Levels of Proof in Lower Secondary School Mathematics

MIKIO MIYAZAKI

LEVELS OF PROOF IN LOWER SECONDARY SCHOOLMATHEMATICS

As Steps from an Inductive Proof to an Algebraic Demonstration

ABSTRACT. The purpose of this study is to establish levels from an inductive proofto an algebraic demonstration in lower secondary school mathematics. I propose that wecan establish six levels of proof in lower secondary school mathematics as steps from aninductive proof to an algebraic demonstration on the basis of three axes (contents of proof,representation of proof, and students’ thinking). To reach this conclusion, I firstly examinethe meaning of ‘demonstration in lower secondary school mathematics’ and ‘proof in lowersecondary school mathematics’, and show the relationships between them. Secondly, I setout four basic levels of proof, as seen from two aspects (contents and representation ofproof). Thirdly, I subdivide them into six levels from the third aspect of students’ thinking.Finally, I illustrate my discussion with a 7th grader’s activities.

INTRODUCTION

Demonstration1 has always been an important component of school math-ematics, and its importance will not change in the future (Hanna, 1995).Learning a demonstration has some educational value. For example, stu-dents learn to make a logical explanation not only in mathematics, but alsoin other activities.

In Japan, after the course of study was revised in 1958, demonstra-tion is taught in the 8th grade for the first time. Although many teachersand educators have been improving the teaching of demonstration, manylower secondary school students cannot recognize the necessity of demon-stration, cannot construct it, nor can they appreciate its value.(Kunimune,1987; Elementary and secondary education bureau of Monbusho, 1985;Koseki, 1978, 1980).

The cause of students’ undesirable achievement may be the age 8thgrade at which demonstration is taught in Japan. However, even if studentsare past 8th grade, they remain in the same undesirable achievement inother nations (De Villiers, 1991; Senk, 1983; Fischbein, 1982). Rather, ashift from an inductive proof to a demonstration should be focused on,because below 8th grade, lower secondary school students have alreadylearned to prove numerical or geometrical propositions inductively with

Educational Studies in Mathematics41: 47–68, 2000.© 2000Kluwer Academic Publishers. Printed in the Netherlands.

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constructions, measurements, operations etc. In the lessons about demon-stration, students are taught the insufficiency of an inductive proof and thenecessity of demonstration. However, in many cases, most students2 can-not understand the above. Unfortunately, students learn a demonstrationonly as a kind of efficient ritual in school mathematics. As a result, studentsdo not attempt to make a demonstration in situations outside school math-ematics where a demonstration would be useful and/or profitable for them.On the other hand, among some kinds of proof, there may be ‘germination’of demonstration. If students could reach a demonstration through ‘grow-ing’ such a special proof, they would deeply understand the necessity ofdemonstration.

PURPOSE AND METHOD

The purpose of this study is to answer the following question:What kinds of levels can we establish as steps from an inductive proof toan algebraic demonstration in lower secondary school mathematics?

There are two significant points to answer the above.The first is related to the practice of teaching a demonstration. Assum-

ing that we could establish some levels of proof, teachers can evaluateeach student’s level, and can establish more appropriate goals so that theirstudents will be able to achieve a demonstration. Moreover, the prerequis-ites to shift between the levels can lead teachers to examine more easilywhat kinds of teaching are appropriate for their shifts. Accordingly, withteachers’ efforts, we can hope that more students become able to reach alevel of demonstration.

Another significant point is related to proof researches in mathematicseducation. So far, we have made a simple distinction of proofs on the basisof reasoning (such as induction, deduction, analogy and so on). Since de-ductive proofs are superior to inductive proofs, with respect to a generalityor universality of proposition, deductive proofs tend to be emphasized inlower secondary school mathematics. On the other hand, many researcheshave improved the lessons of demonstration and/or proof from students’positions. For example, one research identified some essential aspects forstudents to achieve a demonstration exactly (Galbraith, 1981). Some re-searches focused on meaningful proofs for students regardless of qualityof assumptions, reasoning and/or representation (Action proof (Semadeni,1984); Illuminating examples (Walter, 1986); Inhaltlich-anschaulicher Be-weis (Wittman and Müller, 1988); Preformal proving (Blum and Kirsh,1991); Kunimoto, 1994, and so on). Other researches devised types or

LEVELS OF PROOF IN LOWER SECONDARY SCHOOL MATHEMATICS 49

modes to classify/arrange proofs (Balacheff, 1988a, 1988b; Braconne andDionne, 1987).

