Above are some videos that show a variety of ways to use folding.  A list of these videos is to the right.  In them, I use Tabula to demonstrate the folding process, but paper and sometimes scissors and a straightedge are all that you need.  You can also view these videos at the  Numeracy Works YouTube Channel.

Throughout my career as a math teacher I’ve had students cut, fold, tape and manipulate paper in a number of ways.  The goal was to learn geometric properties in an active, hands-on, and usually collaborative means.  I always felt a great deal of success in this approach.  Recently, I’ve been developing a computer program called Tabula that works with virtual shapes much as one works with paper shapes.  My interest and knowledge in this kind of manipulation, in particular folding, has grown. I’ve created this web page to share what I’ve learned.  Below is a somewhat cursory review of relevant research with links.  Also is a list of resources that I’ve found useful. 

Benefits of Paper Folding

What are some of the benefits of paper folding?  It is a manipulative.  The effectiveness of manipulatives is well documented by research.  It is accessible.  Both ordinary paper and patty paper are cheap.  It tends to be a positive experience.  Coad makes a good case for the positive affect associated with paper folding.  All traditional compass and straight-edge constructions can be performed plus additional constructions such as the trisection of an angle.  There are also many resources available.  Perhaps the best known resource in education is Michael Serra’s Patty Paper Geometry.

Approaches To Paper Folding

One approach to paper folding is origami.  Origami has the benefit of a long tradition that includes an artistic aspect.  The research on paper folding is primarily on origami. Research by Norma Boakes, for example, suggests origami is an effective approach in the math classroom.  Another approach to paper folding I’ve seen is a formalized/axiomatic one.  Basic folds are established as axioms which sets up a system conducive to proofs and rigor. Olson and Coad provide good examples of this.  One more approach, and these approaches are by no means exclusive, is an investigation approach.  This is the approach I took. Folding is done to manifest properties and justification can be systematic or sometimes intuitive. 

Dynamic Geometry Software and Paper Folding

What about dynamic geometry environment (DGE) and paper folding?  I will shy away from saying anything definitive.  Jones notes in his journal article that “DGE cannot provide a self-contained environment but that other activities are needed for students to make progress in mathematics.”  Perhaps paper folding could be one of the other activities.  In the progression through the Van Hiel levels, paper folding as well as other manipulatives, would seem appropriate at an earlier stage than DGE.  DGE is a great tool, but there are issues to consider such as the amount of procedural knowledge and student readiness.  Perhaps for many, the best relationship for DGE and folding might be a complimentary one.

I’d be grateful for any comments, corrections, or contributions to this webpage.  If you would like to know more about DGE software, I have created a comparison page.  Also, I have a website for Tabula if you are interested.

David Hartmann (david@numeracyworks.com)

Research Bibliography

1. Sowell, E. (1989). Effects of manipulative materials in mathematics instruction. Journal for Research in Mathematics Education, 20(5), 498–505. This 1989 meta-analysis appears to show “math achievement improves through the long-term use of concrete instructional material but with conditions. (I’ve read the abstract only. Link is to the abstract.)
   

2. Suydam, Marilyn N.; Higgins, Jon L. (1977). Activity-Based Learning in Elementary School Mathematics: Recommendations from Research Information Reference Center (ERIC/IRC), Ohio State University - This review of K-8 research conclude “lessons using manipulative materials have a higher probability of producing greater mathematical achievement than do non-manipulative lessons.” 
   

3. Boakes, Norma J. (2009). Origami Instruction in the Middle School Mathematics Classroom: Its Impact on Spatial Visualization and Geometry Knowledge of Students. Research in Middle Level Education, v32 n7 p1-12 -  A quasi experimental study on the effectiveness of origami use versus traditional instruction.  Provides a good discussion of the pedagogical concepts involved in origami that can be generalized to paper folding and manipulation generally.  The results of the study suggest that origami is as beneficial as traditional instruction.
   

4. Jones, Keith (2002) Research on the use of dynamic geometry software: implications for the classroom. MicroMath, 18, (3), 18-20.
   

Other Folding and Manipulatives Resources

1. Mathematics of Paper Folding @ Wikipedia - A short description with many links.
   

2. Patty Paper Geometry by Michael Serra - A classic resource from Key Curriculum Press containing dozens of “open investigations” involving folding. Serra provides a complete lesson with each investigation.  Patty paper is cheap and great for folding but the material in the book pertains to folding in general.
   

3. Proving the Area Formulas by Paper Folding by Yun Quan Zhang - The area formulas for triangles, trapeziums,  and kites are derived by folding.  (To access this paper you’ll need to use a service.)
   

4. Mathematics Through Paper Folding by Alton Olson, University of Alberta - An excellent collection of paper folding exercises for different ages.  Diagrams are well done and appropriate points of discussion are posed.
   

5. Improving Mathematics Teaching by Using Manipulatives by James W. Heddens, Kent State University - This short paper targets elementary level manipulatives.  The advice on using manipulatives and assessment can be generalized to higher level manipulatives.
   

6. Paper Folding in the Middle School Classroom and Beyond by Lance Coad - Well written description of paper folding uses in the classroom. Unlike most resources on paper folding, this is not focused on origami.  The author sets up paper folding axioms and approaches activities systematically.

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8. Last Updated 6/27/2011