# analyzing 3D-printed artifacts to develop mathematical modeling strategies: While working on the modeling activity, the students also experienced the benefits of teamwork and persistence.

Introduction and Background

Algebra has consistently been identified as a gatekeeper subject because of the role success in algebra has on students' ability to graduate from high school, their readiness for college-level mathematics, and their opportunities for employment (Loveless, 2013; Rech & Harrington, 2000). The importance of algebra has led to efforts to reform mathematics curriculum in schools across all grade levels so that algebraic reasoning is encouraged beginning in early elementary grades (e.g., National Council for Teachers of

Mathematics, 2000; National Governors Association Center for Best Practices, 2010), In addition to algebraic reasoning, the National Mathematics Advisory Panel (2008) specified "fitting simple mathematical models to data" (p. 16) as one of the major topics of school algebra.

Mathematical modeling can be found in algebra curricula standards at both the national and local levels. The National Council of Teachers of Mathematics (2000) specifies the importance of mathematical modeling in the Algebra Strand of its Principles and Standards for School Mathematics: "One of the most powerful uses of mathematics is the mathematical modeling of phenomena. Students at all levels should have opportunities to model a wide variety of phenomena mathematically in ways that are appropriate to their level" (p. 39). The use of 3D printing in schools provides a fruitful context for exploring algebra in a more authentic setting; in particular, 3D-printed artifacts can be used as the basis for students' mathematical modeling activities. Mathematical modeling is a cyclical process requiring repeated testing and revision during the development of models, a process that is intimately related to the design process as described in Standards for Technological Literacy: Content for the Study of Technology (ITEA/ITEEA, 2000/2002/2007).

Mathematical Modeling

Mathematical modeling is the process of representing real-world situations using mathematics as a way to understand and solve a specified problem (Daher & Shahbari, 2015; Bliss & Libertini, 2016), and the model itself is the mathematical description of the real-world situation (Lesh & Lehrer, 2003). A model-eliciting activity (MEA) is "a problem-solving activity constructed using specific principles of instructional design in which students make sense of meaningful situations and invent, extend, and refine their own mathematical constructs" (Kaiser & Sriraman, 2006, p. 306).

The purpose of MEAs is the modeling process itself rather than the application of known procedures to produce a final solution. As a result, MEAs can accomplish several goals. By emphasizing the modeling process, MEAs encourage students to think mathematically and provide them with an opportunity to showcase their mathematical capabilities (Daher & Shahbari, 2015). MEAs also provide students with multiple entry points to a problem because they encourage authentic problem solving. Problem solving is defined as, "engaging in a task for which the solution method is not known in advance" (National Council of Teachers of Mathematics, 2000, p. 52). Because there is not a prescribed procedure for MEAs, mathematical modeling tasks are open-ended, and the final models themselves can vary (Bliss & Libertini, 2016).

Methods and Materials

The Laboratory School of Advanced Manufacturing (Lab School) is a joint venture among the Make to Learn Lab at the University of Virginia, Charlottesville City Schools, and Albemarle County Public Schools. With support from the National Science Foundation, the Make to Learn Lab, working with the Smithsonian Institution, is developing a series of Invention Kits that allow students to reconstruct historical inventions (e.g., solenoid, motor, generator, speaker). Technological literacy requires an understanding of technology as a human creation, its influence on historical inventions and innovations, and its relationship to other fields of study (ITEA/ITEEA, 2000/2002/2007). By reconstructing historical inventions (e.g., solenoid, motor, generator, speaker) using modern manufacturing techniques, students will be able to develop a deeper understanding of the relationships among technology, science, engineering, and mathematics.

The Deriving Ampere's Law task, which is the realistic model-eliciting activity used in this study, was developed to extend the mathematics connections associated with the Solenoid Invention Kit. A solenoid is a coil of conductive wire; when electric current flows through the wire, the coil generates a magnetic field (Figure 1).

