Engineers and Other People: Communicating Engineering Results ...

Session ETD 435

Proceedings of the 2013 Conference for Industry and Education Collaboration Copyright

©2013 American Society for Engineering Education

Engineers and Other People:

Communicating Engineering Results to a Larger Audience

Don Ploger, College of Education, Florida Atlantic University, Davie Campus, Davie, FL

[email protected]

Ravi Shankar, Center for Systems Integration, College of Engineering and Computer Science,

Florida Atlantic University, Boca Campus, Boca Raton, FL

[email protected]

Agnes Nemeth, Florida Atlantic University Schools, Boca Campus, Boca Raton, FL

[email protected]

Steven A. Hecht, Fischler School of Education, Nova Southeastern University

[email protected]

Contact Information: Don Ploger, [email protected]

Abstract

Because the subject is so complex; engineers use complex terminology in communicating with

each other. This specialized language becomes a barrier to the broader audience. To be effective,

they need to describe the content so essential details are conveyed to non-engineers and experts

in other fields as effectively as possible. Concepts in mathematics that are essential to

engineering and also can be learned in a meaningful way may bridge the gap between expert

engineers and novices. The requirements of this less-technical mathematical language is well

within the grasp of adolescents and provides a way to understand more specialized concepts used

by engineers. The goal is not to communicate all the technical details, but rather to communicate

the essential ideas in a way that promotes interest in, and respect for, engineering.

This paper focuses two topics that are centrally important to engineering: the Pythagorean

Theorem and the concept of real world error. This paper examines interviews of high school

students who had built robots and used those robots to simulate problems in mathematics. The

results indicate that students were able to solve the mathematical problems and also account for

real world errors in the simulations.

The end result will be an example of how technical knowledge can be conveyed to other people

– the 99% of the population who are not trained as engineers. Particular emphasis will be on

how engineering artifacts can be used to enhance mathematics education for the wide range of

students.

mailto:[email protected]





Session ETD 435



Introduction

There is a paradox. There is a very high demand for the products of engineering, but a much

smaller interest in the intellectual process that leads to those remarkable products. Engineering

has improved human life in so many ways: from computers and cell phones to safe drinking

water and an adequate food supply. Because engineering is so complex, engineers require a

means of communicating with each other so that the essential details are conveyed to other

experts as effectively as possible. The specialized language that engineers share has contributed

to the stunning accomplishments in the field of engineering.

The language that engineers use to communicate with each other can serve as a barrier to a

broader audience. One source of difficulty that novices have with understanding basic

engineering principles is lacking a conceptual understanding of the mathematical concepts

underlying basic engineering principles. The requirements of relatively less-technical and highly

accessible mathematical language are very different from the specialized language used by

engineers. The goal is not to communicate all the technical details needed to create a product,

but rather to communicate the essential ideas in a way that promotes interest in and respect for

engineering. Students who are taught bridges between mathematics and basic engineering

principles have a chance to gain an appreciation for engineering principles that emphasize these

mathematical concepts.

Communicating with the general public is a well-established issue in engineering education:

As an engineer, it is hard to hold back our egos (and tongues). It is hard to accept

that even though we are able to create complex objects that could save lives or

millions of dollars, it is all worth nothing if we cannot communicate our ideas or

work with others. Having real-world skills differentiates the good from the great.

… Engineers are creative, the source of ideas, but must live with and respond to

the “real-world” presence of others. The “great” engineers accept this necessity

of communication and, by implication, the presence and impact of other people

on the creation of “complex objects” of great value. In this extract,

communication and working with others is equated with “real-world skills.” [1]

This study examines the important issue of communicating engineering principles to the public

in the context of a research project designed to bring robotics into the mathematics

classroom. This paper focuses on two mathematical topics that are central to many principles

that engineer use in their daily profession: the Pythagorean Theorem and the concept of Real

World Error. This paper will describe prior knowledge that non-expert have that can be used as

a basis for learning. The end result will be an example of how technical knowledge can be

conveyed to other people – the 99% of the population who are not trained as engineers.