Remarkably, Balacheff (1987, 1988a, 1988b) theoretically establishedthree levels of proof (pragmatic proofs, intellectual proofs, demonstration)on the basis of three poles (nature of conceptions, formulation, valida-tion). The lowest level ‘pragmatic proofs’ means “those having resourceto actual action or showings” (1988a, p. 217), which includes two typesof proof (naive empiricism, crucial experiment). For example, in provingthe conjecture “the number of diagonals of polygon is n(n-3)/2”, a pairof students checked it with the 20-gon on the basis that the conjecture wastrue for any case, if it would be true for a 20-gon. This comes under ‘crucialexperiment’. The middle level ‘intellectual proofs’ means “those which donot involve action and rest on formulations of the properties in questionand relations between them” (1988a, p. 217), which includes two typesof proof (generic example, thought experiment). For example, a studentdescribed the reason why each vertex had (n-3) diagonals and the reasonwhy n(n-3) was divided by 2. This comes under ‘thought experiment’. Themost advanced level ‘demonstration’ requires a specific status of know-ledge which must be organized in a theory and recognized as such by acommunity: the validity of definitions, theorems, and deductive rules issocially shared (1987, p. 30).

I will adopt Balacheff’s idea that it is possible to establish some levelsbetween proof and demonstration through which students can and shouldshift from a present level to a more advanced level, and finally reach a levelof demonstration.

Although Balacheff’s framework is exciting and farsighted across allgrades, supplementary levels of proof are needed with reference to hisresearch ideas. For, as he mentioned (1987, p. 158), generic example hasan important role to shift from pragmatic proofs to intellectual proofs.However, it is unclear what can be a ‘germination’ of generic example.Therefore, it is necessary to find the ‘germination’ in pragmatic proofs,and to give it a theoretical position as a level of proof. If that will be real-ized, the new levels of proof will be more useful to develop and evaluatestudents’ proving abilities.

PREPARATION

Demonstration in lower secondary school mathematics

Among educational rationales to learn a demonstration, it is necessary tofocus on the accomplishment that students can show others the reason why

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a proposition is true, because this rationale is a core of proof in schoolmathematics, and a prerequisite to accomplish others (Hanna, 1995). Inorder to clarify the rationale, the following considers what the truth ofproposition means in this study.

The objective to learn a demonstration in lower secondary schoolmathematics

Until the end of 19th century, mathematics was covered by the absolutedoctrine that only propositions deduced from a priori self-evident propos-itions were true. However, the doctrine changed from the absolute to therelative through the foundation crisis in mathematics, which originated inthe appearance of non-Euclidean geometry and the compatibility of it withphysical space. The relative doctrine means that the truth of a proposi-tion depends on axioms, which mean tentative assumptions to establish atheory.

Is it possible to adopt the relative doctrine of truth into a demonstrationin lower secondary school? Fawcett (1938) initially highlighted the relativedoctrine in mathematics education, and endeavored to adopt it into uppersecondary school mathematics of those days. Although his research ideasand results have been admired, Sugiyama (1985) pointed out that it wasdifficult to apply Fawcett’s plan and practice to lower secondary schoolmathematics in Japan. For, students in Fawcett’s research were primarily10th and 11th graders, and it takes too much time to carry on his plan forlower secondary school students as it is.

Although it is hopeless for lower secondary school students to find tent-ative assumptions in looking out over an entire theory like mathematicians,students have learned to make or select necessary assumptions to examinewhether propositions are true or false, even before learning a demonstra-tion. Therefore, when attempting to adopt the relative doctrine of truth intoa demonstration, students are expected to make necessary assumptions toexamine whether a proposition is true for himself or herself, and they areexpected to show others deductive reasoning of the proposition.

The existence of others is essential for a demonstration, which accom-panies social activities in nature (Hanna, 1989). We usually change typesof proof or demonstration according to others. Furthermore, demonstrationcan be a ‘game’ to show others reasons, when we share codified rulesof demonstration in a community. On the other hand, it is enough fora demonstration in lower secondary school mathematics to satisfy stu-dents’ consciousness of others’ existence. With excessive emphasis ondemonstration as a ‘game’ from the outset, students cannot understand thesocial effectiveness of demonstration. As the grade goes up, it is necessary


to enrich the consciousness of others’ existence gradually in learning ademonstration.

Consequently, we can establish the following objective to learn a demon-stration in lower secondary school mathematics: “A student can show oth-ers the reason why a proposition is true for himself or herself.”

Prerequisites for demonstration to achieve the objective

What prerequisites should a demonstration satisfy to achieve the above-mentioned objective? A demonstration consists of contents (What does itshow?) and representation (How does it show?). With respect to prerequis-ites of contents, demonstration needs to satisfy a condition that a studentdeduces a proposition from assumptions true for himself or herself. Withrespect to prerequisites of representation, a demonstration is representedmainly with language. As deductive reasoning will be practicable with achain of propositions, a language of demonstration needs to satisfy thefollowing conditions. (If a language satisfies the conditions, it is called ‘afunctional language of demonstration’ in this study.)

– It has symbols and rules of their arrangement to represent objects,their properties, and relations between them.

– It has terms and rules of their arrangement to represent a proposition.

– It has sentences and rules of their arrangement and abbreviation torepresent a chain of propositions.

