Solutions to Homework Problems

Getting Students to Practice Expert Thinking

Nobel-winner Carl Wieman will be speaking at FSU next week; we want to fit in one last message citing his practical advice on teaching, which is applicable across disciplines. We’ve condensed a bit, but the original is available here (and includes both references and strategies for minimizing the burden of grading).

Creating good homework problems

by Carl Wieman
Homework is a vital part of learning in university courses, arguably it is where most of the learning takes place. Doing homework is typically where the student will put in the most hours of intense thought about the subject, and such intense thought has been shown to be essential for developing mastery.  However, the sheer amount of effort expended is not sufficient to ensure mastery. Students have completed thousands of practice physics problems and still do poorly on problems involving the use of simple physics concepts.

The essential features for making homework support learning are:

  • there is challenging and prolonged practice of the components of expert thinking the student is to learn
  • the student is motivated to invest the necessary time and effort
  • the student gets feedback on their practice that gives them guidance as to how they can improve

We have shown large learning gains by analyzing the thinking a physicist uses, and then having students practice and get feedback on this type of thinking in class. The same approach can be applied to the design of homework problems in science and engineering. We start by considering the components of expert thinking that one wants the students to develop and the shortcomings of typical “back of the textbook chapter” problems.

Generic components of expertise in all fields of science and engineering. Be able to:

  1. Identify what concepts are useful for solving the problem and have criteria to use to decide which concepts are relevant and which are not;
  2. separate surface features from the underlying structural elements that determine what concepts apply;
  3. identify what information is needed to solve the problem and what is irrelevant;
  4. look up and, as appropriate, estimate values and/or deduce information that is needed but not given;
  5. make appropriate simplifying assumptions;
  6. break down a complex problem into appropriate pieces;
  7. plan a solution;
  8. use multiple specialized representations of information and move fluently between them to gain new insights, and identify criteria for deciding which representation is most useful in a given situation;

  9. carry out routine frequently-needed solution procedures quickly and correctly, and have criteria for choosing when a specific procedure should be used;
  10. articulate and suitably apply a set of criteria for evaluating if a solution or intermediate result makes sense.

If one considers a typical back-of-the-chapter homework problem, it is clear that few of these components are practiced….The typical problem will result in the student primarily practicing and getting feedback on carrying out routine procedures that are given in textbook. In addition, most standard problems…will strip away the context, to avoid complicating the routine practice. Such idealized artificial problems provide no motivation for students put in effort or see any value in mastering the subject.

A criteria for any homework problem should be that it can pass the “Why should anyone care about the solution to the problem?” test. (Anyone other than the instructor.) The best problems are ones that the students can see provide solutions that are obviously relevant and useful to their immediate surroundings or intended future careers, but admittedly that can be a high bar. However, it is not so difficult to find some meaningful task a person in some occupation would need to carry out that utilizes the knowledge and skills involved in the problem. If you find yourself unable to find any such context, you should ask yourself why you are bothering to teach this material. I have dropped more than one topic from my courses after confronting such a realization.

As you incorporate the design features a)-j) in your homework problems, make them an explicit part the solution that students are required to produce (or stand-alone problems for specific features). For example, a part calling for a list of the concepts that apply and what features of the problem determine that choice, a part asking for what quantities/information are needed and estimates for values of any quantities not given, what procedures or techniques will be employed (Fourier transformation, use of multiple equations with multiple unknowns, …), approximations to be used, justification for why the answer makes sense and the criteria used to support that justification. This means a solution will involve far more than something like a simple number or fact. It provides them with far more explicit practice and feedback on how to develop and improve expert skills than does producing a single number which they find out is correct or not.

Having students attend to feedback on homework

Students attending to feedback and using it to improve is vital for learning. We have found three simple ways to encourage students to do this.

  1. For any question on which a student loses points, give them the option of getting some fraction (1/4-1/2) back by turning in an explanation of what was incorrect about their thinking that resulted in the error.
  2. Have each homework set contain a “reflection” problem such as, “Review your previous homework and the solution set, and then list all the problems you did incorrectly, what you did that was incorrect on each of those problems, and what you need to do differently on future problems of this type. If you did all the problems correctly, identify how you could improve a solution or which problem was most difficult and explain why.”
  3. Have the exam problems be very similar to homework problems and advertise to the class that this will be the case. It is easier to do this if you have good explicit learning goals for the course where it is clear that both homework and exam problems are testing the achievement of the learning goals.


Tuesday, February 14, 11:00, College of Medicine Auditorium

Lecture by Carl Wieman: “Taking a scientific approach to science education.”
Guided by experimental tests of theory and practice, science has advanced rapidly in the past 500 years.  Guided primarily by tradition and dogma, science and engineering education has remained largely medieval.  Research on how people learn combined with classroom experiments is now revealing much more effective ways to teach and evaluate learning at the university level than what is in use in most science and engineering classes.  This research is setting the stage for a new approach to teaching that can provide the relevant and effective science education for all students that is needed for the 21st century.


Student Consultants

If you’d like feedback on how your class is going, and you’d like a student’s perspective, you can now invite our trained student consultants to visit your class and gather data. There are multiple options:

Informal evaluation: The student consultants can come to one of your classes to interview students about how well they are learning and how they perceive the class. The interview usually takes around 20 minutes. The student consultant then compiles the student responses into a report and meets with you to discuss the results.

Observer/Note-taker: The student consultant records in writing what happened in your class (e.g., chronology of classroom activities; time spent in questioning, board work, small group discussion; and so on). If you wish, they can use the COPUS. The student consultant describes rather than evaluates, and meets with you to present and discuss the report.

Primed student: Prior to class, you inform the student consultant what he or she should watch for. Examples: How often do certain students respond? Are the students discussing course material among themselves? What seems difficult for the students? What are the students in the back rows of the class doing? The student consultant writes his or her observations in a report to share with you.

To schedule a visit by a student consultant, contact Fabrizio Fornara at