Principles guiding MFM topic selection
Prerequisites and objectives for the MFM course
Contrast to related existing courses
Contrast to a functions-and-modeling course
Contrast to a quantitative-literacy course
Why not integrate the material into vocation-specific courses?
Why not cover the material in a pre-engineering course?
Contrast to a standard statistics course
Contrast to measurement in science laboratory courses
Contrast to a standard trigonometry course
Other topics for which materials were developed and used
One of the analytical viewpoints found most useful in assessing possible topics for the Math For Measurement course is described by Seymour Papert's "constructionist" principles (Papert, 1980, p. 54) for standards that mathematics topics should meet:
Continuity
Principle: The mathematics must be continuous with
well-established personal knowledge from which it can inherit a sense of warmth
and value as well as "cognitive" competence.
Power
Principle: It must empower the learner to perform personally
meaningful projects that could not be done without it.
Principle
of Cultural Resonance: The topic must make sense in terms of a
larger social context.
One might characterize these principles as guidance for connecting to a student's past, present, and future, respectively. For MFM students, the “well-established personal knowledge” will be based in practical measurement experiences (mostly derived from work situations), the “personally meaningful projects” will be related to either current work or to increased capability in related vocational courses, and the “larger social context” to be considered is the network of activities (scientific and cultural as well as directly occupational) that support and/or use the work of vocational sectors with which students are involved, especially communication about quantitative matters.
But this is not just a matter of translating mathematics into perhaps-limited domain-specific language. There is also a flow of sophistication back into the mathematics from the practical sectors. The idea of an approximate number is just as mathematical as that of a transcendental one (and more sophisticated in some ways). Topics were selected for the MFM course much more on their perceived potential for supporting sophisticated practical thinking, of the kind typical of engineering practice, than for their mathematical interest per se.
The desire for a one-semester,
low-prerequisite, high-external-relevance course whose goal is to build
mathematical sophistication in several areas places severe constraints on the
size and mathematical sophistication of the set of topics that can be included. On
the other hand, in many cases the omission of material present in the standard
courses is seen as a positive advantage, enhancing the conceptual clarity of
the MFM course and promoting authentic mathematical experiences, rather than
being simply a side effect of a limited time budget.
The development of the course objectives, and of the tactics used to achieve them, came mainly from critical analysis (for relevance to the goals and target audience of this course) of each of several standard mathematics and science courses, as well as domain-specific vocational courses. A compressed recapitulation of that analysis will address the question of how the MFM course is not like standard math courses (e.g., statistics, modeling) that touch on some of the same topics, and why the needed result cannot be as well obtained via non-mathematics courses such as science labs, pre-engineering courses, or vocation-specific courses.
The prerequisite mathematics-related skills for the topics covered in the MFM course are modest – basically through the first year of high-school algebra. Scientific calculators are used extensively in the course, but prior familiarity with the transcendental functions supported by them, or with the exponential notation that such calculators sometimes use, is not required.
Because of the stand-alone nature of the MFM course, there is substantial flexibility in the choice of specific instructional objectives to support the overall goal of promoting connections between mathematical and practical thinking. While several topics follow naturally from the selection of a theme of measurement, the balance and arrangement of the topics, as well as the particular instructional tactics, has been modified after each of the initial offerings, and will require further development to reach optimal form. However, here is the current statement of the main specific student abilities that the MFM course is intended to produce:
[1] To distinguish between exact and approximate quantities, based on knowledge or a description of the process producing them.
[2] To produce a numeric description of the appropriate approximate quantity when an over-precise value and a rounding precision (generally a power of 10) are specified. This entails the ability to use (and convert between) direct significant-digit representation, scientific notation, and engineering notation.
[3] To produce an appropriate numeric description of the approximate quantity implied by a set of measurement values. This entails formation of a suitable average and use of either a significant-digit-based representation or of a direct statement about typical or maximal variation, as appropriate for the situation or request.
[4] To identify systematic errors in a measurement situation, to assess their impact on the accuracy of measured values, and to both produce and make use of correcting calibration methods. The analyses should distinguish offset and scale-factor errors for cases where the relationship between nominal and actual values is linear.
