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Math manipulatives are used in early grade school to help the children understand counting and numbers with something they can touch. Buying simple plastic...

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Math Formulas that Students can Hold in Their Hands

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Math Formulas that Students can Hold in Their Hands

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A Penn State Hazleton professor is using 3D printing in education in a unique way: creating solid representations of mathematical formulas that help students better understand mathematics.

Alfredo Jimenez, associate professor of mathematics at Penn State Hazleton, teaches calculus to engineering and science students, and said he is always looking for new ways to teach. "I have been teaching for about 35 years, and I am always interested in learning more about teaching, as I think there's an enormous amount of knowledge that can improve my teaching," Jimenez said. "So I think one of the theories I became hooked on is the theory of constructivism, where students construct their own knowledge, so as a result of that I have been constantly searching for ways I can help students construct their own knowledge in the courses that I teach."

Due to his interest in constructivism, Jimenez was intrigued when he saw Joseph Ranalli, assistant professor of engineering at the Hazleton campus, give a presentation on 3D printing. This got him thinking. "I thought at that time it would be really neat if I could produce some solids with the programs I have been producing using (math computation program) Mathematica, to give students a visual aid for math," he said.

Jimenez knew at the outset that he wanted to use 3D printing of models in his classroom, he just needed help figuring out how to get Mathematica to export some clean, printable files. With the help of Penn State's Teaching and Learning with Technology unit, which recently partnered with the University Libraries to open the Maker Commons 3-D printing lab, Jimenez was able to accomplish this to create test models to work out any kinks. Ronald Harman Jr., information resources and services supervisor/manager from Hazleton's campus library, did post-processing on the models to enable them to be printed properly, then printed the actual final models that Jimenez used in his class.

"It was really nice, because for the first time I was holding in my hand a solid, 3D version of something that I had been drawing on a board or showing images to my students for so many years," Jimenez said. "That was very useful for me, because it made it easier to explain the mathematical concept, it gave me a real visual. I thought 'whoa, this could definitely have great possibilities for use in my calculus class.'"

An example of how Jimenez has used 3D printing in his class is to help explain an important concept in calculus, the definite integral to compute volumes of solids. "So what I did was to produce a series of about seven to eight different solids," he said. "I came to the classroom and divided the class into groups. I gave each group a solid with a simple instruction, to compute the volume of the solid.

"I think this is very interesting because normally what I would do is give the students some direction, some equations, and then the students follow those directions. But this time around, they have the solid in their hands, a visual to guide them in what they are doing. Plus now they have to find the equations to describe the solid on their own."

Jimenez said that sometimes students have a great deal of difficulty trying to visualize a solid represented by a given set of equations. But now, he notes, they can hold the solid in their hands.

"I think that having the solid really helps them understand," Jimenez said. "I can actually give them a representation of the approximation process because I give them the solids. This is something that is very easily done and it greatly enhances the learning process."

Jimenez said this could be taken one step further for advanced students. He said an interesting project for higher-level calculus students would be to produce a program to model a particular solid and then print it to see if the model produces the desired solid. He said students could even propose their own program and equation, 3D print it, and see if it is what they expected.

"It is changing the parameters tremendously," Jimenez said. "As math teachers we are teaching concepts and mathematical ways of thinking, which can be very difficult for some students. Having a concrete, tangible object that represents some of these concepts and ideas is a very positive new development that is worth exploring, because it makes mathematics a real thing that they can see before them."

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3D printing creates math formulas that students can hold in their hands | Penn State University

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A Penn State Hazleton professor is using 3D printing in education in a unique way: creating solid representations of mathematical formulas that help students better understand mathematics. Alfredo Jimenez, associate professor of mathematics at Penn State Hazleton, teaches calculus to engineering and science students, and said he is always looking for new ways to teach.

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Visualizing Mathematics with 3D Printing

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The book is available from the publisher, Johns Hopkins University Press, with a 30% discount by using the promo code HNAF. You can also mail or phone your order, using this flyer. Alternatively, you can also buy it from Amazon.com.

