resource materials for high school and college teachers— an illustration in calculus and analytic...

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Resource Materials for High Schooland College Teachers�an Illustration in Calculusand Analytic Geometry

Nira Hativa

The development of resource mate-rials designed to facilitate teacherlesson preparation and to enrichclass presentation while saving timeand effort is illustrated here by aspecific set of experimental mate-rials�a handbook for teachers ofcalculus and analytic geometry. Al-though these materials were writtenwith the aim of improving the in-struction of teaching assistants in auniversity, the approach is suitablefor teachers in other school levels (inparticular for the upper high schooland college level), and for other sub-ject matters (in particular mathe-matics and the natural sciences).

The Rationale for Preparing Resource Materials for High School andCollege Teachers

There are two major objectives for preparing the described resource ma-terials:

School Science and MathematicsVolume 85(2) February 1985

Resource Materials 137

(a) to facilitate teacher work in lesson preparation while increasing lessonquality and effectiveness, and

(b) to preserve and disseminate material related to pedagogical issues of pre-sentations of specific topics.

The following is an elaboration of these two points:(a) Research shows that students perceive thorough lesson preparation

as one of the most important qualities of their teachers. For example,studies that investigate college students’ perceptions of good teaching(e.g., [I], [2]) consistently show that lesson preparation significantly ac-counts for students’ satisfaction from the course and for instructor’s rat-ings on overall performance. But preparing effective, rich presentationsis very time consuming. To achieve this aim teachers need to read theclass textbook, to rely on their own teaching experience, to read other re-lated books or journal articles, to prepare beneficial handouts, to designthe use of visual aids, to identify some related practical applications, andto plan the incorporation into the presentation of teaching strategies thatmake the material attractive, clear, and relevant to students.Since preparing all these materials and looking through all kinds of

sources require extensive time and effort, the burden of practically doingthe more time-consuming preparations and literature search should betaken from the teacher. I suggest that commercial publishers or teams invarious interested departments cooperate in preparing handy resourcematerials, articles, and teaching aids to enable teachers to prepare in arelatively short time good effective presentations.

Indeed, this type of resource materials has been prepared by commer-cial publishers for the elementary school level. In mathematics, for ex-ample, many methodology books and teaching aids for the elementaryschool teacher are available: materials to make abstract concepts con-crete (e.g., materials for math labs); master sheets with problems andtests easily reproduced for all students in class; transparencies with layersready to be used on an overhead projector, etc. But, the higher theschool level, the fewer the commercial teaching aids that are available.For the high school and college level there is a scarcity particularly inmethodology books. While there are still some methodology books thatprovide ideas for presenting specific topics at the high school level (e.g.,[3] through [8]), very few of these discuss pedagogical issues related tospecific topics such as: creative ideas for presenting the topic, ways foradjusting the presentation to students’ different learning patterns and


138 Resource Materials

abilities, suggestions for overcoming students’ problems in mastering thematerial, enrichment materials that enhance students’ learning, etc.Moreover, for the college level, such commercial material almost doesnot exist. While materials related to some of the pedagogical issues listedabove may be found in "teaching tips" or in other articles published inprofessional journals (e.g., in [9] through [13].), they are spread outamong many journals. The regular teacher, though perhaps interested inenrichment materials for a specific topic, cannot afford the complicatedand time consuming search through the many various sources for iden-tifying and sorting related articles.

(b) There are many wonderful high school and college teachers whoprovide superb lessons to their students, who use creative ideas in pre-senting specific topics, and who prepare excellent handouts and otherteaching aids for their classes. But teaching is an isolated profession.Usually, teachers do not convey their creative ideas nor do they dissem-inate their beneficial teaching aids to other teachers. Only a very smallportion of these ideas and materials is presented at conventions or is sentto be published in a professional journal. Thus, the majority of originalcreative teaching ideas and materials is not transmitted to new and in-experienced teachers and is being lost when the instructor stops teachingthat topic. In comparison with research, the other task that some collegeand university faculty are engaged with, new research is always basedupon a foundation laid by other researchers, and upon knowledge ac-cumulated from previous studies. Since communication of new findingsis essential for scientific progress, there is an extensive effort to dissem-inate knowledge accumulated in research. This is done at professionalmeetings and in all types of publications. In contrast then, the dissem-ination of knowledge related to pedagogical issues is indeed very minor.To remedy this lack of information transference, it is suggested here

