1

Engineering-Based Modeling: Bridging School Mathematics and Real-World Problems

Lyn D. English and Nicholas G. Mousoulides

Mathematics Teaching in the Middle School (Vol. 20, no. 9, 2015, pp. 532-539)

As they go about their daily lives, our students are faced with increasingly complex,

powerful, and dynamic systems of information. Just searching the Internet exposes them to

comprehensive data sources, some of which are contradictory and potentially unreliable.

Careful and critical attention is needed if these sources are to be used wisely. For many years

we have been working with classroom teachers in creating modeling problems involving

comprehensive data sets that require this type of detailed analysis (e.g., English and

Mousoulides, 2011).

Our modeling activities are engineering-based and provide a rich source of meaningful

situations that capitalize on and extend students' routine learning. By integrating such

activities within existing curricula, we have been able to assist students in appreciating how

their school learning in mathematics and science applies to problems in the outside world.

Furthermore, the problems are open to multiple approaches and solutions, and encourage the

creative, critical, and flexible thinking that is often lacking in more traditional textbook

examples. We discuss here one of the problems we implemented in 6th-grade classes,

namely, a bridge design activity based on the collapse of the 35W Bridge in Minneapolis.

Engineering-Based Modeling Problems

In our activities engineering serves as a rich source of appealing, authentic contexts that draw

upon school mathematics, science, and technology. Our engineering problems are examples

of model-eliciting activities (Lesh and Zawojewski, 2007). They are realistic, client-driven,

multifaceted, and idea generators. That is, the core ideas are not presented "up front," rather,

they are embedded within the problem and must be elicited and operated on in producing a

solution model.

2

By constructing our engineering-based problems as model-eliciting activities we can

promote design processes that apply across STEM domains (Science, Technology,

Engineering, and Mathematics). One example of these design processes appears in Figure 1.

Students are encouraged to: Ask (What is the problem? What are the constraints?), Imagine

(What are some possible solutions?), Plan (What diagram can you draw?), Create (Follow

your plan; create a model; test it out), and Improve (Discuss what works; modify your design

to make it better) (Cunningham and Hester 2007).

Figure 1. The cyclic design / redesign process

As students engage in these processes, we see them progress through important learning

cycles. These cycles are repeated until students are satisfied that their model(s) meets the

problem constraints. Lesh and Zawojewski (2007) describe these cycles as:

Understanding the context of the problem and the system to be modelled,

Expressing / testing / revising a working model,

Evaluating the model under conditions of its intended application, and

Documenting the model throughout its creation.

Implementing an Engineering-Based Modeling Problem

Create Improve

Plan

Ask

Imagine

3

In these activities we collaborate with the classroom teachers in developing and

implementing the learning experiences. In the present case, 6th-grade mathematics and

science teachers worked with the bridge design activity, and conducted it in two of their

classes (12-year-old students) across three sessions. The activity (adapted from Guzey,

Moore, and Roehrig, 2010) focused on the 2007 structural failure of the 35W Bridge in

Minneapolis, Minnesota, in which 13 people died and more than 144 were injured.

The activity required students to develop a model for: (a) calculating the cost for each

one of the four bridge types (see Table 1), and then (b) for selecting the best possible bridge

type for the reconstruction of the collapsed bridge. All possible factors related to bridge type,

materials used, design, safety, and cost were to be taken into consideration.

In Session 1 (35-45 minutes), we introduced the problem context with an introductory

activity comprising: (a) a newspaper article about the 35W bridge collapse, adapted from the

Mn/Dot Bridge website (http://www.dot.state.mn.us/bridge/), and (b) a video clip of the

collapse (www.youtube.com/watch?v= C31IlOHNzbM ). A second article presented students

with the main characteristics of the four types of bridges. Next, the students studied the

articles and the video, and individually answered questions about the articles to ensure they

understood the core ideas, such as the factors believed to be involved in the collapse.

Excerpts of the articles and the readiness questions appear in Figure 2.

In Session 2 (80-90 minutes), the students studied the problem scenario (see Figure 3).

During this session, the students worked in mixed-ability and mixed-sex groups of three to

four. Each group of students was expected to deliver one (or more) model for solving the

given problem, reflecting the ideas and work of all students. Two tables of quantitative and

qualitative data accompanied the problem text (Tables 1 and 2), together with examples of

the main bridge types (e.g., the Smithfield St. Truss bridge in Pittsburgh). The students then

worked on generating, refining, and documenting their models. The students had to draw on

4

the given data in: (a) developing a model for calculating the cost for each bridge type, and (b)

using the cost model in conjunction with other bridge characteristics (e.g., safety, materials,

design) in proposing the best possible bridge type for the reconstruction of the collapsed

bridge in Minnesota.