[5] To make reasonable estimates of the precision of results formed by combining or computing with approximate quantities. This includes estimation of worst-case deviations due to combined or scaled effects, as well as the estimation of most-likely deviations from the expected partial cancellation of combined random variations.
[6] To identify the sine,
cosine, and tangent functions, and to compute them and their inverses on
request (with a calculator) from adequate information about the sizes of the
angles or sides of a right triangle.
[7] To compute missing
information about the sizes of the angles or sides of a right triangle from
adequate supplied information.
[8] To compute the area of right
or general triangles, given adequate information.
[9] To compute the length of any
side of a general triangle, given information about the sizes of one side and
of two angles.
[10] To distinguish between simple cases of mathematical proof and other forms of assertion. This entails awareness of the basic methods and concepts of mathematical proof, including the necessity of initial assumptions, the dependence of the truth of a proof on the truth of its assumptions, and the inadequacy of particular examples or approximate measurements to establish a mathematical proof.
[11] To use scientific calculators robustly. This is addressed primarily in the contexts of practical trigonometry and the expression of approximate numeric quantities. The issue of the significance of calculator-derived numerical results is given particular attention. Because of the salience of calculators as practical tools, mathematical explanations that relate to calculator usage command student attention, and scientific calculators have sufficient complexity that a good case can be made that understanding will be a more dependable guide to correct usage than memorization of a usage pattern for each type of problem.
Other topics dealt with in the
MFM course (but using materials not written by the thesis author):
In the initial sections of the MFM course, some further trigonometry topics such as the unit circle and the law of cosines were covered using other published materials, which also supplied a set of applications problems from which examples suitable to the covered materials were chosen.
The main additional topic area covered was an introduction to functional modeling. The concentration was on linear and exponential functions, with the main point being to show how the functional form is a consequence of the understandable aspects of the situations that give rise, respectively, to constant-addition and constant-multiplication conditions.
The Mathematics For Measurement course has many common elements with the new “functions and modeling” courses that are being developed and offered at both the secondary and college levels. Almost half of the time in the initial MFM classes was devoted directly to this topic area, and such courses have great potential to promote the expansion of mathematical thinking and perception that is the goal of the MFM course.
The differentiating factors of the MFM course are [1] the lower demands made on mathematical (especially algebraic) dexterity, as reflected by the limited variety of functional forms addressed, [2] the greater demands made on non-mathematical experience, as reflected in its recurrent appeal to external practical situations that can be productively examined from a mathematical perspective, and [3] the use of a central theme, measurement, that both anchors the course in non-mathematical experience and provides a context for connecting a variety of distinct strands of mathematical thought (functions, statistics, trigonometry, and approximation) both to each other and to existing student concepts and competencies.
It is certainly reasonable to see the MFM course as simply one instance of a variety of efforts in course design of which both functions-and-modeling courses and certain other reform initiatives in mathematics and science education are examples. The particular instructional materials and efforts reported on in this thesis, however, are those elements of the initial MFM classes that are not addressed in current typical modeling courses.
One standard variant of the math for liberal arts course is designed to ensure that students understand, and to some extent can use, the main quantitative techniques to which non-specialists are exposed in contemporary society. The topics are typically drawn from logic, statistics, probability, financial calculation, and discrete mathematics.
This type of course is an expression of a welcome trend toward relevance and general accessibility in mathematics education, and as such is clearly compatible with the goals of the MFM course. The main differences between the courses are that the MFM course is much more focused in its topic selection and is oriented toward the expansion of work-related capability, and toward the establishment of the relevance of mathematical thinking to such capability, rather than toward general familiarity with a broad range of quantitative techniques.
One strategy used for connecting mathematics with practical knowledge is to add appropriate mathematical applications (e.g., dosage calculations for nurses) into career-specific courses. This establishes relevance and utility, and provides opportunities to familiarize students with the particular math-related methods and reference materials normally used in the profession.
