PaPer taBLe Emerging Engineers Design a

w362 MatheMatics teaching in the Middle school ● Vol. 18, No. 6, February 2013

With the advancement of special-ized middle schools and high schools focusing on the arts, communication, engineering, mathematics, and sci-ence, many students who attend tra-ditional schools miss out on valuable learning opportunities—in particular, when it comes to learning mathemat-ics. Mathematics classrooms can be fi lled with real-world applications and concepts for all learners, regardless of the school environment. This notion is supported throughout the Common Core State Standards for Mathemat-ics (CCSSM) with Mathematical Practices centered on—

The engineering design process challenges students to work together and communicate as they construct learning.

Mary C. Enderson and Melva R. Grant

• making sense of problems and persevering in solving them; and

• constructing viable arguments and critiquing the reasoning of others.

When teachers present middle school students with problems grounded in engineering principles, students will become active learners and will view mathematics as stimu-lating and useful. They will also learn that through mathematics, they can be innovative. We present an engi-neering project called Paper Table, an investigation into building a sturdy table out of newspaper and cardboard. P

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The engineering design process challenges students to work together and communicate

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leg length

364 MatheMatics teaching in the Middle school ● Vol. 18, No. 6, February 2013 Vol. 18, No. 6, February 2013 ● MatheMatics teaching in the Middle school 365

Throughout this project, the engi-neering design process is implement-ed and connected to middle school mathematics concepts.

the engineering design Process for Middle school Math Engineering design projects provide an environment in which students can be active, inquisitive, and creative. However, engineering is the one area in STEM education that appears to be overlooked in traditional school environments. This component of STEM is likely ignored in schools be-cause it is not included in most state standards. This situation is unfortu-nate because engineering provides a real-life application of mathematics.

The engineering design process is similar to mathematical problem solv-ing (NCTM 2000) and includes the steps shown in figure 1. These steps are rarely executed in a linear fashion, and designers often toggle back and forth among them as progress is made toward a final solution (Science Bud-dies 2011).

On the basis of the geometry content in the Common Core State Standards (CCSSI 2010) and the en-gineering framework discussed above, we believe an activity called Paper Table (WGBH Educational Founda-tion 2008) is a viable task for middle school students. This activity is part of the Design Squad Nation® program that is broadcast on PBS.

The show presents real problems that center on building an item and then testing it against requirements. The show’s companion website (pbskids.org/designsquad/parents educators/resources/paper_table .html) presents challenging problems and tasks for students, who then analyze and design solutions. We believe that such tasks can be used to develop a greater understanding of mathematics.

the PaPer table Project Producing a paper table can lead to explorations of geometry and mea-surement. CCSSM grade 7 standards in these two areas include—

• drawing and constructing poly-gons;

• using a protractor to measure and draw specific angles in a polygon;

• finding the area of polygons; and • finding the surface area of three-

dimensional figures.

The Paper Table project chal-lenges students to build a strong table that is 8 inches tall using newspaper, masking tape, and cardboard. The design requirements specify that the table must be sturdy enough to hold a heavy book (WGBH Educational Foundation 2008). To help math teachers envision how to work the project into the classroom, we recom-mend a three-part framework. The math concepts are discussed, which students can then apply to the project. Last, new ideas may emerge from the investigation, leading to extension activities.

In part 1, students explore the properties of polygons and make con-jectures about the strength of various shapes to ascertain if they could serve as table legs. Students work in groups to build different polygon models using rolled-up paper (see fig. 2) and test each polygon’s strength. Finally, they must come to consensus within their groups to determine which poly-gon is strongest.

As students build each polygon

model, they must provide specific properties, such as side and angle measures and orientations. To test for strength, students push down on the polygon sides and vertices using their hands to get a feel for the amount of weight the polygon can hold (see fig. 3). To reach agreement, group members need to develop convincing arguments to support their choices for the strongest polygon.

We suggest that the initial focus for exploration be on a family of polygons such as quadrilaterals. As shown in figure 2, students are explor-ing convex and concave quadrilaterals. For homework, they build, identify, and test other polygons. The goal is to encourage students to consider the strength of polygons to support a sturdy table and make connections between strength and polygon proper-ties, thus allowing students to think about producing structurally sound table designs.

