Motivation for developing the course
Differences between liberal-arts and practical-arts students
Goal: A “math for practical arts” course
This thesis report has five major narrative sections, in addition to appendices that provide copies of the instructional materials developed for the course it describes. The Introduction explains the motivation for the Mathematics For Measurement course (henceforth “MFM”), describes its target audience of “practical arts” students, summarizes the history of MFM development to date, and outlines the primary themes used in planning, designing, and critiquing the course. Subsequent sections address related practice, related theory, reports on the initial design and classroom results of the modules that I contributed to the pioneer MFM sections, and plans for revision and extension of the course.
The centerpiece of the section discussing related practice is an analysis of the insufficiencies of existing courses (in mathematics and other disciplines) for addressing the goals of the MFM course. In addition to giving a recapitulation of the main motivation for development of the course, this section provides a context to illustrate how the MFM treatment of a variety of topics differs from that in related courses such as introductory statistics or physical-sciences labs. These contrasts with existing courses are preceded by a statement and discussion of the initial MFM course objectives.
The theoretical discussion draws ideas from a variety of sources. In addition to the relevant strands of standard educational discourse (especially those related to non-school and adult learning), these include my experiences in devising industrial measurement systems and some ideas from evolved-systems theory, not commonly used in an educational context, that can be used to support and inform the ideas of Vygotsky and others.
The section describing the instructional modules on approximate numbers, trigonometry, and measurement theory that I developed for the pioneer offerings of the MFM course summarizes the design motivation and the implications of the initial classroom trials for each of the twelve handouts collected in the appendices.
A final section (prior to the endnotes and the appendices containing copies of the instructional materials developed) discusses the planned future development for the course. The main themes are integration of measurement-related material with the modeling portion of the course and integration of computer-based methods of various kinds.
The Mathematics for Measurement course was developed jointly by me and Mary R. Parker, Professor of Mathematics at Austin Community College and Senior Lecturer in mathematical statistics at the University of Texas at Austin.
I designed and wrote all the materials shown in this report, extending and modifying them as the course developed in the spirit of a “design experiment” (Brown, 1992). Portions of a few sections closely follow material I wrote earlier as part of my graduate coursework. I also provided measurement-practitioner-view critiques of the parts of the MFM course for which I did not develop instructional materials, but these are mentioned here only in relation to the initial design efforts for the course, which were about equally shared between me and Dr. Parker.
Dr. Parker determined the overall course structure after consultation with the vocational-program faculty at Austin Community College for whose students it is designed, and has taught the initial sections of the course. She has a long history of participation in national mathematics and statistics curriculum-reform planning and activities under the auspices of the Mathematical Association of America, the American Statistical Society, and the National Science Foundation.
My direct teaching experience is limited to three years of secondary-school math and science in Nigeria during Peace Corps service in the 1960’s, but I have had many years of experience developing measurement devices for scientific and industrial uses (and some experience in informal science education) prior to my recent graduate studies in math/science education.
Dr. Parker provided extensive feedback after each class on course materials I developed, which were used for about three weeks of the Fall 2000 section and six weeks of the Spring 2001 section. The remaining time in those classes were based on other published materials selected by Dr. Parker.
Revision and extension of the material written specifically for the MFM course is envisioned in successive semesters until a full semester of integrated and tested material has been produced. The materials shown here are considered to be only exploratory, and extensive additional work will be needed before the course fully reflects its stated design principles and goals, which have themselves evolved in response to consideration of the experiences with the pioneer sections.
The wider objective for the Mathematics For Measurement course is to acquaint its students with the power of mathematical approaches in aiding analysis, description, and performance of the practical tasks that are encountered by skilled workers in all fields. The purpose is thus not training in useful vocational techniques (although practical utility is used to attract and hold student attention) or preparation for further mathematical studies (although the course may well raise that prospect in the minds of students to whom it would not otherwise have occurred). The main purpose is to make connections between mathematical thinking and the sophisticated practical thinking of which students are already capable. It is felt that such connections are essential to developing the flexible problem-solving capabilities that are required to deal successfully with the dynamic work situations that contemporary students will encounter in almost all fields.
The urgency of this goal stems from the deep alienation from mathematics that the majority of students feel by the time they enter college. That this is not just a result of student immaturity or intellectual incapacity is indicated by the fact that this alienation is shared by most successful adults (including most college teachers, except in the minority of disciplines that use advanced mathematics). Even among people who perform well in mathematics courses in college (including the calculus course that is commonly required for a much wider range of majors than would seem indicated by the content of those fields), only a small minority retain competence or interest in mathematics a decade later.
