A Note to Students
Student visitors to The Physics Classroom website come from a variety of backgrounds. Most are enrolled in a physics course; some are studying physics as part of a home-school program. Many come as frustrated students seeking help to get them back on the proverbial track. And some are simply enthusiastic or conscientious and are supplementing their usual successful studies with the use of online resources. Students come from a variety of backgrounds and for a variety of reasons. But nearly all these student visitors have one thing in common - they are confronted with the frequent challenge of solving physics word problems.
Solving physics word problems can be frustrating. For many, it is the most frustrating part of a physics course. At the same time, there is a great sense of accomplishment associated with reading a challenging problem, identifying the proper formula, performing the correct mathematical operations and ending up with the correct answer. Unfortunately, the emotional reward associated with getting it right is often an infrequent experience for many students. Many students become frustrated, lose heart, and give up, eventually concluding that I'm just not any good at physics; the math always gets in my way. These sentiments of frustration and loss can turn a physics course into a long ordeal. Fortunately, these frustrations do not have to ruin your physics experience. As many students have found, frustration and physics do not have to be synonyms. With some hard work, some learned habits, and some practice, even the most defeated student can become a relatively effective problem solver. The Calculator Pad seeks to improve student problem solving abilities and provide the guided practice which is needed to boost their confidence in this area.
What Is At The Calculator Pad?
Most physics textbooks are accompanied by end-of-chapter pages filled with physics word problems. And most teachers assign at least a portion of these problems to students. The Calculator Pad is in part the equivalent to those end of chapter problems. Most textbooks have answers to a portion of these end of chapter problems at a location in the back of the book. Often times, the answers to all odd numbered problems are listed on a collection of pages sandwiched between the Appendix sections and the Index. Like these textbooks, The Calculator Pad provides answers - answers to each and every problem. But what is lacking in most textbooks is a solution. It is presumed that a student who misses a problem will revisit their solution, identify errors, correct errors and repeat the process until their answer matches the answer in the answer section. Certainly the correcting of an error, the revision of a solution, and the repeating of a problem is a good practice. More physics is usually learned from missing a question and correcting it than from getting the same problem correct on the first try. Yet for many students, the process of solving the problem the first time was a painful enough experience; who wants to inflict even greater punishment?And for other students, the solution to the problem didn't make it all the way to an answer. They got stuck very early in the solution process. Or worse, they got stuck before they got started. The Calculator Pad attempts to intervene in the process, providing solutions to every problem. The solution takes the form of a .mp3 audio file, much like the ones which play on an iPod or mp3 players. The audio files provide a detailed solution to each problem, explaining the process of solving the problem in a detailed manner. The audio files were recorded by an experienced physics teacher who is familiar with the common mistakes which students tend to make when solving such problems. In addition to explaining the detailed solutions to each problem, the mp3 files also describe the habits which can help any student become more successful at solving physics word problems. By slowing down and taking the time to adopt such habits as their own, any student (even those who are frustrated) can improve their problem-solving abilities and more frequently experience that sense of accomplishment associated with getting a problem correct.
How Can I benefit from Using The Calculator Pad?
If you are enrolled in a physics course which emphasizes the solving of physics word problems, then it is in your best interest (or at least in the best interest of your grade) to improve your ability to solve such problems. Such improvements are the result of practice. Yet practice isn't always the pathway to improvement. For many students, practice leads along the pathway to frustration. The Calculator Pad makes an attempt to lead students along the proper and most profitable pathway - the pathway to improved problem solving ability and increased confidence in one's habits of solving problems.
Students need feedback when solving physics word problems. They need to know if they got the correct answer. Every problem at The Calculator Pad is accompanied by an answer. By default, the answer is hidden. Clicking a button reveals the answer. When a student misses the answer, it is important that an effort is made to review the solution and identify the location of error. Self-correction of errors is one of the most impressionable experiences that can contribute to one's learning. For many students, finding one's errors is difficult and frustration often sets in before errors are found and corrected. Every problem at The Calculator Pad is accompanied by an audio file which identifies common errors, points out frequent mental obstacles, describes occasional misconceptions, and explains the proper way of thinking about and solving the problem. By following the suggestions in the audio file, students can more easily solve the problem. But more important than solving a single problem correctly, a student who listens carefully can begin to adopt habits which allow them to think about solving physics problems the way a physicist solves a problem.
Many beginning students of physics occasionally or regularly have difficulty getting started on a problem or simply get stuck once they have started. At the heart of their frustration is the fact that they have spent five minutes on a problem and have little more than a blank sheet of paper as a product. Getting stuck before getting started or simply getting stuck is common. Even expert problem solvers experience sticking points. One difference between an expert problem solver and a frustrated problem solver is that the expert knows what to do when stuck and has the confidence in his/her ability to get unstuck. The mp3 files at The Calculator Pad explain the solution to the problems by emphasizing the habits which expert problem solvers employ as they approach physics word problems. While every problem is different, each problem is approached in the same manner. The ultimate goal of The Calculator Pad is help students to improve their approach to problem-solving - to learn how to get unstuck when they find themselves in sticky situations. The goal is to teach students to solve problems the way a scientist solves problems: by carefully analyzing the physical situation, identifying what is known and what the problem demands, and combining conceptual knowledge and mathematical relationships to think through a strategy to allow one to proceed from known information to unknown.
For such students, there is a need for help in order for them to sustain the individual application of problem-solving in Physics. The Calculator Pad is intended to meet that need for individual help in order to get a student started or un-stuck. Each problem set is accompanied by a collection of .mp3 files for each of the problems. The audio files provide explanations, suggestions, hints, diagrams and at times animations for solving the problems. The ultimate goal is to help students become more confident in their problem-solving abilities and more independent in their approach to solving problems.
