Formal Reasoning and School Mathematics

640

Formal Reasoning and SchoolMathematics

A. Dean Hendrickson

The work of Jean Piaget first drew theattention of educators involved withearly childhood education and elemen-tary education. Several groups attempt-ed to utilize the stage development theo-ry of Piaget in designing curriculum.These products emphasized classifica-tion, seriation and the establishment ofrelationships. Some of the better knownexamples of these curricula include theScience Curriculum ImprovementStudy, the Elementary Science Study,the High Scope Early Childhood pro-gram, and the Nuffield projects in En-

gland. Few secondary teachers saw Piaget’s work as relevant to their instructionand most assumed their students capable of abstract thinking.

Inhelder and Piaget (1958) made the first major contribution to focusing at-tention on formal operational reasoning patterns. Soon after that, several re-searchers in the Geneva school, in England, and a few in North America beganto examine the extent to which students age twelve and up really were able to useproportional reasoning, combinatorial reasoning, it-then reasoning and analogyin thinking, and to manipulate and control the effects of variables. Some of theseinvestigators found surprisingly high numbers of students in secondary schoolsand in colleges who were still primarily concrete operational in their perform-ance on a variety of tasks that require formal operational thought processes.Many of these results indicated that a full use of formal operational reasoningdevelops at a much later age, if at all. Some of these investigators include Lovell(1969. 1971), Karplus et. al. (1970, 1974, 1977, 1978, 1979), Arlin (1976) andRenner et. al. (1976). Renner and Stafford (1976) concluded that three fourths ofthe students in secondary schools are primarily concrete operational in theirthinking, and that instruction in science would have to be modified to take thisinto account. They also found 73% of the high school sophomores tested to beincapable of the abstract thinking required in the typical geometry course andrecommended that the development of reasoning patterns be an important objec-tive for students thirteen years of age or older. They concluded these studentsneeded active participation in hands on learning in mathematics and science and

School Science and MathematicsVolume 86 (8) December 1986

Reasoning and Mathematics 641

that concrete experiences should be used to introduce all new topics. McKinnon(1970) found that over fifty percent of the college students, in a sample that wastaken from seven colleges of different kinds and sizes, gave concrete operationalresponses to tasks involving proportional combinatorial reasoning. He then de-signed a course in science that gives specific experience with these processes andfound it was possible to accelerate some students into formal thinking.

Students who fail at reasoning tasks, although assumed bright and capable ofthis type of thinking, may follow a normal course of intellectual developmentslower than that described by Piaget, or they may lack appropriate experiences inthe curriculum, or they may combine these factors. A curriculum K-12 that em-phasizes development of these reasoning processes as opposed to acquisition ofcontent and skills has never been developed and tried, either in mathematics orscience.

In response to these results, in 1977, the Lawrence Hall of Science facultyspearheaded the development of a series of five workshops for secondary teach-ers and college teachers of general science, earth science, biology, chemistry andphysics. In these workshops, participants learned some Piagetian theory andwhere, in these disciplines, formal operations were needed. Ways were suggestedto introduce topics in such a way as to lead into formal thinking with early con-crete experiences. The learning cycle�exploration, concept introduction and de-velopment, concept application�was suggested as a model for this. Many of thepersons who developed these workshops contributed to the Lochhead and Cle-ment (1979) summary of research findings on thinking skills. No comparableworkshops have ever been developed for secondary and college mathematicsteachers.

This paper contains examples of it-then and combinatorial reasoning situa-tions that have proved successful with college students and that can be used withsecondary students.

Logical reasoning can be introduced by activities with attribute materials.Those described here were developed by the Elementary Science Study and areone of the units in that program. They consist of 32 blocks with 4 values ofshape, 4 values of color and 2 values of size. First, similarity and differencegames should be played to familiarize the students with the materials, and the at-tributes and values of attributes present in them. Then these materials can beused to emphasize the meaning of and, or, not, and/or, if-then, if, only if, some,and all. Since 75% of the college students in classes taught by this author do notunderstand the meaning of these terms, it is safe to assume that most high schoolstudents don’t either. Many of these college students have had three years of highschool science and mathematics.Here is one example of an activity to accomplish the development of meaning

for these terms:


642Reasoning and Mathematics

Ask the students to loop a cord and collect inside the loop all squares in their set ofblocks. Then have them collect all red pieces in a second loop. It may take awhile forall to realize that the loops must be arranged as shown in Figure 1 in order to proper-ly place the pieces that are red and square.