Using Robotics To Enhance Mathematics Education

This project has multiple goals: make the course interesting so they would want to continue with

engineering; explain the concepts in terms of simple Mathematics and Physics concepts that they

could relate to and reason with; provide robust software and hardware building blocks so that

they had to use only the most fundamental troubleshooting techniques [2] (using visual checks,

Session ETD 435



multimeter for measuring resistances – short circuits and open circuits, and trial runs to check on

their conceptual errors [3]); and an appreciation of the errors involved. We also had to create an

environment where they could leverage each other’s strengths and make progress. Out of this

reasoning came the concept of teams. This also served to channel their energies together towards

a common goal. We also felt that some students may feel threatened by all the high-tech

gadgetry that goes into the robots. All of this led to an incremental approach to building,

programming, and testing the robots, a systems design approach that is sound and widely used by

engineers in the real world. There was also concern whether these young students would have

difficulties in assembling the mechanical parts of robots and in soldering electronic components.

So, our options were to give them pre-assembled robots [4], [5] to program and apply to

Mathematics lessons or to let them assemble a robot and use it to solve a problem that they

proposed.

We decided to have students build their own robots for several reasons:

These were declared as pre-engineering majors and thus expected to do engineering

tasks; we told them that they could help other teams and get extra points, and that the

grading would be based on absolute scores, so the environment was not competitive but

collaborative.

Engineering and other real-world projects are conducted in a team environment. This

happened to be their first such semester long team project, but they learned quickly to

depend on each other to make progress. Further, this is expected in engineering program

by the ABET Accreditation Board [6]

Open source tools provided an intuitive and simple way to bring in system approaches.

We used open source high level hardware components and software libraries supported

by Arduino [7]. We explained the concepts on a JIT (just-in-time) basis, which was

coupled with building and testing their robots in smaller steps (see below). This boosted

their confidence and excitement about the final product.

The robotic platform was first built without any electronics. They used it to measure

distance traversed in terms of number of full and fractional turns. Then they added the

microcontroller and used it to drive LED (light emitting diode) displays, before

connecting it to motors to drive the wheels of the robot, again using a simple process (of

pulse-width modulation, with no feedback). In later steps, they incorporated ultrasound

and infrared sensors, and eventually optical encoders (for feedback and better control

of distance traversed or angle rotated) . This also allowed multiple pathways to complete

their projects. In fact several of the teams did neither use the US and IR sensors nor the

encoders in their final system, but still managed to complete their project. However, other

teams did utilize these additional features

The students could focus on a problem solving task from start to finish. The final project

was the plotting of a geometric shape on a large 6’ x 6’ canvas by their autonomous

robot. Since this was proposed by the group early on, all their thinking focused on

solving this particular problem. The student projects ranged from drawing large size

letters (“FAU”) to concentric triangles, to Stars [8]. All of them required the teams to

program their robots to cover precise distances in various directions, change angles by

precise amounts, and automatically stop once the drawing was completed. Student blog

sites listed at [7] give proof of this

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The real world concept of error made better sense to students. Note that we used the word

‘ precise’ not ‘accurate’ in the previous bulleted item. This was because we could use

robots to explain the difference between precision and accuracy. We explained the real-

world constraints of using robots to do practical Math. They understood the concepts of

errors due to resolution (due to the hardware used), estimation (in programming), and

actual measurement (due to the finite size of the foot print of the robot). We estimated

that the total error was less than + 10%. So, while this was large enough to surprise

them, they soon learned to factor this in in their logical reasoning.

This also gave an opportunity to build in other skills that ABET recommends: on problem

solving, documentation, presentation, communication, and team and project management.

The student blog sites [9] give ample proof that the students acquired these skills.

Embedded Robotics Course

Our experiences from fall 2011 has allowed us to draw up a lesson plan that involves the

introduction of one component at a time: Draw a line (use of stepper motors); vary PWM (pulse

width modulation) to slow and speed up the motion; draw a straight line on a narrower grid (with

the aid of parallel IR (Infra-red) reflective walls and side-mounted IR range sensors on the

robots; make the robot stop after a given distance with the front-mounted US (ultrasound) range

sensor and a reflective wall in the front; and draw a rectangle, one line at a time.

