Model-Eliciting Activities: A Case-Based Approach for Getting Students Interested in Material Science and Engineering
Tamara J. Moore
Department of Curriculum and Instruction, University of Minnesota, 230A Peik Hall, 159 Pillsbury Drive SE, Minneapolis, MN 55455, USA; email@example.com
Attracting students to engineering is a challenge. In addition, ABET requires that engineering graduates be able to work on multi-disciplinary teams and apply mathematics and science when solving engineering problems. One manner of integrating teamwork and engineering contexts in a first-year foundation engineering course is through the use of Model-Eliciting Activities (MEAs) - realistic, client-driven problems based on the models and modeling theoretical framework. A Model-Eliciting Activity (MEA) is a real-world client-driven problem. The solution of an MEA requires the use of one or more mathematical or engineering concepts that are unspecified by the problem - students must make new sense of their existing knowledge and understandings to formulate a generalizable mathematical model that can be used by the client to solve the given and similar problems. An MEA creates an environment in which skills beyond mathematical abilities are valued because the focus is not on the use of prescribed equations and algorithms but on the use of a broader spectrum of skills required for effective engineering problem-solving. Carefully constructed MEAs can begin to prepare students to communicate and work effectively in teams; to adopt and adapt conceptual tools; to construct, describe, and explain complex systems; and to cope with complex systems. MEAs provide a learning environment that is tailored to a more diverse population than typical engineering course experiences as they allow students with different backgrounds and values to emerge as talented, and that adapting these types of activities to engineering courses has the potential to go beyond “filling the gaps” to “opening doors” to women and underrepresented populations in engineering. Further, MEAs provide evidence of student development in regards to ABET standards. Through NSF-funded grants, multiple MEAs have been developed and implemented with a MSE-flavored nanotechnology theme. This paper will focus on the content, implementation, and student results of two of these MEAs.
Keywords: problem-based learning; modeling; cooperative learning/teaming
Mathematical modeling in engineering is a foundational ability for professional engineers. Gainsburg1 studied structural engineers at work and found they perform significant mathematical modeling in their day-to-day activities. One conclusion from Gainsburg’s research was “modeling-transforming hypothetical structures into mathematical or symbolic language for the purpose of applying engineering theory-is the heart of the profession.” At the Curriculum Foundation Workshop in Engineering, professors from several disciplines of engineering all stated that mathematical modeling was a central skill in their professions2, and in Smith’s book, Teamwork and Project Management, he devotes an entire section on modeling in the engineering profession3. In fact, Smith finds modeling so central to the engineer’s work that he co-authored an entire textbook on modeling4. The need for the graduating engineer to be able to create mathematical models is evident; therefore, teaching undergraduate engineering students to model is essential. The purpose of this paper is to introduce a particular kind of modeling activity, Model-Eliciting Activities (MEAs), describe their nature and construction, and demonstrate how they have been used with a nanotechnology theme.
MODELS AND MODELING
In order to understand Model-Eliciting Activities, it is important to have a common definition of models and modeling. The researchers who work within the theoretical framework of models and modeling take broad view of modeling. Simply, a model is a system that explains, describes, or represents another system. A model has elements, operations, and relations that allow for logical relationships to emerge. Many times, the model is not sufficient to completely describe the system it represents, but if it is a useful model, it closely approximates the system in a manner that people can use when working with the system without being unnecessarily complex.
MODEL-ELICITING ACTIVITIES OVERVIEW
Model-eliciting activities (MEAs) are one method with which to teach modeling to undergraduate engineering students. An MEA is a complex problem solving task set in a realistic context with a client, characteristics that place MEAs in the authentic assessment category5,6. Solutions to MEAs are generalizable models, which reveal the thought processes of the students. The models created include procedures for doing things and more importantly, metaphors for seeing or interpreting things. The activities are such that student teams of three to four express their mathematical model, test it using sample data and revise their procedure to meet the needs of their client. MEAs set in engineering contexts have been developed through the NSF-funded Small Group Mathematical Modeling for Gender Equity in Engineering (SGMM) project7 and an NSF- Nanotechnology for Undergraduate Education8 grant, and are described at length by Lesh, Hoover, Hole, Kelly, and Post9. They are also a key component of the book Beyond Constructivism: Models and Modeling Perspectives on Mathematics Problem Solving, Learning, and Teaching by Lesh and Doerr10. The MEA framework provides a means to not only deliver more open-ended engineering problems (engineering content) but also address multiple ABET criteria, especially those that are problematic to integrate in engineering courses11. They are being expanded through an NSF - Course, Curriculum, and Laboratory Improvement Phase 3 grant entitled “Improving Engineering Students Learning Strategies through Models and Modeling.”