The strength of the magnetic field produced by a solenoid (B) is dependent on the number of wraps of wire that comprise the solenoid (N), the length of the solenoid (I), and the electrical current (I). This relationship is known as Ampere's Law, B = [mu](N x I/L), where [mu] is a constant dependent on several factors. The Deriving Ampere's Law task addresses several learning objectives across the STEM disciplines, including electromagnetism, direct and inverse variation, and problem solving (Table 1).

Development of the Deriving Ampere's Law Task

In order to produce magnetic field strength measurements that would allow a wide range of students to be successful with this task, a set of prewrapped solenoids was designed and calibrated. Solenoid tubes of varying length were 3D-printed (Afinia H79 and uPrint SE) and then hand-wound. The final prewrapped set included six solenoids: four solenoids with 50 wraps of wire, but varying lengths (1 inch, 2 inches, 3 inches, 4 inches) and two additional solenoids measuring 2 inches, but varying wraps of wire (100 wraps, 150 wraps). A variable power supply was used to vary electric current, and PASCO's PASPORT Magnetic Field Sensor was used to measure magnetic field strength. The sensor connected to the accompanying SPARKVUE software via Bluetooth using the PASCO AirLink interface (Figure 2, page 14).

The Deriving Ampere's Law task was pilot-tested in April 2016. During this pilot test it was discovered that a slight shift in the placement of the coils of wire would drastically alter the magnetic field strength generated by the solenoid. While students were able to successfully derive Ampere's Law, the inaccuracies in data collection did cause some confusion. Following this pilot test, the solenoids were recalibrated, and epoxy putty was used to secure the coils on each of the solenoids. With the newly recalibrated solenoids, the data was collected under laboratory conditions (Table 2). The power supply used for this task reliably held 3.16 A, which is why the solenoids were calibrated using this current value.

The Deriving Ampere's Law task was implemented in June 2016 during the Summer Engineering Academy, an annual enrichment program hosted at the Make to Learn Lab in the Curry School of Education at the University of Virginia. A total of twelve rising eighth-grade students from two different local middle schools were selected by their principals to participate in the two-week-long Academy. These students worked alongside their science and engineering teachers, Curry faculty, and doctoral students to manufacture solenoids, a generator, a motor, and a speaker in order to understand the science behind these historic inventions.

Prior to participating in the Deriving Ampere's Law task, students used a rudimentary tangent galvanometer to explore the differences in the magnetic field strengths generated from a straight wire versus a coiled wire. Students developed a qualitative understanding that coiled wire generated a stronger magnetic field. They also spent time making and wrapping their own solenoids, which were later used to construct a motor, a generator, and a speaker.

Of the twelve students who participated in the Engineering Academy, six were purposefully selected to participate in the Deriving Ampere's Law task. The students were grouped based on the mathematics coursework they had completed during seventh grade. The three students in the first group had already completed algebra (algebra group), and the three students in the second group had not yet taken algebra (pre-algebra group). The algebra group consisted of one male student and two female students. The pre-algebra group consisted of two male students and one female student.

The Deriving Ampere's Law task was presented as four separate investigations (see Sidebar, page 17). In the first investigation, students were asked to relate the strength of the magnetic field produced by the solenoid to the number of wraps of wire, with the other independent variables held constant. Similarly, in the second investigation, students were asked to relate the magnetic field strength to the length of the solenoid. In the third investigation, students were asked to relate the magnetic field strength to the electric current. For each of the first three investigations, students were asked to develop a model that could be used to predict magnetic field strength. Once students had developed separate models for each of the independent variables, the fourth investigation challenged students to create a new model that related magnetic field strength to the three independent variables.

Results

The students required varying levels of scaffolding to complete the task; however, both groups were able to successfully derive Ampere's Law. Students were able to relate the number of coils of wire, length of the solenoid, and current output to the strength of the magnetic field both qualitatively and quantitatively. All of the students recognized the pattern in the data they collected (e.g., as the number of coils increases, the magnetic field strength increases; as the length of the solenoid increases, the strength of the magnetic field decreases).