For example, in the following demonstration about the proposition thatif the sum of digits of a number is a multiple of 9, then the number isa multiple of 9, each sentence to show a proposition is arranged in theform of two columns connected by a symbol ‘=’. The terms of reasoningrules are put in parentheses. The left columns repeated with previous rightcolumns are abbreviated.

n∑k=1ak · 10k−1 =

n∑k=1ak · (10k−1− 1)+

n∑k=1ak [The distributive law]

=n∑k=1ak · (10− 1)(

k−1∑j=1

10j−1)+n∑k=1

ak [Factorization]

= 9 · {n∑k=1ak · (

k−1∑j=1

10j−1)} +n∑k=1

ak

[The communicative law/The distributive law]

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The meaning of ‘demonstration in lower secondary school mathematics’

A demonstration consists of human activities which include the followingfactors: ‘actors’, ‘objects of action’, ‘objectives of action’. These factorsare restricted in this study. Firstly, ‘actors’ mean students who are intendedto learn a demonstration. Secondly, ‘objects of action’ mean propositionsdealt with in lower secondary school mathematics. Thirdly, ‘objectives ofaction’ mean the above-mentioned objective: “A student can show othersthe reason why a proposition is true for himself or herself.” To achievethe objective, the contents of demonstration needs to involve deductivereasoning from assumptions true for himself or herself, and the represent-ation of demonstration needs to be completed with a functional languageof demonstration.

Consequently, ‘demonstration in lower secondary school mathematics’means the following human activities.

In order to show others the reason why a proposition dealt with in lowersecondary school mathematics is true for a student,

– he/she deduces the proposition from assumptions true for him/herself,– he/she represents a chain of deductive reasoning with a functional

language of demonstration.

Proof in lower secondary school mathematics

TABLE I

A meaning of ‘proof in lower secondary school mathematics’ will beexamined, comparing with a meaning of ‘demonstration in lower second-ary school mathematics’. Actors, objects of action, and objectives of actionare the same as those of ‘demonstration’ in order that students finally reacha demonstration. The contents of proof consist of logical reasoning, such as


deduction, induction, and analogy in lower secondary school mathematics.Therefore, the contents of proof need to be logical reasoning from assump-tions true for himself or herself. The representation of proof consists oflanguage/ drawings/ manipulable objects (Lesh, Post & Behr, 1987). Con-sequently, ‘proof in lower secondary school mathematics’ means humanactivities in Table I. In short, ‘demonstration in lower secondary schoolmathematics’ is a specialized ‘proof in lower secondary school mathemat-ics’, and the former has the most desirable contents and representation ofthe latter.

ESTABLISHING LEVELS OF PROOF IN LOWER SECONDARY SCHOOL

MATHEMATICS

Significance of relative truth to establish levels of proof

The levels of proof will be established in order that students reach a demon-stration. Constructivism considers it desirable to accept various proofswhich are accessible to a demonstration. As described above, accordingto the relative doctrine of truth, it is important that each student exam-ine assumptions true for him/herself. And then, more kinds of proofs canbe acceptable in lower secondary school mathematics. In other words,the relative doctrine of truth makes it possible to enlarge a concept ofproof qualitatively. Therefore, the doctrine is significant for this study toestablish the levels of proof.

Axes to establish levels of proof: contents, representation, and thinking

The contents and the representation of proof, and students’ thinking arethree axes to establish levels, because there are crucial differences betweena proof and a demonstration in the lines of ‘Contents’ and ‘Representation’of Table I. And, a student’s thinking makes it possible to know why andhow he/she examines the contents or representation of proof.

Basic levels of proof

The distinction of proof with contents and representationTwo categories are provided in each axis of contents and representation.One category of an axis should have nothing in common with the othercategory so that each level is discrete from others. In the contents axis, thetwo following categories are provided: ‘inductive reasoning’ and ‘deduct-ive reasoning’. In the representation axis, the two following categories areprovided: “a functional language of demonstration” and “a language other

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than a functional language of demonstration, drawings, and/or manipulableobject”.

By combining two categories in each axis, four basic levels are estab-lished as shown in Table II.

TABLE II

Examples of proof A, B, C and DAbout the proposition “the sum of three consecutive numbers is three timesthe middle number”, the following proof can be classified into the mostadvanced level, ‘Proof A’.

The proof consists of deductive reasoning from assumptions to the pro-position to be proved. And, it is represented with an algebraic formallanguage that belongs to a functional language of demonstration.

The following proof can be classified into ‘Proof B’.

.

The transforming process includes deductive reasoning from assumptions.Assumptions (e.g. conserving the number of marbles) are true for a studentwho makes the proof. The representation of proof includes manipulableobjects, their transformation, and some sentences without a functional lan-guage of demonstration.


The following proof can be classified into ‘Proof C’.

This proof consists of inductive reasoning, and it is represented with nu-merals (‘9’, ‘570’, etc.), operational symbols (‘+’, ‘·’, etc.), relational sym-bols (‘=’), and terms of arithmetic or mathematics (‘consecutive numbers’,etc.). Equal signs are not used as a functional language of demonstration,but only as the symbols to prompt calculation.

The following proof can be classified into ‘Proof D’.

The difference of numerical symbols represents the difference of number.The universal proposition is induced on the base of three cases at least.Although some sentences (e.g. ‘a+b+c=3b’) fit a rule to arrange symbols,the proof does not use rules to arrange or abbreviate sentences.