Figure 1: A robot constructed in the class this semester.

In a course in Fall 2011 engineering students explored ways to create geometric shapes using

low cost robotic systems. Figure 1 shows the robot being used this semester with 9th grade

students (more details below).

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There were many engineering challenges for students, such as limited range sensing, slippage on

paper, drag due to a heavy pen, four-wheel drive, battery drainage, etc. Students were given

suggestions for use of debouncing, interrupts, and feedback. [13] We have learned from these

experiences and now have improved the robot (see Figure 1) and accessories to overcome them.

Figure 2 shows a right triangle as drawn by a robot.

Getting a robot to draw a triangle requires much more knowledge than that typically acquired by

high school students in a geometry course.

Figure 2: A triangle drawn by a robot.

Conceptual Errors

In learning, we can look at mistakes in a systematic way. Conceptual errors involve a

misunderstanding of a mathematical idea, such as incorrectly believing that the hypotenuse of a

right triangle can be longer than the sum of the two legs. When students examine conceptual

errors, the process enhances mathematical reasoning and lead to deeper learning [10], [11].

Student errors are an inherent part of the learning process, and analyzing errors can provide

learning opportunities. Errors cannot be removed simply by teaching the correct information; the

student needs to understand what has gone wrong. Instructional approaches should capitalize on

errors by focusing directly on analyzing and discussing conceptual errors. Students should be

exposed to correct answers and to conceptual errors.

Students have an interest in errors, especially when those errors are committed by other people.

A previous study examined how a mathematical error could be used to enhance elementary

school mathematics instruction [12]. Mark Twain wrote that Huckleberry Finn “could say the

multiplication table up to six times seven is thirty-five.” This quote was used to help children

understand mathematical ideas more deeply by examining mathematical errors. The children

Session ETD 435



appreciated the joke, and they also were able to use that error as a basis for learning

mathematics. We can say much more about the statement “6 x 7 = 35” than “That answer is

wrong: 6 x 7 = 42.” But they learned much more than that. They realized that the product must

be even because an even number times any whole number gives an even product. The only way

to get an odd product is to multiply two odd numbers. Mathematical errors are interesting to

children. In some cases, examining those errors can lead to rich learning experiences. Because

we appreciate the errors of others more than our own, this study provided such an example.

However, students can learn to recognize any incorrect answer, not simply because it does not

match the answer in the back of the book, but based upon sound mathematical reasoning.

The Pythagorean Theorem

The Pythagorean Theorem is centrally important to mathematics, and prominent in the

curriculum. It is taught to all high school students. Mastery of the subject matter is another

issue.

The Pythagorean Theorem has an interesting role in popular culture. In the

classic motion picture, The Wizard of Oz, immediately after he gets his brain, the

Scarecrow says:

“The sum of the square roots of any two sides of an isosceles triangle is equal to

the square root of the remaining side. Oh joy! Rapture!” [13]

That answer is completely wrong. In a parody on the TV show, the Simpsons, Homer quotes

the Scarecrow’s (incorrect) statement.Someone overhears and admonishes Homer for that error,

Co-worker: “That’s a right triangle!”

Homer: “D’Oh!” [14]

The example is more thoroughly analyzed, with a full correction in [Pythagorean Theorem]

The corrected version states: “The sum of the squares of the legs of a right triangle is equal to the

square of the remaining side.” [15]

Although the Scarecrow’s error has been noticed and corrected by mathematically sophisticated

observers of popular culture, the general public is at a loss. In an episode of “Who wants to be a

millionaire,” a contestant was asked [16]:

Which of these square numbers also happens to be the sum of two smaller square

numbers? 16, 25, 36, 49

The contestant did not know the answer and asked for the audience to be polled. Only 30% of

the audience gave the correct answer (25). The most common answer was 16, given by 50% of

the audience. [Only 22 % of the AOL audience was correct; their consensus was also the

incorrect answer of 16.] The contestant went with the majority and lost $16,000 and the game.

