# Modeling Applied to Problem Solving (MAPS)

Since 2008, we have been creating and studying a new pedagogy –Modeling Applied to Problem Solving (MAPS) that is focused on helping students become more expert problem solvers. MAPS consists of two complementary parts: students learn how to organize their physics knowledge utilizing a hierarchy of *core physical models* that summarize mechanics. Simultaneously, they learn to plan their solution to problems by *modeling* the system and the relevant interactions as a prelude to applying one or more of the core physical models. As students learn Modeling Applied to Problem Solving (MAPS) – their attention and dialogue shift from algebraic and numerical manipulations towards planning the solution based on MAPS approach of emphasizing the system and interactions as a route to understanding the underlying conceptual structure of the problem at hand and planning a solution. MAPS incorporates many of the modeling ideas developed by Hestenes, Wells, Etkina, and others.

In practice, MAPS requires students to apply a universal problem solving approach [System, Interactions, Model (S.I.M.)]: identify the systems and interactions and then choose the appropriate model (PRI 09d)*.* This approach gives students a systematic way to start unfamiliar problems, and thereby to solve complex problems (e.g. involving more than one concept either simultaneously or sequentially). It is successful even when the problems are not broken up into a series of logical parts a), b), c)… – a practice that discourages students from learning problem solving skills since they can do well by mastering only the factual and procedural knowledge involved in the separately prescribed parts. From a cognitive perspective, MAPS pedagogy specifically develops students’ strategic knowledge (the ability to look at a problem and determine which factual and procedural knowledge can be applied to obtain its solution). Students’ development of strategic knowledge is facilitated by the Model Hierarchy in which the five basic core models each specify their allowed systems, relevant interactions, and mathematical law of change (e.g. F=m dv/dt or E_{f }= E_{i} + Non-conservative Work).

This pedagogy is the foundation of the material presented in our supplemental Mechanics Wiki resource, and has been implemented in our current online course (http://RELATE.MIT.edu/learn-physics). The pedagogy has been employed in three-week, intensive review courses for D-students in both 2009 and 2010. It has been strikingly successful in improving both student attitudes and student problem solving performance. is a simplification of the Modeling Physics approach (Hes87, WHS95, Hes96)

## Assessment of MAPS Pedagogy

MAPS pedagogy has been employed in three-week, intensive ReView courses for students who received a D in the Fall version of 8.01, the MIT introductory mechanics course. Data from both 2009 and 2010 show that it has been strikingly successful in improving both student attitudes and student problem solving performance. We have assessed the students’ problem solving skills and the expertise of their attitudes in several ways:

- The MIT final exam – largely composed of multiple-concept problems
- The Colorado Learning Attitudes about Science Survey (CLASS)
- Transfer of problem solving skills to the following course (8.02, Electricity and Magnetism)
- From students free response to questions about what they learned overall

In addition, we are developing a Mechanics Reasoning Inventory (ref) on which scores have been observed to improve from 7-10 at the start to 13-14 over the three weeks.

**MIT final examination**

The MIT final examination in Introductory Mechanics (8.01) is largely composed of multiple concept problems that are not broken into a series of questions.

Students who received a grade of D in our standard introductory course, 8.01, our three-week ReView course using MAPS pedagogy improved their MIT final exam score by over one standard deviation of the freshman class on average.

**Expertise of Attitudes About Science**

Attitudes of students taking our three-week ReView course using MAPS pedagogy became significantly more expert-like on the Colorado Learning Attitudes about Science Survey, in contrast to the students in our 801 course whose attitudes became less expert-like as is typical for introductory mechanics courses nation-wide. The largest increase was observed in Problem-Solving Sophistication, which mainly concerns self-confidence and ability to solve problems without using a formula sheet.

**Transfer to subsequent E&M course**

After taking our ReView course, students rejoined the main sequence and took 8.02, Electricity and Magnetism, showing Improved Student Performance In Electricity And Magnetism Following Prior MAPS Instruction In Mechanics.

Exam scores in 8.02 are highly correlated with those in 8.01, as shown by the solid line fit to the data (up to z = +2, not shown on this plot). The ReView students (red) perform ~1/2 standard deviation above this line, whereas the control group (purple, from 2008 when weak students were sent forward without benefit of ReView). It is notable that students below -1.5 do not significantly improve above expectation; even when required to take another semester of 8.01 (blue) their 8.02 grades are best predicted from their 8.01 grade the *first *time they took it.

**Student Comments**

These comments are taken from end-of-course free responses to questions involving what was learned and/or whether the students feel prepared to move forward to E&M.

“I now know better how to organize the material in my head using the rubric…”

“I liked…looking at the basic solving steps that are normally assumed in 8.01 TEAL.”

“The S.I.M. rubric was extremely helpful. It should be taught in 8.01 the first time around.”

“I enjoyed systematizing my approach to problem solving…”

“The S.I.M. system…helps me plan my attack of the problem.”

“The systems, interactions, models…makes you think before jumping in.”

“I now feel I understand the concepts and I am no longer just plugging numbers into equations.”

“…the rubric on how to solve physics problems is very beneficial and keeps me on track while I am doing my work.”

“The material aside, the way we were taught to approach problems is extremely valuable.”