Instructional-modules overview
General criticisms of the instructional materials
Overview of approximation topics
Module A1 – Exact & approximate numbers
Module A2 – Rounded numbers; combining approximate numbers
Module A3 – Scientific and engineering notation
Module A4 – Expressing measurement accuracy
General conclusions on approximation materials
Overview of trigonometry topics
Module B1 – Right triangles: side/angle measurements, trig ratios
Module B2 – Calculating angle size in right triangles
Module B3 – Relationship between right-triangle sides
Module B4 – Area of triangles; Mathematical proof
General conclusions on trigonometry materials
Measurement-theory topic group
Overview of measurement-theory topics
Module C1 – Calibration of a measurement process
Module C2 – Mathematical effects of random measurement errors
Module C3 – Error sensitivity of calculation processes
Module C4 – Graphical interpretation of error sensitivity
General conclusions on measurement-theory materials
The following sections of the thesis are organized to parallel the sequence of topics using materials from the thesis author as offered in the two initial Mathematics For Measurement classes. The material examined is broken into three topic groups: approximation (4 handouts for the Fall 2001 class), trigonometry (4 handouts starting the Spring 2002 class), and measurement theory (4 further handouts for Spring 2002). Associated homework or test problems developed by the thesis author are included with the corresponding handouts; these were supplemented by others provided by the instructor, which are not included in this report.
An overview and set of general conclusions is presented for each topic group; these comparatively concise sections surround a statement of design intention, a list of key instructional messages, and a critique for each module in the topic group. In order to preserve the layout of the student handouts, they are collected in the appendix; for detailed understanding of the modules, each should be examined prior to reading the associated remarks. In future work, these design and critique comments will become the basis of teacher’s-edition notes for the more successful items and an outline of revision efforts for the difficulties and failures identified.
While analysis of the strengths and weaknesses of specific modules, and of each of the three major module groups, are given as part of the summaries listed below, there were several problems and issues uncovered in the testing of the materials that applied in many places. These general criticisms indicate some of the main directions of planned future development (described in more detail in a later section) and thus may help in clarifying the imperfectly-expressed intentions of some of these initial instructional modules.
[1] In many cases, the material as presented went too fast for the students to fully assimilate it. This was not generally a question of too much material, but rather indicates a need to spread treatment of each topic over more instructional days, with other topics interspersed so that about the same total amount of material can be covered. While multi-day presentations for the material in a module were sometimes done by the instructor on an ad hoc basis, this was not adequately supported in the material as written, except for the next-day effect of dealing with the homework problems assigned on a topic.
[2] While some practical examples were supplied in almost all modules, many more are needed, both for homework problems and to indicate the range of potential application. These will need to be carefully constructed, however, since the students, as anticipated, were adept at detecting any inauthentic problem descriptions.
[3] Improvement is needed in the balance between math-centric and application-centric exposition. In particular, more use of clearly-separated reviews of foundational math topics (such as linear graphing or scientific notation) would help avoid a recurrent situation in which weaker students were confused by having to re-master an old math topic in the context of a new measurement-theory concept.
[4] Direct student classroom experience in measurement needs to be more carefully structured, so that the variation intrinsic to measurement does not confuse students into thinking that mathematical definitions such as trig functions are also approximate rather than exact.
[5] A more extensive and persuasive set of measurement experiences common to all students is needed. This will probably entail both formal student laboratory activities and some use of demonstration equipment to permit many measurement-related concepts to be illustrated in a more concrete form. Attempts to use homework problems and lightly-structured classroom activities to provide data from which various characteristics of measurement can be investigated were not successful.
The overall objective for material I wrote for the Fall 2001 MFM class (the first one offered) was to enable students to develop a mature view of the mathematical issues associated with numerical designation of approximate quantities. This material covered about 15% of the course schedule; the rest of the Fall 2001 course was based on externally-written materials on trigonometry and functional modeling and is not discussed in this thesis.
An essential feature of any measurement process (other than counting, which is taken here as being a different type of process) is that it produces approximate rather than exact information. This is in striking contrast to elementary mathematics, whose hallmark is an exactness based in logical distinctions. This contrast is a natural source of confusion when attempts are made to use mathematics to describe measurement. Showing how numbers are used in well-defined ways by practitioners in different sectors to express either exact or approximate knowledge thus both prepares students to resolve any such confusion and serves as an example of mathematical description of practical methods.
