Piaget and Middle School Mathematics

Piaget andMiddle SchoolMathematics

William Juraschek

Nowhere is the teaching of mathe-matics more challenging than in themiddle or junior high school. Theearly adolescents who make up thissegment of the school populationmanifest mind-boggling emotional,physical, and intellectual diversity.Those who teach this age group, of-ten puzzled and frustrated in tryingto deal effectively with their stu-dents, continually search for mean-ingful, motivating curricula and ef-fective teaching methods. One fruit-ful source of a better understandingof adolescents and, consequently,more insightful teaching, is thecognitive development research ofJean Piaget. Certainly, his theoryprovides a good perspective forsome fresh approaches to mathe-matics teaching in the middleschool.

Piaget (Piaget & Inhelder, 1969)theorizes that our cognitive develop-ment proceeds through an invariant

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Piaget and Middle School Mathematics 5

sequence of four stages: sensorimotor (birth to about 2 years), preopera-tional (2 to about 7 years), concrete operations (7 to about 11 years), andformal operations (about 11 to adult). The stages are continuous andbuild on previous stages, each marking the active acquisition of newcapabilities. The concrete and formal operations stages are most relevantto middle school, so let’s take a close look at some of the differences be-tween concrete and formal operational thinking as manifested on threePiagetian tasks.

Concrete and Formal Thought

One indication of the acquisition of formal operations is the ability torecognize all variables in a situation, generate hypotheses about the rolesof the variables, and finally manipulate them systematically to ascertainwhether or not the hypotheses conform with reality. For example, whenpresented with the appropriate apparatus and asked to figure out whatdetermines the period of a simple pendulum, the formal operationalyoungster can isolate the relevant factors and systematically control eachto discover that the pendulum’s length determines its period. While theformal operational youngsters know they must devise appropriateexperiments that hold constant all but one factor to detect its effect, theconcrete operational child will vary two factors simultaneously and in-correctly conclude that both affect the period of oscillation. The childwho is not formal operational simply is not aware of all the logical possi-bilities and combinations to be tested.Formal operations also enable the construction and manipulation of

second order operations, or relations between relations. This is revealedin the capacity to use proportions, equalities between ratios which arethemselves relations between whole numbers. The development of thiscapacity can be seen in performance on the following two tasks.

In the first, called Equilibrium in the Balance (Inhelder & Piaget,1958), the child is given an equal-arm balance with equally spaced pegsand asked where various weight combinations may be placed to balancethe beam. At best the concrete operational child realizes that greaterweight compensates for less distance from the pivot point, but does notsuspect a general rule. The formal operational thinker, if not intuitivelyaware of a general rule involving proportions, can, and will, experimentwith various weight placements and discover the rule. Again the moregeneral nature of formal operations motivates the search for a generalrule, in this case one involving proportions, or relations between rela-tions.


6 Piaget and Middle School Mathematics

A third task that illustrates the attainment of these more powerfulcognitive schemes involves intuitive notions of chance (Piaget, 1975).When shown two sets of marbles, each containing blue and whitemarbles, and asked from which set one is more likely to select at randoma white marble, formal operational students will base their decisionsupon a comparison of ratios�relations between whole numbers�ratherthan upon a comparison of simple whole numbers, as does the concreteoperational child. Suppose one set contains two white and two blue mar-bles, while the other contains three white and four blue. The concreteoperational child is very likely to predict the better chance for the secondset because it contains more white marbles, or perhaps the first set be-cause there are fewer total marbles there. In neither case is the decisionbased upon ratios and proportions. The formal youngster will base apreference for the first set upon a comparison of ratios, in terms of com-

"... the development of new mental operations throughmaturation, experience, and social transmission, is the directresult of the self-regulation process."

mon fractions or percents, and may wonder why anyone would ever useany other strategy.