Grading proofs according to contents and representation‘Proof A’ is equal to a demonstration in lower secondary school mathem-atics, and it is the most advanced level. Therefore, ‘Proof A’ has the mostadvanced categories in both the contents axis and the representation axis.‘Proof C’ has the lowest categories, and then it falls in the lowest level. Onthe other hand, ‘Proof B’ and ‘Proof D’ are intermediate between ‘ProofA’ and ‘Proof C’, because one category is equal to that of ‘Proof A’, andanother is equal to that of ‘Proof C’.

Additional distinction of basic levels of proof with students’ thinkingcategories of students’ thinking: concrete operations and formaloperations

Piaget’s theory (Piaget, 1947) of thinking is appropriate for this study

Reason 1: Piaget’s theory has genetic interpretations about thinking, andemphasizes the need of interactions between a subject and its envir-onment to develop one’s thinking. The interpretations and the need of

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interactions fit the teaching and learning situation of lower secondaryschool students.

Reason 2: The most advanced thinking, ‘formal operations’ is necessaryto make a ‘demonstration in lower secondary school mathematics’.

Reason 3: The concept of ‘décalages verticaux’ theoretically backs up thenecessity to shift from a lower level to the most advanced level ‘demon-stration in lower secondary school mathematics’ in order.3

Among Piaget’s five levels of cognitive equilibrium, concrete opera-tions and formal operations need to be two categories of students’ thinking,because most of lower secondary school students ought to have reachedconcrete operations.

The relation between students’ thinking and the previous distinctionIf students’ thinking could be independent of the contents and/or the rep-resentation of proof, eight types of proof could be distinguished as il-lustrated below. However, students’ thinking takes part in examining thecontents and the representation of proof. Then, it is necessary to checkwhether or not one with concrete operations or formal operations achieveeach type of proof.

As ‘Proof A’ is represented with a functional language of demonstration,students’ thinking has to treat propositions. The treatment can be accom-plished only with formal operations. Therefore, in order to achieve ‘ProofA’, students’ thinking has to reach formal operations.

Concerning ‘Proof C’, both with concrete operations and formal opera-tions, one can reason inductively. However, the essential characteristics of


formal operations involve deductive reasoning with propositions. There-fore, ‘Proof C with formal operations’ is identified with ‘Proof C withconcrete operations’.

‘Proof D’ has to be devided into ‘Proof D with concrete operations’"and ‘Proof D with formal operations’ theoretically, because one sentenceof ‘Proof D’ cannot always have the same meaning. For example, in thecase of ‘a+b+c=3b’ with concrete operations, the symbols, ‘a’, ‘b’, ‘c’ takeroles of ‘place holder’ for which the numbers for justification are substi-tuted, and the symbol ‘=’ means to prompt calculation. On the other hand,the same symbols with formal operations mean variables, and the symbol‘=’ means an equal sign, and then, the sentence can represent a universalproposition.

However, there are practical limitations of theoretical distinction of‘Proof D’. Firstly, it is difficult to find out which a proof belongs to ‘ProofD with concrete operations’ or ‘Proof D with formal operations’, becauseboth proofs are represented with a functional language of demonstration.Secondly, it seems hard for students to shift from ‘Proof D with formaloperations’ to ‘Proof A’ meaningfully. In order to get deductive contents,students should go back or go through ‘Proof B’.

The difference between ‘Proof B with concrete operations’ and ‘ProofB with formal operations’ is essential for this study. Students’ expectedactivities will be specified theoretically, and will be illustrated with a 7thgrader’s real activities.

Students’ expected activities concerning ‘Proof B with concreteoperations’The following proof about the proposition “the sum of two odd numbers iseven” is an example of ‘Proof B with concrete operations’.

The proof consists of the following consecutive actions on manipulable ob-jects. An ‘action on manipulable objects’ means an intentional transform-ation and a viewpoint change of manipulable objects. The proof includes

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deductive reasoning at the back of the consecutive actions on manipulableobjects as follows.

.One with concrete operations can internalize actions on manipulable ob-jects. The internalization makes it possible to repeat consecutive actions,and to reverse them in mind beyond a time restriction. Therefore, for ex-ample, on the basis of the internalization, students can construct ‘View-point 3’ from ‘Viewpoint 2’ by means of transformation on manipulableobjects.

One with concrete operations can reason {Viewpoint 1, then Viewpoint5}, because one with concrete operations can unify two consecutive actions(Piaget, 1947, p. 169), on the basis of the following reasoning: {Viewpoint1, then Viewpoint 2}, {Viewpoint 2, then Viewpoint 3}, {Viewpoint 3, thenViewpoint 4}, {Viewpoint 4, then Viewpoint 5}.

Suppose that students on ‘Proof B with concrete operations’ could verb-alize the reasoning between viewpoints (e.g. {Viewpoint 1, then Viewpoint2}) into a proposition. However, the verbalization would be impossible,because one with concrete operations cannot deal with a proposition itself.This means that the reasoning between viewpoints can be done as a kindof ‘theorem-in-action’ (Vergnaud, 1982, 1988). Therefore, students cannotmake a deductive proof only with statements.

