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## Giving Meaning to the Numbers

**1/23/2007 - Jeff Marshall, Bob Horton, and Joyce Austin-Wade**

When learning, students yearn for meaning, challenge, and relevance. Integrated learning fulfills these desires by limiting the compartmentalization of learning—providing a more coherent learning environment. Too often, mathematics and the physical sciences are taught as separate entities. Yet, many commonalities exist, especially between chemistry and Algebra II and between physics and precalculus (including trigonometry).

I (Jeff) am a science teacher, and my quest to integrate science and mathematics learning began over a decade ago when I participated in a two-year National Mathematics and Science Fellows project sponsored by the National Science Foundation and the Coalition for Essential Schools. As a science teacher, I welcomed the opportunity several years later when a math teacher also had a desire to integrate application-based learning. We created an integrated math/science course that specifically relates physics and precalculus concepts. I collaborated with my coauthor Bob to make sure the mathematics aligned with the most current mathematics standards and concepts.

Context

An alternative education program provided the context for this learning experience, but this course would also work in more traditional settings. In the featured alternative program, students take one or two courses at a time. Upon completion, students start the next course. For more traditional settings where classes are 55 minutes or 90 minutes long, students’ classes can be blocked together (e.g., first-period physics and second-period mathematics or vice versa). Blocking student schedules provides longer exploration opportunities and flexibility to reconfigure groups so individual needs are met (e.g., a group of students can work on the computer to learn specific mathematics skills while another group gathers data to be analyzed).

Framework

The *National Science Education Standards* (NRC 1996) and the *Principles and Standards for School Mathematics* (NCTM 2000) share a significant number of standards. Figure 1 (p. 38) provides several examples where physics and mathematics standards logically parallel one another. Our integrated course unites recommendations from the College Board for Physics (2006), both the National Science and National Mathematics Standards, and the state standards. Although this is not an Advanced Placement class, being aware of the College Board’s expectations helps lay the foundation for those taking a second year of physics.

Given a textbook and a list of standards, teachers too often start at the beginning of the textbook and “cover” as much material as possible until the year ends. To avoid this, our approach when developing the course began by planning backwards. First, we identified the skills and knowledge base in each discipline that students should attain by the end of the school year. Then, we condensed each discipline’s list to the most critical things students should know and be able to do by the end of the course (McDonald 1992; Wiggins and McTighe 1998). Figure 2 (p. 39) details the list for physics and precalculus. The core of the integrated course focuses on commonalities and complementary ideas between the two courses. Standards that do not link well between disciplines are taught using an unlinked teaching approach. For instance, proving key trigonometric identities is a mathematical skill that does not have a parallel physics concept. We believe that trying to integrate marginally linked content would ultimately weaken the curriculum for one or both of the disciplines.

Because students receive credit in math and in physics, integrated assignments receive grades for both courses. Each teacher assesses different predetermined aspects of integrated assignments. For example, during inquiry investigations the math teacher places higher value on the data and results section, while the science teacher places greater emphasis on the ability to study a scientific problem, collect meaningful data, and draw accurate conclusions based on findings.

A tremendous benefit to an integrated math-science course is that redundancy is greatly reduced. This frees up time for exploring learning in greater depth or working longer on difficult-to-understand concepts. Additionally, the synergistic effect of working with individuals outside one’s field can have a powerful impact on teaching and learning. In the past, students rarely made connections between physics and math; for example, using trigonometry in math class to study vectors in a coordinate plane was seen by students as different from applying vectors to two-dimensional motion problems in physics.

In the physics course, the integration of math concepts related to Newtonian mechanics (e.g., kinematics, motion, forces, and energy) receives the largest emphasis, followed by electricity and magnetism; waves and optics; and nuclear, thermodynamics, and modern physics. Trigonometric, polynomial, rational, exponential, and logarithmic functions receive the most attention in the mathematics course. The curriculum consists of various inquiry experiences, problem sets, computer instruction, quizzes to test concept mastery, small group instruction, projects, and a culminating portfolio. The final portfolio encompasses application of knowledge in seven different areas, such as graphing and motion, energy and momentum, and careers that use mathematics and physics.

I use full-class instruction infrequently in the science classroom—primarily to introduce a new unit, introduce major project assignments, and help pull major concepts and ideas together. Though inquiry-based instruction is becoming more common in mathematics classes, the math teacher uses full-class instruction—in particular direct instruction (Cruickshank, Jenkins, and Metcalf 2006)—much more frequently, as students learn and practice mathematics skills and concepts. Three specific examples of the integrated curriculum are detailed here.