FIGURE ]

Then have students describe the pieces in each of the four regions shown usingand, or, and not. Encourage students to generate such statements for each re-gion. Also ask for sentences describing the blocks in each region using all andsome. By having them describe region IV using not and describing those that arenot in region II, the De Morgan laws can be concretely shown. The De Morganlaws are used to relate disjunction, conjunction and negation:

1. NOT (A or B) is equivalent to NOT A and NOTE, and2. NOT (A and B) is equivalent to NOT A or NOT B.

A Truth Table showin

A

T

T

F

F

g the truth v

B

T

F

T

F

alues for AN

NOT A

F

F

T

T

’JD, OR and Is

AANDB

T

F

F

F

iOT follows:

A ORB

T

T

T

F



This indicates that a disjunctive statement can only be negated by negating allclauses, and a conjunctive statement by negating only one of its clauses. Particu-lar attention must be paid to distinguishing between and/or (inclusive or) andeither-or (exclusive or). Truth tables for and/or, and, not and either-or should bedeveloped during this activity.

This activity can be followed by one where three conditions for membershipthat involve values of the attributes of the blocks can be imposed. Figure 2 showssuch an example.

FIGURE 2

After the blocks have been sorted out into the 8 regions, have the students de-scribe the blocks in each region using several different sentences that have and,or and not in them. Discuss these and reinforce the concrete models of De Mor-gan’s laws. One that really presents difficulty to college students is one like thatshown in Figure 3. Students have difficulty seeing the fact that the contradiction


644 Reasoning and Mathematics

FIGURE 3

in requirements in some cases, and the additional restriction in other cases,change the possibilities for membership. In this example no color is possible inregion II and region III, and only two colors are possible in region IV and regionVII.

Conditional statements in if-then form can be introduced using the same mate-rials. Place a set like that shown in Figure 4 on the overhead projector. Transpar-ency versions of these blocks can be made easily from colored transparency film.Ask the students to generate statements in if-then form that are true for this setof objects. Two examples of such statements for the set given in Figure 4 are "Ifgreen, then circle," and "If square, then blue." When all possible true state-ments have been generated, select one and ask the students what piece could beadded to the set or taken away that w^ould make this statement false. Point outthat the only way this can be done is to keep the if part true and deny the thenpart. Add or subtract blocks to the set to make the //part false and then to make


Reasoning apd Mathematics 645

FIGURE 4

BLUE

the then part false. This does not change the truth value. Do this again so the ifpart becomes false and the then part true. The statement still does not changetruth value. Summarize these results in a truth table.At this point the truth tables for and, or, not and if-then can be compared for

given truth value pairs of if-then and truth tables can be developed for if-thensentences with compound premises and conclusions. The major emphasis shouldbe on cases where the converse has the same truth values as the original and whenit doesn*t, what it takes to disprove something, and on the contrapositive pairsthat have the same truth values�inverse-converse and original-contrapositive.Give examples that illustrate how it sometimes is easier to use the contrapositiveform because of the way information is given or known. Point out the differencebetween if-then reasoning and cause-effect reasoning, and between if-then rea-soning and correlations.

After these ideas have been well established with concrete materials, introduceverbal propositions and work with these using truth tables. Truth values can beassigned to each part of the conditional and the effect on the truth value ofchanging the form of the sentence can be analyzed. At this point the differencebetween //and only //should be emphasized and related to the ideas o! sufficien-cy and necessity. Applications of and, or and not to electronic circuitry and simi-lar binary decision making situations can be pointed out. The use of if-then totransfer control in linear sequences such as BASIC programs can be analyzed. Ifcomputers are being used, the built in logic of most versions of BASIC can be il-lustrated. For example in Applesoft, <<!" will be printed for true and "0" forfalse when conditional are evaluated.