Once the connections between shape and strength are established, students move into part 2: creating sketches and building the actual paper tables using newspaper. Once students go through the engineering design process to build a paper table and share their table designs and results, the teacher can decide whether to move on to the extension activities that include concepts of area and sur-face area, which comprise part 3.

iMPleMenting the PaPer table ProjectEngaging seventh-grade mathemat-ics students in the Paper Table project requires a little more than two class periods (50 to 60 minutes each) and includes assignments completed outside class. The Paper Table outline (see fig. 4) is provided to assist with planning and implementation.

Consider introducing the prob-lem during the end of class (15-20 minutes) the day before students build

their paper tables. Give students in-formation about the supplies they will be given that will need to hold a heavy book: one piece of cardboard (8 1/2 in. × 11 in.) as the tabletop, eight sheets of newspaper, and mask-ing tape. We recommend opening the discussion with a brief description of the problem. As part of the introduc-tion, include these four phases of the project:

1. Research questions2. Hold a design team meeting3. Build, test, and redesign4. Present design team results

Research QuestionsDiscuss the predesign research process that students completed for homework and the sample questions that are online. To support students’ research into idea generation, consider providing a sample stick-figure sketch of a paper table, so that they under-stand that their sketches need to con-vey design ideas and polygon proper-ties, not works of art (see fig. 5).

Encourage participation by help-ing students understand that they are engineers for this design project. In the real world, engineers solve problems

by analyzing project requirements, ap-plying their understandings to design solutions, then testing and redesigning until requirements are met or exceeded. When engineers are given new design projects, they often begin by doing research to familiarize themselves with new objects and concepts related to the project. This is where the mathematics

stages of the design Process1. Define the problem.2. Do background research.3. Specify requirements.4. Create alternative solutions.5. Choose the best solution.6. Do development work.7. Build a prototype.8. Test and redesign.9. Communicate results.

PaPer table inVestigationYour group is responsible for building a paper table that is strong enough to hold a heavy book (like your math book). Since you are the design engineer, prepare for the project by doing a little research, which will enable you to contribute to your design team. The following elements are part of the engineering process:

1. Research questions 2. Hold a design team meeting3. Build, test, and redesign the table4. Present design team results

(Note: A fully formed outline is online. See the More4U informa-tion at the end of the article.)

fig. 1 These steps in the engineering design process provide structure but are rarely executed in a linear fashion.

fig. 4 This outline provides an overview of the Paper Table project.

fig. 2 Making polygon models out of rolled-up paper helps students explore structure and strength.

fig. 3 Math and engineering were put to the strength-push test on two different quadrilaterals.

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engage in complex problem solving in mathematics. If the teacher encounters haphazard design approaches, he or she can intervene with these questions: Why do you think the table collapsed? If you rebuilt the table, what thoughts do you have to make it stronger? What polygons seem to be good candidates for holding more weight? Why? These types of questions will encourage stu-dents to think more deeply about their designs and will support them as they build arguments to defend their ideas and critique other groups.

We worked with middle-grades students during a family math night. During their build-and-test phase, they shared several insightful com-ments about the table structure:

• “When pieces were taped together, there was not enough support to stabilize the table to hold a book.”

• “Looking at polygons to serve as the legs is important. We need more strength so they will not bend.”

• “There was some difficulty in getting the taller table to hold the heavy book. When it was shorter, it was easier to hold the book.”

These emerging engineers used reasoning and discussed results from the engineering design process to find a solution that met the requirements. Here are examples of student findings:

• “Just using the separate rolls as legs and taping them to the cardboard isn’t a great idea. By building it, we saw there wasn’t enough support, so we need to rethink how to stabi-lize things better.”

• “The legs are too wobbly. We need more strength so they won’t bend.”

• “In using a quadrilateral, we need to have something like a brace to support the middle of it, sorta like the diagonals of a quadrilateral or a trapezoid.”

These comments suggest reflec-tions about the table design after a failed test, followed by thoughtful ideas for a redesign to meet require-ments. This type of generative discourse is what the teacher wants to hear from design teams. If this level of discussion is not occurring, we sug-gest asking questions to help students enter into an iterative design-and-test cycle: (a) retest (b) reflect on design failure, (c) collaboratively consider thoughtful ideas for refining the design, and (d) build a new table or adapt the current table if it was not destroyed during the test.

