374 School Science and Mathematics

3 4 n = r2 = rs = 1, r4 = 2 and in + in + i = rn + 4

4 3 n = r2 = r3 = r4 = 1 and rn + rn+ i = rn+4[ "�J i �

4 4 rn = 2 l 4 J

Teachers and their students are invited to seek other sequences result-ing from further variations of the conditions of the original Fibonacciproblem.

A Unit Box For Secondary School Mathematics Teachers

David E. KullmanMiami University

Oxford, Ohio 45056

The "unit box approach" used by Mitchell E. Batoff’ in elementaryschool teacher preparation can also be a successful component of amethods course for secondary school mathematics teachers. This writerhas been using it with prospective high school mathematics teachers forseveral years under the name "resource unit."A resource unit is a collection of elements used in teaching a particular

mathematical topic. Putting such a unit together has proved to be an ex-cellent means of involving prospective teachers in the various activitiesassociated with mathematics teaching. Instead of merely reading andtalking about various methods and materials, the students roll up theirsleeves and join in the process of selecting instructional objectives, con-structing test items, review devices, audio-visual materials, and the like.By doing all of these things in the context of a single mathematical topic,they end up with a comprehensive assemblage of items which can actual-ly be used in teaching a course. They also get a foretaste of the carefulplanning and hard work which go into the development of a successfulinstructional program.The specific items which make up a resource unit include:

1. Content outline.2. List of objectives.3. Resource slips.4. Motivational activities.5. Sample teaching strategies.6. Models and other visual aids.7. Individualized activities.8. Drill and review activities.9. Test items.

10. Optional resources.

1. Batoff, Mitchell E. "A Description of and Progress Report on The Unit Box Approach: A Novel Facet of Elementary School Science and Mathematics Teacher Preparation." School Science and Mathematics, 74 (December, 1974), 667

679.

A Unit Box for Secondary Math Teachers 375

Early in the methods course students are given an outline of the itemsrequired for a resource unit, along with a list of suitable mathematicaltopics. Since these are prospective high school teachers, the topics arechosen from algebra, geometry, or general mathematics. The studentsmay suggest topics of their own, subject to the approval of the professor.No two students in any given term are allowed to work on the same topic.This encourages independent work and also lessens the competition forcertain library resources. The students are given a week to choose theirtopics, and they must have their content outlines and objectives formu-lated about three weeks later. This forces them to get an early start, andit gives the professor an opportunity to react to their initial thinking andmake suggestions for improvements. The entire unit is due at or near theend of the course.

Content Outline. The content outline for a resource unit is divided intotwo major sections. The first is a list of all topics and associated subtop-ics which are likely to be taught in the unit. The students consult severalcontemporary high school textbooks before preparing this outline. Thesetextbooks generally provide the "core" topics, but students may excludethose which they feel are inappropriate and add others which are notfound in traditional courses.A second major part of this outline consists of statements describing

the various mathematical concepts listed in the first part, along with im-portant relations between them. Such statements would include defini-tions, theorems, formulas, examples, and applications. The entire out-line is organized mathematically, with no consideration being given atthis time to strategies for teaching the material.

Objectives. The mathematical statements in the content outline do notspecify the nature or depth of learning that is to take place relative to agiven concept or principle. For this reason, a list of objectives or goals tobe achieved by the pupils comes next. This writer believes that behavioralobjectives have a place in mathematics education but that they are notadequate for specifying all the learning that we want to take place. Theprospective teachers preparing their resource units are urged to state cog-nitive objectives in behavioral terms when possible, but they are also in-structed to include some goals from the affective domain and higher lev-els of the cognitive domain which do not readily lend themselves to beha-vioral language.

Resource Slips. All teachers should try to keep abreast of professionaldevelopments in their field. The sheer volume of publications todaymakes this a formidable task. This is where a personalized file of re-source slips pays off. These resource slips (usually in the form of filecards) contain examples of and/or references to any of the following cat-egories of items:

376 School Science and Mathematics

Applications of mathematics.Historical notes.Mathematical anecdotes and stories.Enrichment problems and problem solving strategies.Derivations and proofs.Examples illustrating various concepts and principles.Computational techniques.Motivational activities.Math lab activities.Suggestions for student projects.Audio-visual materials and teaching aids.Teaching strategies.Other interesting ideas.

Each student in the methods course begins accumulating a file of suchresource slips, organized in whatever way he feels will be most useful tohim as a teacher. It is not uncommon for a student to have accumulatedfifty or more slips by the end of the course, and we hope that they willcontinually update their files throughout their teaching careers. Each re-source slip includes topical headings, indicating both the mathematicalcontent and the type of teaching/learning activity involved. When appro-priate, the source of the idea is also given so that more details can be re-covered later.

Motivational Activities. When mathematics teachers are asked to de-scribe the most serious problems that they face in the classroom, motiva-tion always is found at or near the top of the list. Since motivation seemsto be a universal problem, each resource unit must contain one or moremotivational activities w^hich can be used in connection with the topic inquestion. Typical motivators have included such things as: applicationsof mathematics; challenging problems; the history of a particular mathe-matical idea; anecdotes and personal stories related to mathematics; therelationship of mathematics to games, sports or hobbies; and recreation-al games, puzzles, or tricks.