Particular-application training of this kind seldom contributes to the development of mathematical thinking, however, since its emphasis is usually on following standardized procedures and applying standard formulae (whose value in part is precisely that they reduce the need to think about their area of application) rather than on analyzing a situation to extract information from it by mathematical techniques. This tendency is reinforced by the limited mathematical sophistication of many of the teachers of such courses, but its root cause is that career-specific courses already have a full agenda and cannot afford to expand on the mathematical aspects of their subject.
The premise of the Mathematics For Measurement course is that, rather than trying to add the expansion of mathematical sophistication to the goals of vocational courses, it will be more effective to provide a separate mathematics course that is designed to make useful mathematical thinking visible, and then to get the vocational teachers both to provide authentic application problems suitable for use in the MFM course and to make use of some of the techniques covered in the MFM course in advanced problems in the vocational courses.
Topics such as calibration, measurement-noise analysis, and process modeling are actually best characterized as engineering concepts rather than as either scientific or mathematical ones. Further, engineers generally comprise the technical leadership of the sectors in which practical-arts students will work, and command their respect as role models. This suggests that a course specifically identified with engineering might be best situated to help practical-arts students transcend their mathematics alienation.
The main difficulty with making use of this suggestion is that the few explicit pre-engineering courses that are currently offered are designed to prepare students to enter college baccalaureate engineering programs, and thus require much more math than the practical-arts students for which the MFM course was designed will have mastered (or need to master). Most secondary schools do not distinguish between preparation for scientific and engineering programs (on the probably-correct assumption that both need the same good math and science foundation), and the increasing emphasis on integration of application-related material into advanced math and science courses gives little motivation to change this situation.
A new “general engineering” course offered at about the level of a secondary-school general-science course (or perhaps integrated with one) would be an excellent place to deal with much of the MFM material, although it would have to be adapted to reduce dependence on existing practical sophistication. But by the college level, the educational paths of engineers and of the technicians who will support them have already diverged, with differences in mathematical-thinking mastery one of the main distinguishing factors. But the connection is still quite relevant — an important goal of the MFM course is to enable practical-arts students to communicate productively with engineers in their area, much more than with mathematicians.
Statistics is the only standard mathematics course in which the issues of measurement are addressed at all, and it provides an essential basis for any extended investigation of the subject. But the emphasis of statistics courses on the description of parameters of populations (whose distributional forms are empirical rather than mathematical) obscures the simpler concept of measurement of an actual value. There is an important epistemological difference, reflected in easier accessibility to authentic student comprehension, between the concept repeated approximate measurements of the same value that is focused on the in the MFM course and the statistical concept of collection of a sample from a heterogeneous population of values. The fact that the first case can be handled mathematically as a special case of the second, while an interesting result for an advanced course (and essential to the full analysis of complex measurement situations), should be omitted from the first examination of measurement concepts.
Because measurements are presented as being somewhat noisy reports of a single actual value from a continuous range, and because almost all measurement processes are approximately normally distributed in some natural choice of parameters, the statistical issues associated with measurement sets can be greatly simplified. The simplifications used for the MFM course are listed below.
The arithmetic mean (trimmed if outliers are suspected) is taken as the best estimate of the central tendency of the measurements. The possibility of bias and the calibration methods to detect and correct for it are addressed promptly. There is no need to discuss the mode statistic (which is an unstable construct in any case), and the median can be presented as a case of extreme trimming of the mean. Since the natural concept of a “most typical value” corresponds quite well with the idea of a trimmed mean of samples from a unimodal population, this is a case of providing a mathematical label and computation method for an existing student concept, a connection that is emphasized in instruction.
The stability of a measurement process is described in the MFM course by a “typical deviation value” computed by a method (the deviation encompassing half the sample values) that obviously produces a value consistent with the natural-language meaning of that phrase. The goal is to enable students to quantify the concept that they already hold about variation rather than to develop a new related concept (such as the standard deviation) that will be optimally convenient for math-theory discussions. This entails little or no loss of information, because measurement-of-a-single-object processes generally produce data whose distribution is close to normal, and for normal distributions all possible measures of variation are related by constant factors and are thus equivalent when used as weighting factors in computing error propagation (including the important special case of averaging of multiple measurements of the same object).