How Should I Use The Calculator Pad?
The Calculator Pad was designed to provide students an opportunity to practice solving physics word problems. The audio files are intended to not only step students through the solution to the problem, but also to help improve the problem-solving habits of students. Thus, students will benefit most if they first take the time to solve each problem carefully, using the effective habits discussed on these pages. It is strongly suggested that students get get out a sheet of paper, get organized, take their time, and work out their solution on paper. Once solved, the answer can be checked. If incorrect, an effort should be made to trace through their solution to find an error pertaining to their solution strategy, their physics, or their mathematics. If the solution cannot be self-corrected, then students should take a moment to listen to the audio file. Of course, the student should take advantage of the play/pause/continue features of the file, pausing it to check over their mathematics whenever needed. If a comment in the audio file helps to re-direct the solution process, then it is suggested that a student continue the solution from the point of re-direction. The less dependent that a student can become on the audio files, the more beneficial that The Calculator Pad becomes.
For some problems, a student might find that they get stuck before they even get started on the problem. A sincere effort should be made to diagram the problem, to identify the known and the unknown quantities and to think about relevant equations. After making this effort of getting started, the audio file should be played through once from beginning to end. After listening to the audio file, an effort should be made to solve the problem without the use of the file. Of course, if the solution is still not possible, the audio file is still there to guide a student through the solution process.
Students having frequent difficulties with the problems should take the time to read (or re-read) through the information provided in the Overview page for each problem set. These Overview pages provide background information about the physics which is involved in the problem sets. Further and more extensive information may be needed. This information is found at The Physics Classroom Tutorial. Links are provided from each Overview page to specific and relevant pages in the Tutorial. Often times, insufficient understanding of the underlying physics concepts is the cause of difficulty with solving physics problems. A good physics understanding is often an essential prerequisite for effectively reading a problem and interpreting the meaning of the words.
Ways Not to Use The Calculator Pad
The Calculator Pad has the potential of being a very useful resource to students using The Physics Classroom. But like any useful tool, improper use can render it virtually useless. The most improper uses of The Calculator Pad are those which involve inactivity. Problem solving in physics requires an active engagement in the process. Far from a spectator sport, physics problem-solving requires that a student become involved in the process - doing the reading, preparation, thinking, strategy-plotting, algebra, and calculator work. Simply listening to the audio files will do little to make a student into an effective problem solver. The audio files must be combined with practice; only then can a student have reliable feedback as to whether or not they are understanding a problem or a set of problems. Many physics students become frustrated because they thought they understood the instruction of their teacher, but did little about the instruction until the day of the test. Understanding the instruction of a teacher may only indicate that the teacher understands the material and be able to communicate that understanding in a simple and meaningful manner. Yet the understanding of the material only emerges within the student when they do something about the material which was presented. The Calculator Pad, like many of the materials at The Physics Classroom website, were designed with the intent that the student would do something about the material in order to check their understanding.
A Note about Significant Digits
Some physics teachers place a strong emphasis upon following the rules of significant digits when dealing with numerical quantities in laboratory work or problem solving. While some calculators have a Significant Digit feature that can be enabled and disabled, most students are unaware of its existence. As such, a student who performs the operation 5.3 divided by 9.8 on the calculator will receive the answer 0.5408163265. While the calculator is performing the mathematical operation properly, it is failing in the area of significant digits. Consistent with the accepted rules for performing multiplication and division operations, the number of significant figures on the final answer is equal to the least number of significant figures on the two numbers which are used in the operation. Thus, 5.3/9.8 is equal to 0.54. The amount of precision on the answer is limiting by the amount of precision on the two numbers used to determine the answer.
In the same manner, a student who performs the operation 0.06795 subtracted from 1.28 on the calculator will receive the answer 1.21205. Again, the calculator is performing the mathematical operation correctly but failing in the area of significant digits. Consistent with the accepted rules for performing addition and subtraction operations, the number of decimal places on the answer is equal to the least number of decimal places on the two numbers which are used in the operation. As such, 1.28-0.06795 is equal to 1.21. The amount of precision on the answer is limiting by the amount of precision on the two numbers used to determine the answer.
Throughout The Calculator Pad, every effort has been made to follow the rules of significant digits. Once an answer is determined, it is rounded to the proper number of significant digits and reported to that degree of precision. In some instances, when it is perceived that students may have difficulty recognizing such a rounded answer, a brief note is included within parenthesis indicating that the reported answer was obtained by rounding the numerical value reported on the calculator. Such an answer might be expressed as
370 N (rounded from 373 N)
In instances in which several mathematical operations are performed before the final answer is obtained, the rounding to the proper number of significant digits has been performed after the final step. Every effort was made in The Calculator Pad to carry through unrounded numbers from the intermediate steps until the final answer was obtained.
Perhaps the most important note to make about significant digits is to warn students (and teachers) not to major on minors. The proper number of significant digits is NOT the main thing. As the saying sometimes goes: the main thing is to make the main thing the main thing. And the main thing is to learn effective problem-solving habits which help students to become more successful and more confident problem-solvers. The comments included here regarding significant digits are in part intended to help students recognize that their calculated answer of 373.24981 N is the same answer as 370 N whenever the mathematical operation include two digits of precision. The difference existing between the answer on their calculator panel and the answer reported at The Calculator Pad does not indicate an error on the part of the student; the difference represents the rounding of a calculated answer to the proper number of significant digits. Students will need to learn to not take their calculated answers literally.