Figure 2. The newspaper article and the readiness questions

In Session 3 (40-50 minutes), each group of students presented their developments and

key findings to their class peers. Each group prepared and presented a poster, in which they

provided documentation for the appropriateness of their solution. In a concluding class

discussion, the key ideas and relationships that the students had generated were explored.

5

Figure 3. The bridge design problem scenario

6

Table 1. Characteristics of the four major bridge types

Type Advantages Disadvantages Span range Material Design effort

Truss bridge Strong and rigid framework Work well with most applications

Cannot be used in curvesExpensive materials

Short to medium Iron, steel, concrete Low

Arch bridge AestheticsUsed for longer bridges with curvesLong life time

Abutments are under compressionLong span arches are most difficult to construct

Short to long Stone, cast iron, timber, steel

Low

Suspension bridge

AestheticsLight and flexible

Wind is always a concernExpensive to build

Long Steel rope and concrete

Medium

Cable-stayed bridge

Cables are economical Fast to build, Aesthetic

Stability of cables need to be considered for long span bridges

Medium Steel rope and concrete

High

Table 2. Examples of four major bridge types

Name Type Total length (feet) Car Lanes Constructability Cost (Present value)

Hennepin Ave Suspension 1037 6 Easy $100 million

Golden Gate Suspension 8981 6 Difficult $212 million

10th Ave Arch 2175 4 Difficult $9 million

Stone Arch Arch 2100 Bike/Pedestrian Difficult $15 million

Greenway Cable-stayed 2200 Bike /Pedestrian Easy $5.2 million

Arthur Ravenel Jr. Cable-stayed 13,200 8 Easy $62 million

John E. Mathews Truss 7736 4 Difficult $65 million

Eagle Point Truss 2,000 2 Difficult $2.5 million

7

Generating Solution Models

Although a few student groups remained bogged down in the data and were unable to

produce an appropriate model, the majority were able to do so with varying levels of

sophistication. The models varied in the number of problem factors students used (cost per

surface unit of bridge deck, aesthetics of the different bridge types, design effort, difficulty

level of construction, and length), and in the different approaches to dealing with these

factors. For example, some groups did not rank the different bridge types but only provided

paired comparisons between the types, while some other groups developed quite

sophisticated procedures for ranking the different bridge types according to their cost. In the

remainder of this article, we provide two examples of the students' models, one which

considers only mathematical factors, and the other, both mathematical and engineering

aspects of the given problem.

Mathematics Models

One of the groups who produced a model focusing only on mathematical factors decided that

a cable-stayed bridge would be the most appropriate to build. The students based their

decision on the swiftness with which this type of bridge could be constructed. They

maintained from the outset that this bridge type was the most effective. Considerable debate

ensued, however, as to whether the cable-stayed bridge was a better option than the

suspension bridge. Some students commented that the latter type was too expensive, so the

group decided to develop a cost model for each type of bridge in an attempt to base their

selection on more criteria than only qualitative information. Their deliberations included:

Student 1: “A cable-stayed bridge is the best choice […] it can be built fast, it is a beautiful one, and it

is not very expensive, like the suspension one.”

Student 2: “The suspension has the same characteristics […] it is not more expensive […] well, it is

difficult to conclude if we continue to work like this.”

8

Student 1: “OK. First things first. We cannot compare as they are […] they are not the same (refers to

bridge’s length).”

Initially, this group developed a quite simple division model by dividing cost (at

present value) by the total length of each bridge and then calculating the average cost for each

bridge type. The interactions within their group helped them realize that the width of each

bridge given in the data was not the same:

Student 3: “Something is wrong […] this is not reasonable.”

Student 2: “Why not? How do you know it is not correct?”

Student 3: “Well, we took into account the length, but there are big bridges with many lanes and also

small ones for pedestrians […] this is my concern.”

Student 1: “Exactly, we need to use length and number of lanes.”

The students then moved into a second developmental cycle, creating a model for the

cost based on using both the length and the width of bridge deck (money per square feet).

However, this was not a straightforward process. The students faced difficulties in calculating

the width of the different bridges, since the given data referred to car lanes. After some

assistance in determining that the typical width of a lane was 12 feet, the group used this

figure to further improve their model, as shown in Figure 4. The group also made an

estimation of the width of bike and pedestrian bridges by using a fixed width (30 ft) in their

model.

Based on their results, the students confirmed that the best possible bridge type was the

cable-stayed bridge. They further supported their decision by emphasizing that this bridge

type was the cheapest, according to their results. However, they did not question the great

variation in their results, especially the large differences between bridges within the same

type. In the next example, the student group did take into account this variation.