The Accreditation Board for Engineering and Technology (ABET) Engineering Criteria 2000 created the need for change in the way we teach engineering. ABET Criterion 3 requires that accredited engineering programs demonstrate that their graduates can meet standards such as an “ability to function on multi-disciplinary teams,” an “ability to communicate effectively,” and an “ability to identify, formulate, and solve engineering problems”12. In addition to eliciting models, MEAs require teamwork, design processes, and communication. Because MEAs address these ABET criteria and are accessible to all levels of engineering students, implementing MEAs in courses where the instructor wishes to develop these professional skills, as well as problem solving abilities, makes sense.
The model-eliciting activity was originally designed as a research instrument with three purposes in mind: (1) to be thought revealing in the sense that the product of student teams would reveal the thought processes of the students, (2) to be simulations of real-world applications, and (3) to identify abilities of students that are not measured by standardized tests. The first two purposes are achieved by careful construction of each MEA. In order to guarantee careful construction of an MEA, there are six principles that guide the design9,13,14. These principles will be described in full later in this paper (see The Six Principles of MEA Development section). A well-constructed MEA assures that the students reveal their thinking. MEAs needed to be thought revealing so that researchers could understand the problem solving processes used by students. To study the problem solving process, MEAs were created to elicit models in a manner that required students to present their thinking in writing. Students are put into realistic situations in MEAs. Researchers wanted to understand the problem solving process in the context of real-world applications. The use of MEAs within realistic contexts helped researchers to identify ways of thinking that were not being revealed through other texts on problem solving. The third purpose is achieved through the identification of students that have abilities that were not being identified on standardized tests. Thorough study of students during the problem solving process and student solutions to MEAs allow researchers to identify people who can work in teams, complete complex multi-stage tasks, and adapt to continually changing tools (spreadsheets with graphs, calculators, etc.). The importance is that these qualities emphasize a much richer mathematics than the typical problem solving activity - the mathematics of descriptions and explanations as much as computation and derivation.
Although model-eliciting activities were not originally created as instructional tools, it has been found that they work very well in the classroom. The purposes originally set out to be research goals also parallel as instructional goals. The thought revealing aspect of MEAs allows instructors to understand what students are thinking while working on a problem. Instructors can use this information to shape other instruction. The real-world contexts of MEAs allow students to have an authentic problem solving experience in a classroom setting. When the authentic context is tied to a profession, the students are exposed to an aspect of the profession that is usually not possible in the classroom. Because MEAs can identify strengths that other types of assessment do not test, MEAs are useful for identifying students with inclinations toward certain fields of study. Model-eliciting activities as instructional tools, within a material science and engineering context, have been described at length by Diefes-Dux, et al.14 and Hjalmarson, et al.15 MEAs are now being designed as instructional tools that are meant to complement the content of a course in conjunction with other instructional tools.
Research with Model-Eliciting Activities
Model-eliciting activities have been the subject of many research projects. However, the majority of these research projects have focused on middle school students. For example, Lesh and Harel16 studied middle school students and their abilities to form proof structures while solving MEAs, and Richardson17 studied the diffusion of ideas while working on MEAs in a middle school classroom. Lesh18 also studied middle school students’ development of representations in model-eliciting activities.
Assessment of model-eliciting activities has been the focus of a few studies on MEAs. Carmona19 designed an assessment tool to describe middle school students’ mathematical knowledge. Lesh and Clarke20 investigated assessment of the quality of responses to model-eliciting activities.