In summary, while the concrete youngster can apply simple logic toactions on familiar, perceivable things of only modest abstraction, theformal youngster can go beyond the immediate and perceivable and rea-son about more abstract entities, such as relations and verbal proposi-tions. Concrete operations are bound by the immediate and "real";formal operations are not. The formal youngster can also perform moresophisticated mental operations, intuitively using proportions to explaincompensations and devise valid experiments to test for the possible ef-fects of variables. The key difference, and the source of the above stagedistinctions, is that, with concrete operations, reality dominates thought,but with formal operations, possibility dominates thought. Awareness ofthe stage distinctions naturally leads one to wonder about what causesthem, that is, what contributes to cognitive growth.



Factors in Cognitive Development

According to Piagetian theory, cognitive development results from theinterplay of four factors: physical maturation, experience, social trans-mission, and self-regulation (Piaget & Inhelder, 1969). Physical matura-tion refers to organic maturation of the nervous and endocrine systems.Experience consists of both physical experience involving actions on ob-jects, and logico-mathematical experience involving mental coordinationof the physical actions. Social transmission involves linguistic and educa-tional experiences, as well as social interactions.

Self-regulation is the process of actively generating new mentalschemes when existing schemes are insufficient and comprises the com-plementary phases of assimilation and accommodation. Recall the prob-ability task, described earlier, where one is asked to predict from whichof two containers of marbles there is a greater likelihood of randomly se-lecting a white marble. If the scheme of basing predictions upon the com-parison of only the number of white marbles in each set produces satis-factory results, youngsters are not motivated to change their reasoning.But when confronted with two new sets, one composed of two white andtwo blue marbles, the other containing three white and eight blue mar-bles, the child may feel uncomfortable choosing the set with more whitemarbles�it just doesn’t feel right. This cognitive discomfort, or disequi-librium, forces the realization that the existing scheme is insufficient.Ideally, through self-regulation, the child restores equilibrium by accom-modating and develops a new reasoning pattern, in this case, one involv-ing the use of proportions to make predictions. For Piaget, the develop-ment of new mental operations through maturation, experience, and so-cial transmission is the direct result of the self-regulation process.

So far we have examined the characteristics of concrete and formal op-erational thought, together with the factors that influence intellectual de-velopment. Now let’s explore the implications for teaching mathematicsto early adolescents.

Implications/or Middle School Mathematics

Although Piaget himself has made only general remarks about themeaning of his research for education, there have been many educatorswilling to list implications of his theory for practice in science and mathe-matics classrooms. (See Adier, 1966; Case, 1973; Karplus, 1977; Lamb,1977.) Most of these focus on the elementary school. However, I hopethe following will persuade the reader that Piaget’s theory is equally rele-vant for the middle school.



A primary consideration is the recent accumulation of evidence thatonly about one-half of all fifteen-year-old youngsters have reached thestage of formal operations (Blasi & Hoeffel, 1974). This implies that thetypical middle school class most likely contains a few students who areconcrete operational and a few who are formal operational, with the ma-jority being in some transitional phase between the two stages. What if,as the reader may have already conjectured, formal operations are neces-sary for the assimilation of many of the important mathematical con-cepts in the middle school curriculum? In this regard, Piaget has assertedthat not until children reach the stage of formal operations are they capa-ble of genuine understanding of proportions, similarity, and probability.(Inhelder & Piaget, 1958). Other empirical evidence supports thisproposition.

Lovell (1971-a) reported studies indicating that an understanding offormal proof and an intuitive grasp of negative numbers are not evidentuntil the formal operations stage. Lawson and Wollman (1980) and Lov-ell (1971-b) found that formal operations appeared necessary for successwith proportions problems. Collis (1975), on the basis of several interest-ing studies, concluded that not until this stage can one understandmathematical structures in terms of properties and relationships betweenoperations. He found, for example, that unlike formal operational stu-dents, those who were concrete operational had trouble understandingthe inverse relationship between addition and subtraction when workingat the symbolic level.