Some examples are:

Print NOT 0Print 1 and IPrint 45 > 3Print 1 OROPrint 4 >3 AND 4 = 4Print 1 OR (0 AND 1)Print NOT NOT 1

Print 1 AND (1 ORO)Print NOT 1Print NOT 4 > 3Print 0 OROPrint 0 OR (1 AND 0)

These all Yield "1" because thestatements printed are "True"

These all Yield "0" because thestatements printed are "False"

Students should also have experience with syllogistic chains of true statementswhere the conclusion of one becomes the premise of the next. Consider the fol-lowing exercise:The following statements are all true:

I will have had a sauna only if I drink a can of beer.It is sufficient for me to work in the shop to get dirty.It is necesssary for me to stay indoors for it to have rained.Being dirty is sufficient for me to take a sauna.I will work in the Shop if I stay indoors.

a. Rewrite each of these in IF-THEN formb. Chain these: a -"b, b -*� c, c -» d, d -^ e, e -> f: a�f.c. Write the IF-THEN sentence that links the initial premise with the final conclusion.

Several of the publications from Midwest Publications and others listed in theResources give other activities that can be done to give experience in logical rea-soning.

Combinatorial reasoning and use of the fundamental principle of counting, ordetermining of permutations, are often confused by students. A situation that re-quires both kinds of thinking is even more troublesome.Consider this problem:

A committee is to be selected from student, faculty and parent nominees. Two stu-dents are to be selected from six nominees; three parents are to be selected fromseven nominees; and two faculty are to be selected from five nominees. In how manydifferent ways might this committee be constituted?

Combinatorial reasoning is needed to find each subgroup for the respectivenominees, but the fundamental counting principle is used to find the number ofways that these subgroups could coexist with each other on a committee, sincestudents and parents and faculty are on each possible committee in given num-bers.



Students should have much experience with a variety of situations and prob-lems to be able to determine when and how each kind of thinking is used. One in-volves forming subsets of a given size from a choice set, and the other involvesthe placing of elements in a given set in ordered arrangements.

Finding the number of subsets of several possible different sizes given somechoice set involves "or," which is additive thinking. Using the fundamentalcounting principle or making ordered arrangements involves "and" or multipli-cative thinking. Rarely is this distinction pointed out to students.One very visual and concrete way to help students learn to use these different

kinds of thinking appropriately is through an introduction using colored cubes.To get them to see how combinatorial thinking is used, have the students make

combinations of color cubes. Start with cubes of two different colors, then add acolor to have three different colors, then add another color to have four differentcolors and so on.

Given one red (R) cube, and one blue (B) cube, have students build the two setsthat have only one color in them, the one set that has both colors in it, and theone set that has no color in it�{R}, {B}, {RB}, and { } or 0.Have the students record these numbers in a table like that in Table 1. Add a

Table 1The Number of Sets of all Sizes Possible From Different Numbers of Choices.

NUMBER OF COLORSAVAILABLE

ONE

rwo

THREE

FOUR

FIVE

SIX

NUMBER OF COLORS PRESENTIN SETS OF EACH SIZE

0

1

i

1

1

1

1

1

1

2

3

4

5

6

2

1

3

6

10

15

3

1

4

10*

20

4

1

5

15

5

i

6

6

1

TOTAI

2

4

8

16

32

64

"The Knight’s Move to generate new rows on the table.



third color, white (W), for example, and increase the number of cubes of eachcolor to enable the building of one of each of the possible sets. These will be {R},{B}, {W}, {RB}, {RW}, {BW}, {RWB}, and 0, all but the latter having one cube ofeach color to show the presence of that color. Record the numbers of these in thetable also. Point out how the new sets made with the additional color can bemade by adding this new color to each of the old sets, including the empty set.The old sets are all still possible, including the empty set. The latter can bethought of resulting from a choice of no color. Since this choice can always bemade in one way, it does not matter how many elements are in the choice set. Acolumn for 0 size set can be added to the table and "IV inserted in that columnin every row.As students build sets from colored cubes that increase in number of colors by

one each time and make entries into the corresponding rows and columns of thetable, they should be asked to observe the following:

The effect on the existing sets of adding one or more color to the choice set.