The general strategy was to make students aware of the variety of situations under which numbers are used for both exact and approximate specification, with the goal of enabling them to see through particular representations to the operational meanings of statements using such numbers, and thus to the consequences of operations with such values. The process of rounding was selected for initial emphasis since it provides a well-defined method of producing approximate numbers of exactly-known precision. Attempts were then made to extend the idea of approximate numbers to show that a measurement is described by its precision as well as its average value and to show some of the consequences of computations involving approximate numbers.
Establish the logical distinction between exact and approximate numeric quantities. Show that while the form of a number often gives an indication of whether the quantity it represents is exact or approximate (especially if it has been written down by a careful practitioner), the determining factor is the nature of the process that produced it. Promote student consideration of the variety of sources of exact and approximate numerical quantities.
Exact numbers come from processes that produce an unchanging value: (i.e., counting, mathematical relationships, definition, or decision).
Approximate numbers come from either rounding processes that shorten a more-exact value, tolerance specifications that directly state the range of approximation, or measurement processes that give numerical information about something but don’t repeat exactly.
Overall, this unit was seen by the instructor as being successful, although it revealed some of the difficulties that arise when there is an attempt to introduce new concepts and make extensive use of them on the same day. Even so, students were able to give a great variety of examples of approximate and exact numbers, including several sophisticated practical cases whose exact classification would depend on context. The topic thus succeeded in one of its main goals, which was to show the students that they, and their background of practical experience, had something to contribute to discussions in a math course.
The students were particularly comfortable with the idea of approximate numbers (perhaps indicating that many of them see all numbers from the continuum as fuzzy, which is a quite defensible stance for references to the physical world). This competence extended even to the somewhat sophisticated tolerance-specification numbers.
There was less certainty and agility in dealing with some forms of exact numbers, although counting numbers did not cause problems. There was resistance to the idea that a fractional number (such as 0.01 for a blood-alcohol limit) could be the result of a decision or definition process (and thus arbitrary to some extent), but still be exact. Student comments indicated that much of their discomfort here was with the idea that a numerical value used in a rule could be simultaneously arbitrary and exact; there was a tendency to deny one or the other, even if that led to giving more authority to a number than it merited (such as a 0.95 significance level, to mention an example from another sector). There is in fact more depth to this question than I first thought, and the student reactions may lead to a recasting of how it is presented.
The category of exact mathematical numbers (e.g., the square root of two) did not cause much student confusion, but that was because it was largely ignored. This area needs substantially more attention in the curriculum than it received in this initial version. An appropriate place to deal with it would be in the trigonometry module, where the trig functions provide plenty of examples.
Design
intention:
Prior to dealing with the complexities of the normal random-error variation of measurement processes, take advantage of the rule-driven nature of the rounding process to introduce and illustrate most of the ideas associated with approximate numbers. Also discuss the different possible rounding rules to introduce the idea that the mathematical rules applied to a situation are often chosen by practitioners based on the needs of the situation rather than being ordained by mathematicians.
Introduce the idea that the uncertainties associated with rounded numbers can be increased when different rounded numbers are combined. The emphasis is on the size of the possible errors rather than on the likely error. This best-case/worst-case approach used also lays the groundwork for later perturbation analysis of measurement-based calculations.
Key
instructional messages:
Rounded numbers are limited to multiples of a rounding precision, and thus differ by a rounding error from the more-exact values from which they are formed.
The most generally useful rounding method rounds numbers to the nearest of the neighboring rounded values (usually an unbiased process), but values may be systematically rounded preferentially in one direction in situations where the costs of overestimates are very different from the costs of underestimates.
The accumulation of rounding error can introduce significant uncertainty into totals or other computations made with rounded quantities.
The range of possible variation of the result of a computation using rounded numbers whose precision is known can be computed by computing the results for extreme cases. It is often larger than the variation of either of the numbers.
Retrospective
critique:
The materials on rounding methods represent the extreme among the materials reported in this thesis of a detailed didactic approach – effort was made to avoid it in subsequent modules. While the essential material about rounding precision and rounding error was mastered by the students, the detailed discussion of different methods of rounding and of tie-breaking rules left most of them confused. The intention had been to take a topic (rounding) with which students were already familiar and subject it to a multifaceted theoretical analysis and description, showing how a familiar concept can be generalized and extended. This failed because it did not make it clear that the extensions to other rounding methods was of secondary importance compared to the core ideas of rounding precision and error.