Consequently, students’ expected activities concerning ‘Proof B withconcrete operations’ are specified as follows.

1: Students can construct a viewpoint from others by means of actionson manipulable objects.

2: Students can reason {Viewpoint X, then Viewpoint Z} from {View-point X, then Viewpoint Y} and {Viewpoint Y, then Viewpoint Z}.

3: Students cannot make a deductive proof only with statements.


Students’ expected activities concerning ‘Proof B with formal operations’The following proof is an example of ‘Proof B with formal operations’.

The proof has two remarkable characteristics. One is related to a prin-ciple of sufficient reason. The other is related to a generality and a decon-textualization of representation.

Concrete operations cannot deal with propositions directly. For studentswith this operation, things such as marbles are nothing but physical objects,and manipulations of things are nothing but physical movement. It meansthat ‘Proof B with concrete operations’ remains in the world of a principleof causality. On the other hand, formal operations can deal with propos-itions, and can deduce a new proposition from others. For students withthis operation, things and their arrangements mean graphical embodimentof numerical relations, and manipulations of them correspond to deduct-ive reasoning. It means that ‘Proof B with formal operations’ reaches aprinciple of sufficient reason.

Verbalization can be a clue to recognize reasoning between viewpointsas a proposition, because verbalization seems to be helpful in order toreflect or objectify one’s thinking, and, as a result, a proposition becomesobvious in mind.

There are three necessary elements of an action on manipulable to beverbalized: the way of an action (transformation or viewpoint change), theviewpoint before the action, and the viewpoint after the action. After verb-alizing the three necessary elements, students need to combine them into aproposition. The prior viewpoint of an action corresponds to an antecedentof proposition, and the subsequent viewpoint corresponds to a consequent.The way of an action corresponds to a reason between the antecedent andthe consequent. Students with formal operations can arrange them into aproposition, which may remain particular, rather than universal or general.Furthermore, after translating each action into a particular proposition,they can organize a set of propositions into a chain of them.

Although one with formal operations can drift away from a particu-larity, and move toward a generality, ‘Proof B with formal operations’cannot be decontextualized completely. For example, the above proposi-tion of ‘Action 2’ remains particular. The italic parts show it; “Gathering

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.

paired marbles, if the white marbles consist ofonepair of marbles withone remaining, and the black marbles consist oftwopairs with one marbleremaining, then, we can seethreepaired marbles and two remaining.” Onthe other hand, in the first example of ‘Proof B with formal operations’,this proposition is represented more generally; “Gathering paired marbleslike this, (if) both consist ofsomepaired marbles with one remaining,(then) we can seesomepaired marbles and two remaining.” However, thisproposition includes some contextual words; ‘gathering’, ‘marbles’, and‘like this’. It means that the proposition is drifted away from a particularity,but cannot be decontextualized. Therefore, it is may be that students can-not write correct expressions, or cannot simplify the process of numericalexpressions according to formal rules.

Consequently, students’ expected activities concerning ‘Proof B withformal operations’ are specified as follows.

1: Students can translate actions into propositions, and can organize theminto a chain of propositions.

2: Students can use a language with a generality of representation.3: Students cannot decontextualize a proof completely.

7th grader’s real activities concerning ‘Proof B’The following data of a 7th grader’s activities were collected with a methodof teaching experiment. Toshiki was a 7th grader of public lower secondaryschool in a village. He was a high-achieving student relatively, and gotfourth grades in the five grades evaluation system of mathematics. An ob-


server prepared many systematic procedures (questions, instructions, etc.)according to Toshiki’s remarkable behaviors. In the teaching experiment,Toshiki was expected to conjecture a proposition “the sum of five con-secutive numbers is five times the middle number”, and to make someproofs.

Actually, Toshiki calculated eight cases of five consecutive numbers,and induced the previous proposition for quick computations with the ob-server’s help. The observer asked, “How can you show your friends that itis always correct in any case?” Then, Toshiki said as follows while usingfive colors of some marbles in a wide whiteboard.

The observer asked, “Can you always move them like this in any case?”Then, Toshiki added one marble to each row, transformed the arrangementas follows, and said, “Even if the numbers go on increasing, I can countthe sum in the same way.”

In considering Toshiki’s activities, it is clear that Toshiki had his viewpointof each arrangement, and that he could construct a viewpoint from otherones by means of actions on marbles. The first arrangement with five colorsand Toshiki’s saying about the last arrangement “So, five times three isthe answer” show that he could reason {1+2+3+4+5, then 5·3} which isequal to the proposition about five consecutive numbers. And consecutiveactions include deductive reasoning from assumptions to the proposition.Furthermore, after adding one marble to each row, Toshiki’s saying “Even

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if the numbers go on increasing, I can count the sum in the same way.”shows that he could see a generality of his proof.

