Aids to Develop Formal Operational Thought Chester VanderZee Sioux Valley Schools, Volga, SD 57071 A survey of college chemistry instructors in New Mexico reeardine incomine freshmen showed that what thev considered most important was that "high school teachers produce students that ran think" ( 1 ). I'ersonallv. I believe that the one thing that a school can d o better than most social structures is to provide an unique opportunity for a person to learn how to think. Too many people in this TV age have the mistaken notion that being conscious and thinking are synonymous! Incidentally, one does not have to invoke the name Piaget to be doing useful work in this area-any activity that attempts to connect the abstract with the concrete & that fosters systematic, sequential thought processes is helpful. If i t is granted that learning to think is the crux of the matter, maybe we should review briefly how this takes place. Language is a system of symbols, for the most part abstract.. . i.e., the sounds of the words or their appearance in print do not resemble the objects they designate. No eat lwks like a e-a-t nor sounds like kuh o t . . . In other words, to arrive at the meaning of a concept, one needs images of particulars that have properties common to the class.. . Learning to read and speak means connecting sounds or marks on paper with images of particular things and actions and relations. The images may come from actual sensory experience or by combining images in the mind, i.e., hy the exercise of the imagination. From the day of birth the mind is heing stocked with images. . . It is this imagic-conceptual store that is activated whenever we read or speak or listen to speech. We comprehend with these resources (2). T h e overworked phrase "hands on" implies that to experience is to learn. "But concrete exoeriences are not ~ a r t i c u larly useful if all the student does is touch, smell, Hee, and manipulate without being forced to think about what he is doing" (3).It is of course true that "we always need to describe and order experience (concrete reasoning) before we explain experience (formal reasoning) (4). Forcing oneself to write down each step in a la11rrport t ~ computational r exercise is a descrit~ingand ordering anivity. Jack Imkhead has found it helpful to require "detailed verbalizations of' each step in the thought process." One of his students expressed the prohlem in this way: "They should have a course to teorh you how to learn-here, all they have is courses on what to learn" (5). Exacerbating this is our tendency as science educators to omit the simoler thines and eet on to the fancier thines. As Patricia A. ~ o r ~ says, a n ''wlhat ever happened to t h e idea that science means the intellieent observation of ohenomena? . . . We may now have a generation of students who substitute meter readings for ~ e r c e ~ t i o and n . who exchange ability to solve sophiHticated problemifor ability to deciae what the problem is" (6). Some of your students will he frustrated while working the exercises provided below. However, ". . .disquiet with the apoarent facts will be an encouraging sign . . . it will not be th; experience itself that gives rise to understanding, but the person's own attempt to coordinate the knowns and unknowns of the occurrence with previous knowns, thus leading to new organization in reasoning" (7). The following are examples of exercises that I have used with my students to help them "learn how to t h i n k
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Using a metric ruler, measure the diameter (d) and circumference (c) of two Petri dishes of different sizes and of the top of alab stwl. Make a table of the data. Then calculate these four quantities far each set of measurements: d c, c - d, e X d, c t d. Enter these calculations in the table. Can you deduce anything which you believe is true for all circles? If so, write a relation (formula)which is true for all circles, based on your work above. Be sure to show all data and work in an orderly manner on your paper (adapted from ref. ( 5 ) ) . In a certain African village there live 800 women. Three percent of them are wearing one earring. Of the other 97 percent, half are wearing two earrings, half are wearing none. How many earrings all together are heing worn by the women? Show/explain how you arrive at your conclusion (8). Smith gave a hotel clerk $15 for his room. When the clerkdiscovered that he had overcharged by $5, he sent a bellboy to Smith's room with five ane-dollar bills. The dishonest bellboy gave three to Smith, keeping two for himself. Smith has now paid $12. The bellboy has acquired $2. This accounts for $14. Where is the missing dollar? ShowJexplain how you arrive at your conclusion (8). Ten full crates of walnuts weigh 410 pounds. An empty crate weighs 10 pounds. How much do the walnuts themselves weigh? Show/ explain how you arrive at your conclusion (9). Without using the proportion method, explain the two steps one uses to solve this problem: If three-fourteenths of a field can he plowed in six hours, how long will it take to plow the entire field? 'You wish ta measure out 7 qts. of Liquid from a large vat. You have a 5-qt. container and a 3-qt. container. The containers are irregular in shape with no volume marks on them. Using only these two containen and the vat of liquid, how are you going to measure out the 7 qts. needed? If allsynthetic fibers (nylon and polyesters)were replaced by cottan, it would take 40 million acres-nearly the total arable land in Mississippi, Arkansas, Louisiana, Alabama, and Texas. Suhstituting wool for synthetics would demand a billion acres of grazing land, equal to all the agricultural land in the US. How many times greater is the acreage needed for all our clothes to be made from wwl compared to the acres needed if d l our clothes were made from cotton? A number of ewelfs . . at the bottom of apit. There were half - are lving a$many wells yntrrdny,and tomorrow there will he twirem many sinrewelfsdouhlerverydny. At thkrare, the pit will be full in60 days. In how many days will it be halifull?
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In summary, to foster thinking one must work t o combine orecise descriotions of accurate observations with an attempt to synthesizemeaning-significance4 of the experience. Once attention to detail is begun while wbrking with the simpler everyday kind of examples, a pemon is more likely t ( ~ d e v e l o ~the selfhnfidence and satisfaction that acrompwies success in doing abstract thinking.
Literature Cited (1) Stuart, TrsevC..J. C ~ MEouc., . 54.373 (1977). (2) Broudy, Harry %Phi Delta Koppon,348 (Jan.1979). (31 Herran, J. D.,J. CHEM. Eouc.,52,150 (1975). (41 Herron, J. D., J. CHEM. EDUC.,55,166 (19781. (51 Laekhead, Jack. "Teaching Studen* How to ham: Physics Dept., University of Massachusetts. (61 Morgan, PaVvia A.. Chrmbfry. 2 (Nov. 19781. (71 Roth. Jane B.."Pra&iesiApplications of Rew~ewh,"PDK.2 (Ds.1978). (8) Gardnu. Martin,Reoder'* Digest, I39 (Jan. 1978). (91 Whimbey, Arthur, "Teaching SequentiaiThought . .. CPDK, 255 (Ds.1977).