Motion

Early in the year, students investigate linear motion—specifically, the concepts of displacement (distance), velocity, and acceleration. Guided- and open-inquiry labs (Martin-Hansen 2002) initially use stopwatches, metersticks, and graph paper for collecting data—thus making learning concrete and tangible. Computer simulations and calculator-based labs are added later to incorporate technology into learning experiences; in addition, they speed up the rate and accuracy of data collection.

One of the first motion labs provides students with 10 m of string, a timer, graph paper, a textbook, and a calculator; groups of four to five students prepare a report for the city council that answers whether speeding is a problem on the roads near the school. Because students conduct their investigation near a public road, parents sign a permission slip that allows their child to participate. Further, students are instructed to always stay a safe distance from the roadway. After discussing safety concerns, teams devise their methodology, which must be approved and signed by the teacher before starting. After collecting and analyzing the data, students give their response to the speeding issue. Visuals are later posted and referenced to help remind students of the meaning behind distance versus time or velocity versus time plots. Data analysis, problem-solving, and mathematical reasoning are also addressed by this lab.

In math class, statistics concepts—including mean, median, range, and standard deviation—now have purpose and meaning. Polynomial functions, including both linear and quadratic functions, are used for modeling the collected data. By exploring first- and second-order finite differences in the data, the seeds are sown for first- and second-order derivatives that occur in calculus, while giving specific meaning to slope.

Students must prepare presentations in which they defend why they use concepts such as mean or median or choose a particular function to model the speed of the cars. They quickly realize that one or two outliers (e.g., someone who is excessively speeding) can greatly skew the mean value. Finally, this investigation allows students to demonstrate Mastery of the Science as Inquiry standard that requires students to develop their abilities to do scientific inquiry as well as the process standards emphasized by National Council of Teachers of Mathematics (NCTM 1989, 2000).

Additionally, deeper conceptual understanding with other mathematical ideas is achieved when studying and graphing motion problems in laboratory investigations. Critical precalculus ideas such as rates and limits are clearly seen and applied. Investigating something that appears trivial, such as studying a bouncing ball, provides a concrete way for students to grasp the meaning associated with otherwise abstract concepts such as distance versus time, velocity versus time, or acceleration versus time graphs.

Overall, investigation of automobile speeds also helps transform students’ understanding of mathematics as they learn to see math as a tool for modeling real-world phenomena. In absence of a concrete investigation such as this, slope merely becomes a rise over run algorithm to be completed. Limits become critical to the discussions and analyses as students are confronted with additional questions such as, “Is a bouncing ball always in motion?” and “What evidence supports your claims?” These discussions force students to extend beyond average rates of motion to address instantaneous rates of motion and understand that even a moving ball can have moments when its velocity is zero.

As students move on to calculus, they will be able to calculate more complex characteristics of motion using differentiation and integration. Integration allows a learner to calculate the area under the curved velocity versus time graph to determine the distance traveled. In precalculus settings, students use methods such as limits or summations of geometric shapes to calculate the areas under a curved velocity versus time graph.

Critical thinking

Four times throughout the course, specific critical-thinking investigations integrate mathematics and science. Each assignment has a different focus: logic and reasoning, science and math in everyday life, evidence and proof, and physics at the movies. In physics at the movies, the laws of physics are applied to action scenes from popular movies. Brief clips from some movies include the following (with applied concept): *Chain Reaction* (nuclear fusion), *Michael Jordan’s Playground* (kinematics), *Indiana Jones and the Last Crusade* (impulse, momentum), *Lord of the Flies* (optics), *1492* (harmonic motion), *The Fugitive* (Newtonian mechanics), and *Speed* (Newtonian mechanics).

Mathematical functions are incorporated throughout. For example, using a clip from the movie *1492*, students explore sinusoidal functions as they work with periodic motion. For another assignment students assume the role of a science correspondent for the 20th Century Fox movie studio for the making of the movie *Speed*. To make the movie more realistic, students draw and detail the conditions that would allow the bus to “safely” jump the 50 ft section of uncompleted highway. Additionally, using details provided in the movie, students use their knowledge of physics and mathematics to determine whether the bus could have made the jump; they must justify their conclusions and state all assumptions.

The jumping scene involves parabolic motion, which requires using physics concepts, algebraic functions, and trigonometric functions. Specifically, students must apply their knowledge of free-body diagrams that are central to any general physics class kinematics unit. Figure 3 diagrams the free-body layout for the *Speed* jump scene. Trigonometry is required to solve the resultant vectors needed to determine the conditions for a successful jump. For instance, the net force for the bus traveling on the ramp is the applied force less the friction force and the opposing component force of the gravitational force. The opposing component force of the gravitational force is calculated by *mgsinÈ*. These values reinforce triangle trigonometry. Students must approximate or find values for the following before they can solve the problem: launch angle, mass of the bus, and height of landing area in relation to launch point. This open-ended investigation integrates math and science content while allowing for unique solutions to be justified.

