from Inquiry, Volume 1, Number 2, Fall 1997, 31-34
© Copyright 1997 Virginia Community College System
Abstract
The purpose and application of
a fundamental strategy for students to use in order to
successfully solve word problems encountered in any entry level
science course are reviewed.
Sciences demand that students learn the specialized and voluminous vocabularies of each discipline, master their intricate conceptual models, and become well versed in the procedural intricacies of laboratory work. These difficulties, however, are taken in stride (more or less) by students mainly because most have, over their academic history, developed strategies for coping with them. Although the sciences approach their particular problems and support their various insights through mathematical relationships couched in word problems, students seldom have developed any systematic coping ability to handle such challenges. Word problems pose special difficulty to students for a variety of reasons, but the most fundamental is that they do not know how to attack word problems in a methodical and logical manner.
Perhaps the students' earlier educational experiences failed to teach them how to approach word problems from the basis of associated dimensional units. The author's experience has indicated that students are more likely to succeed with solving word problems if they know how to “read” the problem for subtle hints, determine what “tricks” to look for, and recognize when it is necessary to reach “outside” the problem for specialized information such as identities and definitions. However, they also need enough supervised experience with a general set of relevant problems to be able to methodically assimilate needed skills and to arrive at correct answers. Lacking both strategies and experience, students tend to muddle along, uncertain as to their real abilities, hopeful that with experience they might stumble onto some intuitive insight as to how to solve word problems, but too often they are content if they can simply “get by.”
I completed Self-Paced Study of Strategies Useful for Solving Word Problems in The Physical and Biological Sciences in 1995 with the support of a Virginia Community College System Professional Development Research Grant. This book wrote itself, in a manner of speaking, in the guise of the author's experience helping community college students learn to solve word problems found in course materials of entry level science courses. The book emphasizes strategies and methodologies demanded by the dimensional units associated with the problem.
The purpose of this book is to present the neophyte with a brief but reliable set of strategies for solving word problems through a format that will encourage self-confidence in successfully solving typical classroom problems. A self-paced format is used to meet individual needs for study, practice, and immediate feedback. Since this study program is intended to help students facing the realities of difficult science courses, it is brief and will require no more than six or seven hours to complete, even if every problem is worked through to completion.
The overall format of Self-Paced Study is to present instructions and worked out examples before guiding students, step by step, through a series of problems that provide instant feedback of correct answers. The topics are divided into five Study Sessions, which are subdivided into sub-units, each having its own logic and sample problems. The sixth and final Study Session is a series of word problems (with solutions) for students to practice their newly acquired skills.
Problems in the practice units are included to illustrate various facets of solving “real” problems taken from the disciplines of chemistry, physics, and biology. They range in difficulty from average to demanding. Because problem solving using dimensional analysis often can lead to the correct answer even when one has no familiarity with the subject, many problems have been selected that involve terms completely unfamiliar to the students. However, the author's experience has indicated that students take particular pride in their work when they succeed in solving such problems. This experience helps them gain confidence in using their newly learned problem solving strategies.
Students usually approach word problems with the misconception that the goal is simply to arrive at a solution which will be a number. This goal is difficult to unlearn; yet it must be replaced with the belief that numbers are nothing but passive entities associated with the problem and are “carried along” (in the correct association with the appropriate dimensional units) until the problem is correctly solved. One can, after all, readily tell if the problem is solved correctly without having the final number. Thus, one objective in the first Study Session is to re-orient students to “the number”— the most ubiquitous and self-defeating aspect of solutions to word problems. Of course, the required arithmetical manipulations are necessary to arrive at final answers and can not be neglected. However students must learn what professionals know: the pathway to the solution of any word problem lies with the dimensional units in the problem, and the “test” for the correct solution also lies in the dimensional units of the answer.
Before that, however, students have to learn what dimensional units are and become skillful in knowing how to look for them in terms of measurements such as weight, length, volume, and temperature. Often more than a one-dimensional unit is associated with a problem, and the student must learn how to read and cope with these. Also dimensional units are used sometimes in problems in indirect codes that must be recognized. For example, gases may be confined under “standard conditions” which must be interpreted in the dimensional units of atmosphere of pressure and degrees Celsius. To convince students that this process really leads to a successful strategy is necessary at the outset; therefore, the first Study Session presents examples of how dimensional units can be used to solve very simple problems.