that resource materials be made available to teachers to facilitate lessonpreparation, enhance topic presentation and student learning, and pre-serve teaching ideas and teaching aids, thus promoting their dissem-ination.One model of resource materials designed to achieve these objectives is

described below. These materials were developed in 1982 for a short, in-tensive (four two-hour sessions) program of training teaching assistants(TAs) in the Department of Mathematics at Stanford University. TheTAs to be trained taught in all undergraduate calculus and analyticgeometry courses designed for non-mathematics and non-science majors.



The materials were prepared by a team consisting of the trainer�amathematics educator�and a few TAs of that department, chosen onthe basis of long experience in teaching these courses and of high eval-uation on teaching effectiveness by their students.

A Handbook for Teachers: an Illustration in Calculus and AnalyticGeometry

Preparation of the handbook consisted of the writing and of the collect-ing of various resources related to the teaching of specific topics in cal-culus and analytic geometry. The handbook is organized so that for eachtopic to be presented (or for each section of the textbook) there are someteaching aids or teaching ideas sorted into five categories. The followingis the description of these categories with an example from the handbookfor each.

CATEGORY I. "Suggestions" for teaching the course topics. Thesesuggestions were written by the TAs on the team who conveyed to thehandbook users their own experiences with teaching the materials. The"suggestions" include teaching tips such as: how to connect new mate-rial with past and future topics; how to apply the material; how to solveproblems that undergraduate students face in learning specific topics,and how to approach these problems. As an illustration, the following isthe introduction to a set of theorems to be taught consecutively. Theseare: Continuity, Rolle’s Theorem, Mean Value Theorem, FundamentalTheorem of Calculus, and Differentials. The handbook also includes adetailed discussion for each of the theorems separately but this discus-sion is not included in this article.

Introduction to the Set of Theorems

This two week section returns to theorems that were skipped earlier in thecourse. It is almost universal that students in courses (who are non-mathematics and nonscience majors) dislike theorems, so we suggestsome deep thinking and discussion with fellow TAs and faculty coordi-nator about how to approach this area. The following are a few sug-gestions:

1. The value of presenting a "batch" of theorems is that you can empha-size the connections between them: how each new theorem is just a small step



above the preceding one. You can tell your students that mathematicians arevery fond of such "elegant" reasoning, and that it can be very powerful.

2. Remember, your students are not mathematicians and therefore youshould keep things simple. The point you should work hardest on is gettingyour students to understand the statements of these theorems. In addition,because of the esoteric nature of theorems, it is more important than ever todemonstrate concrete examples.

3. Because the students worked around these theorems in the previouscourse in the sequence of calculus, they have already used many of them im-plicitly. Take advantage of this by using examples which make clear this con-nection, and by assigning homework from different sections of the textbookalready covered that relate to the theorems discussed.

4. Prepare a good answer for the time when your students ask, "Why dowe have to study these theorems?" One way to answer is as follows: Thelogical method behind these theorems is typical of mathematics. Stanford is aliberal arts college, not just a technical school. In addition to the concreteskills students pick up as undergraduates, they are supposed to come to ap-preciate an understanding of what principles guide the different disciplines inthe arts and sciences. In other words, these theorems illustrate the "culture"of mathematics.

5. When you talk about theorems in class you may want some way of test-ing your students on their understanding: true-false questions, (e.g., "If f iscontinuous on (a,b) it attains its maximum"), or exercises like: "Draw a pic-ture of a continuous function that^s not differentiable."

CATEGORY II. Short journal articles related to the course topics. Afew journals ([9] through [13]) and books ([14], [15]) were searched forarticles that include teaching tips, enrichment material, anecdotes, or his-torical notes�all related to calculus and analytic geometry. For example,the journal articles [16] through [22] deal with the topic "Applications tomaxima and minima." In the handbook, these articles are reproduced infull (with compliance to copyright laws).

