Harvard calculus courses use 3D printed models to engage students

Harvard calculus courses use 3D printed models to engage students

  • University
  • Mathematics

What is the best way to teach concepts like surface area and volume relating to mathematical objects in a three-dimensional world? The curricular design team at Harvard University thinks they have the answer: put those objects in students’ hands. Literally.

Math 21a: Multivariable Calculus is Harvard’s largest math class, and it serves as a prerequisite for courses across the STEM disciplines. It is most students’ first math course involving significant three-dimensional spatial reasoning and visualization, and students must engage with the concepts they learned in single-variable calculus in the context of three-dimensional coordinate geometry. While most students in Math 21a are able to learn the procedures needed to do calculations in the course, some students either find the visualization component to be a significant impediment, or fail to attach geometric meaning to what they had learned.

In 2017, the curricular design team decided to add a new element to the course to target these student struggles: active learning lessons involving interactions with 3D models of the objects of study in the course. While over 500 students take Math 21a every year at Harvard, the course is not taught in a large lecture hall. Classes are taught exclusively in sections of thirty or fewer students so that students can work in small groups during class time. Our goal was to produce enough copies of a dozen different 3D models so that each small group could have one.

Manipulative-Calculus

The project was dubbed Manipulative Calculus. While Senior Preceptor Janet Chen and I spent the spring semester of 2018 prototyping models and piloting lessons with small groups of students, I contacted Ultimaker, and they agreed to collaborate on the project. A few weeks later, their fleet of 3D printers had produced half of the models we needed. The rest of the models had some intricacies that required dissolvable supports, and to print these, we used two Ultimaker 3 printers in Harvard’s Instructional Physics Lab.

Ultimaker-3

With the help of our project assistant and experienced maker Dishen "Dixon Magister" Yang, we kept the Ultimaker 3 printers churning out the rest of the models. The exceptional reliability of the machines and generous print bed allowed us to produce over 250 models in time for the end of summer.

parts
cura
printing-parts

The project “went live” in the fall 2018 semester. Hundreds of students sat down for the first day of Math 21a and were handed 3D printed models. In part of this first lesson, students had to figure out on paper the shapes of the cross-sections of a certain surface, then illustrate those cross-sections by drawing on the models themselves. Each model had been finished with chalkboard paint over the summer, so students could draw on the models then erase before the next class.

class

We used the models throughout the fall semester to let students engage with and attach geometric meaning to calculations involving functions and equations with several variables, volume, surface area, and more. We had designed the models together with a suite of curricular materials, including worksheets, lesson plans, related homework problems, and computer interactions. The idea was to make these lessons easy to use for both our teaching team, which includes a number of graduate student instructors, and by those outside our department.

class2

We are in the process of assessing the effects of the project with the help of the Bok Center for Teaching and Learning, but anecdotally, we have found that not only do students develop a better understanding of the material with the 3D models in hand, but that we as instructors are better able to assess students’ conceptual knowledge when they explain their thinking using the models as props. Several of our undergraduate course assistants (who had taken Math 21a in prior semesters and are now part of the teaching staff) have told us that they wish they had had these models when they were first learning the material.

We plan to continue the use of these models in Math 21a next semester and into the future, hopefully applying what we’ve learned to other courses as well. We hope that our project will demonstrate that using 3D printed models for active learning instruction on this scale is now feasible thanks to precise, fast, affordable, and reliable printers like the Ultimaker 3.

Read more education blogs

  • College of the Desert 3D printing club

    Feel the Action: Learn about how a college is changing lives in ways we can touch!

    A 3D printing club emerges at College of the Dessert to help make ideas accessible to all students

    Testing Quadcopter

    How the tools we use influence the designs we make

    Michael Delaney writes about his iterative process of integrating electronics and 3D printing in his programmable quadcopter project.

    Ultimaker Met

    Bringing 3D printing to the Metropolitan Museum of Art

    Part of the mission from Ultimaker North America's Community team is to help educators get up and running with 3D printing and to help them find meaningful ways to integrate the technology into their curriculum while celebrating their successes. When we w

  • Smithsonian Learning Lab

    3D printing as part of the Smithsonian Learning Lab

    Pioneer Christopher Sweeney shares the work he did with 3D printing and the Smithsonian Learning Lab to create collections of educational resources, organized and structured for teaching and learning.

    Harvard calculus courses use 3D printed models to engage students

    Harvard calculus courses use 3D printed models to engage students

    What is the best way to teach concepts like surface area and volume relating to mathematical objects in a three-dimensional world? The curricular design team at Harvard University thinks the...

    Mathematica Tea Light Holder

    3D Design in Mathematica: Tea Light Holders

    If it’s happening in Ultimaker’s world, you can find out about it here. 3D printing stories about inspiring moments, original 3D printed projects and much much more.

  • Code your 3D designs with Tinkercad’s new Codeblocks app

    Code your 3D designs with Tinkercad’s new Codeblocks app

    Tinkercad released Codeblocks last June, and now that it's out of Beta, Pioneer Rob Morrill tells us how he uses it.

    printingUltibot

    Modifying an STL with Tinkercad

    When Ultimaker Pioneer Alex Larson contacted me about the Ultibot-D project, I was super excited. As a teacher and parent, I am a huge cheerleader for risk. I printed the base models at ⅕ scale and gave it to my students to explore

    Ultimaker at Camp

    Math Camp: Having fun doing stuff

    Pioneer Dr Toni Szymanski writes about summer fun at camp with 4th and 5th graders, math, and 3D printing.