The other important element of the design-and-test phase is to describe in writing design flaws revealed from the test and thoughtful ideas for refining the design of the last model, includ-ing a new sketch. These notations will serve the team well during its presen-tation preparation as well as help them communicate their findings more accurately. Documenting the design process might be a challenge for many students, so we recommend giving

concepts and properties of polygons come into play. Having some knowl-edge of polygons—both convex and concave—allows students to consider all possibilities and how and why the figures can support weight.

Hold a Design Team MeetingThe next day, convene design team meetings (10 to 15 minutes) of groups of three or four students. The first task is to determine who will serve in the roles of (1) lead designer, (2) building supervisor, and (3) materials man-ager—all positions that are aligned with the engineering design process. Several tasks are assigned to the lead designer, as shown on the activity sheet, but overall his or her role is to help the team maintain focus, ensure that the design process is documented, and that all members have equitable opportunities to contribute ideas and participate. The building supervisor

manages the project by making sure that there is an equitable distribution of labor during the process. The ma-terials manager checks that supplies are on hand as needed. If technology is available, consider allowing stu-dents to capture digital records (i.e., photographs and video) during the build-and-test phase that they can use in their final presentation. If this op-tion is possible, the materials manager would be responsible for handling the digital data collection.

Before transitioning to the build-and-test phase, design teams need to show evidence of understanding the problem and planning (i.e., a sketch) for their initial paper table. We rec-ommend that the teacher visit each group during the design team meet-ings to judge whether teams are ready for the next phase. The teacher can ask students to restate the task and the problem requirements if teams

lack clarity. The teacher should ask to see design sketches, but he or she need not validate designs or become a bottleneck to the design process flow.

The engineering design process will efficiently help designers iden-tify poor designs through testing. If a heavy book collapses the table, it is obviously not strong enough. This situation is profitable because it gives students the opportunity to learn about poor design (or poor use of polygons) and to identify ideas for improving or adjusting the design (by selecting other polygons).

Build, Test, and RedesignA large portion of class time should be allocated for the build-and-test phase (45 to 50 minutes). Given the time factor, we anticipate design teams building two or three paper tables each. If the teams are in-structed to reuse the cardboard, then the amount of materials needed can be calculated based on the number of design teams. Allot only one roll of masking tape to each team. If the teams solve the problem too quickly, adjust design requirements. For example, if students are asked to build a paper table capable of holding two or more books, allot three additional pieces of newspaper.

Throughout this phase, the teacher’s primary goal is to ensure that design teams are working collabora-tively to refine their designs. They should not be haphazardly building paper tables with little reflection and thought about what is causing the design to fail. The teacher can observe students analyzing properties of poly-gons and encourage students to focus on how situations change or remain the same as one moves from using a convex to a concave polygon.

The engineering design process is a cognitive endeavor when implemented appropriately and helps learners develop transferable skills needed to

each team a structured recording sheet with steps identified so that design teams can self-assess their written accounts of the designs and processes (the steps are listed in fig. 1).

About 5 to 10 minutes before the end of the build-and-test phase, direct design teams to stop working and capture their current design status, as shown in figure 6. For example, the teams explicitly describe where they are in the design process, comment about what they anticipate happening during their next test, and why. These reflections can be used as evidence when preparing the final presenta-tion. By the end of the build-and-test phase, design teams should have a presentation performance rubric and be given time outside of class to pre-pare for presentations.

Present Design Team ResultsThe final phase, the design team presentations (50 to 60 minutes or more, based on the anticipated average length of time per design team), is arguably the most important component. Students’ learning and understanding of the engineering de-sign process are on display, as is their reflective advice to peers. Minimal guidelines for this phase are included on the activity sheet. Consider invit-ing people to the Paper Table student presentations, including other stu-dents, administrators, parents, and

fig. 5 Sketches of a paper table must precede construction and be functional, not “works of art.”

fig. 6 A little success in holding up a relatively light book shows students that the design must be revised to support more weight.

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The engineering design process is a cognitive

endeavor when implemented appropriately and helps learners develop transferable skills needed

to engage in complex problem solving in

mathematics.


community members. If outsiders are invited begin with an overview of the Paper Table project, including the short (2:15 minutes) video from the Design Squad Nation website.

extensions of the PaPer table ProjectFrom a mathematical point of view, the Paper Table project revolved around properties of polygons and mathematical processes and practices. This task bridges other mathematics, such as finding the area of polygons. After producing models, students can ponder ways to find the area of the polygons that comprise the legs of the table. In some instances, students can measure sides, heights, and angles; in others, they may think about cover-ing the area with square tiles to get an approximation.