CATEGORY III. Practical applications to the course topics. All the ab-stracts of UMAP (see Note) related to calculus and analytic geometrywere reproduced and sorted into topics. For example, for the topic"Applications to maxima and minima," the abstracts of the followingapplications have been printed in the handbook: "The Design of Honey-combs" (application to biology, Module 502); "Listening to theEarth: Controlled Source Seismology" (Application to earth sciences,physics, and oceanography, Module 292), "The Human Cough" (appli-cation to physics, biological and medical sciences. Module 211), and



"Five Applications of Max-Min Theory from Calculus" (application toeconomics, physics, and biology. Module 341). Although teachers do notusually have enough time to discuss these applications in class, they maymention the applications for motivational reasons and refer interestedstudents to the department library where they can find the completeoriginals.

CATEGORY IV. Handouts and tests related to the course material.Many handouts and tests that had been prepared by experienced TAs fortheir classes were collected and sorted into topics. As an illustration,Figure 1 presents a handout that belongs to the topic "Theorem Rela-tions." The theorems are those referred to in the illustration in CategoryI.

CATEGORY V. Pictures and transparencies related to the course mate-rial. Transparencies to be used on an overhead projector, pictures, andillustrations on paper ready for preparing transparencies were prepared.These illustrations provide aesthetic visual applications for the material.For example, most pictures printed in the journal article [23] which pro-vide fascinating illustrations of applications of conic sections to physicsand astronomy were arranged on two pages and transferred into twotransparency sheets. It takes only a few short minutes for the teacher todiscuss the application and at the same time to demonstrate with trans-parencies on an overhead projector the concepts involved, but it largelyimproves students’ understanding and appreciation of the material.

These are the five categories that make up the handbook for teachersof calculus and analytic geometry. A teacher preparing a topic for pre-sentation may look in the handbook under the name of that topic andfind there teaching tips, anecdotes, handouts, journal articles, sheets forpreparing transparencies that demonstrate practical applications, andabstracts of practical applications, all related to that specific topic. In ad-dition to materials related to specific topics, the handbook includes somegeneral materials for helping the TAs effectively prepare their tests andlessons. For example. Figure 2 presents the form that teachers are en-couraged to use for lesson preparation. It is designed to help incorporateinto class presentations effective teaching strategies identified in educa-tional research and especially in study [24].


Resource Materials

THEOREM RELATIONSCONTINUITY(1) A continuous Function on a closed interval attains its maximum and

minimum valuesMAX MAX ^MAX

(2) if f is continuous on [a,b] and if f(a) < L < f(b), then there is a point cin [a,b]at which f(c) = L.(INTERMEDIATE VALUE THEOREM)

^.(3) A continuous function on the closed interval [a,b] is uniformly contin-uous

DIFFERENTIABILITY* (1) A differentiable function is continuousy(2) At a local maximum or minimum the derivative of a differentiable func-

tion is zero

continuous on [a,b]

^ROLLE’S THEOREM: If f is continuous on [a,b] and differentiable on(a,b) and if f(a) = f(b) = 0 then there is a point c in (a,b) at whichf’(c) = 0

b-^

’MEAN VALUE THEOREM: If f is differentiable on [a,b] then there is apoint c in [a,b] at which f’(c) = [f(b) - f(a)]/(b - a)

C2 b

L’HOPITAL’S RULE: If the limit as x approaches a of f(x)/g(x) is inde-terminate, and if f and g are differentiable at a, then the value of thelimit is the same as the limit as x approaches a of f ’(x)/g ’(x)

1 INDEFINITE INTEGRALS are unique up to a constant

INDEFINITE INTEGRALS: definition as the limit of areas of rectangles

’FUNDAMENTAL THEOREM OF CALCULUS: If F is any indefinite in-tegral of f, then J" f(t)dt = F(b) - F(a)

FIGURE 1



A LESSON PLANINTRODUCTION1. Lesson outline or objectives: state/write on board/handout2. Place of topic in list of past/future topics of course3. Review of material related to the main topics4. Motivation/real life application for the main topics5. Assigning homework

MAIN TOPICS�REVIEW, MOTIVATION/APPLICATIONS1. _____________________________________2. _____________________________________3. _____________________________________

MAIN TOPICS SUBTOPICS (reviews, motivations, anecdotes, historical notes,graphs, interrelations, comparisons/contrasts, mnemonics, algorithmsexamples/illustrations)

3.

