Innovative Education: Early O.R. education

By Kenneth Chelst and Thomas Edwards

Imagine a U.S. revolution in K-12 mathematics education that emphasizes mathematical modeling in real-world contexts. Envision cohorts of graduating high school students trained to interpret results of models and then justify their conclusions. Finally, visualize an incoming college class that has absorbed the concepts of probabilistic reasoning from a K-12 mathematics curriculum steeped in probability, statistics and data analysis. This revolution is coming to every local school district throughout more than 45 states. It is called the Common Core State Standards Initiative and is approved for rapid implementation in all but a handful of states (www.corestandards.org).

The first visible signs of this revolution will appear in the 2014-2015 academic year. This is the target date for implementing statewide standardized student tests that are developed around open-ended questions in context. These will require students to explain the process used to obtain their answers. Most states and school districts have already begun planning to require a fourth-year mathematics course for graduation. They are searching for appropriate curriculum suitable for a broad range of students. This brave new world of K-12 mathematics education makes Project MINDSET’s high school course in operations research an ideal fourth-year capstone course. (For a discussion of how the MINDSET course is aligned with the new standards, see a file posted on the MINDSET home page, www.mindsetproject.org).

Common Core State Standards Initiative

The Common Core State Standards Initiative is the result of an effort at the state level coordinated by the National Governors Association Center for Best Practices (NGA Center) and the Council of Chief State School Officers (CCSSO). The standards were developed in collaboration with teachers, school administrators and experts from higher education, business and industry to provide a clear and consistent framework for curriculum. As of June 2012, 45 states, three U. S. territories and the District of Columbia have adopted the Common Core State Standards [NGA Center & CCSSO, 2010].

The Common Core State Standards Initiative is proposing an ambitious timeline for implementation. Because assessment often drives what occurs in classrooms, a companion group, Smarter Balanced Assessment Consortium, is at work on assessments to be used to evaluate students who are learning the Common Core State Standards mathematics curriculum (www.smarterbalanced.org/smarter-balanced-assessments). The consortium expects to have viable examinations ready to administer in 2014. Without such a jump-start, it is likely that the Common Core State Standards would go the way of most proposed reforms: some implementation that is not very widespread followed by a gradual disappearance.

CCSSO presents two sets of standards – one for English language arts and the other for mathematics. These standards define the knowledge and skills students should gain from their education in kindergarten through grade 12 so that they will graduate from high school able to succeed in entry-level, credit-bearing academic college courses and training programs for entry into the workforce. The standards reflect the best of the most effective curriculum standards models from across the country and around the world.

Key Points in the Mathematics Standards

Kindergarten through grade 5 standards are designed to provide students with a solid foundation on which to build more demanding mathematics concepts and procedures and allow for consideration of applications in later years. The Middle School Standards provide a coherent and rich preparation for high school mathematics. A strong foundation will allow students in grades 6 and 7 to engage in hands-on learning in geometry, algebra, probability and statistics. Students who have mastered the content and skills in grades 6 and 7 will be well prepared to study algebra in grade 8. The High School Standards emphasize mathematical modeling, which the standards define as the use of mathematics and statistics to analyze problem situations in order to understand them better and improve decisions (www.corestandards.org/about-the-standards/key-points-in-mathematics).

Perhaps the most provocative aspect of the document is the Standards for Mathematical Practice. These practices build upon important processes and proficiencies that have long been topics of research and theory in mathematics education. The first of these are the process standards proposed by the National Council of Teachers of Mathematics (NCTM): problem solving, reasoning and proof, communication, representation and connections [NCTM, 2000]. The second are the strands of mathematical proficiency specified in the National Research Council’s report “Adding It Up”: adaptive reasoning, strategic competence, conceptual understanding, procedural fluency and productive disposition. This set of mathematical practices also provides the clearest connection between the Common Core State Standards for Mathematics and the Operations Research (O.R.) professional community.

The eight Standards for Mathematical Practice require that students:

1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.

