engineering student who likes algebra: there are a lot of places people who like algebra can learn calculus for free. Our audience is the folks who didn’t like it taught that way, or avoided it because they didn’t like the lead-up classes to it, or who think they “stink at math” (but don’t). Our goal isn’t (necessarily) making mathematicians. We want to make more math-literate and more math-intuitive people who maybe will take their newfound intuition back into traditional math, or maybe into inventing cool things. In other words, giving people tools they might not be able to use otherwise.

you may be able to geometrically demonstrate the limit of sequences, i.e. 1 + 1/4 + 1/16 + … as a set of squares of diminishing size, sitting next to one another, placed on the RHS of the last, in succession, then draw the line passing through the midsection on the RHS of each square… food for thought anyway.

Cool idea you’ve got going there. Keep it up.

Replying to e: Yes, limits are interesting to visualize. Stay tuned on that one. A mathematician friend has been suggesting some chaos theory ideas, although functions that go kablooie with small changes from a stable point create some interesting 3D printing issues…

I have to agree with the European approach. Evey student who had difficulty with calculus had weak algebraic skills. The public schools in the US just brush past most of the important topics and don’t believe much in practice. I think Feynman said that being in the school math club and having speed contests factoring quadratics and things like that were a big aid to him later.

I have some Russian high school texts (In English. There was a little science store in Palo Alto in the 1980’d that had things like that.) that are really impressive, both in the level of sophistication and the number of good problems. They are also nice and small, printed on high quality thin paper and no color. Illustrations are well done and there are no multi-cultural social engineering photos and problems. Modern texts in the US are gigantic and full of extraneous bullshit printed in the extra wide margins and in the text and examples. They are too heavy to hold while reading.

My grandfather’s geometry book from 1915 is also very complete and compact, and with wonderful example applications.

What an insightful comment. The absence of forced multi-cultural social engineering is working out really great for Russia.

Calculus made a lot more sense to me when I took Physics.

“…relationship between speed and acceleration…”

That should be velocity, not speed.

Speed is scalar. Velocity is vector. Acceleration is vector. If you only have magnitude, it is scalar. Though if you have a direction too it is vector.

Not a math major, but I did love calculus and physics.

“Speed is scalar.” <– English major requesting a definition of "scalar". (I can look it up, but how do I know I'm using the same definition you are?)

"Velocity is vector." <– repeating request for definition. AFAIK, velocity is speed and direction. So is a "vector" a force/event with two aspects/dimensions?

"Acceleration is vector." <– See above.

"If you only have magnitude, it is scalar. Though if you have a direction too it is vector." <– when did magnitude enter this discussion? Why was it included?

Not trying to troll, but I really want to understand. (Calculus was one of the classes I flat-out failed in college, and it's always bothered me.)

Digital Corpus is right strictly speaking (although “acceleration” is commonly used for both the vector and scalar in my experience), but we were going for using common words at this stage and getting into those distinctions later.

For mb, if we are talking about how something is moving, we can talk about it in two ways: either broken up into how much it is moving in each of three directions (which in the project,for now, we call up-and-down, front-and-back, and side-to-side- that will become z, x, and y later on.) If we keep track of those separately – and which direction, say, front instead of back – that is a vector.

So we might say we are moving 3 feet per second forward, 2 feet per second downward, and 5 feet per second left. If we keep track like that, it’s velocity. If instead we add up those using a 3D version of the Pythagorean theorem and just get the overall added-up speed — here, the square root of (3*3 + (-2)*(-2) + 5*5) or about 6.2– that is a scalar- just a number, with the direction components lost. An accelerometer measures the components, and in Log #1 we build a device to display two of the components. (We throw out one to get rid of gravity, which accelerates downward.) Keeping track is useful if, for instance, you want to do something like ignore gravity which always pulls the same direction, and you just want to find out if your sparkle skirt should light up because you’re dancing hard enough. :-)

MB

Magnitude is an amount for example 10 grams, 50 miles per hour, 3.3Volts, etc.

Direction can be in 3 dimensions x,y,z, north-south/east-west/up-down, etc.

Speed is a scalar meaning it only refers to a magnitude. How fast are you going? 50mph.

Does that mean you are getting closer to work or farther away? Direction answers that.

For the whole solution you need both magnitude and direction which is the definition of a vector.

To recap:

Something that is a scalar has only magnitude.

Something that is a vector has magnitude and direction.