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Here is a cool project list for teachers.

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Here is a cool project list for teachers.

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3D Printers | Feature

7 Hands-On Projects That Use 3D Printers

You can see more great feature articles in the latest issue of our monthly digital edition.

3D printing dates to the 1990s. The earliest 3D printers were designed for rapid prototyping for industry. They were very accurate and very expensive. Early companies like Stratasys have a great reputation in the industry for making machines that are used in commercial and high-end educational (largely college) environments. If this was where the technology had stopped, we wouldn't be talking about 3D printers in K-12 schools today.

The big change happened around 2008, when the RepRap project resulted in open-source designs that allowed anyone to build their own self-replicating 3D printer. These primitive systems could be built for a few hundred dollars, which opened the market to hobbyists who wanted their own printers. Since that time, inexpensive, fully assembled printers have come to the market that have expanded the user base beyond hobbyists to include schools.

There are two kinds of 3D printers: those that heat plastic filament and push it through an extrusion nozzle to build a design layer by layer, and a less popular type that uses a laser beam to harden a liquid resin to create the layers of the finished object.

There are currently more than 100 3D printers on the market that might be appropriate for educational uses. With a list this long to choose from, the challenge of finding the right printer for your district can be daunting. Printers range widely in price, and some inexpensive printers are as good as some of the more expensive models. (See the sidebar below for some buying tips that might be helpful.)

Teaching in 3D
The fact is that I'd be hard-pressed to think of an academic subject that can't be enhanced by the use of 3D printers. Of course, the easy (and obvious) subjects like physics, math and engineering come to mind. With all the current talk about STEM education, inexpensive and reliable 3D printers have hit the market at the perfect time. But there are many other subjects that can be taught with 3D printers.

Fine arts: My 7-year-old granddaughter was visiting recently, and I thought she might enjoy making decorative ceramic tiles using her own designs. You can see a short video of the process here. I helped her use the free TurtleArt software to create a geometrical pattern to her specifications. We then printed the pattern on a plastic stamp that could be pressed into wet clay to leave behind a design. We baked the clay and my granddaughter painted the tiles. I can think of numerous art projects that use 3D printers, some of which you can find in The Invent To Learn Guide to 3D Printing in the Classroom, a book I co-authored.

Literature: Suppose a student reads a book. She can build a model of one of the scenes from the book using characters and settings that she designed and printed herself. Finished parts can be painted with acrylic paint. This is important for this example because most 3D printers only work with one color of plastic per part.

History: Imagine the insights students can get into the ancient Greeks when they design and build models of temples like the Parthenon. In the course of this project, students would likely learn about the Golden Mean and its appearance in classical architecture.

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the invent to learn guide to 3d printing in the classroom

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7 Hands-On Projects That Use 3D Printers -- THE Journal

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3D Printers | Feature You can see more great feature articles in the latest issue of our monthly digital edition. 3D printing dates to the 1990s. The earliest 3D printers were designed for rapid prototyping for industry. They were very accurate and very expensive.

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This article is a general introduction to printing across many subjects.

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This article is a general introduction to printing across many subjects.

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Printing in 3D is steadily transforming the world through innovation, speed, localised manufacturing and empowering the creativity of the individual.

Where computers and mobile technology changed the world and the landscape within education, 3D printing will add another dimension to learning and give rise to the next generation of engineers, designers, entrepreneurs, artists, and industry innovators.

The concept of industrial 3D printing has been around for many decades, used within industries such as aerospace, automotive, product development and engineering. However, since the rise of the RepRap project in 2009, desktop 3D printing has taken leaps and bounds resulting in a technology now suitable for offices, classrooms and households.

The importance of 3D printing is not just focused on the desktop version, which traditionally uses materials such as plastic, but also focuses on medical and bio 3D printing. The Geno Project has played a significant role in creating 3D printed organs, cell tissue and blood vessels through to innovating cosmetic and reconstructive surgery, for example.