Session ETD 435



These videos demonstrate that cultural artifacts can arouse a sense of humor and irony yet also

describe a central problem that engineers have in communicating complex concepts to the lay

person.

We can safely assume that engineers understand the Pythagorean Theorem. Based upon the

Wizard of Oz and Who wants to be a millionaire?, we can safely assume that the general public

does not understand the Pythagorean Theorem. In this study, we posed a mathematical problem

requiring the application of the theorem, and then asked students to explain it so that it would be

easy to understand.

Method

During the spring of 2012, a semester long elective course was offered at Florida Atlantic

University School to 9th grade students. Under the school-wide STEM initiative, the goal of the

course was to enhance students’ exposure to, and comfort level with STEM disciplines so they

can migrate towards a curriculum that allows them to pursue their passions while taking

advantage of STEM advances and opportunities. Seventeen ninth grade students in groups of

three assembled and programmed robots to draw geometric art on a large (6’ x 6’) canvas. The

use of open source software and hardware helped high school students focus on the experience,

rather than dealing with lower level engineering details. At the end of the course, groups

presented their geometric art in class.

Five students who participated in the robotics course volunteered for an interview. This paper

contains the interview conducted with two high school students.

Having completed Pre-Algebra in 6th grade, algebra in 7th grade, Geometry on line during 8th

grade and Algebra 2 in class during 8th grade, Karl is in an advanced level Pre-Calculus in 9th

grade. He is at least two years ahead in math compared to the regular 9th grade cohorts. Evan,

also at a somewhat accelerated math track has completed Pre-Algebra in 6th, Algebra 1 in 7th,

Geometry in 8th grade and was scheduled in Algebra 2 in 9th grade while in the robotics course.

Interview questions:

Students were first told the following:

“Let’s start off with a simple problem. Please explain how you would solve it. How would you

explain it to someone who is just learning the topic.”

Once these initial instructions were understood, they were given the following problem:

Tamara leaves school and walks 6 blocks east. Then she turns left and walks 8 blocks

north. How far did Tamara walk?

Tamara has a pet carrier pigeon that is trained to fly directly from Tamara’s house to the

school. How far does the pigeon fly?

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Now imagine that this is going to be simulated by a robot moving on a map of Tamara’s town.

The robot is going to trace the path that the carrier pigeon flies. What distance do you expect the

robot to travel?

Suppose the robot travels 9.3 units.

What would you say?

Suppose the robot travels 17.1 units.

What would you say?

Prompts

Do you expect the robot to travel exactly 10.0 units?

If the robot travels 9.3 units, is something seriously wrong?

How would you define real-world error?

If the robot travels 17.1 units, is something seriously wrong?

How would you define an error in mathematical understanding?

Results

Evan and Karl worked as a group. The problem was read and then Evan was asked to solve the

first part of the problem.

The following is a verbatim transcript of the interview. When necessary, clarifications are added

to clarify the meaning, and these clarifications appear in italics.

Interviewer: Tamara leaves school and walks 6 blocks east. Then she turns left and walks 8

blocks north. How far did Tamara walk?

Evan: 10

Evan gave the direct distance.

Interviewer: OK [Pause]

Evan: 14

I thought that was what you were looking for.

Interviewer: You answered the next question.

The Interviewer is noting that Evan first gave the direct distance, 10, and then gave the walk

distance, 14.

Evan: She walked 14 blocks

Interviewer: Correct. You answered the next question.

Tamara has a pet carrier pigeon that is trained to fly directly from Tamara’s house to the school.

Session ETD 435



How far does the pigeon fly?

Evan: 10 Blocks

Interviewer: Great.

Thank you Evan, that’s wonderful.

Now..

That’s the math problem.

Imagine that the robot is going to simulate this.

Karl: In blocks?

Interviewer: We will call the length of a block, ‘one unit’

So 10 units.