His statements are not sufficient for a deductive proof. In order to besufficient, these statements need to include numerical relations betweenrows of the first arrangement; e.g. “Setting the center as standard, thefar left is two less, the adjacent left is one less, the adjacent right is onemore, the far right is two more.” The observer thought that Toshiki mightpossibly not express the relations, but might understand it. Then, timeagain and again, the observer asked about the reason why he could getthe last arrangement with consecutive actions. But, Toshiki could not sayanything about the numerical relations, and only repeated to say how tomove marbles. It means that Toshiki remained in the world of a principleof causality. The following scene is an example.

Observer: You can get a correct one (while pointing to the last arrangement ‘5·4’). Canyou say the reason why you can get five rows of four marbles?

Toshiki: Www. . ..Observer: Tell me anything whatever you found.Toshiki: Back to the first (making the arrangement ‘2+3+4+5+6’), the center row has four,

and cut off this part (separating three marbles). Then, move them in this way, and getit.

Observer: At that time, why can you change the left end row and the second row into four?Toshiki: Www. . ..

Finally, observer asked him to compare the number of the center row withthe others. Then, Toshiki said as follows.

Oh, I got it. The left end row has two marbles less than the center. The opposite row hastwo more. In the same way, the second one has one marble less, and the opposite has onemore. So, this right part (pointing the three separated marbles) is equal to that left part.Then, moving this part, I can get five groups of four marbles.

The observer asked to write a proof, then he wrote as follows.


[Setting up 4 as standard, taking a step from the center row toward theright, the row has five marbles. So, the number is one more. Taking onestep from the center toward the left, the row has three marbles. So, thenumber is one less. By moving the one more to the one less, the number ofeach row changes into the standard. In the same way, taking two steps fromthe center row toward the right, the row has six marbles. So, the numberis two more. Taking two steps from the center toward the left, the row hastwo marbles. So, the number is two less. By moving the two more to thetwo less, the number of each row changes into the standard.]

These statements have a chain of propositions, but retain a particularity;for example ‘Setting up4 as standard’. The observer asked, “If you areasked to show your friends the reason why it is true in the case of largenumbers, such that (258, 259, 260, 261, 262)?” Then, Toshiki rewrote aproof with a letter ‘x’ as follows.

.

[258, 259, 260, 261, 262From a standard (x), toward the right, the first row has x+1, the secondrow has x+2. Toward the left, the first row has x-1, and the second row hasx-2. Then, the left rows have x–3, and the right rows have x+3. Then, thereare five rows of x.]

This proof shows that Toshiki could move away from a particularity,and move toward a generality. However, this proof could not be decon-

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textualized completely, because the expression ‘next to theright/left’ and‘There are fiverows of x’ can be understood only by those who see thesame arrangements of marbles.

You may think that Toshiki could make a demonstration with an algeb-raic language, if asking him to make it. However, Toshiki could not makea demonstration by himself. Actually, he wrote the following proof, afterthe observer taught him how to translate the arrangements of marbles intoalgebraic expressions very carefully.In the first line of his proof, hecould not use two parentheses‘( )’ correctly. Possibly, hewanted to mean that each par-enthesis corresponds to eachrow of the first arrangement.The third line has incorrectexpressions; ‘x+–1+1x’. Con-cerning ‘–1+1x’, he seemed toforget a sign ‘+’ between‘–1+1’ and ‘x’. As a whole, he could not simplify the whole of express-ions on the basis of literal expression rules.

CONCLUSIONS

I propose the following conclusion.We can establish six levels of proof in lower secondary school math-

ematics as steps from an inductive proof to an algebraic demonstrationon the basis of three axes (contents of proof, representation of proof, andstudents’ thinking).


Among six levels, ‘Proof B with concrete operations’ corresponds to a‘germination’ of generic example in Balacheff’s research framework. Al-though generic example has an important role to shift from pragmaticproofs to intellectual proofs in Balacheff’s framework, it was unclear whatcould be a ‘germination’ of generic example. Among six levels, ‘Proof Bwith concrete operations’ corresponds to the ‘germination’, because thisproof has deductive reasoning in action, and it can ‘grow up’ to be ademonstration ‘Proof A’ through ‘Proof B with formal operations’. Fur-thermore, ‘Proof B with concrete operations’ indicates that we can/shouldselect a ‘germination’ from inductive proofs through seeing students’ prov-ing activities more carefully.

The six levels justify that we need to develop curricula of demonstrationincluding ‘Proof B with concrete/formal operations’ in addition to induct-ive proof, before learning a demonstration. The shifts between the levelsshould proceed in the following order; ‘Proof C’< ‘Proof B with concreteoperations’< ‘Proof B with formal operations’< ‘Proof A’. Especially, inthe shift between ‘Proof B with concrete/formal operations’, verbalizationof actions can be helpful for stundents to reflect or objectify one’s thinking,and, as a result, a proposition becomes obvious in mind. On the other hand,‘Proof D with concrete/formal operations’ can be useful for teachers andresearchers to grasp some covert problems students often encounter in theirproving activities. For example, ‘Proof D’ suggests that it is possible tomake a superficial shift to ‘Proof A’ via ‘Proof D’. However, it seemshard for students to make a meaningful shift from ‘Proof D with formaloperations’ to ‘Proof A’. In order to get deductive contents, students shouldgo back or go through ‘Proof B’.

