Category Archive for: ‘Fabrication’
This video is one of the final requirements for Dr. Glen Bull’s MacArthur Digital Media and Learning Competition submission. The submission requests funding for the development and creation of Fab@School laboratories that feature curriculum adapted for digital fabrication in elementary schools as well as the establishment of an online, open-source library for sharing and disseminating digital designs.
After viewing the video, I encourage you to register your vote for the project by posting comments! Your comments will be instrumental in helping this become a funded reality. Even if you already commented in the previous round, you may comment again incorporating your reaction to the video.
Post Comments at http://dmlcompetition.net/pligg/story.php?title=630
Deadline for comments/feedback: Thursday, April 22, 10 AM EDT
I also posted a number of other videos that might be of interest to people wanting to know more about digital fabrication and the Curry School of Education’s commitment to this STEM initiative.
There is a great post on Scientific American about “Juice Box Geometry” that relates to this post. I encourage readers of this post to check it out!
This post is a redesign of Kris’s Cereal Box digital fabrication project. Although the tweaks that I describe move Kris’s elementary lesson into the middle school and high school realm, it could be altered to be more of a discovery-type experience for elementary students familiar with digital fabrication and ModelMaker.
I came across a lesson this weekend that very simply stated that companies like General Mills and Post must have some reason for creating cereal boxes in their current dimensions other than “bottom line” economic decisions. The rationale was simple: If companies that make cereal really wanted to maximize their profits, then they would reconsider the way in which cereal was packaged. Cereal companies could sell the same amount of bran flakes and crunchy goodness (volume) by redesigning the packaging (surface area) so that it resembled a cube. Less surface area (packaging materials) and equal volume = lower manufacturing costs. Plus, there would be ecological benefits.
This afternoon I pulled out something that I have not touched for a truly mathematical purpose since high school: a ruler. I wanted to test this lesson’s underlying ideas. I grabbed my box of Total Cinnamon Crunch off the refrigerator, and calculated the following dimensions using my ruler:
- Length: 7 5/8 inches
- Width: 2 inches
- Height:10 1/8 inches
I had to do some fraction-decimal conversions in order to re-create the 3D model below. That was worthwhile as a standalone activity if readers are thinking about mathematical procedures, processes, and algorithms. The following image is a 3D representation of my cereal box (without graphics) in FabLab ModelMaker.
I have to admit that I did not spend the time calculating the cereal box’s surface area and volume. I actually used ModelMaker’s properties to give me the answer for surface area and volume.
- Surface Area: 225.406 inches squared (rounded)
- Volume: 154.406 inches cubed (rounded)
To test the hypothesis, I took the cubic root (I forget the mathematical terms) of the volume, 154.406 inches cubed. The rounded answer was a length, width, and height of 5.365 inches (rounded), the dimensions for my cube. The result? A package that has relatively similar volume (disregarding variations due to rounding) and a much smaller surface area!
If I was going to engineer a cereal box, I would not begin with off-the-shelf examples as a starting point. Greater surface area for a given volume equals higher manufacturing costs and greater waste, a practice that has to have some origin… Why do companies like General Mills and Post continue to produce cereal boxes that are clearly economically and ecologically inferior? The answer to this question is likely an entirely new learning strand that relates to social studies…
Is there truth to the idea that packaging relies more on the Golden Mean then the economic bottom line?
Cross-posted on the Fab@Home blog.
Personal manufacturing, digital fabrication, and 3D printing are terms and processes that aren’t foreign to Fab@Home users. With the publication of Chris Anderson’s Wired article entitled In the Next Industrial Revolution, Atoms Are the New Bits, a wider swathe of the general public now has these less-than-palpable ideas in their vocabulary. Yet, if Anderson is correct with his assertion that society is at the precipice of a “democratization of manufacturing,” then I imagine that 3D printing and personal manufacturing will not only be semantically accessible and commonplace but directly actionable.
For a number of elementary students in Charlottesville, VA, personal manufacturing and digital fabrication are already entrenched facets of school and learning.
My work at the University of Virginia’s Center for Technology and Teacher Education focuses on how personal manufacturing can support, extend, and transform elementary students’ understanding of mathematics. More specifically, geometry and spatial visualization. Led by Dr. Glen Bull, we are currently using 2D fabricators (Aspex’s FabLab ModelMaker) with a pilot group of local elementary schools.
Students aren’t learning about personal manufacturing or assisting an expert; they are designing and fabricating objects as active users in the experience. The low entry-level skills needed to create 3D designs in FabLab ModelMaker coupled with the inexpensive 2D fabricators, Silhouette, remove the barriers for implementation and heighten the learning opportunities.
Make no mistake: The students are learning, and the teachers are uncovering previously unseen misconceptions in students’ thinking about geometry, shapes, and visualization.