Karl: Ideally, it should go 10 blocks.

Interviewer: Now, suppose the robots odometer reads 9.3 units.

Is there something wrong? What would you say?

Karl: Figure out the percentage?

Interviewer: Yes, please.

Karl: OK.

93…. Minus 7 per cent.

NOTE: Karl is considering 10 units to be 100 per cent, and 9.3 units is 93 percent.

Interviewer: Does that indicate a serious problem?

Is something seriously wrong?

Karl: I don’t believe seriously.

Slightly.

NOTE: Karl is responding to the question, “Is something seriously wrong?“.

Interviewer: Good.

Now, suppose the robot’s odometer reads 17.1.

Karl: It is seriously.

Interviewer: And if someone checked that and said it’s OK, that person is missing a step, right?

Karl: (nods)

Interviewer: Do you expect the robot in the real world …

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The answer to the math problem is 10, you got that right away.

In fact, you got it before I asked it!

But do you expect the robot to travel exactly 10.0 units?

With robots from commercial parts.

The goal of the project is to get this out as affordably as possible.

So under those constraints, do you expect it to be exactly 10.0 units?

Karl: No.

Interviewer: Great. And if it travels 9.3, is something seriously wrong?

Karl: No.

Interviewer: Good

How do you define real world error?

Karl: There are always certain factors that would affect the robot movement.

For example, the temperature of the air would affect ultra-sound.

The error is acceptable, but it depends upon what you are doing.

Interviewer: Now if someone checked it and said the robot traveled 17.1 units, is something

seriously wrong?

Karl: Yes.

Interviewer: Can you explain that?

Karl: Because it is almost double the distance.

Interviewer: You shouldn't have that much error?

Karl: Yes.

Interviewer: So you’re saying: 10 percent error, that’s OK, but 70 percent error..

Karl: (nods)

Interviewer: Is there another way…You did the Pythagorean..

Please write down the three sides of the triangle.

Karl: Triples.

Interviewer: Can you say something else about them?

Karl: Well, they do follow the Pythagorean Theorem.

Session ETD 435



Interviewer: Do they share a common factor?

Karl: 3, 4, 5

Karl noted that the triples were 6, 8, 10. He implicitly realized the common factor was 2, and

then stated that this is a 3, 4, 5 triangle.

Summary: At this point, the students realized there was a simpler way to solve the problem that

agreed with their answer. They realized that their answer was correct, but that it could be

enhanced by demonstrating another method.

Discussion

Both students interpreted the question as a mathematical problem, which required the

Pythagorean Theorem. In each case, the student gave the correct solution, 10 units. The

problem did not specify that the pigeon’s flight had the same initial and final altitude, and that

there were no intervening trees. In of the five interviews, no student mentioned these

details. That is, they correctly interpreted the mathematical problem and provided the correct

solution. In fact, one student anticipated a question, and gave the correct answer, before the

pigeon was mentioned.

When discussing the robotic simulation, they all recognized that if the robot odometer read 9.3,

there was no serious problem -- that is real world error. However, if the odometer read 17.1,

there was a big problem. One student made the interesting comment that perhaps the robot

traced the path twice and the odometer was not re-set.

Students were readily able to understand the difference between conceptual error and real world

error. They knew how to solve the mathematical problems, and how to examine the simulations

of the robots. They were able to coordinate both types of knowledge into a meaningful pattern.

There were relevant aspects of the problem that the students did not immediately provide. For

example, they calculated the hypotenuse of the (6, 8, 10) triangle using the Pythagorean

Theorem. This was, of course, correct. However, they did not realize that this problem could be

also solved using specialized knowledge that the given triangle was a special case of the ratio

3:4:5. With prompts, the students did make this connection, and extended their knowledge of

the mathematical subject matter and of problem solving.

The students were also able, with prompts, to realize that their work could be used as an

instructional tool to a wider range of students. They became aware that they had formed a group

where there was much tacit knowledge – shared by the group, but not common to the average

high school student. There were ways to communicate this to a broader audience and they

became interested in doing so.