It is planned to completely recast the module, and follow a quick introduction of nearest-neighbor rounding by examination of the consequences of rounding in a variety of particular situations. A few of these (not in the first group) will be situations with asymmetric error-consequence costs (such as balancing a checkbook), which will show why an asymmetric rounding method might be useful. This will be included in a “variations on the basic method” section along with some other subtle effects such as the even-integer tie-breaking rule and the prudence of deferring any rounding until the final answer when possible. These variations would be presented as having evolved from practical considerations rather than as being mathematical rules.
In contrast to the excessively didactic way in which the discussion of rounding methods was introduced, the topic of error accumulation was introduced without preamble through the homework, setting up a more constructive student experience. This was generally successful, although it would have been better to start with some simpler problems that each posed only a single question, and to deal extensively with nearest-neighbor rounding prior to any problems using the variant techniques. The homework also inadvertently included one subquestion (about how changing to nearest-neighbor rounding would change the expected error of the total) that was poorly posed and could not be solved with the techniques yet covered. But well-designed expansion of the homework problem set, and better integration of the homework problems with topic strands spread over several days, is planned as a central tactic in the improvement of this module.
Design
intention:
The discussion of scientific notation is designed to show that the motivation of scientific notation is to make it easier to work with numbers that are very big or very small, and to demystify the exponential notation that students will encounter on their calculators. The transition into engineering notation (powers of 1000 rather than of 10) and the corresponding engineering-unit conversions, also related to their calculators, is both useful information in its own right and provides a context for demonstrating (mainly by example in the problems) that the different forms of representation are each convenient in different situations and can be interconverted as needed.
Key
instructional messages:
Scientific notation avoids all insignificant zeros by expressing a number as a mantissa between 1 and 10, multiplied times an exponential term that is an integer power of 10.
Numbers less than one are expressed with negative powers of 10, which are the same as dividing by the corresponding positive power.
Engineering notation uses powers of 1000 rather than powers of ten, with each step associated with a corresponding metric prefix that can be combined with engineering units such as meters to permit expression of quantities of all normally-encountered sizes in a way that avoids both insignificant zeros and any exponential terms.
Retrospective
critique:
This module would benefit from being broken up, with the scientific-notation section presented as review material earlier in the course and followed by some practice in exponential-notation conversions (with special attention to how such numbers are handled on calculators). The discussion of engineering units and metric prefixes is also basically review material, although in this case the opportunity should be taken to build bridges from math terms to practical use (by collecting actual references to prefix/unit combinations, for example) and to generalize the process for conversion by reminding that only the prefixes need be examined to determine the conversion factor. A touch of mathematical generalization can be added by posing playful questions such as “how many kilozaps in 2.3 gigazaps?”, which (by forcing the question to be addressed without the distracting connotations that real units sometimes evoke) might actually be easier to answer than a more practical one.
The real point of this module for the measurement theme is to ensure that students can use the ways that have been developed to easily express approximate-number quantities without either leading or trailing insignificant zeros. Note that while this task is handled with scientific notation in the relatively few sciences that need to express a very wide range of numbers, the great majority of approximate-number references are in engineering or practical situations, which almost always use engineering units rather than any form of exponential notation.
Design
intention:
Connect the existing student competence of rounding with the theme of measurement by introducing the problem of stating a rounded number that conveys most of the information implied by a set of slightly-varying measurements of the same underlying value.
Key
instructional messages:
Keep the “significant” digits that are common to all measurements, drop the “guesswork” digits that vary randomly. Note that an ending zero can be significant.
For maximum accuracy, retain a digit that varies only over a part of the range, but mark it as transitional (typically by using parentheses).
A number written to represent a measurement value includes, by the number of significant digits it contains, an assertion about the stability of the measurement process that produced it.
A variety of methods are used to describe the stability of a measurement process, and the one chosen for a particular case should be matched to the intended audience and to the use to be made of the measurement.
Retrospective
critique:
This module was reported as working pretty well. The only area in which significant student concerns were expressed was about the reason for recording the transitional digit. This should be discussed more fully, and revisited during the subsequent discussions of error accumulation and propagation that show its benefit.
Some actual demonstrations of the production of measurement-data sets would be beneficial here to validate the consistency with reality of the printed measurement lists that will be used in problems. For efficiency and clarity, these measurements should be done by the instructor or an already-skilled student, however.