Earlier this month, Jeff Lipton and Hod Lipson traveled to Charlottesville for the unveiling of the Children’s Engineering Center (CEC) at the Curry School of Education. The CEC features a number of fabrication stations equipped with computers, scanners, design software, and 2D fabricators. The cornerstone machine in this new facility is a Fab@Home 3D printer.
The Fab@Home 3D printer will play a prominent role in not only the pilot group of elementary classrooms but also undergraduate and graduate students preparing to enter K-12 schools. The Curry School is planning on embedding engineering and digital fabrication within elementary math methods courses, and software like FabLab Model Maker as well as the Fab@Home unit make creating physical models and manipulatives both accessible and easy.
The text below is a “rough” literature review connecting digital fabrication systems/techniques to elementary students’ understanding of geometry. I decided to create a post with my work because I think that it could be a valuable resource for people other than my professor and me.
I have included links in appropriate places. Direct links to references are not included because of copyright restrictions.
Enhancing Students’ Understanding of Geometry Through 3D Construction of Shapes
Geometry is a field of study that provides students with ways to interpret and react to shapes and space, and it offers opportunities to reflect on one’s immediate physical environment (Clements, 1998). Apart from being a recognized mathematical strand, geometric understanding and visualization is evident in other math-related topics including algorithmic thinking, optimization, functions, limits, and trigonometry (Goldenberg, Cuoco, & Mark, 1998). Geometry also connects to discrete disciplines like design and engineering.
Despite evidence of geometry’s importance and its applicability to other domains, American mathematics instruction currently places little emphasis on geometry as compared to other mathematical fields of study (Porter, 1989). The scope of geometry teaching in schools generally consists of an initial exposure to recognizing and identifying shapes in grades pre-K through 8 and optional courses covering axiomatic, Euclidean geometry in high school (Clements, 2003). Even when formalized curricular materials exist in upper elementary and middle school classrooms, teachers spend no time directly teaching geometry or teach to an “exposure” level of understanding (Porter, 1989). The cursory coverage and the length of time between formal instruction create conceptual holes such that elementary students often possess a degree of formal understanding of shapes that middle and high school students lack (Clements, 2003).
Recent research suggests that traditional geometry curricula at the elementary level compound the problems of coverage and time. According to Porter (1989), typical instruction focuses on “such topics as recognizing and naming geometric shapes, developing skills in using a compass, ruler, or protractor to make geometric constructions, and plotting points on a two-dimensional graph” (p. 11). Conversely, the National Council of Teachers of Mathematics (NCTM) suggests that geometry teaching should incorporate experiences that include manipulating concrete models, creating drawings, and using dynamic software (Standards for school mathematics: Geometry). Although a number of districts and schools have since addressed the inconsistency between NCTM’s recommendations and classroom practice though adoption of progressive curricula like Investigations in Number, Data, and Space, American elementary students’ proficiency with geometric concepts remains low in comparison to international students of equivalent age (Gonzales et al., 2007).
Effective elementary instruction is important because children begin developing notions of special representations and building geometric ideas at an early age. Piaget and Inhelder’s (1967) theory of space conception posits that children construct mental representations of shapes’ connectedness, enclosure, and continuity by actively manipulating physical objects in their immediate environment. Through play that includes building, drawing, and perceiving, children initialize cognitive frameworks that contain rudimentary geometric principles, and this often occurs prior to explicit geometry instruction (Clements & Battista, 1992).
Children’s self-developed spatial and geometric understanding are primarily characterized by visual recognition of rudimentary shapes (van Hiele, 1986). Although children are unable to apply geometric terms to groups of shapes, they are able to group and classify objects. For example, an upright rectangle is similar to a door which is a physical object that is touched (Clements, 2003). Absent from whole object recognition are shape properties and attributes; horizontally orienting a rectangle results in misidentification because the shape does not look like commonly encountered objects.
van Hiele (1986) contends that the next phase in children’s understanding of geometry and shapes is characterized by description and analysis. The descriptive-analytic level of comprehension is reached when children are able to recognize and identify shapes by inherent properties like number of sides and internal angles. Children rarely attain descriptive-analytic thought processes without explicit instruction, and many students fail to recognize shape properties until middle or high school because of traditional geometry curricula (Clements, 2003). However, children’s capacity to cognitively master shape description and analysis at an early age is possible if properly scaffolded with effective teaching methodologies.
The type of instruction that builds elementary children’s descriptive-analytic understanding is one that incorporates exploration, investigation, and playful manipulation of shapes. van Hiele (1999) suggests a five-phase sequence of activities that include:
- an inquiry phase during which materials lead children to explore and discover certain structures;
- a direct orientation phase that involves tasks with shapes that are presented in such a way that the characteristic structures appear gradually;
- an explication phase during which the teacher introduces terminology and encourages children to use it in their conversations and written work about geometry;
- a free orientation phase during which the teacher presents tasks that can be completed in different ways and enables children to become more proficient with what they already know; and
- an integration phase that requires children to pull together what they have learned (p. 316).