Figure 3. Diagram for the bus scene, modified from the movie Speed.[ Note: Careful viewers of the film Speed have noted that the take off point for the bus was a flat section at the top of the bridge, making such a jump impossible (since the launch angle would be zero). The actual stunt was performed with a stunt bus, using a short take-off ramp hidden from view in the final editing of the film.] |

Our setting allowed us to work with students collaboratively for three-hour blocks. Students were often not aware of when they were doing science or when they were doing mathematics. When math and science classes are taught separately, planning is essential to align the specific concepts and skills to be taught in both classes. In either situation, the skills that are being applied in physics class are then studied in greater depth to explore the underlying mathematical principles.

Solar racers

Miniature solar racers (Marshall 2004) provide an excellent opportunity to synthesize many of the concepts and ideas studied throughout the year in physics (motion, forces, electrical circuits, conservation of energy) with key mathematics content (trigonometric functions, accuracy, error, modeling, and problem solving). Although racing the mini solar cars provides motivation for learning, the majority of the learning transpires before the race even occurs.

For example, the power output of a solar array is directly correlated to the angle of incidence of the sun as it strikes the solar panel. When the sun is low in the sky such as early morning or late afternoon, the power output of the panel is greatly reduced when compared to the sun striking the panel perpendicularly. For instance, the power output is only half when the sun forms an angle of 30° with respect to the panel—assuming no other factors are involved such as partial cloud cover or reflections off other surfaces. This can be supported conceptually as well if you look at the area that a flashlight illuminates when shone directly overhead versus when it is lowered 60°. The larger circle shows that the light’s intensity is more spread out when compared to the overhead position.

In the real world, situations are even more complex. To determine how long it takes a vehicle to get to the finish line, students test speed, accuracy, friction, different circuit combinations, and energy outputs. Variables such as angle of incidence of the light source, intensity of the light source, gear ratio used, mass of vehicle, and friction are critical factors in the overall success. The process of formulating and testing conjectures is just as important in math as it is in science. Specifically in math, students too often look at the dependent variable as a function of a single independent variable [y = f(x)]. Here students realize that the dependent variable is a function of several independent variables. This understanding is essential for students if they are to connect math class to the mathematics of the real world.

The race becomes a time to celebrate learning. Further, students quickly learn that the process and continuation of learning is valued over the product (the race results). After all, automotive corporations are successful after numerous prototypes have been tested—always learning from prior experience.

The reality

Strong relationships with colleagues, the willingness to plan with others, and flexibility are all requirements for successful integrated teaching. However, collaborative efforts are fraught with challenges. Working with counselors to schedule students is often a tricky endeavor, so administrative support is essential. Each school, program, and discipline will have its own challenges that will require creative solutions. If these challenges can be overcome, the reward becomes learning that more meaningfully applies to students’ lives. The goal of integrated teaching and learning is realized when students begin questioning whether the current investigation is math or science.

If an entire integrated class is not feasible, coordinating one major concept per quarter with another discipline provides a great start. As you begin working with one or several teachers on making explicit connections between courses, the possibility of developing a more integrated curriculum becomes a reality. Find a math, history, literature, or art course that most of your students currently are enrolled in and seek to understand the standards and objectives for that course—you will be surprised the number of authentic connections that can be made with a little effort. When we willingly step beyond our own content areas, the meaning and relevance of learning becomes apparent for our students.

* Jeff Marshall* (marsha9@clemson.edu)

*is an assistant professor of science education and*(bhorton@clemson.edu)

**Bob Horton***is an associate professor of mathematics education, both at Clemson University in Clemson, South Carolina;*(jaustin@francistuttle.com)

**Joyce Austin-Wade***is a mathematics teacher at Project HOPE in Oklahoma City, Oklahoma.*

References

College Board. 2006. Physics course description. http://apcentral.collegeboard.com/courses/descriptions.

Cruickshank, D.R., D.B. Jenkins, and K.K. Metcalf. 2006. *The act of teaching*. 4th ed. New York: McGraw-Hill.

Marshall, J. 2004. Racing with the sun. *The Science Teacher* 71(1): 40–43.

Martin-Hansen, L. 2002. Defining inquiry. *The Science Teacher* 69(2): 34–37.

McDonald, J. 1992. Steps in planning backwards: Early lessons from the schools [Electronic Version]. CES national web. www.essentialschools.org/cs/resources/view/ces_res/121.

National Council of Teachers of Mathematics (NCTM). 1989. *Curriculum and evaluation standards for school mathematics*. Reston, VA: NCTM.

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

National Research Council (NRC). 1996. *National science education standards*. Washington, DC: National Academy Press.

Wiggins, G., and J. McTighe. 1998. *Understanding by design*. Alexandria, VA: ASCD.