While it is not difficult to argue that O.R. addresses each of the eight standards, we believe that O.R. connects most strongly with standards 1, 3, 4 and 5. Making sense of problems and solving them is the life-blood of O.R. Mathematical modeling is typically the vehicle by which O.R. problems are conceptualized and explored. Moreover, O.R. practitioners use technological tools to obtain solutions to problems and perform sensitivity analyses. Finally, sensitivity analysis, the quintessential form of explaining and interpreting results, is a clear demonstration of understanding both the nature of the problem and its solution.

The high school standards emphasize mathematical modeling.

What Developers of Standards Did Not Know: O.R.

The leading mathematics educators, state mathematics coordinators, teachers and other partners in the development of the grade-by-grade skill sets seemed not to know of operations research. Surprise! Surprise! Their problems placed in real-world contexts were more interesting than what is found in current textbooks but still very limited in scope, depth and broad relevance.
Every operations researcher should be excited to see the following sets of algebra standards.

Students should be able to:

1. Represent constraints by equations or inequalities, and by systems of equations or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.
2. Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.

Although these examples are placed in context, they cannot realistically help the teacher answer the question: When will I ever use this? Because they were limited in their vision of solution procedures, the examples are restricted to two decision variable problems that can be solved graphically. They were unaware of how easy it is to tackle more realistic problems using Excel Solver.

In our experience, it takes no more than a few hours to teach high school students how set up a model in Solver and run with it. Once the computer solution is introduced, the size and array of realistic problems grows dramatically. Students can then handle minimization and maximization, integer variables and the transportation problem, binary variables and the assignment problem. Equally important, the use of Excel Solver opens up the world of sensitivity analysis. Students can discuss and explore the impact of constrained resources or estimates of profit margins. An assignment that asks students to explain their answers and justify their solution plan becomes a natural part of the topic of mathematical programming. In addition, the areas of application are endless, touching on every profession from truck driver to oncologist, from a part-time worker in a fast food restaurant to a planner or purchasing agent responsible for the supply chain of a global consumer goods company.

Another Standard indicates that students should:

Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm’s law V = IR to highlight resistance R.

The standard is laudable but the example would likely generate, “So what?” The same skill can be developed in the context of queueing theory. Students can be asked to manipulate L=λW, as well as equations for L_q and W_q. The queueing context is broadly relevant to any organization that offers a service with limited capacity.

Operations researchers can similarly enhance the development of the more realistic and relevant of probabilistic reasoning skills. Shouldn’t every student learn how the Poisson distribution impacts the management of emergency services? Why not apply the binomial distribution to overbooking policies?

In our opinion the biggest limitation in the current planning of a curriculum to meet the new standards is the size and depth of the examples. In general, the contexts are usually described in a few sentences or single paragraph. Textbooks will still have dozens and dozens of examples in each chapter with little meaningful variation. It is usually difficult to create a sense of real decision-making with problems of this size. In contrast, introductory college and MBA operations research textbooks develop realistic cases that run on for pages with numerous intermediary steps. These problem types are better able to address and develop the first Standard for Mathematical Practice: “make sense of problems and persevere in solving them.”

Project MINDSET Update

Project MINDSET is a five-year NSF funded $3.3 million project to develop, refine and implement a fourth-year high school mathematics curriculum based on operations research modeling techniques. The course will be evaluated in terms of its impact on student attitudes toward mathematics and their ability to tackle multi-step problems. Over the course of the first three years, we developed and piloted in classrooms 19 chapters – nine based on deterministic modeling and 10 on probabilistic decision models. Our texts are appropriately named “When will you ever use this? Volume 1, Deterministic Modeling” and “Volume 2, Probabilistic Decision Modeling.” In year four, the curriculum was piloted in a half-dozen classes as full-semester courses. In the fifth year, 2011-2012, 50 high schools across North Carolina, Michigan and Georgia taught Project Mindset courses to more than 1,500 students in diverse school districts. We are in the early stages of analyzing Project MINDSET survey and test data from both participants in our courses and from a control group in traditional fourth-year mathematics courses. We will be working during an extension year to collect data from a much larger sample.