The idea that measurement stability can be stated in a variety of ways (no one of which is intrinsically superior in all cases, including standard deviation) is a good candidate for the tactic of spreading out mention of an idea over several modules, with multiple descriptions given for the various measurement sets that are examined for other reasons. This topic would also benefit from many more supporting exercises, which should include both descriptions of situations for which the students are to choose appropriate stability-description methods, and assignments to find practical instances of the various methods (or of ones not mentioned).
Many of the strongest conclusions were those that have already been identified as general concerns. The material on approximation needs to be spread out over the semester and to be integrated more closely with the other topics. Any needed review materials should be covered prior to depending on their concepts or techniques in the introduction of new material. A greater number of well-crafted problems and examples are needed, combined where possible with diagrams and other features to avoid complete dependence on careful reading of text. Most of these conclusions simply reflect the untidiness of the development process, especially when the materials are being written by a novice at that task, even if well-advised by an experienced instructor.
The question of greatest interest to the course developers was how the students would relate to materials developed (however imperfectly) to reflect our theoretical analysis of the learning situation of practical-arts students. In general, the students justified our expectation that they would be able to display sophisticated practical thinking when given an opportunity to relate it to the content of a math course. There were certainly enough interchanges of this kind to provide us with feedback to improve the course, and to refine our ideas of the attitudes and capabilities of practical-arts students.
The initial section of the course was conducted with 15 students. The section offered the following semester filled to its class limit during registration, and was conducted with 29 students.
Determination of all the information about a triangle from an appropriate subset of that information is both an important practical application and an opportunity to introduce several mathematical topics (trigonometric functions, inverse functions, mathematical proof) that can substantially extend mathematical sophistication.
Direct measurement is used as an initial tool in calculating the values of trigonometric ratios, grounding their definition in experience (and incidentally demonstrating the lack of exact repeatability of measurement processes), and validating the obscure-origin but precise values that calculators give for direct and inverse trigonometric ratios.
The primary emphasis is on right triangles, culminating in the demonstration (via examples rather than proof) of the sin2+cos2=1 “Pythagorean Identity”, which is shown to imply the a2+b2=c2 Pythagorean Theorem.
Investigations of the method of determining the area of triangles are used to extend attention to general triangles, with partition by an altitude being shown as a technique that both shows how area can be computed and can be used to connect angle and side measures in a general triangle.
The derivations of the formula for triangle area are also used as the basis for a discussion of the idea of a mathematical proof, which is then further illustrated with an area-based proof of the Pythagorean theorem.
Except for the definitions, the Pythagorean Theorem, and the simple tan=sin/cos and sin(A)=cos(90º-A) identities, the MFM course avoids reference to trigonometric identities and generally minimizes the algebraic aspects of trigonometry.
The course also avoids reliance on memorization of the function values for the special-case 30º/60º/90º and 45º/45º/90º triangles. It is felt that these cases distract students from the general techniques, and are now (due to calculators) not appreciably faster to use. However, this position is being reexamined (but not abandoned), as is discussed in the “conclusions” section for this module group.
The trigonometric material shown in this report was used as the primary printed instructional materials for about 20% of the class in the Spring 2002 section, starting at the beginning of the semester. (The immediately-following 20% of the course was supported by the author’s measurement-theory materials discussed in a later section of this report. The rest of the course was supported by textbooks and other materials chosen by the instructor and used without close interaction with the thesis author.)
Design
intention:
The goal is to both build familiarity and facility with the sine, cosine, and tangent functions, and to validate against direct experience the trig-function values produced by calculators. A secondary goal is to provide a classwide experience with physical measurement (especially with the lack of exact repeatability of measurements of what is clearly an unchanging object) that will be referred to later in the course.
One important aspect of the direct-measurement efforts is the validation that the ratio-of-sides definition of trigonometric ratios gives the same result for the same angle, even for right triangles of different size. This is an experience-based fact, due to the flatness (or at least the developability) of the surface in which the triangles are imbedded, not a logical necessity.
Key
instructional messages:
Definition of sine, cosine, and tangent functions for right triangles.
On flat surfaces, trigonometric ratios depend on angle size but not on triangle size.
Similar right triangles can be used to solve problems by using measurements of the more accessible triangle to infer a missing measurement from the less accessible one.
Retrospective
critique:
While this module was generally successful, the student reactions indicate that it would benefit from restructuring in two ways:
The definition of the trigonometric ratios needs to be preceded by a separate review discussion of the properties of similar triangles, with an accompanying set of simple proportionality problems set in that context. The critical need here is to separate these ideas, which are general in their application, from the new concepts about right triangles.
