Mathematical modeling in the K-12 curriculum

The Common Core State Standards (CCSS), created by the National Governors Association Center for Best Practices and Council of Chief State School Officers and adopted by 45 of the 50 states, the District of Columbia, and four U.S. territories, outline minimum standards for K-12 student progress in English Language Arts and Mathematics. Given their prevalence in all states except Alaska, Texas, Minnesota, Virginia, and Nebraska, most educators regard them as the future of education in grades K-12.

Dr. Todd Abel, a Mathematics Education expert from Appalachian State University, spoke to the Colorado School of Mines Applied Math and Statistics Department on one challenge of the CCSS, integrating mathematical modeling. He explained that one of his greatest interests in speaking was , “I don’t think math educators and mathematicians get enough chances to talk.”

Abel first had to define the CCSS and mathematical modeling as well as their respective purposes. The standards are designed to combat a traditional criticism that U.S. math education is “mile wide, inch deep,” said Abel. One of the main focuses of the standards is a conceptual understanding as well as computational skill.

Abel explained that this understanding, specifically in the form of modeling was important, because “Real world situations are not organized and labeled for analysis.” Modeling standards require students to analyze situations and form their own processes to solve for unknowns.

Abel represented the CCSS modeling process with a flowchart, wherein students begin with a probe into a question, formulate a solution, compute the answer, interpret their answer, validate their process, and report successful results or return to the formulate step with unsuccessful results and repeat the process.

However, the integration of modeling into K-12 curriculums is faced with a number of challenges, the largest being the math teachers themselves. “If we want students to be involved in this creative process, we need teachers who can do it…most teachers are really not prepared for that,” said Abel.

One challenge in preparing teachers to think more creatively about math problems is the preconceptions of teachers themselves. Abel reported on a classic study, which found that high school geometry teachers believed that if a student could not solve a problem within five minutes, they could not understand mathematics. Students, perhaps picking up on this attitude, are highly unlikely to continue on a problem after between three and four minutes of effort, Abel reported. A general belief in unchangeable statuses as “good at math” or “bad at math” is also prevalent. As an additional challenge, most American math teachers “have few experiences with rigorous mathematics” relative to their international peers.

Another specific challenge Abel has met with in educating teachers is the question “But, when do I teach?” Abel explained he received this reaction after he showed a group of educators methods for integrating modeling into their curriculum. He ascertained that the educators were viewing teaching as a chalkboard lecture and were not open to working problems alongside their students.

Abel said many of these problems could be combated by either pre-service or in-service courses for teachers, such as the Discrete and Continuous Math Modeling and Statistics for Teachers courses offered for education students at Appalachian State University and summer institutes and conference workshops for current teachers.

“If students have some hook for understanding [a math problem], they’re much more likely to be able to answer the problem,” said Abel. He outlined a three act formula for teachers to integrate into their lessons. In Act 1, the hero is faced with a problem. In Act 2, he or she must overcome adversity, and in Act 3, the hero reaches a resolution. By integrating a classic narrative structure into mathematics, students are able to understand the problem more clearly.

Abel concluded by soliciting discussion from the audience and saying optimistically of the mathematics education situation, “We’re having a moment here.”