Abstract:

It is widely accepted that effective teaching should provide both the necessary notions and the skills to apply them to analyze and actually solve real problems. How to nurture problem-solving capabilities, however, is the most difficult part. A good way to achieve that is to focus on analysis and decomposition. The teacher’s decomposition of the original problem into a chain of linked sub-problems offers guidance to the student’s solving activity. Being actively involved in this activity, the student learns how to cope with non-elementary problems and how to build well-organized solutions.
The capability of analyzing problems and decomposing them is at the basis of problem-solving skills.
It is not easy to turn a worked out example, even if constructed step by step with the students, into a fully interactive learning experience. We propose a way to overcome this difficulty with spreadsheets, using a feature which is not available with a paper-and-pencil approach. We exploit the fundamental difference between a solution (the procedure, formula or algorithm which solves the problem as it depends on the initial data) and results (the expected, and typically numerical, answers to a given problem). The teacher prepares a spreadsheet where his own solution is hidden. This solution, however, is continuously used to compute and show to the student only the corresponding results. The same results are expected, ultimately, to be obtained by the students’ solutions. The availability of all results provides a non-ambiguous description of what each (sub)-problem asks for. A misunderstanding of what is actually requested is often one of the major causes of students’ failure in problem-solving activities. In this setting, the teacher’s results provide a feedback about the correctness of student’s solutions.
The check of correctness, based only on the expected results, implies that the student is not forced to follow the teacher’s solution, but he is free to think of and pursue his own, provided that it always supplies the same results as the teacher’s.
This approach, based on decomposition of the original problem and the accessibility of all results, supplies some benefits which are not generally available in standard problem solving approaches:
• The student may use the teacher’s results from a previous step as input data to test and validate the solution he has given for a subsequent step. Thus all steps become independent and can be individually tackled. In this way the student is not blocked if he is not able to solve a particular step.
• Using teacher’s results, the student can also stop error propagation, a problem which typically invalidates, and sometimes inhibits, all steps following one containing errors.
• Having all steps independent, the student is also free to tackle the whole problem as he desires, thus self-tailoring its difficulty to his own expertise. Moreover, he can approach all the separate sub-problems with different methodologies (such as top-down or bottom-up).
The tool we present in this paper uses spreadsheets to make the approach described so far concrete: spreadsheets interactive capabilities and their vast tool-set to analyze and present data allow us to turn problems into engaging interactive learning experiences.
The authors have used this teaching approach for some years at the Naval Academy in Livorno. In the paper we also present and discuss the results of the pilot study.

Keywords:

technology, educational software, learning and teaching methodologies.