9

Figure 4. Group’s cost model based on money per square feet of deck

Mathematics and Engineering Models

One of the most sophisticated models was developed by a group who took into consideration

mathematical, engineering, and societal factors. The group commenced the problem by

excluding a truss-type bridge because:

Student 1: “The collapsed bridge was a truss one […] it is here (points to a frame of the provided

video.”

Student 3: “We should not choose this one (refers to the bridge type).”

[…]

Student 4: “Selecting the truss type bridge would make people feel insecure and bring back all those

bad memories.”

The students decided that a cost model for ranking the different bridge types was needed, but

their subsequent reasoning was more refined than the previous group's.

Although they initially developed a similar model to the previous group’s example, the

students realized that calculating the average cost (money per square feet of deck) for each

bridge type was not the best possible solution. They concluded that the great variation in their

10

results for bridges of the same type (two examples per bridge type were provided) could be

corrected by integrating within their model more factors. They reported:

Student 3: “Our calculations are correct. There is nothing wrong. The cost is very different.”

Student 2: “There are other things (factors) that are important and influence the cost […] for those

(bridges) that are close to sea it is more difficult.”

Student 3: “Yes, like in the Golden Gate bridge. It is so expensive and it is not that long.”

Student 4: “[…] Cost is not proportionally related to the surface of the bridge (deck), but also the level

of difficulty in constructability, just like in the Golden Gate, is an important factor.”

On using the data provided in Table 1, however, the group concluded that all bridge

types had their advantages and disadvantages, and therefore they could not determine a

recommended bridge type from the first table. The group then entered their next cycle of

model development by taking into account further factors. Their discussion on calculating the

cost model (see above) also contributed in refining their perceptions of the engineering and

societal factors that should be integrated in their model.

These factors included the necessary extra lanes for bridges, bikes, and pedestrians, as

well as the difficulty level of each bridge construction, as shown in Figure 5. The last factor

was determined by dividing the final cost estimate by 1.5 for the bridges listed. Identified as a

“difficult constructability” factor, the group specifically created this approach (dividing by

1.5) to provide the same basis of comparison for all bridge types. The group's model

presented a ranking of cable-stayed, arch, truss, and suspension bridge. However, the students

finally selected the arch type as the best possible solution, since they were still concerned

about the stability of a cable-stayed type for long span bridges (as indicated in Table 1).

11

Figure 5. A mathematics and engineering based model

Concluding Points

Decision making in solving complex problems is not an easy or straightforward process,

which the students realized in the present activity. This created some sense of frustration at

the beginning, especially for groups of students with limited interactions between them.

However, as the activity progressed, more and more students got actively involved, and

highly appreciated their developments along the way. Working in mixed-ability groups

provided a safe environment for students to express their ideas. For the teachers, the inclusion

of both individual work (session 1) and mixed-group work in the remaining sessions enabled

useful insights into their students’ thinking. As one student commented, "I really enjoyed the

activity. It was great. I was frustrated at the beginning and did not like it, but then we

gradually realized what we had to do […] it was so pleasant that we discovered this on our

own […] I really liked using maths in solving such a difficult problem." Snapshots from

students’ work in solving the bridge problem appear in Figure 6.

12

Figure 6. Students collaborate in their groups in solving the problem

The activity helped students appreciate the nature of problem solving beyond the

classroom. Dealing effectively with problems in daily life often necessitates combining many

factors, some of which may be ambiguous and conflicting. There are frequently multiple

objectives that need to be satisfied, complex data to be analyzed carefully, and more than one

acceptable solution to be considered. Engineering provides a rich source for creating learning

experiences that reflect the challenges of real-life problems.

References

Cunningham, Christine and Kate Hester. "Engineering is elementary: An engineering and

technology curriculum for children." In Proceedings of the 2007 American Society for

Engineering Education Annual Conference & Exposition, 1-18. Honolulu, Hawaii:

American Society for Engineering Education, 2007.

13

English, Lyn D. and Nicholas Mousoulides. "Engineering-based modelling experiences in the

elementary classroom." In Dynamic Modeling: Cognitive Tool for Scientific Enquiry,

edited by Myint S. Khine and Issa M. Saleh, 173-194. Netherlands: Springer, 2011.

Guzey, S. Selcen, Tamara, J. Moore, and Gillian H. Roehrig. "Curriculum development for

STEM integration: Bridge design on the White Earth Reservation." In Handbook of

Curriculum Development, edited by Limon E. Kattington, 347-366. Hauppauge, NY:

Nova Science Publishers, 2010.

Lesh, Richard and Judith S. Zawojewski. "Problem solving and modeling." In Second

Handbook of Research on Mathematics Teaching and Learning, edited by Frank K.

Lester, 763-804. Greenwich, CT: Information Age Publishers, 2007.