Even when sensitization of students to the possibility of non-normal measurement processes[1] is desired, this goal is best approached in a course of this kind by presenting the problem of detecting whether a set of measurements has been produced by mixing values from more than one normal process with different means and/or stabilities. Consideration of such situations, which are the source of most outliers or abnormal distributions in actual measurement data, provides students with mental tools to understand the sources of variation for a measurement process of interest, rather than to merely describe its pattern of variation. The goal of the MFM course is to produce problem-solvers (or at least problem-recognizers), not recording clerks.
One way of looking at MFM’s limited subset of statistical topics is as using from statistics the products of one of its historical roots: the mathematics developed about 200 years ago to calibrate, assess, and reconcile astronomical observations. The stars are constant in their courses (on the time scales and resolutions accessible to circa-1800 technology), so the position of each star is simply a stable fact – it is the process of measurement itself that must be abstracted to explain and handle the differences in measured positions, and that abstraction readily yields a very useful set of mathematical tools. Neither the descriptive-statistics tools for general population description, inferential-statistics topics such as hypothesis testing, nor impressive theoretical results such as the central limit theorem are needed to accomplish this task, and they should not be permitted to distract attention from it.
The other type of standard course in which measurement is often explicitly included is the laboratory portion of science courses, especially in physical sciences such as chemistry and physics. These can be extremely valuable experiences for students, and are probably currently the venue of most school-inspired illuminations about measurement, especially about the nature and characteristics of measurement bias and noise.
But the focus of most school science experiments is validation of the scientific laws that summarize simple fundamental interactions, or demonstration of effects that will serve as occasions for which scientific explanation will be provided (or evoked from the student). The lab work, especially the detail of the measurement process, is firmly subordinated to the basic goal of student mastery of the scientific concepts (e.g., molecule, reaction, force, energy) required to enable scientific thinking, and of the basic relationships between those concepts. Students are being trained to look through the bias and noise in their measurements (and even through the measurements themselves), not at them.
While science-course demonstrations of the ability of mathematical laws to summarize and predict scientific relationships are in accord with the MFM agenda of showing the potential of mathematical thinking for practical use, the typical effect of science courses on practical-arts students is more intimidation than inspiration. Because of the prestige of science and the conflict of some of its most famous results with ordinary experience, most people want to be assured that “it’s not rocket science” before attempting to think about a problem – science, especially in aspects that require mathematics, is seen as too hard for ordinary people. While this opinion can and should be challenged, the MFM course offers a way to advance ordinary-person mathematical sophistication without waiting for success in the parallel efforts in science education.
There is also a widespread belief that high-culture tools such as science and mathematics can only be used in situations in which all causal factors are understood or are arranged with controlled-experiment tidiness. There is some truth to this, in the sense that naïve application of scientific analysis to practical activities seldom succeeds in fully achieving the knowledge that has been accumulated empirically by practitioners, and usually produces an oversimplified model that would be disastrous to depend on blindly. Thus the skepticism of engineering practitioners about overreaching theoretical analysis is well-based. It is not surprising that this skepticism is shared, and extended to mathematical thinking, by practical-arts students for whom such practitioners are role models.
On the other hand, the history of the last century or two has shown that the combination of empirical knowledge with mathematics and scientific theory can greatly advance understanding and capability in almost all fields, especially when the theory is used to organize and illuminate practical experience rather than to supplant or discredit it. To date, the productive interactions based on this approach have been primarily at the engineering-design level, involving a limited number of people. The MFM course is designed to make the approach more available to the people who will actually be doing the day-to-day work.
The MFM course includes both practical trigonometry (setting up and solving triangles, given sufficient information) and examination of sine, cosine, and tangent (and their inverses) both as ratios and as general functions. It also covers the Pythagorean theorem and its associated sin2 + cos2 = 1 identity, and the simple (and easy to prove) tan = sin/cos and cos(A)=sin(90º-A) identities, but excludes any further work with identities and does not discuss formulae such as those for the trigonometric functions of sums and differences of angles. Thus the MFM course omits the algebraic manipulations and functional transformations that are among the main activities of a standard trigonometry course.