If we take a moment to consider learning - innovating and exciting the classroom, transforming learning processes, and introducing the ability to simplify complex design processes can all be achieved quite simply with a 3D printer.

Design and technology

Complex designs and products can be printed and manufactured quickly and efficiently. Printing in 3D enables a more seamless process, allowing the user to modify the design, tailor sizing, add additional features and use a variety of materials ranging from traditional plastics in solid or transparent colours to wood, chalk and rubber. The entirety of this process creates a great foundation for higher education.

3D printing robots

Robotics and 3D printing are closely becoming aligned, with leading robotic androids now featuring 3D printable parts.

The importance of robotics within education remains in its infancy although sectors such as warehousing, manufacturing, engineering, finance and technology have been transformed. It seems fair to say that the next generation shall use robotics in a variety of forms and situations within the home and work.

3D printable robots can educate students in a variety of key skills, such as engineering, mechanics, electronics and programming. For example, BQ has introduced PrintBots, a simple 3D Printable Robot kit aimed at all ages with 10 easy steps.

The exciting aspects of kits of this nature ensures the process does not stop once the robot has been built. Instead, the process can develop further to programming, additional printable parts can be downloaded for print, and those students who want to push the boundaries and challenge their computer-aided design (CAD) skills can create their own unique parts for their robots.

Art

3D printing art has challenged the traditional concepts and processes in art and we now have a new generation of 3D printing artists and enthusiasts. It has helped empower the creativity of students to create complex and unique pieces, from jewellery, abstract shapes and concepts to replicating traditional art formats such as statutes, vases and interior design pieces. 3D printing can also be used to simplify complex designs and introduce 3D CAD into art lessons.

Science

The ability to innovate in science can be seen with the rise of industry bio and medical 3D printers and the introduction of 3D printers into science at an early stage will give the first foundation of how this technology is changing the landscape of medicine, healthcare and science. 3D printing can be used in a variety of formats, such as printable dissection kits - the dissection of a frog printed in plastic to illustrate key organs and discuss how they work and fit together. Moveable joints, working mechanical hands and a pivoting solar system can also illustrate the subject area, creating and necessitating group projects and activities.

History, geography, maths

3D printing can have a place in all subject areas. Learning can come to life by turning back time and creating dinosaurs, fossils and historical features such as pyramids, Roman helmets and castles. Designs can be versatile and be printed as a whole or in stages to demonstrate key syllabus objectives and to further educate the student of the workings of 3D printing, thus allowing a multitude of skills to be learnt and developed simultaneously.

In maths, questions such as how parts interact with each other and the regularities and symmetries of various parts may come to mind. The boundaries can now be explored further by printing tangible parts, such as moveable cogs, a working abacus and mathematical shapes.

Geography lessons can also benefit significantly, recreating complex structures such as volcanoes and sediments in parts or stages to illustrate the inner workings or create 3D terrains of various world locations.

Even drama

From Games of Thrones to Hollywood sci-fi blockbusters, 3D printing has transformed film sets with leading prop-makers using the technology. Drama students can use this outlet to be creative and work with various 3D printing materials to create a range of props and allow productions to become one step closer to realism.

Conclusion

With all new technologies, the concept can seem at times confusing and complicated. However, we are now entering the third generation of desktop 3D printing and the process is as simple as plug and print. The design process, which comes in tandem, could also be considered daunting - however, with software such as Google Sketchup and TinkerCad, 3D CAD has also become a simpler process.

The cost of 3D printing and printers has also decreased significantly with a variety of education-friendly 3D printers now available from around £550. This downward cost trend will continue, making it an accessible tool within schools.

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Ideas for 3D printing across the curriculum

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Printing in 3D is steadily transforming the world through innovation, speed, localised manufacturing and empowering the creativity of the individual. Where computers and mobile technology changed the world and the landscape within education, 3D printing will add another dimension to learning and give rise to the next generation of engineers, designers, entrepreneurs, artists, and industry innovators.