  • 3D printing for glass

    GlitchCraft

    Guest blogger Astrida Valigorsky writes about combining the old and the new at Timothy Belliveau's GlitchCraft class where students combined 3D printing and glass blowing.

    basecampProject

    Using Basecamp to manage your classroom/makerspace projects and print queue

    Pioneer Andrew Woodbridge uses Basecamp to organize his students' projects, and he explains how you can too.

    3D printing in math and chemistry

    A unique 3D printing collaboration between mathematics and chemistry faculty

    Passing it on. After integrating 3D printing into her own math courses, Kristen Schreck helps spread 3D printing across disciplines at Saint Xavier University.

  • World monument project

    Living world monuments assignment

    Pioneer Joanne Barrett shares about a middle school project that combines 3D printing, Augmented Reality, History, and Art.

    parts in cura (1)

    My reintroduction to Netfabb

    A recent NetFabb workshop convinced the writer that they can't live without this application. See why it's time to take another look at Netfabb.

    hero wrenchs

    Wrench Engineering

    Inspired by NASA printed in space wrench, Pioneer Rob Morrill gave his fifth graders a design challenge to design their own real-world tool.

  • goblet project

    Integrating 3D printing and The Goblet Project

    Pioneer Chris Hanusa shares one way he integrated 3D printing into his Integral Calculus class

    Finland’s mobile libraries

    3D printers in the public library: Finland ahead of the curve

    US Professor Joshua Pearce, sponsored by Fulbright Finland for Research, shares his first impressions of Finnish libraries.

    Happiness

    Lessons in letting go - releasing student energy speeds creation and use of our art and innovation lab

    Pioneer John Nordell enlisted his students to put the Art and Innovation Lab together. The result was one of the best experiences he has had as an educator.

  • toni teaching fusion

    Teacher training for 3D printing

    Pioneer Dr. Toni Szymanski thought that 3D printing could engage students in math classes. To test this out, she had to learn all about 3D printing first.

    ceramic stamps

    Digital fabrication informs ceramics decorative process

    Pioneer Young Kim blends his classical arts background with digital fabrication to create a ceramic project that incorporates 3D printing.

    buckeyball

    Modeling rocks at iCREATE

    iCREATE's Tiffany Huang shared an iCREATE success story about Jackie Zheng.

  • Empathy map

    Elementary-College Engineering Design partnership

    Pioneer Matthew Wigdahl writes about how his fifth graders and local undergraduate engineering students learn from each other.

    014-SchoolHero

    Introducing Ultimaker’s Core Lessons Set for STEAM Educators

    Designed by the Ultimaker North America Community Team, our new Core Lessons: STEAM Set is a resource for educators who need inspiration and ideas when they bring 3D printing into their classrooms.

    MakerGirl

    Empower one MakerGirl, empower the world

    how the group traveled 10,000 miles around the country to help bring 3D printing workshops to over 1,000 young girls. In this week's post we catch up with the organization that is still working to create a new generation of confident, creative women leade

  • Georgia Connections Academy

    Building a mobile maker space: part 2—up and rolling

    So what exactly does a Mobile MakerSpace look like? That's the question I had to ask myself once I began the project of bringing 3D printing technology to virtual students. When we started this project, we knew we needed something that would be easy to

    Building a mobile makerspace: part 1— getting started

    Building a mobile makerspace: part 1— getting started

    Pioneer Wendy Aracich is putting together and implementing a mobile MakerSpace for her virtual school of 4000 students spread across the state of Georgia.

    LibraryLyna

    Tactile Problem/Solution Bank Community Project

    3D modeling and printing should be accessible of every educator so that they may offer their students tactile means to understand spatial concepts.

  • Design Engine Box

    Design Engine community project

    We want to challenge educators and students to help evolve the Design Engine game. We want to see how you're using or modifying the game with your students, and we want to incorporate your ideas into the next edition.

    coca cola upcycle

    Upcycling community project

    Upcycling challenges students and educators to use their creativity and 3D printing skills to breathe new life into a few familiar objects.

    pecha-kuchas

    Construct3D to Kamehameha Ed Tech Conference

    Last year Pioneer Greg Kent traveled from Hawaii to North Carolina to attend Construct3D 2017. We thought we'd share his reflections with you now since we recently announced Construct3D 2018

  • Plastic Ocean by Kevin Krejci

    Ocean Plastic Community Project

    The Ultimaker Community Team will be launching a series of interdisciplinary projects over the next few months that challenge students to research, explore, design, and 3D print. Ocean Plastic is the first project of this series.

    fractal2

    3D printed fractals at JMU 3SPACE

    Pioneer Professor Laura Taalman, (a.k.a. mathgrrl), reviews a multi-week study of fractals by general education math students in the JMU 3D printing classroom.

    Drone1

    Using drones and 3D printing to develop design thinking during a summer robotics camp

    Pioneer Yuriy Drubinskiy writes about his experience leading a summer program and how creating drones with 3D printing brings form, structure, and design together.

  • The Starter Pack launch event at Digital Harbor Foundation

    Introducing the Ultimaker Design Engine Starter Pack

    Presenting the Ultimaker Design Engine Starter Pack: a game created to provoke, inspire, and entertain students, educators, 3D designers, artists, and engineers of all experience levels!

    first puzzle cube and package to be made on a 3D printer entirely out of PLA

    Davidson Desktop Doohickeys: Puzzle cubes

    Pioneer Adam Davidson writes about a project in his curriculum that is a rite of passage for his high school's engineering program students. He explains at how it started and what caused it to change

    soft robot mold

    3D printing with UMaine Bioengineering students

    If it’s happening in Ultimaker’s world, you can find out about it here. 3D printing stories about inspiring moments, original 3D printed projects and much much more.