The students who had completed algebra referred to both the direct and inverse variations by name. These students also applied their understanding of slope (e.g., for each increase of 50 wraps, there was an increase of 36 Gauss) and used this to generate a linear equation (e.g., y = 0.72x, y = 36/50 x) for the direct variations. Those who had not yet completed algebra did not recognize the types of variation, but were able to numerically predict magnetic field strength for different solenoids, even for those that they did not have (e.g., a solenoid wrapped with 200 coils would have a magnetic field strength of 144 Gauss). With additional support, these students were also able to generate equations for the individual variables (Figure 3). Students worked similarly when analyzing the relationship between current and magnetic field strength. Both groups were able to immediately recognize that the relationship between the length of the solenoid and the magnetic field strength was nonlinear. Those students who had studied algebra were able to generate the equation y = 72/x without much difficulty. Those who had not previously studied algebra had difficulty with this at first, but eventually arrived at the same equation.

Both groups initially struggled when asked to derive an equation that incorporated all three independent variables. The students who had completed algebra initially made several intuitive attempts at combining the equations (e.g., adding terms, averaging them), but found that their equations did not work for their collected data. Once these students were reminded of the direct and inverse relationships they had previously identified, they were able to combine the equations in a way that preserved the types of variation (see Figure 4). The students who had not completed algebra were prompted to discuss the nature of direct and inverse variation prior to beginning the fourth investigation. With the types of variation in mind, all students were able to recognize the structure of the relationship as [wraps x current]/length and they all came up with an equation similar to y = 0.46 ([N x I]/L)

At the conclusion of the task, all students felt a sense of pride and accomplishment that they were able to derive Ampere's Law. One student commented, "We figured out an equation without really any help. We went through the same process that a scientist [Ampere] went through, and we figured it out and we're seventh graders. That's pretty cool." All of the students shared that this task was unlike anything that they had done in their mathematics classes. Unlike typical mathematics tasks, these students were provided with an open-ended problem without a prescribed procedure. They appreciated the opportunity to explore varied solution strategies. As another student explained, "Usually in class, we do worksheets on stuff that the teacher's been talking about on the board. [This] was something different. We had to think using our own brain to try to figure out what to do."

Discussion

The use of the Deriving Ampere's Law task accomplished two content-specific learning goals. First, the students who participated in the activity developed a better understanding of multivariate mathematical modeling, which was the intended goal of this model-eliciting activity. These students were able to apply knowledge that was learned in their pre-algebra and algebra classes (e.g., slope, direct variation, inverse variation) to a contextualized problem. In particular, one of the students recognized that the strategy he used to experimentally derive Ampere's Law (i.e., identifying types of variation and arranging variables accordingly) could be applied more generally to other variables and modeling situations that involve both direct and inverse variation. Second, the students who participated in the activity also developed a better understanding of the connection between electricity and magnetism. While working through the Solenoid Invention Kit, these students came to understand that a current passing through conductive wire produces a measureable magnetic field, which can then be used to recreate a variety of artifacts. As an extension to the Solenoid Invention Kit, the Deriving Ampere's Law task helped students develop a quantitative understanding of how different properties of solenoids (e.g., solenoid length, number of wraps of wire) affected the strength of the magnetic field generated by a solenoid.

Furthermore, working on the Deriving Ampere's Law task provided these students with the opportunity to meaningfully engage in a scientific process. To complete the task, students had to collect and analyze experimental data. Students in both groups realized the importance of testing and verifying their hypotheses against collected data without any prompting, They also utilized multiple representations of their data (e.g., numerical, graphical) when developing their models. For example, when analyzing linear data, some students recognized that the data was linear from their data table and used a graph to confirm that. When analyzing nonlinear data, some students graphed the data to get a better sense of the relationship before settling on an equation.