A few studies were aimed at investigating pre-service educators. A study to examine preservice teachers using experimentation while performing an MEA was undertaken by Koleza and Iatridou21. The goal of this research was to investigate the experimentation that preservice teachers use to solve higher cognitive problem solving tasks and to research the factors that prevent the crossing of boundaries between day-to-day mathematical functioning and school tasks. Another study by Sriraman22 studied graduate students in mathematics education to investigate their understandings of the notion of model-eliciting.
In engineering curriculum, model-eliciting activities have been studied through the SGMM project at Purdue University. Follman, Liguore, and Watkins23 investigated gender interactions in an online user interface. In another study, small group interactions were observed to look at students’ complex mathematical thinking during MEAs24. The use of model-eliciting activities in a first-year required engineering problem solving course have been linked by Diefes-Dux, et al.25 to persistence of women in engineering. Moore26,29 has studied team effectiveness and its role in solving MEAs in engineering.
THE SIX PRINCIPLES OF MEA DEVELOPMENT
The six principles (Table I) are crucial in guiding the development of an MEA. They are the benchmarks to which the developer must always return to verify that the task is being written with the students’ growth of ideas in mind. Here, growth of ideas refers to the students bringing their current knowledge to the situation and transforming these ideas to a more developed and directed state.
|Ensures the activity requires the construction of an explicit description, explanation, or procedure for a mathematically significant situation|
|Also known as the Model Share-Ability and Re-Useability Principle. Requires students to produce solutions that are shareable with others and modifiable for other closely related engineering situations|
|Ensures that the students are required to create some form of documentation that will reveal explicitly how they are thinking about the problem situation|
|Requires the activity to be posed in a realistic engineering context and to be designed so that the students can interpret the activity meaningfully from their different levels of mathematical ability and general knowledge|
|Ensures that the activity contains criteria the students can identify and use to test and revise their current ways of thinking|
|Ensures that the model produced will be as simple as possible, yet still mathematically significant for engineering purposes|
The Model Construction Principle requires that the MEA elicit a model, which is defined here as a system to describe, explain, or represent another system. The procedures which are routinely asked for in an MEA are this model. The solution to an MEA is a process. This is one of the main differences between MEAs and other types of open-ended problems in engineering. Here the problem is not only asking for an answer for the current situation, but also for other similar situations. The process needed to solve the problem is actually the product asked for by the MEA. This ties into the idea of the Generalizability Principle. The Generalizability Principle ensures the MEA requires students to document their model in such a way that others can use it (Share-Ability), and therefore understand the students’ thinking, as well as be able to use it on similar data that is not given in the problem (Re-Usability).
MEAs allow an instructor to be able to observe all students’ thinking in a short amount of time without actually having to do a behavioral observation of each student or team of students. A well-constructed model-eliciting activity assures that the students reveal their thoughts through the Model Documentation Principle. MEAs need to be thought revealing so that instructors can understand the problem solving processes used by students. The MEA is designed to have student teams provide their solutions through the writing of rich descriptions of their model. Students are put into realistic situations through the Reality Principle. It is important to note that the word “realistic” is used in the description of the Reality Principle. In order to allow students to enter into the problem, often the MEA must be idealized to lessen the complexity of the problem. The realistic nature of the problem aids in motivation for solving the problem. MEAs are purposely designed to create the feeling in the students that they are working for a “real” engineering client for a “real” purpose. This can help students learn about the work of engineers in forms more authentic than lecture, yet still in an in-school experience that is less costly than service learning.
MEAs are designed for the instructor or researcher to have a hands-off role during the student team completion of the activity. In order to do this, the Self-Assessment Principle must be well developed. This principle is what will allow students to go through cycles of revision. Once students bring a model for solving the problem to the table, the MEA should present data that will force students to go beyond their current ways of thinking. For example, one MEA, Just-In-Time Manufacturing, has students compare shipping companies in minutes late for delivering an order. The students must create a model to rank these and other shipping companies. Students naturally will try taking the mean of the data. In order to force students beyond this first way of thinking, the developer of the MEA set most of the means of the data within seconds. Therefore, the students would be forced to think of other ideas of centrality and spread to create a procedure to rank shipping companies.