In addition to studying area, the visual appeal of the final paper tables can be addressed. In other words, how much paint would be needed to

cover the surface area? Students could look at models that did not work and compare surface area for earlier models versus later models. It would also allow teachers to connect cost to mathematical ideas.

assessing the PaPer table design ProjectSeveral opportunities are available for analyzing student learning and un-derstanding of the engineering design project including formative, summa-tive, and performance assessments. For example, the research assigned for homework could be collected and used diagnostically. This will ensure that students clearly understand the task and design requirements during the design meetings and connect structural design sketches to paper table models during the build-and-test phase.

Informal assessment can also occur during the build-and-test phase. As the teacher visits the design groups and listens to conversations among team

members, questions can be asked to improve collaboration. During the final visit, a score can be assigned based on the quality of conversation or the prog-ress since the initial intervention. The design team presentation is the summa-tive performance assessment. Scoring is established using a rubric (see fig. 7).

A creativity category could also be added to the rubric for bonus points. This category could encourage stu-dents to think about attempting shapes other than typical convex polygons for the tabletop. Consider allowing peer assessments or adding a peer-question-ing requirement (e.g., each team must ask at least three questions to earn a portion of its grade) to encourage students to pay close attention to other design team presentations.

Finally, to gather evidence of each individual student’s understanding, we recommend asking students to write reflectively, using one or more of these prompts:

• During the engineering design project, I contributed to my design team by. . . .

• Describe the mathematics you used for the design project and what you learned.

• What do you think was your best contribution to the project, and why?

• What did you like most and least about the engineering design project?

doing the MathToday’s classrooms can provide greater learning opportunities for all students to “do the math.” Carrying out ex-periments, collecting data, building models, and testing and revising are important elements necessary for making sense of real-world problems. The Paper Table project gives students a challenge that can increase their interest in mathematics and let them recognize that engineering has many ties to the math that they study and use in class. Such STEM experiences

are likely to excite, stimulate, and mo-tivate students. It is time to include engineering projects in the classroom to allow today’s students to embark on processes of thinking and designing so that they can develop into tomorrow’s leaders in science, technology, engi-neering, and mathematics.

referencesCommon Core State Standards Initiative

(CCSSI). 2010. Common Core State Standards for Mathematics. Washing-ton, DC: National Governors Associa-tion Center for Best Practices and the Council of Chief State School Officers. http://www.corestandards.org/assets/CCSSI_Math%20Standards.pdf.

National Council of Teachers of Math-ematics (NCTM). 2000. Principles and Standards for School Mathematics. Reston, VA: NCTM.

Science Buddies. 2011. “The Engineering Design Process.” http://www.science-buddies.org/engineering-design-process/ engineering-design-process-steps.shtml.

WGBH Educational Foundation. 2008. “PBS Kids Go! Design Squad Nation: Resources: Structures: Paper Table.” http://pbskids.org/designsquad/ parentseducators/resources/index .html?category=structures.

Mary c. enderson, [email protected], teaches mathematics content and methods courses to prospective and in-service teachers at old Dominion University in Norfolk, Virginia. Her research interests include

mathematics teacher knowledge, mathe-matical discourse, and case study research.

Melva r. grant, [email protected], is the graduate program director for the math spe-cialists’ program and teaches mathematics to pre-K-grade 8 teachers and methods courses to prospective teachers at old Dominion University. Her research interests include developing effective mathematics learning environments and making calculus more accessible.

Download a free smartphone app, then scan this tag to access the Paper Table Inves-tigation information that is online at http://www.nctm.org/mtms038.

Performance criteria 2 Points 1 Point 0 Points

Table Design Includes a design sketch with polygon properties, evidence of at least one test and redesign cycle

Includes a design sketch with polygon properties

Includes a design sketch

Paper Table Model

Shows at least one paper table model, and model is able to hold a math book

Shows at least one paper table model, but model is unable to hold a math book or is broken

No model

Mathematics Defines the problem, identifies polygon properties correctly, uses mathematical language

Defines the problem, identifies polygon properties correctly

Defines the problem

Presentation Everyone partici-pates, speaks clearly, and answers at least one question

Everyone participates and speaks clearly

Everyone participates

fig. 7 The rubric for the Paper Table presentation lists point criteria.

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