In addition to the various
improvements to these materials suggested by the initial trials, which are
described in the respective critique sections, the people involved with this
course have discussed a variety of possible extensions to its content or
methods. While it will not be
possible to fully develop all these paths (due to course time constraints if
nothing else), each suggestion has value and will probably leave some trace in
the mature course.
Integration of modeling and measurement
Efforts to apply the approach of these initial modules to the modeling part of the course are among the top development priorities. A major component of those will be to collect instances of function use from a wide range of practical applications, and to then present them to students in groups related by the relevant mathematical form. A goal will be for the students to find further applications and identify needed refinements in their own areas of primary interest.
Integration opportunities among the course topics include using sinusoidal functions in modeling. While the existence of three controlling parameters (amplitude, frequency, and phase) makes these more complex than the two-parameter linear and exponential cases, the sinusoid parameters have the advantage of corresponding well to distinct practical characteristics.
Another potential expansion of the scope of the modeling section would be to the modeling of measurement processes themselves. The main thrust of this would be to examine the characteristics of models that are the sum of two overlapping normal curves. This would provide a basis for discussion of two different measurement situations of particular interest. The first is the combination of a narrow intended-measurement curve with a lower-frequency but wider curve of unintended contributions (e.g., cosmic-ray glitches); this models many typical actual situations, which is why a trimmed mean is so often to be preferred to an uncritical average of all values. The other case is two normal curves of equal width and similar heights whose centers are displaced from each other so that they overlap but do not coincide; this models situations where two varieties of a related object are being measured at once. The goal here is to teach students to recognize situations of this kind and use their knowledge of the practical circumstances that produced the measurements to connect any anomalies back to their physical causes.
An area identified as having great potential for additions to the course is use of the internet and other computer technology to illustrate and extend the ideas presented in the classroom. Immediate opportunities include the provision of animated measurement demonstrations and simulations as web applets, and annotated linkages to extension materials and to examples of actual practical uses of mathematics (with enough variety that all students can find cases relevant to their particular interests). A sufficiently well developed computer-based support structure might make it possible for an MFM course to meet fewer days per week, which would be of great convenience for practical-arts students, who are often combining their college work with a full-time job.
A final area of possible expansion, which has so much potential that it may have to be avoided because it would take over the course, is the inclusion of computer programming by students. Programming takes the MFM guiding principle of “sophisticated use of simple techniques” to extreme lengths. While an adequate discussion of the interaction between programming, mathematics learning, and the mathematical analysis of practical work would require a separate thesis, several opportunities are obvious. A spreadsheet or exploratory-statistics program could be used, with appropriate scripts, to display tabulated measurement data and compute relevant statistics. A modeling program used to construct and display models could be used with scripts to find optimal parameter settings by iterative search. Or scripts could be used to simulate noisy measurement processes or to show error-propagation effects.
If the scripts that control such computer interactions are written by the instructors (or by other non-students) and their source text is not brought to the students’ attention, then they are just invisible parts of the applets or other programs that students use, and not programming in the sense meant here. The point of including programming would be to show students something of the underlying algorithmic mechanisms that cause the mathematical results. This would start with just making it possible for students to see the text of the scripts, but would proceed at least as far as lessons in which students modify existing scripts to see the effects of various changes or to try to reach a particular different result. In the most advanced usage, independent student programming projects on measurement topics could be commissioned.
Such student programming might well be judged worth including in the course simply as the most convenient mode of presentation of certain ideas, and incidental use of programs will certainly increase as both computer power and computer-use sophistication continue to grow. But one of the most compelling reasons seen by the author for incorporation of programming into the MFM course is that computer programs are entities that exist at the same time in the mathematical pattern-world and the practical stuff-world.
Programs are clearly exact, and will always produce the same result from the same input (unless a random value is explicitly asked for, which is done by changing the input with a hidden variable). Programs in appropriate languages can refer to and work with numeric values and functional transformations, although there are a few (exactly-defined) deviations from some of the fine points of standard mathematics. Programming usage of symbols for both values and processes is closely parallel to standard mathematics.
But although similar in their exactness and in their use of sequential exposition, programming and mathematics differ in one feature of fundamental significance for education. Since the essence of post-arithmetic mathematics is its abstractness, questions of mathematical meaning and correctness pose acute intellectual challenges, especially at early stages of mental development or sophistication. What does an equation mean? What makes it true? false? relevant? How can you tell? A computer program, on the other hand, is almost as concrete as a player-piano roll – a program doesn't mean something, it is something and (as part of a computer) does something. The analogous questions in the case of programs (e.g., What does the program do? For what inputs does it work as intended?) are much easier to understand and investigate.
So it may well be that programming is already situated in the middle of the bridge between practice and theory that the Mathematics For Measurement course is intended to provide, and could supply a skill that will connect students productively to both worlds. Fortunately, the best way to investigate the issue is probably incremental expansion of the computer and scripting components of the course, along the lines already discussed at the beginning of this section. But the potential for large gains in student understanding of mathematics itself, not just measurement topics, makes it clear that the potential of using programming in this way should be explored in practice.
The final section of this thesis is the Appendices (Instructional Handouts).