Math is easily one of the least liked subjects in school. In fact, most people who are not majoring in Mathematics or some other STEM field will likely never take a mathematics course beyond Calculus 1. Some don’t even make it that far. And to me, as a math major, that is heartbreaking. But why is math such a disdained subject among students?
Take one commenter’s perspective on the matter in a TED discussion entitled Why Does Everyone Hate Math?
Math is a language. Its a set of symbols and concepts that relate and work together. Why everyone hates math is probably because the language is not spoken from birth and its rarely spoken in a dialect that people can understand. For most people, it would be like trying to learn Japanese; reading it as a native English speaker, and having the instructor pronouncing with a Russian accent.
– John Gianino
One of the main problems many people face is in understanding the strange symbology and obscure theorems presented in math classes. People have troubling forming intuition around an idea when they can only receive information through textual explanations and 2D representations of the idea. Sometimes, you need more. And in an attempt to solve this problem, my partner, Zach Schuhmacher, and I have re-envisioned the Calculus Curriculum at JMU (or any university) to provide more visual aid in understanding the material. So let’s start at the beginning.
Math 235 – Calculus 1
In calculus 1, students are first introduced to the idea of limits and continuity. They begin to learn differentiation, optimization, and integration. One of the places they use these tools is in relation to conic sections. The different conic sections (Circle, Ellipse, Parabola, Hyperbola) can be very confusing to visualize and work with as a student just learning the material. Many do not even understand why they are called conic sections. One user of the online 3D printing community Thingiverse, Karlcrosby, designed a cone, pictured below, that decomposes into the different conic sections.
This allows the student to get hands on practice with conic sections and increase their intuition when working with them. But as students begin to look into the third dimension, 3D printing’s utility to the classroom becomes immediately clear.
Math 236 – Calculus 2
Though Calculus 2 consists largely of the study of sequences and series, students at this level normally start to find the volume of solids of revolutions through integration.
For example, one exercise many students complete is finding the volume and surface area of Gabriel’s Horn. Take the function y= 1/x and revolve it around the x-axis. Through different methods of integration, students will find that this amazing solid has finite volume (namely pi), but an infinite surface area (Check out the linked wikipedia page for more information). But many students will ask, what does this look like? How it this possible? Well, with the combined use of Mathematica and a 3D printer, our very own Laura Taalman recently printed out a whole bunch of Gabriel’s Horns for a math conference in November.
With these, students can actually hold this object, and see 1: the original curve, y = 1/x, and 2: the symmetry the object has around the center. Though this is one example, any solid of revolution can be modeled and printed on a 3D printer.
Math 237 – Calculus 3
Calculus 3, or Multi-variable calculus, is really the place where 3D printing can thrive. The uses for it are so immense that I will list out a few examples.
- Quadratic Surfaces.
- Visualizing surfaces formed by revolving a uni-variate function around a line.
Example: 
Revolution of y=e^x (x<0) around x=0
- Determining continuity. Take f(x,y) = (x^(2)y)/(x^(4)+y^(2)). We can discover that this is discontinuous through various methods, but a simple, intuitive way would be to look at the 3D printed surface and see how at certain points, there are cliffs and spikes.
Summary
As you’ve seen, 3D printing could change the way students learn Calculus, but that’s certainly not the limit of its capabilities. The use of 3D printing can provide intuition to students taking Topology, Geometry, or any other course that examines shapes in a space. Understanding the properties of knots can be difficult when having to deal with a knot diagram, but with the use of our 3D printers, Laura Taalman has been able to actually create some different knots (8_18, 8_19, 10_125).
Even though animations (like in the video below) exist to help in this respect, implementing the use of 3D printers in the math department will help advance the learning of our students.
As promised: A very cool video from the LSU Department of Mathematics.
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