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.

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:

1. an inquiry phase during which materials lead children to explore and discover certain structures;
2. a direct orientation phase that involves tasks with shapes that are presented in such a way that the characteristic structures appear gradually;
3. an explication phase during which the teacher introduces terminology and encourages children to use it in their conversations and written work about geometry;
4. 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
5. 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?

References

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.