# Make math concrete with digital fabrication

For too many students, doing mathematics means just plugging numbers into a memorized formula to get an answer. And because they don’t understand the formulas they’re using, they often fail to use the right one.

Take a look at Isaac’s work below, for example. He is a fifth grade student who tried to find the surface area of a rectangular prism by incorrectly adapting a previously memorized formula for calculating perimeter. He calculated two times the length plus two times the width (2L + 2W) and tried to account for the height by multiplying it by 4, then adding it to the previous sum. Unfortunately, Isaac is not alone in this type of approach. Students who use formulas by rote may never come to see mathematics as sense-making and may never understand the formulas they use. And there are so many formulas to memorize! Teachers who prematurely introduce students to formulas risk denying them opportunities to develop the necessary conceptual foundations for mathematical understanding.

One way to help students like Isaac understand surface area is to present it conceptually using manipulatives. Physical manipulatives — concrete objects such as blocks, chips, unifix cubes and geoboards — are already a common staple of mathematics classrooms. Virtual manipulatives — computer images created by, for example, computer-aided design (CAD) software, Geometer’s Sketchpad or Flash programs — are also becoming more and more prevalent in classrooms. And now, with the increased availability of 3D printers and CAD software in schools across the country, digital fabrication gives teachers an opportunity to incorporate both physical and virtual manipulatives into the same lesson.

### What is digital fabrication?

*Digital fabrication* is the process of creating a physical object from a digital design developed on a computer. You can transform your classroom computers into personal fabrication systems that your students can use to create both virtual and physical manipulatives by installing CAD software and hooking them up to a 3D printer, a die cutter or even a standard inkjet or laser printer.

CAD software, such as FabLab ModelMaker, Fab@School Maker Studio and Autodesk 123D Design, allows students to design, rotate, transform and measure 3D solids. After creating these virtual manipulatives on screen, they can print the shape’s corresponding net on cardstock, vinyl or other types of material. Finally, they can use a die cutter to cut the outer edges of the net and perforate the fold lines to make construction of the physical 3D solid easier, or they can just cut it out by hand.

*Both virtual manipulatives (shown in the photo on the left) and physical manipulatives (printed out as a net, on the right) can help students make sense of surface area.*

### How we used digital fabrication to teach surface area

Our fifth grade students’ used their own personal fabrication systems — which included MacBooks, FabLab ModelMaker, Canon inkjet printers and Silhouette die cutters — to complete the following lessons during their digital fabrication unit:

Initial exploration.Students looked at different models of 3D solids and identified key attributes, such as the number of faces, the shapes of the different faces, etc.

Introduce hardware/software.Students created their own one-inch cardstock cubes, which gave them the opportunity to explore the software. They designed their solids, measured different attributes, looked at different views, and printed physical models from their virtual manipulatives.

Define surface area.We asked students what they thoughtsurface areamight mean, focusing on the relationship between surfaces and faces, their prior experiences with area, and the key attributes of 3D solids they had identified.

Connections between representations.Using the software, students explored the relationship between their printed-out, folded cubes and the cubes’ nets. Students rotated their cubes using the software to explore all six faces. Then they hypothesized how they might calculate surface area.

Rotating and coloring.We asked students to review their definitions of surface area and how it is calculated. Students used the software to rotate and color different faces of their cube and explore the relationship between the colored faces on the cube and its net. After calculating the surface area of their cubes, students constructed other types of rectangular prisms and calculated the surface area of these solids.

### Developing strategies and skills

The digital fabrication unit gave students the chance to develop effective strategies for learning about surface area, such as recognizing the qualities of 3D figures they can’t see in a 2D representation and carrying out multi-step processes.

When students look at a 2D representation of a rectangular prism, they are able to see only the top face, front face and a side face. To find the surface area, they need to understand that each visible face has a corresponding face that is not visible. Throughout the digital fabrication unit, students had multiple experiences with nonvisible faces. The software allowed them to rotate 2D representations of prisms to make the nonvisible faces visible, and students could see both the 2D representation of a solid and its corresponding net, which displayed all of the faces. Students were also able to explore physical models of their solids.

They also developed strategies to keep track of their work, such as listing the areas of the faces, labeling each face with letters to account for all six faces, and annotating faces with calculated areas. Again, students used the software to help facilitate these strategies. Coloring each face and exploring the relationship between opposite faces gave students opportunities to keep track as they calculated face areas. When working with their physical prisms, they used their fingers as calipers to count faces in pairs. They would also label faces with a mark, letter or the area.

Take a look at Isaac’s calculations, below, after completing the digital fabrication unit. His work shows his ability to both consider what was not shown and keep track of his calculations.

### Digital fabrication addresses the ISTE Standards

The ISTE Standard for Students addressing Creativity and Innovationrecommends that students “apply existing knowledge to generate new ideas, products or processes” and “use models and simulations to explore complex systems and issues,” both of which happened in our digital fabrication unit. One ISTE Standard for Teachers, Design and Develop Digital Age Learning Experiences and Assessments, asks teachers to “design, develop and evaluate authentic learning experiences and assessments incorporating contemporary tools and resources to maximize content learning in context.” And digital fabrication promotes the ISTE Standards for Administrators’ Digital Age Learning Culture*, which *encourages school and district leaders to “model and promote the frequent and effective use of technology for learning” and “provide learner-centered environments equipped with technology and learning resources to meet the individual, diverse needs of all learners.”

Digital fabrication also supports the National Council of Teachers of Mathematics’ Principles and Standards for School MathematicsTechnology Principle: “Students’ engagement with, and ownership of, abstract mathematical ideas can be fostered through technology. Technology enriches the range and quality of investigations by providing a means of viewing mathematical ideas from multiple perspectives.”

Digital fabrication technology played an integral role in helping our students really understand surface area. Using physical and virtual manipulatives helped them develop conceptual understandings of surface area and facilitated the development of two problem-solving strategies they will be able to apply to other content areas.

*Kimberly Corum is a Ph.D. student in mathematics education at the University of Virginia. A former high school math and science teacher, her research interests include students’ understanding of 3D measurement and the use of manipulatives in the classroom to support student learning.*

*Joe Garofalo is co-director of the Center for Technology and Teacher Education at the University of Virginia. His primary interests are in mathematical problem solving, use of technology to facilitate mathematics learning and mathematics teacher education.*