The Effective Prototype Principle ensures that the MEA solution provides a useful prototype, or metaphor, for interpreting other structurally similar, but not necessarily contextually similar situations. In other words, this requires that students will likely need to think in this manner again in a different situation. This gets at the idea of transfer. If students are going to do a problem that takes more time than a typical coursework task, then they should be learning something that will transfer to other problem solving situations later. In the Just-In-Time MEA, the ranking of companies uses many useful concepts in the solution. For example, ranking data is an often-used engineering concept, as well as using multiple types of statistics, particularly of different types (centrality, spread, variance, etc.), to create a model.
NANOTECHNOLOGY THEMED MODEL-ELICITING ACTIVITIES
This paper will discuss two MEAs, the NanoRoughness MEA and the Aluminum Bat MEA, both of which focus on nanotechnology as the context for the MEA. These MEAs were designed for a first-year required engineering problem solving and computer tools course. In both cases, the MEAs were implemented as the fourth (and last) MEA in the course.
NanoRoughness MEA and Sequence
The manner in which the NanoRoughness Modeling Series has been implemented has a four-part sequence. The progression consists of: (1) a pre-reading, (2) the NanoRoughness MEA, (3) a Model-eXploration Activity (MXA), and (4) a Model-Adaptation Activity.
The pre-reading was given prior to the classroom time in which students were going to work on the MEA in teams. The purpose of the pre-reading was to focus on vocabulary. This was an individual assignment focused on communicating the purpose and functionality of the Atomic Force Microscope (AFM). Requiring that students have a very basic knowledge of the AFM helps them get into the MEA. The pre-reading discusses how the cantilever is drawn across the surface and the reaction of the materials causes the cantilever tip to move. The students are told that the measurements of the movement of the cantilever are measured and used to create the digital image. Figure 1 shows a diagram given to the students to help them understand how the cantilever works. In order to encourage student reflection and understanding and assess student completion of the pre-reading, the following questions were asked: (1) What are the three modes of the AFM, and what are the advantages and disadvantages of each? (2) What is the AFM resolution limit? and (3) What causes the resolution limit?
FIG. 1. Diagram of AFM cantilever tip interaction with a surface.
The NanoRoughness MEA is the next part in the sequence. The MEA is separated into 2 parts. First, students work individually to orient themselves to the context and the problem, then they get together with their teams to develop a solution to the MEA. In the individual section, the students get oriented towards the concept of roughness by first answering the following:
(1) How do you define roughness?
(2) What procedure might you use to measure the roughness of the pavement on a road?
(3) Give an example of something for which degree of roughness matters.
- For your example, why does the degree of roughness matter?
- How might you measure the roughness (or lack of roughness) of this object?
The students then read a profile about a company they hypothetically work for that develops coatings for orthopedic and biomedical implants. The company, Liguore Labs, specializes in biomedical applications of nanotechnology; it manufactures coatings for implants. Based on its experience with gold-coated artery stents, the company wishes to start producing synthetic diamond coatings for joint replacements. The students are then asked to individually read a memo from the teams’ “boss”, which outlines the team task. The purpose of the individual part of the MEA is to allow all students to have time to process the problem. It has been found that without this individual processing time, faster processing members of the group may start working on the problem before all have engaged. Therefore, the focus of the individual part of an MEA is engagement of all students in the problem.
The team part of the MEA requires students to develop a procedure to measure roughness given AFM images of three different samples of gold. The motivation for developing the procedure is established by using a realistic context in which a company specializing in biomedical applications of nanotechnology wishes to start producing synthetic diamond coatings for joint replacements. The company intends to extend its experience with gold coatings for artery stents to this new application. The teams are asked to create their procedure to measure roughness using images of gold because the company only has one image of diamond at the moment since diamond coatings are still in the development stages. The teams are given the one image of diamond (Figure 2) to help them understand what the diamond will “look like” in the image and to acclimate themselves to the data available from an image. The top down view and the side view are provided to aid in this understanding. But the students are told that the gold images will only come in top-down views. Student teams of three or four are required to establish a procedure for measuring the roughness of gold samples that could be applied to diamond samples as they are created at a later time. The students then apply the procedure to three different samples of gold (one sample is provided in Figure 3) and develop a list of additional information they need to improve their procedure. The team must write a memo to the company describing their procedure and its application to the sample AFM images and listing the additional information needed to improve their procedure.
