KAREN COUCH
KAREN COUCH
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Karen Couch Breitbach graduated from the Adel-Desoto-Minburn High School in 1977 and earned her Bachelor’s Degree in chemistry from the University of Northern Iowa in 1981. She began her career in the Materials Laboratory at the John Deere Tractor Works in Waterloo, Iowa, but it was her 1985 UNI study for her Master’s Degree in chemistry that lit the teaching flame.
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The Lab School hired Breitbach that year for its science faculty, and she immediately embraced the school’s collaborative culture, joining with her colleagues in regular presentations at the National Science Teachers Association’s conventions.
She taught the Lab School high school Physics course, and, in the fall of 1992, she began to talk with Mathematics Department colleague James Maltas regarding a combined course in Advanced Mathematics and Physics. They had noted that the same students, for the most part, took each course and that “overlap” occurred.
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“Many of the applications used in the Advanced Mathematics course involved either a physics problem or were related to one,” explains Maltas. “The Physics course used advanced mathematics skills extensively.”
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Breitbach wanted to fine-tune her use of math language and notation and to expand her use of graphing calculator technology. Maltas hoped to expand the modeling and applications of the physics he used in Advanced Mathematics.
Their discussions led them to set three goals for a potential “Advanced Mathematics and Physics” course:
1. Physics could provide students with concrete examples of abstract mathematical ideas that could enhance the learning of mathematical concepts;
2. Mathematics would enable students to achieve a deeper understanding of physics concepts by providing ways to quantify, explain, and model physics relationships; and
3) Difficult mathematical concepts could be revisited throughout the curriculum to help students develop a deeper understanding.
They decided to create a combined course using both class periods and a team-teaching approach. The double period would allow for more extensive explorations. Team teaching would enable students to observe different approaches to the same problem.
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The MPLS administration was very supportive of this STEM initiative and agreed to schedule the courses back-to-back, have both Breitbach and Maltas listed as instructors, and give both of them credit for two courses in respect to their official course loads.
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The first “Advanced Mathematics and Physics” course was offered in the 1993-94 school year. Immediately Breitbach and Maltas saw the need for more financial resources to support the purchase of more technology, such as various data collection devices that could be used with the graphics calculators and for the development of curriculum materials that were not currently available for this integrated course.
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The two applied for and were awarded UNI Mini-Grants from the Provost’s office in 1994 and 1995 to support these efforts. Some of the grant money was used to share this project throughout Iowa at various Area Education Agencies (AESs) and school districts, including the Southern Prairie, Grant Wood, Western Hills, Cedar Rapids, Loess Hills, and Marshalltown AEAs.
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Breitbach and Maltas were also awarded two Iowa Eisenhower grants, “Modeling Integration in Mathematics and Science” and “Demonstrating Excellence in Mathematics and Science Teaching,” that were used to offer two summer workshops and school year follow-up sessions to acquaint Iowa teachers with the curriculum model and to support them in the implementation of a similar course in their schools.
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The curriculum had three strands. The first strand, and the largest, was the integrated physics and advanced mathematics. This strand covered the traditional topics in physics with data analysis and problem solving using mathematical concepts that spanned algebra, geometry, and calculus. Technology such as graphics calculators with interfacing equipment allowed students to carry out investigations that were not possible with conventional equipment. This use of technology enhanced students’ abilities to collect and analyze data.
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The second strand foregrounded Mathematical Modeling. Models can be physical, but also theoretical. A theoretical model of an object or phenomenon is a set of rules or laws that accurately represents that object or phenomenon in the mind of the observer; it is a mathematical structure, algebraic and/or geometric, that approximates the features of a phenomenon. The modeling process is composed of five main stages:
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Observing a phenomenon, delineating the problem situation inherent in the phenomenon, and discerning the important factors (variables/parameters) that affect the problem.
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Conjecturing the relationships among factors and interpreting them mathematically to obtain a model for the phenomenon.
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Applying mathematical analysis to the model.
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Obtaining results and reinterpreting them in the context of the phenomenon under study and drawing conclusions.
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Testing and refining the model. Are the conclusions usable? Do they make sense? If not, examination of the model's factors and structure is called for and a possible reformation of the model may be necessary.
Some of the modeling activities did not involve all five steps but emphasized only some steps to master these steps.
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The third strand incorporated statistics. A series of activities/investigations accompanied by problem sets were interspersed throughout the curriculum. Students collected, organized, graphically represented, and analyzed real-world data, working from descriptive to inferential statistics during the academic year. At least once a month students were engaged in an extensive statistics lesson/activity. In most cases, the lesson/activity was completed in one day (a single two-period block). However, occasionally, two consecutive days were required to bring the lesson/activity to closure.
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The “Integrated Curriculum in Advanced Mathematics and Physics” teaching materials Breitbach and Maltas created contained these items in the Table of Contents for Displacement, Vectors & Trigonometry:
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Activity I: Locating A Point in Two Dimensions
Activity II: Following Compass Directions
Activity III: Giving Directions/Following Directions
Option B: Treasure Hunt
Activity IV: Indirect Measurement
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Problem Set I:
A. Rectangular and Polar Coordinate Systems
B. Rectangular, Polar, and Compass Coordinate Systems
C. Compass Headings and Cardinal Directions
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Problem Set II: Displacement Vectors (Right Triangle Trigonometry)
Problem Set III: Displacement Vectors (Law of Sines/Law of Cosines)
Problem Set IV: Displacement Vectors (Component Addition/Matrices)
Problem Set V: Angular Displacement
Problem Set VI: Employing Multiple Approaches
Problem Set VII: Trigonometry Challenge
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“I really enjoy the presentation of a situation, a phenomena. [The students] see something and ask themselves ‘What’s going on here?’ What is it that I can explore, investigate?’,” Breitbach explains in the short on-site video below. “Whatever they decide to go into, I want them to be well prepared, so I’m not just training students to go off and take college physics. I’m training students to go out and live the rest of their lives. The technology skills I want to be transferrable. That’s a huge thing to me.”
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