The pattern of building new sets from the old as another color becomes available.

The symmetry that develops on the rows of the table.

Students should recognize how adding one more choice to those available af-fects the old choices to make new choices, and how the old choices are still avail-able. This is the reason for the "Knights Move" often identified as a means forgenerating new rows on this table. The Knights Move in chess is

In summary:

Given choice set (RB), the possible subsets are:

^ 1 with zero elements{R} {B} 2 with one element{RB} 1 with two elements

4 total number of subsetsWith the addition of the third color, the choice set becomes (RWB). The same

four subsets can be made as with two colors available. However, four newr sub-sets are now possible by adding the additional color available to each of the fourlisted above. The result is:

old new(? (?

{R}{B} {W}{RB} {RW} {BW}

{RWB}

1 with zero elements3 with one element3 with two elements1 with three elements

8 total number of subsets,4 old + 4 new



It should be emphasized by reference to the subsets of cubes made and to thetable that of the three 1 color subsets, 2 are old and one is new, and that of thethree 2 color subsets, 1 is old and 2 are new made by adding the third color to thetwo old 1 color subsets. This appears on the table as the Knights Move. Table 1has been completed to show the completion of this activity.The symmetry on Table 1 results from the choosing process. Given five letters

to choose from: A, B, C, D, E. If one chooses A and B to be in a set, C, D, andE are automatically chosen to remain. If one chooses A, B and C, then D and Eare chosen to remain behind. The result is that every time a set of k is made fromn choices, another set of n-k is also made. This is one of the basic features of the

^Good decision making requires the identification of all pos-sibilities . . /’

Pascal Triangle and is generally shown in formula form as

(0-(;.)-So given 5 choices, the number of 1 element choices is the same as the number of4 element choices and the number of 2 element choices is the same as the numberof 3 element choices. This makes it much easier to identify possible combinationsgiven a very large choice set.Good decision making requires the identification of all possibilities as a first

step. Combinatorial reasoning is w^hat is used to identify those possibilities.Determining permutations and linear ordering involves a totally different kind

of thinking. The fundamental principle of counting is involved. When ordering 3things systematically, we realize that the first position in the order can be filled in3 w^ays, the second position is filled from the remaining 2 things and the third po-sition gives no freedom of choice. Students should have the experience of form-ing ordered row^s of colored cubes in a systematic way. Start with 2 cubes�thereare 2 orders possible. With 3 cubes of different colors, there are 6 color orderspossible. As students build these in a systematic way, they begin to see how themultiplication principle differs from forming collections that merely have ele-ments in them without regard for the order in which they were chosen.

Using a tree diagram to illustrate the forming of orders helps students see whythe number of orderings increase so much faster than combinations. Considerordering A, B, and C. We could put any one of these three first as shown:

, (first)ABC


650Reasoning and Mathematics

Then either of the remaining two could follow each of the chosen three, giving atotal of 6:

ABACBABCCACB

At this point there is only one left to fill the third position, thus giving ABC,ACB, BAC, BCA, CAB, CBA. The tree diagram in Figure 5 shows this multipli-cative effect.

FIGURE 5

(1)

(2)

C (3)

A (4)

(5)

(6)

If an ordered process is used in choosing elements for a set from a set of avail-able choices, all possible orderings of the elements of that set result. Consider thefollowing:

Given A, B, C, and D. In how many ways might two of these be chosen?This can be thought of as an ordered process. For example, choose A, fol-

lowed by B. This results in a combination AB. If B is chosen first followed by A,it would still result in the combination AB. The result of the all possible ways ofmaking such an ordered selection are listed below:



A,B B,A C,A D,AA,C B,C C,B D,BA,D B,D C,D D,C

These twelve possibilities result from having four possible letters in the first posi-tion and three possible in the second position. This gives 4 x 3 = 12 orderedpairs. However the two letter combinations are six - AB, AC, AD, BC, BD,and CD. This ordered selection process resulted in the two letter combinationsbeing ordered in all possible ways. Two things can be ordered in 2 x 1 = 2 ways.This ordered selection process results in:

4x3 (choices for the two positions in the ordered pairs)

2 x 1 (ways for the two letter combinations to be ordered)