FIG. 2. Top-down and side view of newly created diamond coatings AFM image.
FIG. 3. AFM gold sample given to students as data to develop the procedure for roughness.
The Model-eXtension Activity (MXA) is given as homework and is both an individual and team activity. For the NanoRoughness Sequence the purpose is to help the students tie their own novel model to an established engineering model of measuring roughness and to get the students to begin to think about using a computer program to measure the roughness. The individual part of the MXA introduces students to the idea of profile through an example at the macroscale and challenges their current definition of roughness through graphs (Figure 4) and the following questions:
FIG. 4. Profile graphs of a topographical image. The x-axis represents the position on the image in the x-direction; the z-axis represents the height.
(1) Of these 4 graphs, which would you characterize as the most rough and why?
(2) What assumptions have you made about these graphs?
(3) If you consider your procedure for roughness from the MEA, would your answer for which graph was the most rough be the same as in the first question? Why or why not?
The individual students are then asked to perform the Average Maximum Profile (AMP) Method of calculating roughness from the image. The AMP Method (Equation 1) is an average of the difference between the heights of the ten highest peaks and the ten lowest valleys on a cross-sectional graph. It is used for evaluating surface texture on limited-access surfaces where the presence of high peaks or low valleys is of functional significance.
where pi represents the ith highest peak and vj represents the jth deepest valley.
Each team member uses the AMP method to approximate the roughness of the three sample cross-sectional graph of gold (Figure 5). Then the team collects all individual measurements for roughness using the AMP method on one cross-section to compare values and answer the following questions:
1. How is Average Maximum Profile similar to the procedure your team produced?
2. How is Average Maximum Profile different from your team’s procedure?
3. How rough are the three samples of gold (A, B, and C) attached to this memo using the Average Maximum Profile procedure? Compare this to your roughness findings using your procedure.
4. In your opinion, which method better quantifies roughness and why?
5. In what ways does the Average Maximum Profile lend itself to the development of a software tool?
6. Does your team’s method lend itself to the development of a software tool? If so, how? If not, why not?
FIG. 5. One of three samples of cross-sections provided for students to perform the AMP Method for quantifying roughness.
The purpose of having students compare and contrast methods and to think about how a computer program might be built from this method is to prepare the teams to create a MATLAB™ program. This part of the sequence is the Model-Adaptation Activity (MAA).
The MAA, in the form for the first-year engineering course, consists of a six-week team project supported by a series of homework assignments. Student teams develop a computer program in MATLAB™ to upload AFM data and implement several industrial methods for quantifying roughness using AFM data. The student teams were asked to understand pixel representations (Figure 6) to implement the AMP Method and a selection of other engineering roughness measures using MATLAB™ code. The curricular goals of the NanoRoughness MAA are to help the students learn to use flow charts, user-defined functions, repetition and flow control structures, 2-dimensional array manipulations, as well as learn to write executive summaries and judge reliability considerations of the program.
FIG. 6. AFM image magnified to demonstrate pixel values to be used for MATLABTM code for the NanoRoughness MAA.
Student Work for the NanoRoughness MEA
The following student work comes from two teams of in-service mathematics teachers. These students are participants in a professional development master’s level course. The students took part in the NanoRoughness MEA as a part of a unit on teaching with authentic assessments in real-world contexts. The student work seen here is the first iteration of the NanoRoughness MEA and the students did not participate in any other part of the sequence other than the MEA. The collection of work from the first-year course could have been used to show similar work; however, the teachers represented here allowed the researcher to collect the hand-written work as well as the typed solution to the MEA providing for a richer data set.
Team Alpha is a team of three female, very early career mathematics teachers. The solution they created for the NanoRoughness MEA gave the following directions to the user:
1. Divide an image into ¼ micrometer by ¼ micrometer regions.
2. Using the grid as a visual guide count how many “peaks” are in a similar range of height in the overall sample. “Similar range” is determined by the four quartiles the range of heights. (This is determined by finding the range of the height bar and dividing it into 4 equal parts.)