The following activities encourage students to shift from ‘Proof C’ to‘Proof B with concrete operations’ (Miyazaki, 1992, 1995).

– Doing various actions with manipulable things to reach a conclusionfrom assumptions.

– Finding invariant properties or relations with a possible suggestion asto common features within the action processes.

– Producing the necessary actions forwardly and/or backwardly with apossible suggestion as to invariant properties or relations.

– Organizing actions from assumptions to conclusion on the basis ofsuccessive performance of manipulations.

– Expanding the range of situations in which to apply the organizedactions (Miyazaki, 1992, 1995).

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The following teachings seem to develop shifts between the levels. Tomake a shift from ‘Proof’ to ‘Proof B with concrete operations’, teachersneed to put a generality of actions into question. To make a shift from‘Proof B with concrete operations’ to ‘Proof B with formal operations’,teachers need to ask the reason why actions of ‘Proof B with concreteoperations’ can justify the proposition to be proved. Furthermore, teachersneed to ask students to make a proof in the case where these actions nolonger execute. To make a shift from ‘Proof B with formal operations’ to‘Proof A’, teachers need to inform the importance of a functional languageof demonstration, and to teach how to translate actions into the language(Miyazaki, 1993).

There are four kinds of questions worthy of future research.

– What kinds of significance can students find out through shifting be-tween the levels?

– In order to shift from a lower level to a more advanced level, whatkinds of activities does /should a student carry out?

– How should teachers assist a student failing to shift?– Is it necessary to subdivide the six levels for more effective instruc-

tion?– Beyond lower secondary school mathematics, what kinds of levels of

demonstration can we establish in school mathematics?

At the end, this study has the following limitation. Since actions onmanipulable objects corresponds to propositions in algebra, we can easilyfind a clear relationship between ‘Proof B with concrete operations’ and‘Proof B with formal operations’, and can translate ‘Proof B with formaloperations’ into a demonstration. However, in geometry, it is not easy tofind such a clear relationship, and to translate actions on manipulable ob-jects into propositions of demonstration. Therefore, I realize the limitationto apply the levels to geometry.

ACKNOWLEDGEMENTS

I thank Dr Nohda, Nobuhiko and my colleague for their suggestions. Ialso thank Mr. Kouichi Inou for his cooperation of data collection withhis students and a meaningful discussion with him. This study was sup-ported by grants-in-aid for Scientific Research of Ministry of Education,Science, Sports and Culture, Government of Japan [PROJECT NUMBER:07801035 (1995∼1996), 09780183 (1997∼1998)].

NOTES1. ‘Demonstration’ means human activities to reason a proposition from assumptions

deductively, and to represent the reasoning with formal language in the study.


2. The word ‘student’ of this study means a lower secondary school student, age 13–15.3. Piaget (1947) pointed out that with repeating activities of concrete operations in formal

operations (‘décalages verticaux’), one can get real meanings of the activities fromconcrete operations (p. 179). Among proofs with concrete operations, a deductiveproof has a relation of ‘décalages verticaux’ with a demonstration.

REFERENCES

Balacheff, N.: 1987, ‘Processus de preuve et situations de validation’,Educational Studiesin Mathematics18, 147–146.

Balacheff, N.: 1988a, ‘Aspects of Proof in pupils’ practice of school mathematics’, inD. Pimm (ed.),Mathematics, teachers and children, Hodder & Stoughton, London,pp. 216–230.

Balacheff, N.: 1988b, ‘Thèse: Une étude des processus de preuve en mathématique chezles élèves de collège’, Université Joseph Fourier, Grenoble.

Blum, W. and Kirsh, A.: 1991, ‘Preformal proving: examples and reflections’,EducationalStudies in Mathematics22, 183–203.

Braconne, A. and Dionne, J.J.: 1987, ‘Secondary school students’ and teachers’ un-derstanding of demonstration in geometry’, in J.C. Bergeron, N. Herscovics and C.Kieran (eds.),Proceedings of the 11th International Conference for the Psychology ofMathematics Education, Vol. III, Université de Montréal, Montréal, pp. 109–116.

De Villiers, M.: 1991, ‘Pupils’ needs for conviction and explanation within the contextof geometry’, in F. Furinghetti (ed.),Proceedings of the 15th International Conferencefor the Psychology of Mathematics Education, Vol. III, Dipartimento di Mathematicadell’Università di Genova, Assisi, pp. 255–262.

Elementary and secondary education bureau of Monbusho: 1985,‘Kyouikukatei jis-sijyoukyou nikansuru sougouteki chousahoukokusyo: chuugakkou suugaku (Generalsurvey on the accomplished status of curriculum: junior high school mathematics)(MESC–3–8517)’, Monbusho (Ministry of Education, Science, Sports and Culture),Tokyo.

Fawcett, H.P.: 1938,The nature of proof: a description and evaluation of certain proced-ures used in a senior high school to develop an understanding of the nature of proof(TheNational Council of Teachers of Mathematics, The Thirteenth Yearbook), AMS PRESS,New York.