Think of a vector as an arrow. If it’s pointed in your direction, but has 0 speed, it’s a good arrow. If it’s pointed towards a hay bail and has lots of speed, it’s good arrow. You need to know both magnitude and direction to figure out if you need to get out of the way.

You may also hear direction being referred to as a “unit vector”. A unit vector is a vector with the scalar component (the magnitude) removed. The largest value in the vector is given the value of 1 and every other vector is some percentage of that. As an example let’s say you have an (x, y) vector on a Cartesian plane with a value of (6, 3). The unit vector would be (1, 0.5) and the scalar would be 6.

mb you are right when you ask if the vector has two or more dimension. The math of vectors is just whatever has to be done to make it work. To add vectors or multiply vectors, etc. There are two common ways to multiply two vectors.

Vector calculus is recent as math developments go, and was done by an American physicist named Gibbs. But today, any series of quantities that are linearly independent can be called a vector. Even some things that are not but can take advantage of the vector math. For example, the 9 degree if freedom data from the inertial sensors in a drone are used in a Kalman Filter that is entirely the math of vectors and matrices. Linearly independent means you can not get info on one dimension from a combination of the others. In a Cartesian XYZ system, you can not determine y from some clever combination of x and z. This is also called orthogonal. Multiplying any two orthogonal vectors (with the method called the ‘dot’ product) will result in 0. This is really handy in a lot of areas, graphics computation being a good one. And the dot product is a simple sum of products, like multiply two vectors by multiplying their x values then add that to the result of multiplying the y values and add that to the z value result. This sum of products is what DSP’s do best, and today the graphics or DSP units in processors have this as the core.

Now, the 9 DOF ‘vector’ does not have orthogonal components. X,Y, and Z accelerations are orthogonal. So are the gyro sensors. However, the three magnetic components change as the others change. The magnetic part is useful to increase accuracy of the others, but in the physics sense, the 9 DOF values are not a vector. Now, I wonder what I started out to write?

So….

In otherwords…

This:

https://betterexplained.com/guides/calculus/

Some interesting ideas in that, but what we’re doing is different. The key part is the hands-on component (which requires a geometrical approach, since algebra does not lend itself to hands-on learning). In particular, some people are tactile or kinesthetic learners (estimates I’ve seen cluster around 10% of the population.) They learn best by making something or handling it. They are often poorly served by the visual-centric standard ways of teaching math. And of course visually-impaired people can only be served with tactile tools. There’s an interesting book that we cite in the project (“Mathematical Mindsets” by Jo Boaler) for those interested in these issues.

Calculus was a rote thing for me until, years after graduating, I acquired Calculus Made Easy by Sylvanus Thompson. That book, written in the early 1900s, explained it in a way that my college professors never had, and I finally understood the basic concepts.

‘Calculus Made Easy’ by Sylvanus P. Thompson (and now with Martin Gardner) is the universally acknowledged, single best expository work on the calculus ever written, and your story is, sadly, almost universal, to wit: even though the book has

, most people never hear of it because it is shunned out-of-hand by the academic community. “Too easy, don’tcha know.” “We’ve gotta whip these freshman into shape with some really hard courses, like CALCULUS! If they flunk, they flunk!” Don’t laugh; this–and a pathetic lack of teachers who can teach calculus–is precisely what’s wrong with students “getting” calculus in the first place.never been out of printfor almost one hundred yearsFrom the preface:

“

Calculus Made Easyhas been damned by every professional mathematician I have asked about it. So far as I know, it is not used in any calculus courses anywhere. Nevertheless, almost a century after its first publication, it still sells briskly in paperback even in college bookstores…”–Julian SimonAt a high level, serious conference at Tulane University in 1986, entitled

Toward a Lean and Lively Calculus,ALL attendees know (knew) of Thompson’s book, and that it is the leanest, liveliest, and most enjoyable introduction to adeepunderstanding of calculus, irrespective of one’s background. A gentleman named “…Peter Renz was the only mathematician at the conference who had the courage to praise the book and list it as a reference.”ISBN 0-312-18548-0

jawnhenry- actually never came across that one. I’ll investigate- thanks!

Has anyone ever seen the PBS series “The Mechanical Universe”? It does an amazingly good job of explaining mathematics including basic calculus. It’s tough to find these days but it’s one of those series that should be shown again.

Calculus really sunk in for me when I started writing programs on my VIC-20 to graph 3-d calculus equations. Physics helped with my understanding of 2-d, but the 3-d stuff was tough to understand until I could actually see the geometric relationships.