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CHALLENGES IN CURRICULUMDEVELOPMENT FOR MATHEMATICALPROOF IN SECONDARY SCHOOL: CULTURALDIMENSIONS TO BE CONSIDEREDYUSUKE SHINNO, TAKESHI MIYAKAWA, HIDEKI IWASAKI, SUSUMU KUNIMUNE,TATSUYA MIZOGUCHI, TERUMASA ISHII, YOSHITAKA ABE

Depuis les Grecs, qui dit Mathématique, dit demonstration (Bourbaki, 1954).

Within the Japanese mathematics education community, we
use two terms for proof: shōmei (証明) and ronshō (論証).
Both terms share the same Chinese character, shō 証. Literally, shō 証 means ‘evidence’ or ‘proof’, mei明 means‘clarity’ or ‘shining’, and ron論 means ‘argument’ or ‘logic’[1]. Sometimes ronshō is seen as a special type of shōmei.
For example, the current categories of research presentations
at the annual conference of the Japan Society of Mathematical Education (JSME) include the category “shōmei,including ronshō and setsumei” (setsumei means ‘explanation’) [2]. Starting from this distinction, we explore culturaland linguistic issues related to teaching proof in Japan andan epistemological perspective on what constitutes proof.
The difference between shōmei and ronshō is similar, but
not identical to the distinction between preuve and démonstration in French. Balacheff uses preuve and démonstrationin the following way:
We call proof an explanation accepted by a given community at a given moment… Within the mathematicalcommunity only explanations adopting a particularform can be accepted as proofs. They are an organizedsuccession of statements following specified rules: astatement is known to be true or is deduced from thosewhich precede it using a deductive rule taken from awell defined set of rules. We call such proofs “démonstrations”. (Balacheff, 1987, p. 148; English translationfrom Reid & Knipping, 2010, pp. 32–33)
In English we find the expression ‘demonstrative geometry’ in texts from the early twentieth century (e.g., Smith,
1919; Taylor, 1930). Japanese translations use ronshō kikafor ‘demonstrative geometry’; kika means ‘geometry’.
Shōmei kika is never used as a translation. As ‘demonstration’ is rarely used to refer to mathematical proofs these days
(Reid & Knipping, 2010, p. 32), and as Balacheff translatesdémonstration as ‘mathematical proof’ we will use the samephrase here, without intending to imply that ronshō means
exactly what Balacheff means by démonstration or what an
English speaker might mean by ‘mathematical proof’. We
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will translate shōmei as ‘proof’, intending to capture its
broader meaning. Already in attempting to specify the topic
of our article, we have evidence for cultural and linguistic
differences related to proof.
We believe that such cultural and linguistic differences
may affect curriculum development in different countries.
Our aim here is thus two-fold. First, we wish to reveal thecultural and linguistic issues that need to be considered in
the development of curricular content and sequencing forteaching mathematical proof in secondary schools in Japan.
Second, we seek to elaborate an epistemological perspectivethat may allow us to understand what constitutes proof in the
curriculum of a given country, so that our results may be
applied in international contexts. This second aim can beachieved by looking back at cultural discussions on Japanand by looking ahead to further research opportunities in theinternational context.
Language and the triplet which composes the
Mathematical Theorem
Let us discuss a bit more the two Japanese terms shōmei andronshō. Some Japanese works distinguish the word ronshōfrom shōmei by referring to their relationship with the system of mathematics. For example, in a book published forJapanese mathematics teachers in the New Math era (JSME,1966), shōmei was described as “mainly related to the truthor falsity of a proposition”, whereas ronshō was described as“mainly related to the truth or falsity of a set of propositions” (p. 366, emphasis added). Some Japanese researchers(e.g., Hirabayashi, 1991; Iwasaki, 1985; Minato, 1974) havealso advocated a similar distinction, such that shōmei isrelated to deriving consequences from premises for establishing the truth of a proposition, while ronshō is related tothe (axiomatic) system in which logical relations between
propositions take place.
These differences in the use of the two words shōmei andronshō reveal something about the Japanese mathematics
curriculum. That is, the use of ronshō indicates an emphasis
on a systematic approach to mathematics. This is probably
because the content and sequencing of the Japanese curriculum in geometry bears a strong resemblance to that of
the geometry in Euclid’s Elements. Miyakawa (2017)
pointed out that the Japanese meaning of shōmei is reserved