3. Using the total number of peaks in a particular quartile, multiply the number of peaks by the median value of the quartile. For example: The median of 90-120nm is 105nm, therefore, if there are ten peaks in that particular range the total peak height is 1050 for that range.
4. Using the peak height totals for each range, calculate the average height, to use as a measure of roughness for that sample.
Team Alpha applied this to Sample A with the following method:
Range 90 -120 nm: 11 peaks Median = 105 11 x 105 = 1155 nm
Range 60 - 90 nm: 36 peaks Median = 75 36 x 75 = 2700 nm
Range 30 - 60 nm: 24 peaks Median = 45 24 x 45 = 1080 nm
Range 0 - 30 nm: 8 peaks Median = 15 8 x 15 = 120 nm
Average height for Sample A: 63.98nm
The work of Team Alpha on Sample A can be seen in Figure 7.
FIG. 7. Team Alpha written work to implement solution for the NanoRoughness MEA.
Team Beta is a team of four members: one female with approximately five years teaching experience and three males all very early career teachers. The solution Team Beta provided for quantifying roughness consisted of the following steps:
1. Obtain sample size of 1 nm by 1 nm with a scale of 0.25 nm.
2. Grid off sample area in 0.25 nm sections (0.0625 nm2 ). The intersections of the grid become the sampling points resulting in 25 samples per 1 nm2.
3. Use the heights of the sampling points as the data set.
4. Compare data set’s inter-quartile range against Liguore Lab’s established acceptable roughness tolerance levels.
Figure 8 shows Team Beta’s hand-written work.
FIG. 8. Team Beta written work to implement solution for the NanoRoughness MEA.
Team Alpha and Team Beta implemented very different solutions. The solution by Team Alpha was quite inventive, but did not take into account the valleys of the images. However, the solution presented here was a type of weighted average, which is a good start to the solution to this problem. Team Beta worked on a method of variability. By using inter-quartile range, Team Beta tried to get rid of outliers and used a sampling method to collect data in a fair manner. Both solutions were good beginnings to a solution to this MEA given that each team was given fifty minutes to complete the task. This work is consistent with the work seen by freshmen engineering students on their first attempt at the problem. At this point of implementation in a first-year course, students would be required to do two more iterations on their model, both peer and instructor feedback would be given.
Aluminum Bats MEA
The Aluminum Bats MEA has student teams create a procedure to measure the average crystal size for aluminum. The Aluminum Bats MEA has a similar sequence to the NanoRoughness MEA. Here the sequence is: (1) the MEA, (2) the MXA, and (3) the MAA. Because these are somewhat parallel problems used in different semesters to cover the same MATLABTM programming content, the MXA and MAA are very similar to the NanoRoughness MEA. For sake of brevity, only the Aluminum Bats MEA will be discussed in detail.
The basic layout of all MEAs is similar. So for the Aluminum Bats MEA, the individual part consisted of having the student teams read a newspaper article and two sections of background information. The newspaper article helped students get into the reality of the need for measuring the average crystal size of aluminum by discussing a local softball coach who had issues with his bats denting during the previous season. He stated in the article that he wanted to find new bats that resist denting. Part 1 of the background information helps students understand aluminum crystals. Figure 9 shows a local traffic light pole with aluminum crystals large enough to be seen without a microscope. Figure 9a shows the crystals at a normal view and Figure 9b shows a close-up view of the crystals with three crystals outlined for emphasis.
FIG. 9. (a) Traffic light pole. (b) Close up of crystals
Part 2 of the background information explains how size of crystals makes a difference in the strength of the material. Since smaller crystals make aluminum stronger, micrographs are also introduced in this section. Using simplified descriptions, this section explains how materials engineers make the crystals more visible.
The team part of the MEA requires that student teams create a procedure for measuring crystal size using three micrograph samples (Figure 10). The student teams are given the following scenario to get them into the context for the MEA. The engineering firm the team works for is competing for a job for a company that produces aluminum and sells it to manufacturers, such as makers of softball bats. The team’s potential client (the aluminum producer) wants to improve quality control over their aluminum manufacturing process. To win the job, the engineering firm needs to propose a procedure for determining the size of aluminum crystals from micrographs. It is each student team’s job to create a procedure for this proposal.