The Deriving Ampere's Law task is a complicated activity, and the open-ended nature of the task provided these students with an opportunity to recognize when and how to work as a team. For example, students in both groups allocated responsibilities during data collection (e.g., adjusting the power supply, positioning the probe). After the data was collected, students in both groups determined when working individually versus working collaboratively would be the most effective for data analysis. At times, the students would first work independently and then share their initial observations. At other times, the students would first collaborate with one another to assign responsibilities (e.g., deciding who would graph the collected data), and once the responsibilities were completed, the students would then regroup.

The Deriving Ampere's Law task also fostered students' stick-to-itiveness. These students persisted for upwards of three hours to complete the task, without any obvious signs of tiring. For example, even when the students had arrived upon a final model after more than two hours of work, they still took the time to test this model against their collected data. It is worth noting that the activity took place outside of a traditional classroom setting and that the Summer Engineering Academy was scheduled during the students' summer vacation. All of the students were motivated to complete the activity, and they voiced a sense of pride and accomplishment when they were able to successfully derive Ampere's Law.

Conclusion

More and more schools are acquiring advanced manufacturing equipment. Activities that incorporate the use of this equipment for the purpose of teaching some academic content can either be designed or redesigned for other educational purposes as well. In the case above, the original purpose of the Solenoid Invention Kit was to help students understand the connection between electricity and magnetism. However, the kit was extended to address mathematical modeling and problem solving through the Deriving Ampere's Law task. An ancillary outcome was that students experienced the benefits of teamwork and persistence. We believe these examples illustrate that 3D printing can be used in schools to address a variety of educational goals.

Deriving Ampere's Law Task: Investigation Descriptions

Ampere's Law, B = [mu] ([N x I]/L) relates the strength of the magnetic field generated by a solenoid to the number of wraps of wire that comprise the solenoid (N), the length of the solenoid (L), and the electrical current (I). This relationship can be derived experimentally by systematically varying the different parameters of a solenoid. One way to make the Deriving Ampere's Law task accessible to a wide range of students is to scaffold the activity into a series of four investigations.

Investigation 1: Relating Wraps of Wire to Magnetic Field Strength

For this investigation, students are provided with a set of three solenoids. The length of the solenoid (2 inches) and the electric current (3.16 A) are held constant, but the solenoids vary in the number of wraps of wire (50 wraps, 100 wraps, and 150 wraps). There is a direct relationship between the number of wraps of wire of a solenoid and the strength of the magnetic field produced. When graphed, the data generates a straight line and the resulting equation, using the data collected under laboratory conditions (Table 2), is: B = 36/50N.

Investigation 2: Relating Solenoid Length to Magnetic Field Strength

For this investigation, students are provided with a set of four solenoids. The number of wraps of wire (50 wraps) and the electric current (3.16 A) are held constant, but the solenoids vary in length (1 inch, 2 inches, 3 inches, 4 inches). There is an inverse relationship between solenoid length and the strength of the magnetic field produced. When graphed, the data generates a curved line, and the resulting equation, using the data collected under laboratory conditions (Table 2), is: B = 72/L.

Investigation 3: Relating Electric Current to Magnetic Field Strength

For this investigation, students are provided with a variable power supply to measure the magnetic field strength of a single solenoid at varying levels of electric current. The other two independent variables (number of wraps of wire and solenoid length) are held constant. There is a direct relationship between electric current and the strength of the magnetic field produced. When graphed, the data generates a straight line, and the resulting equation, using the data collected under laboratory conditions (Table 2), is B = [900/79] I.

Investigation 4: Developing a Final Model for Ampere's Law For this investigation, students are asked to review the models generated from the three previous investigations and come up with one model relating the three independent variables (number of wraps of wire, solenoid length, and electric current) to a single dependent variable (magnetic field strength). Using the data collected under laboratory conditions (Table 2), the final model is: B = 0.456 ([N x I]/L).

References

Bliss, K. & Libertini, J. (2016). What is mathematical modeling? In S. Garfunkel & M. Montgomery (Eds.), Guidelines for Assessment & Instruction in Mathematical Modeling Education (pp. 7-21). Bedford, MA: COMAP, Inc.; Philadelphia, PA: SIAM.