The types of tasks in this sequence can include measuring, folding, coloring, sorting, or cutting shapes in physical and digital form (Clements, 2003).
Manipulatives serve an important role in sequential activities that build children’s analytic understanding of geometry. However, using manipulatives without careful thought of the pedagogical connections often leads to minimal gains in learning (Clements, 2000). A manipulative, whether physical or digital, must be used in meaningful ways that avoid rote application and create cognitive connections between the mathematical concepts and the manipulative (Uttal, Scudder, & DeLoache, 1997).
There is a growing group of researchers examining the use of paper as a manipulative in instructional activities that focus on analytical understanding in geometry. First, Wheatley and Reynolds (1999) describe children’s enhanced spatial sense and property recognition in tasks that involve cutting and folding paper. In addition, Japanese teachers who use multiple representations including two- and three-dimensional objects teach children who have comparatively higher achievement in geometry (Hafner, 1993). Finally, incorporating three-dimensional paper models, shape nets, possibly promotes greater synthesis of shape properties and attributes for children (Nieuwoudt & van Niekerk, 1997).
A relatively large body of research supports the benefits of interactive computer programs and digital manipulatives. Intelligent tutors (The Geometry Tutor), computer games (Shapes), and authoring tools (Logo) have each resulted in children’s increased capacity to understand geometric concepts (Clements, 2003). According to Clements (2000), computer programs offer affordances that are unavailable in tangible manipulatives, namely digitally altering a shape’s size, composing and decomposing shapes with precision, and recording and replaying children’s actions. When combined with pedagogically appropriate scaffolding, interactive computer programs support and enrich children’s analytic understanding of geometry.
There is a dearth of research focusing on interactive computer programs that produce customizable, three-dimensional shape nets through an immersive, CAD-like environment. One such program, Tabs MST, allows for dragging, scaling, and combining digital shape manipulatives. Tabs MST includes an additional feature that enables users to print, cut, and build proportionally accurate paper shape nets of all digital shape manipulatives (Figure 1).
The case study that follows is a qualitative examination of upper elementary students’ descriptive-analytic understanding of geometry after using an interactive computer program, Tabs MST, and electronic die-cut machines (digital fabrication systems) to create digital and tangible three-dimensional shapes. Situated in a curricular experience that aligns with van Hiele’s (1999) five-phase activity progression, the study aims to address one question: To what extent does constructing digital and physical three-dimensional shapes promote identification of shape properties and attributes?
Clements, D. H. (1998). Geometric and spatial thinking in young children. Arlington, VA: National Science Foundation.
Clements, D. H. (2000). ‘Concrete’ manipulatives, concrete ideas. Contemporary Issues in Early Childhood, 1(1), 45-60.
Clements, D. H. (2003). Teaching and learning geometry. In J. Kilpatrick, W. G. Martin & D. Shifter (Eds.), Research companion to Principles and Standards for school mathematics (pp. 151-178). Reston, VA: NCTM.
Clements, D. H., & Battista, M. T. (1992). Geometry and spatial reasoning. In D. A. Grouws (Ed.), Handbook of Research on Mathematics Teaching and Learning (pp. 420-464). New York: Macmillan.
Goldenberg, E. P., Cuoco, A. A., & Mark, J. (1998). A role for geometry in general education. In R. Lehrer & D. Chazan (Eds.) Designing Learning Environments for Developing Understanding of Geometry and Space (pp. 3–44). Reston, VA: NCTM.
Gonzales, P., Williams, T., Jocelyn, L., Roey, S., Kastberg, D., & Brenwald, S. (2007). Highlights from TIMSS 2007: Mathematics and science achievement of US fourth-and eighth-grade students in an international context. NCES 2009-001,
Hafner, A. L. (1993). Teaching-method scales and mathematics-class achievement: What works with different outcomes? American Educational Research Journal, 30(1), 71.
Nieuwoudt, H. D., & van Niekerk, R. (1997). The spatial competence of young children through the development of solids. Paper presented at the meeting of American Educational Research Association, Chicago, IL.
Porter, A. (1989). A curriculum out of balance: The case of elementary school mathematics. Educational Researcher, 18(5), 9.