FIG. 10. Sample micrographs used as data for the Aluminum Bats MEA
The entire collection of the NanoRoughness Modeling Sequence and the Aluminum Bats Modeling Sequence can be accessed at https://engineering.purdue.edu/ENE/Research/SGMM/MEAs_html.
Student Work for the Aluminum Bats MEA
The following student work comes from two teams of in-service mathematics and science teachers. These students are participants in a professional development master’s level course (but a different course than the examples from the NanoRoughness MEA). The students took part in the Aluminum Bats MEA as a part of a unit using engineering as a context to teach mathematics and science. The student work seen here is the teams’ final solution for the Aluminum Bats MEA. The collection of work from the first-year course could have been used to show similar work; however, the teachers represented here allowed the researcher to collect the hand-written work as well as the typed solution to the MEA providing for a richer data set.
Team Delta is a team of two females and one male. Both females teach mathematics teacher at the community college level, one of them is a novice teacher. The male teaches high school physics and geometry and has several years of experience. The solution this team created gave the following directions to the user:
1) Given a sample, you will be creating squares in which you will count the number of crystals. For your square, choose a length such that approximately ten crystals will fit in the box, either as parts or wholes. This is so your box will be large enough to get a decent sample, but small enough to keep the counting to a minimum.
2) Count the number of whole crystals in the box. This includes any crystal that you would not round to ¾ of a crystal (i.e. it is larger than 7/8 of a crystal). Then count the partial crystals, approximating the part in the box as ¼, ½, or ¾ of the entire crystal. Add your fractions to the number of whole crystals to get the entire number of crystals in your box. Note that any enclosed area is considered a crystal. You are attempting to find the approximate number of crystals in the box, and we are including partial crystals in the count.
3) Find the area of the box, and divide it by the number you get in part 2. This is a preliminary value for the average size of the crystals in your sample.
4) Repeat steps 2 and 3 for the same sample at least three more times, and then average your results. This is your final average crystal size. We are doing several smaller samples rather than one large one to make sure a section of the sample is not chosen in a way that skews results to smaller or larger crystal sizes.
Figure 11 shows Team Delta’s application of this procedure to the Sample A micrograph.
FIG. 11. Team Delta written work to implement solution for the Aluminum Bats MEA.
Team Gamma is a team of three members: one female mathematics teacher with approximately five years teaching experience and two males, one mathematics and one science teacher, both with many years teaching experience. The solution Team Gamma provided for quantifying crystal size consisted of the following steps:
- Draw one vertical, one horizontal, and one diagonal (corner to corner) line through each sample. Measure each line in millimeters. Use the scale of each micrograph to convert the measurements to the true length. Reasoning: We used the technique of transecting (used to determine density) to find a linear measure of crystal density in each sample.
- For each line on each sample, count the total number of crystals that makes contact with each line. Record these counts.
- For each sample, calculate the number of crystals per millimeter by dividing each of the three line’s crystal counts by the true length in millimeters. Find the mean of these three values to obtain the average number of crystals per millimeter for a specific sample. Repeat this process for each sample. Reasoning: Three lines were used to take three measurements per micrograph sample. The average was calculated to reduce sampling variability.
- The sample with the greatest average crystal per millimeter count is the strongest sample. Reasoning: The sample with the highest density of crystals is the strongest.
- To find the average length of each crystal, take the inverse of each average density measurement. Assuming each crystal is approximately square, the average crystal length can be squared to determine the approximate average crystal area. The sample with the smallest average crystal area will be the strongest. Reasoning: While the values determined in step 4 provide a justification for the strongest sample, the step 5 values match the criteria requested from the engineering firm to find the size of the crystal in order to compare aluminum strength.
Figure 12 shows Team Delta’s solution applied to Sample A.
FIG. 12. Team Gamma written work to implement solution for the Aluminum Bats MEA.
