For the last two years, we have explored how to create hands-on models to help people learn math and science. Our book 3D Printed Science Projects was published by Apress in 2016, with a second volume in late spring 2017. Each chapter explains a scientific or mathematical principle, and then develops 3D printable models, many of which can be changed to demonstrate different cases. We found that creating these models was difficult and required both deep knowledge of the science and of the best ways to create a 3D printable model.
We now want to take this to the next level and figure out how to restructure learning a math subject completely, aiming at self-learning and at supporting learners who do much better at a subject if they can make and hold something. There are a lot of one-off projects to demonstrate a particular concept (and we have those in our books) but now we want to see how to think differently given the tools available to us.
We talk to many self-taught hackers who are excellent intuitive engineers, but for whatever reason never took calculus. Calculus is a gateway to a deep understanding of physics and more advanced mathematics. There are online classes which can be taken, in some cases for free, but they are traditional algebra-heavy classes as far as we have seen. What is out there for learning advanced math and physics using 3D printing and open source electronics?
There are databases of 3D printable objects but these are not curated and the quality can be uneven. The science or math in free database models can be just plain wrong. Most commonly 3D printable math models are esoteric “math zoos” that look awesome but give no insight into how the model fits into a bigger picture. The lack of curated, rationally-organized 3D printable math and science models, and the difficulty of creating them, has held back use of 3D printing as an educational tool, even where it might be very effectively applied. Electronics projects are even more scattershot in this way.
Kepler's laws in plastic- 3D printed orbits
As an example, our project image is “Kepler’s Laws in Plastic." The models are available for download at our publisher's page (linked above) for the 2016 3D Printed Science Projects book. They depict orbits (Earth, Venus, Mercury in one set, Halley’s Comet to a different scale in the other) with the height being the velocity of the body at that point in its orbit. Kepler developed his Laws about these relationships in about 1609, without benefit of computers, 3D printers (or calculus.) He had to rely on a purely geometrical approach. Although Joan was a JPL rocket scientist for 16 years, when we created this model both of us developed new insights. We want to bring many people those insights and make math as natural as using construction toys. We have been surprised at the emotional response this project has generated in many people (see our narrative in our logs) and we are excited about where this might go.
What Will Our Project Do?
We have gone back and looked at the oldest roots of calculus as we know it - Isaac Newton’s Philosophiae Naturalis Principia Mathematica, usually called Principia. Pages of Newton’s copy with his notes on it (with his papers at Cambridge University in England) can be seen here. Kindle versions of English translations are available starting at 99 cents.
One thing that struck us was that the entire work has only geometry. The emphasis on algebraic forms came later on. There were good reasons for this, but our idea is this:If we go back to the source, knowing what we know now, can we create a mashup across three and a half centuries that Newton himself might have appreciated? We want to find good intuitive starting points to teach calculus-as-physics and then find hackerish ways to teach them in a hands-on way.
Initially we need to define a set of core concepts that best lend themselves to this approach, and organize them into one or more sets that naturally build on each other, to avoid the problem of existing databases that do not lead a user through a good learning path. Then we will create the 3D printable models and/or electronics project, and write supporting text.
For the Hackaday project, we plan to create several modules composed of electronic and mechanical parts, and of course some 3D prints. A module will contain a concept, directions to create a physical model, and various applications of the concept involved.
In the first log entry, we create a simple demonstrator out of an Adafruit Circuit Playground, a rechargeable battery and an (optional) 3D printable case. We are also exploring design options a mechanical device for finding the tangents to a curve. As we have evolved the project, we have moved more and more to 3D printable models. Early feedback was that if we had too many different kinds of demonstrators, very few people would be able to try all the projects. We've focused more on 3D prints in the later parts of the project based on that feedback.
In the longer run, we want to see what kind of structure we can create that is extensible and has modules that people can learn a topic of interest, but also provides a way to have an orderly groundwork to be able to learn some goodly fraction of what is in a traditional calculus class, at least at the intuitive level.
Actually using calculus concepts for sophisticated engineering projects is not easy, and never will be. But a lot of the messy algebra that people fear and bog down in can be done by symbolic programs now like the Wolfram products, much as calculators took away the grind of easier math manipulations. We think if we go back to Newton for inspiration we can unravel all the years of traditional thinking that have baked on to the early insights. We may be able to come up with ways to help people build enough intuition to understand what tools they should bring to bear on a problem, and let the tools take them from there. At a minimum, they may have more insight on how the world works.
Who is This For?
The hacker who wants to learn about the physics of his or her project at a deeper level is the first customer of this material. We have been grateful for the enthusiastic early response and community input on the project as it has evolved.
Students who are struggling with traditional means of learning calculus are the also beneficiaries. We think also that students in vocational programs who are not remotely prepared for traditional calculus and physics could use this tool to become more sophisticated machinists and robotics technicians. Anecdotally we are finding that many science and math PhDs struggled in calculus as it was taught, and years later appreciated the concepts after they redeveloped it geometrically in their heads.
Recently, there has been a lot of discussion of neurodiversity. The book, "Neurodiversity in the Classroom" by Thomas Armstrong has given us insights. Armstrong pointing out that if someone is bad at something, you don't put them in a classroom and have them try to do nothing but what they are bad at- you find a different way to teach. We are grateful to special-education researcher Ting Siu of San Francisco State who played with some of the early models and crystallized this idea for us.