Once students begin to recognize the fundamental importance of dimensional units in solving word problems, they are ready for the second Study Session. In it, students are shown how to manipulate dimensional units during addition, subtraction, multiplication, and division. One of the serendipitous, although minor, benefits of this session is that many students come to understand why areas are measured in “square feet” and volumes in “cubic feet.” (Many assume the terms come from an association with a geometric form!) More importantly, students begin to understand how they can manipulate dimensional units to assure that they can arrive at a correct answer.
Students are shown a methodical approach to solving word problems in the third Study Session. The first step is to write down the correct answer (really only half the correct answer—the dimensional unit—but that is the most important part needed to work out the solution to the problem). At first, they tend to be skeptical about the efficacy of such an approach, but it is the author's experience that once they “catch on” to how one can reliably work backwards from the answer to the correct solution, they become quick converts. Students are most impressed with this methodology when they see solutions evolving from the applications of orderly and logical steps dictated by the dimensional units. They also are often pleased and surprised to find that they can reliably check the answer for correctness before doing any arithmetical work.
The best and most natural examples to follow Study Session 3 are the conversions that usually are introduced at the beginning of every science course: metric units to larger or smaller metric units, metric to English units, and English units to bigger or smaller English units. Therefore, Study Session 4 reviews these thoroughly. Students appreciate the freedom from mind-twisting Greek and Latin prefixes that dimensional analysis gives them and the confidence it encourages. They know precisely where to go in a table of conversion factors for a needed relationship and experience satisfaction in the pleasure of being able to consistently make correct conversions.
More complex problems are encountered in Study Session 5 where mathematical relationships between multiple variables (equations) are presented. The object of this book is not to teach algebra, but students are shown how dimensional units can be used to determine what algebraic manipulations may be required by the problem. Thus problems involving algebraic manipulations, ratios, and percents can be presented as solvable using dimensional units to show the way. As the problems become more involved, physical constants are introduced and dimensional units associated with those constants become the focus. At this point, students begin to appreciate the fact that they can solve problems in chemistry, biology, and physics by following the dimensional unit trail even though they have little understanding of the subject matter of the problem.
Realizing this, most students become true believers in this problem-solving methodology. From their introduction to “word problems” in grade school, students have been told to solve it by “thinking logically” and to check the answer by “looking at the answer and asking if it is reasonable.” This appeal to some sort of intuitive “logic” and a final check with “reason” may find some application for a few smart students when they work with very simple familiar problems dealing with money or other daily dimensions, but this advice leaves other students on very wobbly pins, especially as they move deeper into scientific concepts. How many people ever get an intuitive feel for an oerstad or even a volt or ampere? What does one do with a coulomb much less an abcoulomb? How can one get a feel for a joule? Is a dyne-centimeter something one needs an intuitive sense for like a foot or liter before he or she can solve a problem about such things? If a problem involves a statafarad, is the student doomed to never knowing whether or not the answer is “reasonable”? How can one solve a problem related to even so common a term as density if he or she has no intuitive comprehension of just what the term means? How can any student (or professional, for that matter) confidently solve problems and evaluate their solutions intuitively when the problems involve concepts such as moles, calories, or Reynolds Numbers? However, if students treat these as dimensional units and apply a methodical strategy for handling them, they can solve the problems and check solutions for correctness without ever having to worry about natural “feelings” for the topic at hand.
Indeed, in the sixth and last Study Session students deliberately are given many problems for which they will have no intuitive “feeling.” At first students may be intimidated by these problems. Once they dig in, however, applying dimensional analysis in the logical and methodical manner they have learned, they are pleased (and mildly surprised) to have the answer emerge on cue. This is a confidence builder that can give students the edge that assures success in solving most word problems they may encounter.
This short book is based on the author's successes in helping entering students learn how to approach and solve word problems. It has been reviewed by several members of the science and math faculty at Northern Virginia Community College (Woodbridge campus) and has been “tested” on a few volunteer students to solicit their comments and insights. Reviews have led to a number of useful suggestions, which were incorporated into the manuscript. However, to date the manuscript has not undergone a formal study to determine if it produces a significant improvement in the ability to solve word problems in users compared to similar abilities that members of a control group develop through the traditional classroom experience. Anecdotal feedback, however, has been supportive and encouraging.
George A. Garrigan teaches at the Woodbridge campus of Northern Virginia Community College.