The traditional trigonometry-course emphasis on the special-case 30º/60º/90º and 45º/45º/90º triangles is also avoided, since it is considered to be counterproductive to encourage students to look for special cases rather than to learn the general technique, which calculators make easy to apply in any case. (This position is being reexamined in light of some experiences in the trial sections, but is still felt to be fundamentally correct.) In fact, the MFM course encourages calculator use, using its trigonometry section to ensure student fluency in scientific-calculator usage, with special attention to inverse functions.
On the other hand, the MFM course contains as part of its treatment of triangles some material that would normally be handled in a geometry course. This includes a derivation of the triangle-area formula, with a simple proof that illustrates how some exact relations can be shown to be true by logical argument rather than just being demonstrated by example. Further discussion of the idea of proof culminates in a respectable, although not explicitly axiomatic, proof of the Pythagorean Theorem. The goal here is to help students understand the nature of mathematical proof, including its limitations (such as the assumption of flatness for the surface on which figures are drawn for the proof of the standard area-related formulae), with the target competence being student ability to distinguish mathematical proofs from other types of assertions, not student proficiency in the construction of proofs.
The limited appeals to logical proofs related to triangles are more than balanced by extensive use of direct physical measurements and empirical investigations via calculator-based computation of trigonometric functions for particular values. An example of the tactics used is the distribution to each student of a uniquely-shaped right triangle (cut from the corners of sheets of construction paper), permitting “parallel-processing” (and unique-answer-for-each-student) investigation of conjectures by having each student test the specific case their own triangle represents. This first-day tactic was also designed to reinforce the real-object substrate to which the measurement process is applied, in an attempt to establish that even though MFM is a math course, it is not unwilling to use direct experience for inspiration and guidance.
Certain topics central to mathematical thinking about measurement are not directly covered in any standard courses, although they are sometimes referred to incidentally in the context of practical problems. Thus they are not really ignored, but rather taken for granted, precisely because they reflect the kind of independently-achieved quantitative sophistication that is depended on but not acknowledged by school mathematics. A major goal of the MFM course is to enable students to consciously discuss such capabilities, and thus refine them. The main topic areas of this kind in the initial course offerings were approximate numbers and measurement theory.
A significant portion of the course is directed at increasing student sophistication about the idea of numerical expression of approximate values, including review and higher-level explanation of rounding methods and of scientific and engineering notation. The question of numerical significance is approached both from the vantage of rounded-off numbers (where the possible errors are clearly delineated) and from that of repeated measurements of some object feature, where the lack of exact repeatability indicates the limits on the amount of information a measurement produced by that process conveys about the feature measured.
The measurement process itself is examined, with the introduction of systematic ways of dealing with offset bias and scale-factor bias via calibration. The characteristics of random error are also examined. The effect on expected error of combining measurements is derived for the main different methods of combination, including the important special case of averaging. The culmination of the consideration of measurement error is a demonstration of perturbation analysis, which is illustrated by a graphical approach that shows how error sensitivity can be graphically expressed as the slope of the result computed from a measurement as a function of the measurement value.
[1] The computational convenience that for non-normal processes it is possible to exactly propagate variances (but not other single-value measures of stability) through extended calculations and independent-measurement combinations is of little importance as a practical matter, because if a measurement process is substantially non-normal, almost any rational decision process based on its results will have to be computed by combining an appropriate cost function with a measurement-distribution description that is more complex than the minimal mean-and-variance information. This is one of the reasons that stability measures other than variance are often used in practical situations. From an instructional viewpoint, the example of standard deviation as a measure of stability is best included as part of a group of not-on-the-test refinements that also contains such practical-origin stability measures as those based on percentiles and peak width. Showing how that for normal distributions each of these measures has a constant ratio to the others both anchors the exotic measures by reference to the familiar ones and prepares the ground for using the relative sizes of such measures as tests for normality.
The next section of this thesis is MFM Theoretical Background.