This gives the same result, 6, as would be found from the table developed earlier.Given four elements in the choice set and a subset of three is to be selected, this

ordered selection process would give:

4 x 3 x 2 (choices for each of the three positions)

3x2x1 (ways to order the three elements chosen)

This result, 4, is the number of combinations found on the table.This can be generalized to "n" choices and a set of size "k" to be chosen. This

can be done in (n) (n - 1) (n - 2). . . (n - k + l)/k! ways.Going back to the problem posed at the beginning of this discussion, we see

that determining the possible subsets of students, parents and teachers on thecommittee requires finding combinations or selections from a choice set. This isfollowed by using the multiplication principle to determine the committee com-position possibilities. The result is 10(2 chosen from 5) x 35(3 chosen from 7) x15(2 chosen from 6).When the processes used and the thinking employed in selection and ordering

are well developed in students, they have no need to memorize formulas that theymay not understand. The memorization of formulas can make finding numbersof combinations and permutations easier, but it should only be done after stu-dents have developed an understanding of how and why they work. It-then rea-soning and combinatorial reasoning and the related multiplication principle andresulting permutations are thinking processes that are commonly and frequentlyused in everyday problem solving as well as in mathematics and science learning.Time spent on giving practice with this thinking and to have students solve prob-lems that require this thinking, either by itself, or in combination with the oth-ers, should replace much of the time spent on memorizing formulas and substi-tuting numbers into them. The time spent on developing it-then reasoning andanalysis of situations requiring its use should enable geometry and algebra teach-ers to make proof in mathematics more understandable.



References

Adi, Helen, R. Karplus, A. Lawson and S. Pulos. Intellectual Development Beyond Ele-mentary School: Correlational Reasoning. School Science and Mathematics 78 (Decem-ber, 1978): 675-683.

Arlin, Patricia K. Formal Operations: Do They Constitute Two Distinct DevelopmentalStages? In Piagelian Theory and The Helping Professions: Proceeding of the 5lh Annu-al Conference. Los Angeles: University of Southern California, 1976.

DeBono, Edward. The Mechanism of Mind. Baltimore: Penguin Books, 1969.Debono, Edward. Lateral Thinking. New York: Harper Colophone Books, 1970.Gruber, Gary and E. C. Gruber. Preparation For the Miller Analogies Test. New-

York: Monarch Press, 1972.Inhelder, Barbel, and J. Piaget. The Growth of Logical Thinking from Childhood to Ado-

lescence. New York: Basic Books, 1958.Karplus, Elizabeth, and R. Karplus. "Intellectual Development Beyond Elementary

School: Deductive Logic." School Science and Mathematics 70 (May, 1970): 398-406.Karplus, Robert, E. Karplus, M. Formisano and A. C. Paulsen. "A Survey of Proportion-

al Reasoning and Control of Variables in Seven Countries." Journal of Research in Sci-ence Teaching 14. (September, 1977): 411-417.

Karplus, Elizabeth, and R. Karplus. Intellectual Development Beyond ElementarySchool: Deduct ive Logic. School Science and Mathematics 70 (May, 1970): 398-406.

Karplus, Robert, E. Karplus, M. Formisano and A. C. Paulsen. A Survey of ProportionalReasoning and Control of Variables in Seven Countries. Journal of Research in ScienceTeaching 14. (September, 1977): 411-417.

Karplus, Robert, and B. Kurtz. Intellectual Development Beyond Elementary School: Pro-portional Reasoning. School Science and Mathematics 79 (May-June, 1979); 387-397.

Karplus, Robert, and R. Peterson. Intellectual Development Beyond ElementarySchool: Ratio, A Survey. School Science and Mathematics 70 (December, 1970): 813-819.

Lovell, Kenneth. Lectures delivered at Teachers College, Columbia, N.Y. July, 1969.Lovell, Kenneth, Proportionality and Probability. In Piagelian Cognitive-development,

Research and Mathematics Education. Washington, D.C.: National Council of Teach-ers of Mathematics, 1971.