And finally the visually impaired advanced math student is often pretty out of luck for materials to help them learn. We hope to enable a lot of blind mathematicians and scientists to go on to another level. We are also grateful to friends in the TVI (teachers of the visually impaired) community who followed our 2016 project and who have been commenting in background on this one.
What is Included Here?
The project gives broad philosophy, but also specific projects you can build:
- A "sensor pod" to learn about acceleration (BOM given, Circuit Playground Arduino sketch is in "files" for the project, and the STL (3D printable model) for a 3D printed part is on youmagine)
- In the "Files" section of this project we have included STLs for some representative models to learn about integrals, derivatives, and a PID controller visualization . (See the next section about licensing of these STLs.)
- We have included links to other readily-available published material of ours, where that supports the project.
We had some blind alleys that were ambitious to get done in the time available and/or were probably too complex for most people to build themselves; you'll see those as we go through the logs.
About the 3D printable (STL) files in the FILES section of this project
We have included 3D printable (STL) files in the “Files” section of this project. We are licensing these STLs as CC-BY-SA 4.0. If you print them, please note that these are experimental works in progress, meant only for your entertainment at this stage of our development. Please print and use them with care. The models were created in OpenSCAD and should be credited to Rich Cameron, aka Whosawhatsis. The Arduino sketch in this project’s “Files” section for the acceleration demonstration is similarly in its early stages and for entertainment only as we play with the ideas
Some Facts About Traditional Calculus Teaching
Calculus teaching in particular is ready for a complete do-over. A National Academy of Sciences report by Malcolm and Feder in 2016 states that 25% of students who take calculus at research universities get a D or F grade or withdraw. Women in particular treat this as a sign that they are not cut out for science or math careers. However, we think this is a reflection of how these subjects are taught – with an emphasis on abstruse algebra, and very little intuition about the geometrical and physical roots of calculus in particular
How many students take calculus? According to a report in 2016 by the College Board 308,215 students took the Calculus AB AP exam, and 124,931 took Calculus BC. That is a total of 433,146 students presumably in pre-college programs. For Calculus AB, more students received the lowest grade of 1 (30.7%) than the highest of 5 (24.8%). For Calculus BC (presumably more self-selective) 51.1% earned a 5 score and only 12,847 (10.3%) received a score of 1.
Very telling is that fewer college students than high school ones take a basic college course. The number is thought to be about 300,000 according to a 2014 report by Bressoud, Mesa and Rasmussen. This means that there are potentially about 740,000 students needing varying degrees of help in the US alone. If the site is designed with a focus on 3D models, translation costs for new versions may be minimal (and perhaps can be crowd sourced) so this could be a worldwide resource.
There have been other efforts to look back in history for other means of teaching calculus, and we will look at that literature as well to inform what we are doing.
Recently there has been a push to Career Tech Ed (CTE) training programs at the community college level to train machinists, technicians and similar vocational students. These students are often taught a watered-down math program, but are doing technical work that would benefit from a higher level of mathematical training. We believe our approach will be ideal for people like this who are very visual, hands-on and tactile learners.
We have also been involved in standards development for 3D printable models for blind students, and there is great enthusiasm (but, sadly, no funds) for this endeavor in that community. A set of open-source geometry models we created as a public service for blind students (for our 2016 Hackaday Prize entry) has been downloaded 1,292 times (as of August 13, 2017).
One good online math and science site is the Khan Academy (www.khanacademy.org) which organizes short explanatory videos into sets of lessons that can be watched one after the other. There are simple, easy-to-navigate hierarchies of lessons that can be entered at any point, and they are ideal for a teacher to assign for self-study without posting explicit lesson plans for each possible grade level that might use the material. Khan Academy is a nonprofit and grant-supported, and claims 15 million learners per month. However, the Khan Academy provides materials that align with existing curricular scaffolding. Our intent is that we are developing a new framework to enable self-learning of these topics. We want this to be particularly accessible to the person who wants to learn about something by handling it and with a geometry focus, not by doing a lot of algebra.
How To Read Our Logs
Our logs build up concepts assuming that you read the oldest log (#1) first, then #2, and so on.
Joan Horvath and Rich [Whosawhatsis] Cameron are the co-founders of Nonscriptum LLC (www.nonscriptum.com). Our Pasadena-based consulting and training firm was founded in early 2015 and focuses on teaching educators and scientists how to use maker tech. Joan is an MIT alumna, recovering rocket scientist and educator, and Rich is an open-source 3D printer hacker who designed the RepRap Wallace and Bukito printers. Between us we have written (so far!) six books for the Apress imprint of Springer-Nature publishing. Joan also has an appointment as a Core Adjunct Faculty member at National University College of Arts and Sciences, one of the largest private nonprofit teacher training institutions in California. She also has experience teaching technical subjects at design colleges to audience of artists.
Last year Joan and Rich were Hackaday Prize semifinalists with their project, 3D Prints for the Visually Impaired. That project resulted in the beginnings of an online community, a mix of teachers of the visually impaired and people willing to help create models for them. That experience and community has inspired and informed this project.