Fischbein, E.: 1982, ‘Intuition and proof’,For the Learning of Mathematics3, 9–24.Galbraith, P.L.: 1981, ‘Aspects of proving: A clinical investigation of process’,Educational

Studies in Mathematics12, 1–28.Hanna, G.: 1989, ‘More than formal proof’,For the learning of mathematics9(1), 20–25.Hanna, G.: 1995, ‘Challenges to the importance of proof’,For the Learning of Mathematics

15, 42–49.Koseki, K. et al.: 1978, ‘Zukei niokeru ronsyou sidou nituite: sono 1 (Teaching a demon-

stration in geometry: the first part)’,Journal of Japan Society of Mathematical Education(Mathematics Education)60, 12–19.

Koseki, K. et al.: 1980, ‘Zukei niokeru ronsyou sidou nituite: dai 3 ji houkoku(sono 1)(Teaching a demonstration in geometry: the third report(the first part))’,Journal of JapanSociety of Mathematical Education (Mathematics Education)62, 59–65.

Kunimoto, K.: 1994, ‘Syoumei Gainen no Tayousei (A diversity of the concept ofproof)’, in Proceedings of 27th Research Conference of Japan Society of MathematicalEducation(Kohbe: Hyogo University of Teacher Education), pp. 457–462.

68 MIKIO MIYAZAKI

Kunimune, S.: 1987, ‘ “Ronsyou no igi” no rikai nikansuru hattatuteki kenkyu (A develop-mental study on the understanding of “significance of demonstration”)’, in K. Koseki(ed.), Zukei no ronsyousidou(Teaching a demonstration in geometry), Meijitosho,Tokyo, pp. 129–158.

Lesh, R., Post, T. and Behr, M.: 1987, ‘Representations and translations among repres-entations in mathematics learning and problem solving’, in C. Janvier (ed.),Problemsof representation in the teaching and learning of mathematics, Erlbaum, Hillsdale, NJ,pp. 33–40.

Miyazaki, M.: 1992, ‘Suisokushitakoto ni ippansei ga arukoto wo simesutameni okon-awareru katsudou: Seito ha donoyounisite seiseitekinarei niyoru setsumei wo okonauka?(Students’ activity in order to show a generality of conjecture: how does one student usea generic example to make explanations?)’,Journal of Japan Society of MathematicalEducation (Reports of Mathematical Education), 57, 3–19.

Miyazaki, M.: 1993, ‘Translating a sequence of concrete actions into a proof’, in I. Hira-bayashi, N. Nohda, K. Shigematsu and Fou-Lai Lin (eds.),Proceeding of the 17thConference of the International Group for the Psychology of Mathematics Education,Vol. 3, University of Tsukuba, Tsukuba, Japan, pp. 130–137.

Miyazaki, M.: 1995, ‘Meidai no fuhendatousei no riyuu wo gutaibutsu nitaisuru syokoui deshimesukoto nikansuru kenkyu (A Study on showing the reason of an universal validityof propositions with actions on manipulable objects)’,Journal of Science Education inJapan, 19(1), 1–11.

Piaget, J.: 1947,La psychologie de l’intelligence, Colin, Paris.Semadeni, Z.: 1984, ‘Action proof in primary mathematics teaching and in teacher

training’, For the Learning of Mathematics4(1), 32–34.Senk, S. L.: 1983,Proof-writing achievement and Van Hiele levels among secondary

school geometry students, Unpublished doctoral dissertation, University of Chicago,Chicago.

Sugiyama, Y.: 1986,Kouritekihouhou ni motoduku sansuu/suugaku no gakusyu shidou(Mathematics teaching based on axiomatic methods), Touyoukan, Tokyo.

Vergnaud, G.: 1982, ‘Cognitive and developmental psychology and research in math-ematics education: Some theoretical and methodological issues’,For the Learning ofMathematics3(2), 31–41.

Vergnaud, G.: 1988, ‘Multiplicative structures’, in J. Hiebert and M. Behr (eds.),ResearchAgenda for Mathematics Education: Number Concepts and Operations in the MiddleGrades, Lawrence Erlbaum Associates, Virginia, pp. 141–161.

Walther, G.: 1986, ‘Illuminating examples: An aspects of simplification in the teach-ing of mathematics’,International Journal of Mathematical Education in Science andTechnology17, 263–273.

Wittman, E.C. and Müller, G.: 1988, ‘Wann ist ein Beweis ein Beweis?’, in Bender, P. (ed.),Mathematikdidaktik: Theorie und Praxis, Cornelsen, Berlin, pp. 237–257.

Department of Mathematics Education,Faculty of Education, Shinshu University,Roku – Ro, Nishi-Nagano, Nagano, 380-8544, Japan,Telephone: +81-26-238-4105, Fax: +81-26-234-5540,E-mail: [email protected],homepage:http://math-edu01.shinshu-u.ac.jp/General/Staff/Miyazaki/miyazaki-e.html

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Levels of Proof in Lower Secondary School Mathematics