McKinnon, Joe W. The Influence of a College Inquiry-centered Course in Science on Stu-dent Entry Into the Formal Operational Stage. Doctoral dissertation (Norman, Univer-sity of Oklahoma, 1970)

Osborn, Alex. Applied Imagination. New York: Charles Scribner’s Sons, 1963.Polya, George. Mathematics and Plausible Reasoning. Princeton, N.J.: Princeton Univer-

sity Press, 1968.Renner, John and D. Stafford. The Operational Levels of Secondary School Students. In

Research, Teaching and Learning \vith the Piaget Model. Norman, Oklahoma: Univer-sity of Oklahoma Press, 1976.

Renner, John, D. G. Stafford, A. E. Lawson, J. W. McKinnon, F. E. Friot, D. H. Kel-logg. Research, Teaching and Learning with the Piaget Model. Norman, Okla-homa: University of Oklahoma Press, 1976.

Wollman, Warren and R. Karplus. Intellectual Development Beyond ElementarySchool: Using Ratio in Differing Tasks. School Science and Mathematics 74. (Novem-ber, 1974): 593-611.



Additional ResourcesAttribute Games and Problems. Newton, Massachusetts: Educational Development Cen-

ter, Inc., 1966.Baker, Michael. Syllogisms. Pacific Grove, California: Midwest Publications, Inc. 1981.Barnard, D. St. P. Figure It Out. Greenwich, Connecticut: Fawcett Publications, 1973.Black, Howard, and S. Black. Figiiral Analogies, a series. Pacific Grove, California: Mid-

west Publications, Inc., 1981.Cwirko-Godycki, Jerzi, and J. A. Karczmarck. Visual Logic. Pacific Grove, Califor-

nia: Midwest Publications, Inc., 1982.Dienes, Boltan, and E. W. Golding. Learning Logic, Logical Games. New York: Herderand Herder, 1970.

Emmett, E. R. 101 Brain Pullers. New York: Harper and Row, 1967.Harnadek, Anita. Analogies, a series. Pacific Grove, California: Midwest Publications,

Inc., 1977.Harnadek, Anita, Critical Thinking, Book One. Troy, Michigan: Midwest Publications,

1976.Harnadek, Anita. Mind Benders, a series. Pacific Grove, California: Midwest Publica-

tions, Inc., 1981.Karplus, Robert, A. E. Lawson, W. Wollman, M. Appel, R. Bernoff, A. Howe, J. J.

Rusch, F. Sullivan. Science Teaching and the Development of Reasoning; 1. Earth Sci-ence, 2. Physics; 3. General Science; 4. Biology; 5. Chemistry. Berkeley, California: Re-gents University of California, 1977,

Lochhead, Jack, and J. Clement, editors. Cognitive Process Instruction. Philadelphia,Pennsylvania: The Franklin Institute Press, 1979.

Whimbey, Arthur, and J. Lochhead. Problem Solving and Comprehension. Philadelphia,Pennsylvania: The Franklin Institute Press, 1980.

Wylie, C. R. Jr. 101 Pu^les in Thought and Logic. New York: Dover Books, 1957.

A. Dean HendricksonUniversity of MinnesotaDuluth, Minnesota 55804

GREAT AMERICAN CHESTNUT TREE: WILL THE LEGENDLIVE AGAIN?

Americans sing about "Chestnuts roasting by an open fire ..." They recite"Under the spreading chestnut tree ..." And they have given the name "Chest-nut" to hundreds of streets. Yet few living Americans have ever eaten an Ameri-can chestnut or even seen the legendary all-purpose tree that once dominated theAppalachians. In one of the greatest ecological disasters of all time, a funguskilled 3.5 billion chestnut trees in the early part of this century. However, thechestnut never became extinct. As poet Robert Frost put it, "It keeps smolderingat the roots/And sending up new shoots/Till another parasite/Shall come to endthe blight." Now, it seems that that "another parasite" has come in the form ofa variety of RNA viruses that infect the fungus to the point that the tree is able toheal itself and continue to grow. MSU plant pathologist Dennis Fulbright hasisolated the viruses from mature Michigan trees that survived the blight and is re-searching ways of spreading the most efficient strains to trees still "sending upnew shoots."


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Formal Reasoning and School Mathematics