Daher, W. M. & Shahbari, J. A. (2015). Pre-service teachers' modelling processes through engagement with model eliciting activities with a technological tool. International Journal of Science and Mathematics Education, 13, S25-S46.

International Technology Education Association (ITEA/ITEEA). (2000, 2002, 2007). Standards for technological literacy: Content for the study of technology. Reston, VA: Author. Kaiser, G. & Sriraman, B. (2006). A global survey of international perspectives on modeling in mathematics education. ZDM: The International Journal on Mathematics Education, 38, 302310.

Lesh, R. & Lehrer, R. (2003). Models and modeling perspectives on the development of students and teachers. Mathematical Thinking and Learning, 5,109-129.

Loveless, T. (2013, September). The algebra imperative: Assessing algebra in a national and international context. Brown Center on Education Policy at Brookings. Retrieved from www. brookinqs.edu/wp-content/uploads/2016/06/Kern-Alqebrapaper-8-30 v14.pdf

National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author.

National Governors Association Center for Best Practices, Council of Chief State School Officers. (2010). Common core state standards (mathematics). Washington, DC: Author.

National Mathematics Advisory Panel. (2008). Foundations for success: The final report of the National Mathematics Advisory Panel. Washington, DC: U.S. Department of Education.

NGSS Lead States. (2013). Next generation science standards: For states, by states. Washington, DC: The National Academies Press.

Rech, J. & Harrington, J. (2000). Algebra as a gatekeeper: A descriptive study at an urban university. Journal of African American Studies, 4(4), 63-71.

Acknowledgements

This work was supported by National Science Foundation Grant No. 1513018--American Innovations in an Age of Discovery: Teaching Science and Engineering through Historical Reconstructions. Any opinions, findings, conclusions, or recommendations expressed in this article are those of the authors.

Kimberly Corum, Ph.D., is an assistant professor of mathematics education at the Fisher College of Science and Mathematics at Towson University. She can be reached at kcorum@> towson.edu.

Joe Garofalo, Ph.D., is Co-Director of the Make to Learn Lab at the Curry School of Education at the University of Virginia. He can be reached at garofalo(cO virqinia.edu.

This is a refereed article.

Caption: Figure 1. A solenoid with a 3D printed tube.

Caption: Figure 2. Test instruments used to complete the Deriving Ampere's Law task.

Caption: Figure 3. Pre-algebra students' solutions for first three investigations.

Caption: Figure 4. Algebra students' solutions for the fourth investigation.
```Table 1. Selection of Content Standards Aligned with the

Science (NGSS)               Mathematics         Engineering (NGSS)
(CCSS)

MS-PS-3. Ask            6.EE.C.9. Use           MS.ETS1-4. Develop a
questions about data    variables to            model to generate
to determine the        represent two quanti-   data for iterative
factors that affect     ties in a real-world    testing and
the strength of         problem that change     modification of a
electric and magnetic   in relationship to      proposed object,
forces.                 one another ...         tool, or process such
that an optimal
HS-PS2-5. Plan and                              design can be
conduct an investiga-   HAS.CED.A.2. Cre-ate    achieved.
tion to provide         equations in two or
evidence that an        more variables to
electric current can    represent
produce a magnetic      relationships between
field ...               quantities ...

Table 2. Solenoid Data Collected Under Laboratory Conditions.

Number of    Solenoid      Electric        Field        Constant
Wraps (W)   Length (L)    Current (I)   Strength (B)     ([mu])

50             2 in         3.16 A        35.97 G        0.455
100            2 in         3.16 A        71.87 G        0.455
150            2 in         3.16 A        107.8 G        0.455
50             1 in         3.16 A        71.87 G        0.455
50             2 in         3.16 A        35.97 G        0.455
50             3 in         3.16 A        24.20 G        0.459
50             4 in         3.16 A        18.10 G        0.458
50             2 in         0.79 A         9.0 G         0.456
50             2 in         1.58 A        17.97 G        0.455
50             2 in         3.16 A        35.97 G        0.455
```
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