One source of this alienation is the approach typical of scholastic mathematics instruction, in which a progression of new techniques are taught at each level but applied to simple or abstract problems that do little to develop a student’s ability to think mathematically about the world. The standard school sequence is simplistic applications of increasingly sophisticated techniques, rather than the increasingly sophisticated application of simple techniques that would be much more effective at promoting intellectual growth and in sharpening the pattern-perception skills that underlie mathematical thinking. (Some math courses not on the main sequence, notably elementary statistics and the new functions-and-modeling courses, often avoid this problem to some extent, as do many advanced science courses.)
The MFM course is intended to provide an example of a course that uses an “increasingly sophisticated applications of simple techniques” approach to promoting mathematical insight, and can be used successfully with a broader range of students than existing courses. Note that the sophistication is largely drawn from concepts (e.g., measurement-process stability) that students have already developed in practical contexts – MFM is a course designed for adults, people who have already developed coherent sets of concepts and strategies in many areas. This presents a different set of opportunities and limitations than exist for younger students.
A standard educational tactic for addressing the mathematics alienation of college-level students whose interests are in non-mathematical fields is the “mathematics for liberal arts” course, which selects some topics of social or esthetic interest (e.g., congressional apportionment, fractals) and examines them from a mathematical perspective. Such a course typically has no college-level mathematical prerequisites and has no successor course depending on specific accomplishments. While these conditions leave the rigor of such courses subject to suspicion (since they lack the discipline that ruthlessly reveals shortcomings or gaps in courses imbedded in a sequence), the course has proven valuable and is widespread in American colleges.
However, the standard math-for-liberal-arts course formats have serious drawbacks for students in those college majors that are oriented toward preparation of students for particular areas of technical, but generally non-mathematical, practice. Such students, who might be called practical arts majors, form a significant minority of the enrollment of American community colleges. Examples of such majors (all represented in the two pioneer MFM classes completed so far) include health sciences such as nursing, construction arts such as welding, and mechanical arts such as automotive mechanics.
In general, practical-arts students differ from liberal-arts ones in several ways that must be taken into account in course design if instructional effectiveness is to be maximized. Their learning goals are usually more focused, and their attention to individual topics is more dependent on the perceived relevance of each topic to those goals. There is often an active resistance to abstract generalizations, which are seen as tools of an unrealistic scholastic culture that, for example, values grammar/spelling more than effectiveness of expression and numerical exactness more than how well the source of the number corresponds to the concept it purports to quantify. Although the criticism of schools leveled by such an analysis may be well justified as an overall judgment, the attitude it engenders poses severe challenges to even the best instruction in mathematics, for which abstraction, generalization, and exact statement are central techniques.
On the other hand, it is also common for practical-arts students to have substantial practical vocational experience, which fosters familiarity with “rule of thumb” heuristics (some with an underlying mathematical basis) that are part of the lore of their trade, as well as skill at detecting the oversimplifications that are common in what are typically presented as “applied” problems in mathematics courses. This kind of sophistication makes establishing practical relevance for a math course’s problems simultaneously more difficult (because inauthenticities will not pass unnoticed and authentic domain-specific problems may exceed the scope of the course or the reach of most of the mixed set of students) and more important (because mathematical thinking will be much more likely to take root if it can be connected with already-valued professional thinking patterns).
At one time, it was common for American school curricula to include advanced arithmetic topics that served as the “sophisticated application of simple techniques” that are now missing from most mathematics courses. For example, a problem might be set to compute the economic feasibility of a road from the cost of the materials, the longevity and repair costs for road compositions of various mixes, the impact of the road on the value of the property it serves, and the tax rates that could be assessed to fund it. All the computations involved were simple, but solution of the problem depended on sustained quantitative thinking about matters that had real-world significance, rather than on recognition of a problem type and unreflective application of its associated memorized solution template. The Mathematics For Measurement course is an attempt to recapture this spirit.
The choice of measurement as the unifying theme for a “math for practical arts” course is an attempt to address the difficulties discussed above. Measurement is an activity with which almost all adults will have had some authentic experience and which is relevant in some form, often with subtle refinements, to almost all vocational areas. It is connected to several mathematical areas that are accessible with modest prerequisite skills, are of immediate utility both in practical work and as mathematical tools, and are related to advanced topics that can be identified for potential further learning but not included as part of the course.
The next section of this thesis is MFM Course Topics.