Direct measurements of triangle sides and angles by students, although valuable, need to be usually restricted to situations where the approximate nature of the results is part of the point of the exercise, rather than being used as the leading validation method for mathematical assertions that are in fact exact.
In this case, use of given (but physically correct) side lengths and/or angles for the triangles would convey the needed exactness but still support the use of physical measurement as a check method. This will permit students to concentrate on trig-ratio definitions, while still laying a foundation for later discussion of how accurate such measurements are, and the extent to which measurements can actually prove exact relationships.
The module should also provide more examples in which angle values are given, since one of the main points to be mastered is that the trigonometric ratios can be considered as functions of a single angle rather than as functions of two right-triangle sides. This point will be more easily grasped if student familiarity with similar triangles, and the proportionality of their sides, has already been reestablished.
Design
intention:
The main point is to establish that the size of an angle in a right triangle can be inferred from one of its trigonometric ratios, and that while the relationship is not one of simple proportion, it is regular and can be computed by using the “inverse” trigonometric functions supplied by scientific calculators.
Because of the dependence on calculators, special attention is given to the mechanics of use of inverse functions on a calculator and to the various modes of expressing angles. Degrees are used to express angle size throughout the MFM course.
Key
instructional messages:
Right triangles with the same ratios have the same angles.
A change in the size of an angle will not generally change the trig ratios of that angle by the same proportion.
Calculators can be used to compute the size of an angle from any of its trigonometric ratios.
Retrospective
critique:
The main change in this module indicated by student response is the deletion of the initial section on the effects of halving an angle. The significant point, which is that changes in angle size do not result in proportional changes in the sizes of the trigonometric functions of those angles, can be better conveyed in the context of the tabular listing of trig-function values used to explore the space of potential right-triangle angular values. The focus on the special case of halving the angle is distracting, especially since it is at the beginning of the module.
The section on how to use calculators to compute inverse trig functions went well, although it needs to be adapted somewhat to mention the syntax of graphing calculators as well as that of the simple scientific ones. Because students have a familiarity and comfort with angle size, there was very little confusion about what was being done when an angle was computed from a trig ratio, in spite of the verbal and mechanical complications involved in dealing with inverse trigonometric functions. This is a rare case where the destination in a math course is more familiar than the starting point.
Design
intention:
This module is intended to provide some mathematical generalization by showing the simple identities that follow from the definitions of the trigonometric ratios. The excursion into theory is then promptly rewarded by demonstration (via systematic examples) of the Pythagorean Identity, from which the Pythagorean Theorem is derived by substitution of the definitions and simple algebraic manipulation of the resulting equation. The practical usefulness of that result is emphasized by making it the center of the homework problems assigned for the module.
Key
instructional messages:
The trigonometric ratios of the angles of a right triangle are related in simple ways to the ratios for the other acute angle in the same triangle.
For any angle, the sum of the squared values of the sine and cosine ratios is exactly 1.
The sum of the squares of the lengths of the shorter sides of a right triangle is equal to the square of the length of the hypotenuse.
Retrospective
critique:
This module went very smoothly in class. The balanced progression from manipulation of the definitions, to numerical exploration, to algebraic conversion, and finally to a clearly-useful result is just the kind of mixture of theory and experience and practice that the MFM course is aimed at, and in this case it worked very well.
Design
intention:
The main goal of this module is to provide an example of a mathematical proof of a non-obvious but clearly useful result, the formula for the area of a general triangle. In this context, such a proof can not be based on extended axiomatic development, of course, but this is not felt to be much of a loss in any case, since any proof must depend on some assumptions and in this case these can be better based in geometric intuition than remote theory.
The progression is from the area of squares, to the area of rectangles, to the area of right triangles, and finally to the area of a general triangle partitioned into two right triangles. It is intended as an illustration of the principle that logic consists of answering the question: “If what you already believe is true, what else has to be true as well?” (cf. Becker, 1998). This particular derivation is straightforward – the instructional idea is to have the students fully accept its argument before it is pointed out that they have painlessly followed a mathematical proof.
The essential elements of the mathematical nature of the proof are that it does not depend on direct measurement or particular examples in any way – it is a logical rather than an empirical argument. It is pointed out that a mathematical proof does depend on the validity of its assumptions, however, and that all these particular area proofs are valid only for flat surfaces, for example. The overall point is that both empirical and logical argument methods have strengths and weaknesses, indicating that they can fruitfully be used to check each other.