It does, therefore, seem plausible that real understanding of somemathematical concepts demands formal operations. Couple this with thefact that the majority of middle school students have not fully attainedthis stage of development, and one might infer that there is little hope forthe effective teaching of many basic mathematical ideas in these grades.Such a pessimistic judgment, however, is far from justified when oneconsiders the following.

Juraschek and Grady (1981) administered Piaget’s balance beam taskto over 200 junior high and college students, the majority having been as-sessed, on the basis of other tasks, as being in transition between the con-crete and formal operations stages. They found a significant number ofthe students manifested formal operations "when allowed to experimentand manipulate the apparatus. Suydam and Higgins (1977), after a thor-ough review of the literature on activity-based instruction in mathemat-ics, concluded that "lessons using manipulative materials have a higherprobability of producing greater mathematical achievement than do non-



manipulative lessons (p. 83)." The findings of both of these studies areconsistent with Skemp’s bold assertion in The Psychology of LearningMathematics that <( ... the ’intuitive-before-reflective’ order may bepartially true for each new field of mathematical study (p. 66)." Quiteplausibly, activity-based lessons and the use of manipulatives are of sig-nificant help for the transitional, as well as the concrete operational stu-dent, when trying to learn mathematics.

Unfortunately, as Fey (1979) reported, the activity-based mode is notthe mode. After examining several recent surveys of instructional pat-terns in mathematics classrooms, he reported that the predominantteaching style in middle schools consists of a teacher lecture followed by

tt. . . the middle school teacher must recognize the implica-tions of the diversity of developmental levels among earlyadolescents, as well as the relation between cognitive devel-opment and mathematics learning ..."

student seat-work on paper-and-pencil assignments. If we assume thatone goal of schooling is to maximize each student’s intellectual develop-ment, the middle school teacher must recognize the implications of thediversity of developmental levels among early adolescents, as well as therelation between cognitive development and mathematics learning and,accordingly, provide a curriculum and classroom environment that en-courages the acquisition of formal operations along with mathematicalachievement. In light of the foregoing discussions, it seems appropriateto urge that traditional instructional methods be replaced, or at least aug-mented, by activity-based teaching, incorporating manipulative mate-rials wherever possible. Fully aware that, in many middle school settings,the task may be difficult, I offer the following suggestings for teachingmathematics to early adolescents. The suggestings are not necessarilynew, but I hope that the reader has by this point been persuaded that theybelong in the middle school teacher’s repertoire.

Teaching Suggestions

Proceed from the concrete to the abstract. We all know how frustrat-ing it can be to try to assimilate material that is too abstract for us. Our



first reaction is to plead to see some examples. In doing this we are askingfor something more concrete and familiar for reference and mental ma-nipulation. What is concrete in a particular situation is, naturally, some-what relative. It is probably safe to assume that for most adolescentswhole numbers, the number line, and geometric figures are not too ab-stract or unfamiliar to use as referents. With fractions, however, we mustbe more cautious. The fraction (rational number) concept is relational. Itinvolves a numerator, a denominator, and, most importantly, the rela-tion between these two abstractions. Although early adolescents havegained some facility manipulating the symbolic representations (decimalor common notation), to encourage a better understanding of the con-cept they almost certainly need to work extensively with something moreconcrete such as Cuisenaire rods, fraction bars, paper folding, rulers,grid paper, and geometric sketches. As many teachers already know,these aids can be very helpful, even when used only briefly.When teaching probability and statistics, which is an excellent topic

for this age group because of the meaningful use of ratios, conduct manyhands-on activities and experiments. To ensure familiarity with concretereferents, have students toss dice, draw cards, gather data from theirclassmates, and play simple games before analyzing the experiences anddeveloping any formal subject matter.When teaching geometry use rulers, protractors, and compasses often.

Make polyhedron models and use them for area and volume activities.Use geoboards to explore polygons, the Pythagorean theorem, and area.The students need to get their hands on something more concrete andreal than a picture in the textbook.