Teaching Strategies. A teaching strategy is a sequence of moves used inteaching a given concept, skill or principle. It is a structured procedureleading to a specific objective. A strategy can be thought of as the kernelof a lesson plan. Each student is asked to develop at least two strategiesfor his resource unit. These strategies are to be tied to some objectivesfrom the list formulated earlier.

Teaching Aids. Mathematics teaching falls into the category of an art,and mathematics teachers should employ a great many props in stagingtheir productions. The list of potential teaching aids in mathematics is along one, including such diverse materials as textbooks, colored chalk,overhead transparencies, videotapes, computers, physical models, andeven the daily newspaper. W^hile some of these materials are best devel-oped by educational media specialists, many effective aids can be con-structed by classroom teachers using simple techniques and equipment.

A Unit Box for Secondary Math Teachers 377

Early in the term the students in the methods course visit the Educa-tional Media Department for a brief orientation to production methodsand techniques. Later they are shown examples of homemade modelsand audio-visual materials from previous classes as well as from the pro-fessor’s own collection. Each student is asked to devise at least one goodteaching aid that can be used in conjunction with his particular unit.

Individualized Activities. Individualization is widely discussed in edu-cational circles today, but the term has many different interpretations,not all of which are compatible. While some forms of individualized in-struction require special classroom facilities and management systems,there are many ways that a teacher can bring individualization to the reg-ular classroom in conjunction with other teaching patterns and strate-gies.As part of the resource unit, each student is asked to develop an indi-

vidualized activity that can be used within a conventional school settingand which requires a minimum of classroom reorganization or specialequipment. Such an activity might entail a laboratory experiment usingsimple low-cost materials; manipulative materials with accompanyingworksheets; task cards; or tape-recorded lessons, accompanied by suit-able worksheets. The students are asked to prepare the necessary spiritmasters and other materials, as though they were going to begin the ac-tivity tomorrow. In this way they are forced to think through the detailsvery carefully, and they also get the experience of actually making a spiritmaster and using a duplicating machine.

Drill and Review Activities. Ten years ago the whole idea of drill wasanathema to many mathematics educators. Today, however, we are re-turning to the point of view that drill and review are essential for thelearning and retention of basic mathematical skills. The biggest criticismof drill has always been its tedium. If we could only make it more inter-esting, it would please the teacher and pupil alike.Each student is asked to invent or compose a drill and review activity,

and they have really outdone themselves in showing creativity in thisarea. Perhaps their own memories of monotonous drill sessions havespurred them on. Many of these activities take the form of mathematicalgames. We have seen versions of Old Maid, Bingo, Tic-Tac-Toe, Con-centration, Rummy, Scrabble, and Monopoly�all with mathematicalquestions or problems as the basis for scoring. Other review devices havetaken the form of cross number puzzles, hidden sentence puzzles, dot-to-dot pictures, magic slates, and even good old fashioned flash cards.

Test Items. Evaluation is an essential feature of any educational pro-gram, and students in this course are required to devise test items orother means of evaluating pupils’ progress toward the objectives of theunit. It is suggested that the test items be organized by topic and placed

378 School Science and Mathematics

on file cards so that they can easily be organized into a unit test when theoccasion arises. Students are also encouraged to use a variety of test itemformats, including multiple .choice, matching, free-response, and prob-lem solving. It is more difficult to evaluate the non-cognitive objectives,but some attempt can be made to develop attitude rating scales, interestinventories, etc.

Optional Resources. The items described above constitute the mini-mum requirements for a resource unit. Optional items may take the formof extra motivational devices, audio-visual materials, and mathematicalgames. Also included in this category are posters, bulletin board dis-plays, plans for a field trip, suggestions for student projects, and supple-mentary reference materials for the teacher.

All of the above items are neatly organized, labeled, and placed in ap-propriate folders, envelopes, and boxes. These, in turn, are placed in onelarge box (whence the name "unit box"). Figures 1 and 2 show the col-lected resources of two students. One of these units deals with fractionsand rational numbers, while the other is a unit on area and volume.

In evaluating the methods course, students have consistently rated theresource unit as being very worthwhile, if somewhat time consuming.Accordingly, the unit has been weighted more heavily (up to 40%) in de-termining grades for the course, and other homework assignments havebeen decreased to allow more time to be spent on this project. The coursesyllabus has also been arranged so that topics such as objectives, teachingstrategies, teaching aids, and testing are treated in the same order as theyappear in the resource unit outline. At the students’ request, the last fewclass periods are spent sharing ideas from the various units. The unitbox, which started out as a rather modest assignment, has now becomethe core of this methods course.

DESULFURING COAL

A bacterial process for sulfur removal can render Ohio’s high-sulfur coalenvironmentally acceptable for coal-burning power plants, according to OhioState University scientists. The process utilizes certain bacteria which derive theirenergy by oxidizing the sulfur and iron in coal."We reduced the sulfur content of coal samples from about 5 per cent down to

around 2 per cent in nine days’ time," said Prof. Patrick R. Dugan, a micro-biologist. "We hope to get the sulfur content down to 1 per cent, but even a 2 percent sulfur coal can be blended with a lower-sulfur coal for an acceptablemixture," he said.Dugan declined to project the cost of large-scale removal based on. his

laboratory tests, saying "our approach is to let industry make the calculations.""Some industries are doing this," he said, "and one projection sees this processas less expensive than installing scrubbers or other devices for removing sulfurfrom stack emissions after combustion."