2.12.22.32.42.5

3.13.23.33.43.5

CONCLUSION:�Summary of points covered (looking backward on outline)�Summary and conclusions of content covered�Topics of next lesson

FIGURE 2

To conclude, it should be noted that the handbook described in thisarticle is experimental in form and has been developed for internal useonly. Other schools and departments are encouraged to develop similarresource materials for various subject matters which will suit their spe-cific curriculum, objectives, level and needs of students, etc. It is believedthat the availability of resource materials of the kinds suggested andillustrated in this article can greatly enhance teacher effectiveness andteacher training programs.

POST NOTEI. UMAP: Undergraduate Mathematics and its Applications Project provides modules

of applications of undergraduate mathematics to various subject matters.



References

1. Feldman, K. A. The Superior College Teacher from the Student View. Research inHigher Education, 5 (1976), 243-288.

2. Hativa, N. Good Teaching of Mathematics as perceived by Undergraduate Students.To appear in The Journal of Mathematical Education in Science and Technology.

3. Lehr, H. (Ed.) Learning of Mathematics, Its Theory and Practice, Twenty-first Year-book of NCTM, Reston, Va: NCTM, 1953.

4. Fletcher T. J. et al., (Eds). Some Lessons in Mathematics. New York: Cambridge Uni-versity Press, 1965.

5. Rosskopf, M. F. (Ed.) The Teaching of Secondary School Mathematics, Thirty-thirdYearbook of NCTM, Reston, Va, NCTM, 1970.

6. Butler, C. H., Wren, F. L., and Banks, J. H. The Teaching of Secondary Mathe-matics. New York: McGraw-Hill Book Co. 1970.

7. Banwell, C. S. et al. Starting Points for Teaching Mathematics in Middle and Second-ary Schools. London: Oxford University Press, 1972.

8. Bell, F. H. Teaching and Learning Mathematics in Secondary Schools. Dubuque,Iowa: William C. Brown, 1978.

9. American Mathematical Monthly.10. Mathematics Magazine.11. The Two-Year College Mathematics Journal.12. School Science and Mathematics.13. The Mathematics Teacher.14. Selected Papers on Precalculus; the MAA Publications.15. Selected Papers on Calculus; the MAA Publications.16. Amir-Moez, A. R. The derivative Doesn’t Always Work. School Science and Mathe-

matics, 72 (1972), 723-725.17. Austin, J. D. How Fast Can You Watch? The Mathematics Teacher, 75 (1980), 262-

263.18. Brazier, G. D. Calculus and Capitalism�Adam Smith Revisits the Classroom. The

Mathematics Teacher, 73 (1978), 65-67.19. Goldberg, K. P. Curve Stitching in an Elementary Calculus Course. The Mathematics

Teacher, 77(1976), 12-14.20. Ogilby, C. S. A Calculus Problem with Overtones in Related Fields. American Mathe-

matical Monthly, 65 (1958), 767-767.21. Bussey, W. H. Maximum Parcels Under the New Parcel Law. American Mathematical

Monthly, 20(1913), 58-59.22. Walsh, J. I. A Rigorous Treatment of the First Maximum Problem in the Calculus.

American Mathematical Monthly, 54, (1947), 35-36.23. Baravalle, H. V. Conic Sections in Relations to Physics and Astronomy. The Mathe-

matics Teacher, 65 (1970), 101-108.24. Hativa N. What Makes Mathematics Lessons Easy to Follow, Understand, and Re-

member. The Two Year College Mathematics Journal, 74(1983), 398-406.

Note: References 9 through 15 are incomplete. Oversea’s correspondence with authorfailed to elicit the information on time.

Nira HativaSchool of EducationSharet BuildingTel Aviv UniversityTel Aviv, Israel 69978


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