Standards for school mathematics: Geometry Retrieved 10/4/2009, 2009, from http://standards.nctm.org/document/chapter3/geom.htm
Uttal, D. H., Scudder, K. V., & DeLoache, J. S. (1997). Manipulatives as symbols: A new perspective on the use of concrete objects to teach mathematics. Journal of Applied Developmental Psychology, 18(1), 37-54. doi:DOI: 10.1016/S0193-3973(97)90013-7.
van Hiele, P. M. (1986). Structure and insight : A theory of mathematics education. Orlando, Fla.: Academic Press.
van Hiele, P. M. (1999). Developing geometric thinking through activities that begin with play. Teaching Children Mathematics, 5(6), 310-316.
Wheatley, G. H., & Reynolds, A. M. (1999). ” Image maker”: Developing spatial sense. Teaching Children Mathematics, 5(6), 374-378.
Fast Company’s Jamais Cascio wrote an interesting article entitled The Desktop Manufacturing Revolution a couple of months ago. In the piece, Jamais describes the coming of a digital fabrication revolution when everyday people will be able to take a digital design, import it into a personal computer, and then send the file to specialized hardware that creates a 3D physical object.
Take a design for a simple product–an engine part, for example, or a piece of silverware, and feed it into a computer. Press “print.” Out pops (for a sufficiently wide definition of “pops”) a physical duplicate, made out of materials plastic, ceramic, metal — even sugar. Press “print” again, and out comes another copy–or feed in a new design, for the next necessary object.
It may sound like a scene from a low-rent version of Star Trek, but it’s real, and it’s happening with increasing frequency. This process goes by a few names, but it’s most commonly known as “3D Printing” (the older name, “rapid prototyping,” no longer captures the range of uses, while the other alternative name, “fabbing,” is a little too cyberpunk for the moment). While the process has been around since the mid-1980s, the cost of 3D printers has been dropping quickly, and now range to well under $10,000. [Fab@Home is creating a 3D printing kit that costs approximately $1500.] If that still sounds like a lot of money, you’re right–but don’t forget, it was when laser printers dropped to this price range in the mid-1980s that the desktop publishing revolution kicked off.
Much of my work at UVA’s Center for Technology and Teacher Education deals with the very same digital fabrication techniques mentioned in this article. Although we aren’t using 3D printers (yet), I am supporting a small network of teachers in Albemarle County who are using commercial software (Adobe Illustrator, Google Sketchup, Aspex’s Tabs MST) and die-cut machines to create physical, paper-based 3D models. The movie below is a simple depiction of the process and steps.
The ability to create digital designs like the one in the video might seem overly complex and technologically prohibitive for K12 teachers and students. Yes, this is an advanced example (teachers and students aren’t likely to dream up the parts for a Rack-and-Pinion gear system). However, the creation of 2D “shape nets” with software like Aspex’s Tabs MST and Google Sketchup is relatively easy. Sending the designs to die-cut machines for later construction, although not shown or described in this post, is a matter of learning the procedural steps.
How are the Albemarle County teachers using the software and equipment to digitally fabricate objects?
Elementary school teachers and students are using Aspex’s Tabs MST to create physical models that support geometric curricular topics as well as an emerging elementary engineering strand of learning. Tabs MST enables teachers and students to quickly and easily create shape nets of a user-defined size in a 3D environment that is user-friendly. Drag-and-drop, push-pull, and snap-to features make the rendering of complex designs in tangible form both reachable and practical.
Middle school students are using Adobe Illustrator to design labels and reproducibles for various school projects. For example, the image below depicts monster-like letters (alphabeasts) that middle school students created for elementary students to use when learning the alphabet. The alphabeasts are designed in Illustrator and then printed and cut on magnetic sheets or static-cling vinyl.
As a component of an engineering course focusing on problem solving, high school students are creating digital models in Google Sketchup. The digital designs are then flattened, sent to Adobe Illustrator, and then reproduced in physical, paper-based form. The movie below shows the flattening process in Google Sketchup.
Dr. Glen Bull sums up the educational potential of digital fabrication systems when he says,
Fabrication technologies provide exposure to many mathematical and engineering concepts. Scaffolded practice for students in upper elementary grades and above provides the opportunity to see abstract visualizations- including students’ own drawings and sketches on the computer- translated into physical objects, offering an opportunity to explore these ideas at an early age in a very concrete way.
In later grades, students can study compound machines in science class, use mathematics to design their own digital models, and then fabricate their designs as three-dimensional objects. These types of activities give students direct experience with vectors and geometric transformations that are useful in many STEM-related fields.
Engineers and architects move between the digital world of the computer and the physical world, visualizing the result that occurs when a 2D drawing on the screen is transformed into a 3D object. Students in school can understand mathematical concepts by making connections between virtual and physical representations. These activities can simultaneously prepare them for 21st century careers.
-Glen Bull and Joe Garofalo, Personal Fabrication Systems: From Bits to Atoms
I created a website to support all of the teachers in the digital fabrication network. To learn more, please visit http://www.digitalfabrication.org
Digital Fabrication Image