Base teaching on student activities. One of the most important propo-sitions one can derive from Piaget’s theory is that we learn by performingactions on objects and ideas. This requirement is not satisfied by merelyrequiring homework and lots of drill and practice. The students must beencouraged to manipulate things, as mentioned above, and become ac-tive participants in structured and unstructured settings. For example,when teaching about area, don’t merely tell students what a tessellationis. Have them actually tessellate cardboard regions using various shapesor a transparent grid sheet. For a better understanding of volume, havethem determine the volume of some regular solids by the immersionmethod and compare their results with those found by measuring andusing a formula. Have students determine the area of an irregular penta-gon using a ruler and the formula for the area of a triangle. This forces

School Science and MathematicsVolume 83 (I) January 1983


them to focus on what the b and h refer to in the formulaA = Vibh. Letthem use calculators to explore number relationships and make conjec-tures. Excellent sources of activities are commercially available (see forexample the Creative Publications catalog), and many good ideas appearin this and other professional journals. And, above all, don’t overlookthe fertile imagination of the classroom teacher.Teach for self-regulation. If self-regulation is, as Piaget claims, the

key factor in intellectual development, schooling should provide ampleopportunities for students to grow through this process. Robert Karplus(1977) has developed the concept of the learning cycle, a ’Tiagetian-based" teaching model designed explicitly to encourage self-regulation.It forms the basis for the pedagogy of the Science Curriculum Improve-ment Study (SCIS) units in elementary school science. Karplus’ model iswell known among science educators and is readily applicable to activity-based mathematics teaching.A learning cycle consists of three phases, exploration, concept intro-

duction, and application. During exploration, the learner is involved inexperience with concrete or familiar materials. These experiences, whichcan be very structured or fairly free, are designed so learners encounterinformation slightly beyond their capability for assimilation. When theybecome aware of this shortcoming, disequilibrium occurs, and the teach-er begins the concept introduction phase. The particular concept intro-duced helps the learner accommodate and attain a new level of under-standing. Once they have grapsed the concept, learners are ready for theapplication phase. In this phase they apply the newly acquired concept toother appropriate ideas and information. This provides opportunities forfurther self-regulation, reinforcement, and a broader grasp of the con-cept. Depending upon the concept or concepts, learning cycles can becompleted in one brief lesson or extended over several days or weeks.Consider the following example, based upon the probability task de-scribed earlier.

Target concept: The probability of an event is the ratio of favorable outcomes to totalpossible outcomes.

Exploration: Students are shown two sets of marbles, some blue, some white, and twoaccompanying soft drink cans. The teacher poses the situation: Set A is placed in canA and Set B is placed in can B. Each can is shaken and one marble poured from each.From which can is there a better chance of pouring a white marble? The question isrepeated for various sets of marbles, including some that make the students awarethat predictions based upon simple comparisons of the numbers of white marblesdon’t make sense. This produces disequilibrium.



Concept Introduction: Mention that the best technique for making predictions in allcases is by comparing the ratios of white marbles to total marbles in the sets.

Application: Conduct empirical tests to see if this new technique really provides accu-rate predictions in the long run. Use the ratio definition of probability in other prob-ability experiments. Work appropriate verbal problems.

The reader probably recognizes that many teachers already use meth-ods equivalent to learning cycles, although they don’t identify them assuch. Certainly, many guided discovery lessons implicitly follow thismodel. Nevertheless, the learning cycle provides one more useful modelfor sequencing curriculum and activities, especially when one wants toteach for self-regulation.