Acknowledgements: The authors thank Dr. Ali Zilouchian, Associate Dean, College of

Engineering and Computer Science, FAU, Dr. Tammy Ferguson, Director and Principal,

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Henderson School, FAU, and Dr. Barbara Ridener, Chair of the Department of Teaching and

Learning, FAU. We are grateful to Joseph Gundel for expert assistance in the development of the

robotics course. We thank Kathy Columbus, Charna Rosenfeld, Cassandra Rosa, and Melissa

Bedoya for helpful comments during the development of the project.

References:

[1] Dunsmore, K., Turns, J., & Yellin, J. (2011). Looking Toward the Real World: Student

Conceptions of Engineering. Journal of Engineering Education, 100, (2), 329–348.

[2] Shankar, R., Ploger, D., Masory, O., and McAfee, F.X. (2011). Robotic Games for STEM

Education. ASEE Mid-Atlantic Conference, Philadelphia, PA: Temple University.

[3]Ploger, D., Shankar, R., Nemeth, A. & Hecht, S. (2012) Exporting Engineering Technology

Practice To Enhance Pre-College Mathematics Learning. Proceedings of the ASEE Gulf

Southwestern Section, El Paso, Texas, April, 2012, 236-245.

[4] CMU LEGO (2012). Carnegie Mellon University Robotics Academy:

LEGO Robots. Retrieved October 30, 2012 from

http://www.education.rec.ri.cmu.edu/content/lego/index.htm

[5] CMU VEX (2012). Carnegie Mellon University Robotics Academy: The VEX Robotics

system. Retrieved October 30, 2012 from

http://www.education.rec.ri.cmu.edu/content/vex/index.htm

[6] ABET, “General Criteria 3. Student Outcomes.” ABET website, Retrieved October 30, 2012

from

http://www.abet.org/engineering-criteria-2012-2013/

[7] Arduino, Arduino Open Source Web site, Retrieved October 30, 2012 from

http://www.arduino.cc/

[8] FAU, Website Summary Page on Robotics courses, Retrieved October 30, 2012 from

http://robotics.fau.edu/

[9] FAU, Faculty Summary Page on Robotics courses, Retrieved October 30, 2012 from

http://faculty.eng.fau.edu/shankar/research/robotics/

[10] Kramarski, B.& Zoldan, S. (2008). Using Errors as Springboards for Enhancing

Mathematical Reasoning With Three Metacognitive Approaches, The Journal of

Educational Research, 102 (2), 137-151

[11] Veenman, M. V. J., Van Hout-Wolters, B. H. A. M., & Afflerbach, P. (2006).

Metacognition and learning: conceptual and methodological considerations.

Metacognition Learning, 1: 3–14.

[12] Ploger, D. and Hecht, S. (2012). Meaning, Memory, and Multiplication: Integrating

Patterns and Properties with Basic Facts. Childhood Education, Vol. 88, No. 3 (May-June

2012) issue.

[13] Memorable quotes for The Wizard of Oz. Retrieved October 30, 2012 from

http://www.imdb.com/title/tt0032138/quotes?qt0409923

[14] Homer about triangles. Retrieved October 30, 2012 from

http://www.youtube.com/watch?v=fO1Vhc88QkM

[15] Pythagorean Theorem. Retrieved October 30, 2012 from

http://www.youtube.com/watch?v=fO9EU0w3CrY&feature=related

[16] When not knowing Math can cost you $15,000. Retrieved October 30, 2012 from

http://www.youtube.com/watch?v=BbX44YSsQ2I&feature=related

http://www.education.rec.ri.cmu.edu/content/lego/index.htm

http://www.education.rec.ri.cmu.edu/content/vex/index.htm

http://www.abet.org/engineering-criteria-2012-2013/

http://www.arduino.cc/

http://robotics.fau.edu/

http://faculty.eng.fau.edu/shankar/research/robotics/

http://www.youtube.com/watch?v=fO1Vhc88QkM

http://www.youtube.com/watch?v=BbX44YSsQ2I&feature=related

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