Both teams implemented interesting solutions to this problem. Team Delta estimated the average crystal size in at least three parts of the micrograph and then averaged their estimates. This seems like a reasonable solution for a beginning procedure to estimate average crystal size, but students used estimations of estimations. This leaves room for a lot of human error and interpretation. Team Gamma created a procedure that is very similar to the stereology method that was provided in the homework. The accuracy of Team Gamma’s procedure could easily be increased by adding more than three lines. This is a very advanced solution and it is likely that the science teacher, who teaches large units on density, used a procedure with which he was already familiar and adapted it to this situation. This also gets at the idea of transfer. Both teams did a good job of making an easy-to-implement model that measured what the client needed. However, Team Gamma’s solution was much more elegant and could be easily transferred to a computer program.
Model-Eliciting Activities are innovative curricular tools that have the potential to impact engineering education in a positive way. MEAs have provided engineering educators with authentic assessments that can bring high-level, cutting-edge research to undergraduate engineering classrooms. The NanoRoughness MEA is a good example of an MEA that is appropriate for lower-levels of undergraduate engineering, while at the same time bringing upper-level material science content to these students. MEAs can provide engineering students the opportunity to see vast engineering disciplines in a motivating way, which holds the promise to influence their final career choices. Through research from the SGMM Gender Equity Grant, MEAs have been shown to aid in retention of underrepresented populations of students in engineering, particularly women. Current research is aimed at extending the MEA framework to upper levels of new engineering disciplines, create models that elicit misconceptions, and to help students make ethical decisions as professional engineers.
8. H.A. Diefes-Dux, P.K. Imbrie, and T.J. Moore, First-Year Engineering Themed Seminar - A Mechanism for Conveying the Interdisciplinary Nature of Engineering. (2005 ASEE National Conference, Portland, OR, 2005).
9. R. Lesh, M. Hoover, B. Hole, A. Kelly, and T. Post, Principles for Developing Thought-Revealing Activities for Students and Teachers. In Handbook of Research Design in Mathematics and Science Education (Lawrence Erlbaum, Mahwah, NJ, 591-645, 2000).
11. H.A. Diefes-Dux, T.J. Moore, J. Zawojewski, P.K. Imbrie, and D. Follman, A Framework for Posing Open-Ended Engineering Problems: Model-Eliciting Activities, (Paper presented at the FIE Conference, Savannah, 2004).
17. S.L. Richardson, A Design of Useful Implementation Projects for the Development, Diffusion and Appropriation of Knowledge in Mathematic Classrooms. Doctoral dissertation, Purdue University, W. Laf., IN (2004).
18. R. Lesh, The Development of Representational Abilities in Middle School Mathematics. In I. E. Sigel, Ed., Development of Mental Representation - Theories and Applications (Erlbaum, Mahwah, NJ, 323-350 (1999).
20. R. Lesh, and D. Clarke, Formulating Operational Definitions of Desired Outcomes of Instruction in Mathematics and Science Education. In R. Lesh and E. A. Kelley, Eds., Handbook of Research Design in Mathematics and Science Education (Lawrence Earlbaum, Mahawah, NJ, 113-152, 2000).
23. D.K. Follman, C.V. Liguore, and C.T. Watkins, Work In Progress - Male and Female Engineering Student Roles and Interaction Styles on Online Collaborative Learning Teams (ASEE/IEEE FIE Savannah, GA, 2004).
24. T. Wood, E. Moss, and C. Zapato, Learning from Observations to Establish Researchable Questions. In J. Zawojewski, H.A. Diefes-Dux & K. Bowman, Eds., Models and Modeling in Engineering Education: Designing Experiences for All Students. (Manuscript in progress, 2007).
25. H.A. Diefes-Dux, D. Follman, P.K. Imbrie, J. Zawojewski, B. Capobianco, and M.A. Hjalmarson, MEAs: An In-Class Approach to Improving Interest and Persistence of Women in Engineering (ASEE, Salt Lake, 2004).
28. T.J. Moore, H.A. Diefes-Dux, and P.K. Imbrie. The Quality of Solutions to Open-Ended Problem Solving Activities and its Relation to First-Year Student Team Effectiveness (ASEE, Chicago, IL, 2006).
29. T.J. Moore, H.A. Diefes-Dux, and P.K. Imbrie, Model-Eliciting Activities: A Curricular Tool to Measure First-Year Engineering Students’ Perception of Learning Problem Solving and Teaming (FIE, Indianapolis, IN, 2005)
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