Conclusion

The primary purpose of this paper has been to increase teachers’awareness of Piaget’s theory of cognitive development and its implica-tions for mathematics in the middle school. The reader no doubt detectedthe inevitable oversimplifications and leaps of faith inherent in such aneffort to link educational practices with psychological theory. Piaget’stheory does not, and cannot, explain all mathematics learning. Muchlearning occurs simply through imitation and following rules and is pri-marily association learning. Also, Piaget’s theory is not directed at ex-plaining individual differences, but rather focuses on common develop-mental characteristics. Nevertheless, it is hoped that knowledge of histheory and the cognitive capabilities of early adolescents will compelteachers to adapt strategies that help their students gain more from theirmathematical experiences. As mentioned at the outset, insights from Pia-get’s research can provide the impetus to begin needed innovation andexploration. In this writer’s opinion, we can’t start soon enough.

REFERENCES

1. ADLER, I. Mental Growth and the Art of Teaching. The Arithmetic Teacher. 1966, 13,576-84.

2. BEILIN, H. The Training and Acquisition of Logical Operations. In Piagetian Cogni-tive-Development Research and Mathematical Education. Washington, D.C.: Na-tional Council of Teachers of Mathematics, 1971.

3. BLASI, A. and E. C. HOEFFEL. Adolescence and Formal Operations. Human Develop-ment, 1974,77,344-363.

4. CASE, R. Piaget’s Theory of Cognitive Development and Its Implications. Phi DeltaKappan. 1973,55,20-25.

5. COLLIS, K. A Study of Concrete and Formal Operations in School Mathematics: APiagetian Viewpoint. Victoria, Australia: Australian Council for Educational Re-search, 1975.

6. COPELAND, R. How Children Learn Mathematics. New York: Macmillan, 1979.



7. FEY, J. Mathematics Teaching Today: Perspectives from Three National Surveys. TheMathematics Teacher. 1979, 72, 490-504.

8. INHELDER, B. and J. PIAGET. The Growth of Logical Thinking/row Childhood toAdolescence. New York: Basic Books, 1958.

9. JURASCHEK, W. and M. GRADY. Format Variations on Equilibrium in the Balance.Journalfor Research in Science Teaching, 1981,18, 47-49.

10. KARPLUS, R. Science Teaching and the Development of Reasoning. Journal of Re-search in Science Teaching. 1977, 14,169-75.

11. LAMB, C. Application of Piaget’s Theory to Mathematics Education. In PiagetianTheory and the Helping Professions. Los Angeles: University of Southern CaliforniaBookstore, 1977.

12. LAWSON, A. and W. WOLLMAN. Developmental Level and Learning to Solve Problemsof Proportionality in the Classroom. School Science and Mathematics. 1980, 80, 69-75.

13. LOVELL, K. A Follow-up Study of Inhelder and Piaget’s "The Growth of LogicalThinking." British Journal of Psychology. 1961, 52, 143-53.

14. LOVELL, K. The Development of the Concept of Mathematical Proof in Abler Pupils.In Piagetian Cognitive-Development Research and Mathematical Education. Washing-ton, D.C.: National Council of Teachers of Mathematics, 1971-a.

15. LOVELL, K. Proportionality and Probability. In Piagetian Cognitive-Development Re-search and Mathematical Education. Washington, D.C.: National Council of Teachersof Mathematics, 1971-b.

16. PIAGET, J. Development and Learning. Journal of Research in Science Teaching. 1964,2, 176-186.

17. PIAGET, J. and B. INHELDER. [The Origin of the Idea of Chance in Children] (Leake,Burrell, and Fishbein, translators.) New York: Norton, 1975.

18. PIAGET, J. and B. INHELDER. The Psychology of the Child. New York: Basic Books,1969.

19. SKEMP, R. The Psychology of Learning Mathematics. Baltimore: Penguin Books,1971.

20. SUYDAM, M. and I. HIGGINS. Activity-based Learning in Elementary School Mathe-matics: Recommendations from Research. (ERIC Document Reproduction ServiceNo. ED 144 840).

Dr. William JuraschekUniversity of Colorado1100 Fourteenth StreetDenver, Colorado 80202

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Piaget and Middle School Mathematics