Although we do not have statistical data to report at present, we have anecdotal data that indicate the course meets our expectations and often exceeds the expectations of the teachers who use it. One teacher from western Michigan attended a July 2011 workshop and went back to his principal and said, “We must offer this course next year!” After reviewing the text, they decided the only room in the schedule for the class was after regular class hours. The primary student population would be students who needed an extra class in any subject to graduate. Eight students took the deterministic course each semester. One high school student immediately used the multi-criteria decision model from Chapter 1 to choose the best daycare center for her infant. Another student who lost interest in attending his other classes during his senior year still came to this class after school. In general, to the teacher’s surprise and delight, students who had failed to experience success in previous mathematics courses felt they were really learning valuable skills and had something to contribute. Moreover, some of the students have been able to envision a role for O.R. in their own lives. For example, one student in North Carolina imagined herself as an executive making the decisions described in the problem contexts. Another student in southern California has already announced that she plans a career in operations research.

At the other end of the educational spectrum, a Jewish high school in southeastern Michigan packaged the MINDSET course as part of an entrepreneurial initiative. The program included the MINDSET operations research course, a Jewish business ethics course and an entrepreneurs’ club that met with successful entrepreneurs on a bi-weekly basis (check out the YouTube video www.youtube.com/watch?v=DuhpktNN5W8). Twenty-seven of the 50 graduating seniors chose to take the MINDSET course. Their teacher observed that students approached homework without bemoaning the requirement to solve numerous similar end-of-chapter problems. Furthermore, they looked forward to each new problem context in the homework sections.

Project MINDSET has also established a toehold in southern California and New York City. One high school in each location will be teaching the MINDSET curriculum. In addition, teams of volunteers in Maryland and Texas have begun reaching out to local school officials. Several of the volunteers have already arranged to give short talks at meetings of local mathematics educators. The goal of this outreach is to generate interest in a one-day workshop in each state next year. It is expected that of the dozens of attendees, a handful will be motivated to attend an all-expense paid weeklong workshop the following summer in either North Carolina or Michigan.

What Can You Do Locally?

The Project MINDSET team is ready to help you introduce operations research into local high schools in your area. If you are the parent of a teenager, or soon-to-be teenager, the first contact might be a mathematics teacher or coordinator in your local high school. Alternatively, you might reach out through your social network to identify a school official or mathematics educator. It is important to find answers to the following questions

• What mathematics courses are offered to seniors who are not interested in the calculus track?
• What rules, steps and timeline govern the introduction of a new senior year mathematics course?
• What is the local area group of mathematics teachers called and when do they meet? Would the group be interested in a talk about how algebra or probability is used to make decisions in industry and government?
• Would a school district or county be interested in a talk as part of the regularly scheduled professional development days for mathematics teachers?

Any INFORMS member interested in assisting in this effort will be provided three copies of our textbooks and more copies as needed. The immediate goal of this outreach is to generate enough interest to offer a one-day workshop that introduces operations research as a resource for enriching the teaching of mathematics, particularly algebra and probability.

Kenneth Chelst (kchelst@wayne.edu) is a professor in the Industrial and Manufacturing Engineering Department at Wayne State University and Co-PI of Project MINDSET.

Thomas Edwards is a professor of mathematics education in the College of Education at Wayne State.

References

1. Mathematics Learning Study Committee, National Research Council, eds. Kilpatrick, Jeremy, Swafford, Jane and Findell, Bradford, 2001, “Adding It Up: Helping Children Learn Mathematic,” Washington, D.C.: National Academies Press.
2. NCTM, 2000, “Principles and Standards for School Mathematics,” Reston, Va.
3. NGA Center & CCSSO, 2010, “Common Core State Standards – Mathematics,” Washington, D.C.