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particularly for general statements and includes constructing
the system of geometry. The word ronshō describes more
precisely constructing the system of geometry but is not
used in Japanese textbooks.
Antonini & Mariotti (2008) state “it is not possible tograsp the sense of a mathematical proof without linking itto the other two elements: a statement, that the proof provides a support and a theory, i.e. the theoretical frame within
which this support makes sense.” (p. 403). According to thismodel a ‘Mathematical Theorem’ is a system of statement,
proof and theory. The third element, theory, can be considered as a certain mathematical context that provides anunderlying basis for a mathematical proof within a domainthat can be geometrical, arithmetic, algebraic, and so on. In
these terms the distinction between shōmei and ronshō is at
the level of the theory involved. Hanna & Jahnke (2002) distinguish between ‘small’ and ‘large’ theories. They giveaxiomatic Euclidean geometry as an example of a large theory, suggesting that one way to make sense of ronshō is as
proof in such a large theory. A proof in a small theory,
locally organized and perhaps referring to physical mechanisms, seems more like shōmei.
This difference, at the level of theory, is epistemologicalrather than linguistic. We now consider some linguistic
issues related to statement and proof in Japanese, and thenreturn to the epistemological level of theory.
Linguistic issues: universal quantification
When we turn to the linguistic aspects of statement andproof, we must consider the influence of natural language
used in daily life on the teaching and learning of mathematical proof. Concerning this critical point, Mejia-Ramos andInglis (2011) investigated the semantic contamination ofmathematical proof, where the meaning of a term associated with proof used in natural language influences how theterm is understood by students in the mathematical discourse. Although their study focused on the English usage ofthe term ‘proof’ as both a noun and a verb, we will examine
not only the usage of the term but also the form of the statement to be proven in a certain mathematical domain. In
addressing the linguistic issues faced with respect to statement and proof, different formulations of statement andmultiple meanings of proof can be identified according to
the language used in each country. This perspective allows
us to further discuss some cultural aspects.
In the Japanese national curriculum, which is called the‘Course of Study’, the introduction of proof takes place ingrade 8, and the textbooks usually include chapters on proofin geometry. One characteristic of the statements in Japanese
textbooks is that all statements required to be proven areabout general objects, and statements about specific objects
are not for proving (Miyakawa, 2017). This means that thestatement to be proven is considered as a universal proposition rather than as another type of proposition, such as a
singular or existential proposition. Regarding the treatmentof universally valid statements, some previous studies havepointed out the problem of the relationship between naturaland mathematical language from a logical point of view (e.g.
Cabassut et al., 2012; Durand-Guerrier et al., 2012). We
think that it is also important to discuss this matter by focus-

ing on the ordinary language used in a given country,
because this may affect how students engage with mathematical proof.
In the case of the universal proposition, in many countries the statement is usually formulated as a phrase like
“The sum of the interior angles of any triangle is 180°” or“The sum of any two even integers is even”. While thesestatements are also objects for proving in Japan, universal
quantification using ordinary language and words such as‘any’ and ‘all’ is rarely encountered in secondary mathematics textbooks; that is to say, universality is rarely
formulated in written form. Often, Japanese students (andteachers as well) are required to interpret a mathematicalstatement to be proven as a universal proposition without
any quantification. This is also the case for statements inother domains aside from geometry. For example, in lower
secondary school algebra universal propositions concerning
numerical properties, such as “The sum of two even numbers is an even number”, are formulated and proven without
a quantifier. This is probably due to the fact that the Japanese language does not use articles [3]. In addition, inJapanese grammar, there is no distinction between the singular and plural form of a noun in both writing andspeaking. This matter may also influence understanding of
the words related to quantification. However, we would
not like the reader to misinterpret this as meaning that theJapanese language does not have the words to express quantification. In fact, there are ordinary Japanese words that
correspond to English words such as ‘any’, ‘arbitrary’, and‘all’. These Japanese words might be orally used by teachers in the classroom. However, these words are used only
if the quantification is consciously intended in the proving
process. Therefore, most Japanese students have few opportunities to think about quantification in their own language.
Most of them do not encounter quantifiers formulated inmathematical words or symbols such as ‘∀’ or ‘∃’, based on
predicate logic until they reach the undergraduate level ofuniversity.
The same statement could be formulated in different
ways, such as in diagrams, ordinary language, and mathematical language. As far as the linguistic formulation ofuniversal quantification is concerned, there is no explicit
progression from implicitly quantified statements throughverbally quantified statements to formally quantified statements in the Japanese curriculum. It may be a matter of thesemantic contamination between natural language and mathematical discourse that may affect how teachers and students
engage with proof and proving of a general statement (seeMejia-Ramos & Inglis, 2011). A previous study targeting
prospective Japanese primary teachers found that most of
the participants could not evaluate the validity of solutionsof a proving task in terms of a universal aspect of the statement (Shinno et al, 2012). This is probably because mostprospective primary teachers, who were not mathematics
majors at university, do not have opportunities to learn how
to use universal quantification in their proving activities.
Of course, we cannot conclude that Japanese students’ orteachers’ difficulties concerning universal quantification
are due solely to linguistic issues. Rather, what we would
like to suggest here is that there is a gap between the levels
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of linguistic formulation of statements with regard to thetreatment of the universal quantifier in the Japanese curriculum. It seems that this gap may affect the teaching andlearning of mathematical proof across the grades.
An exceptional case: proof by mathematicalinduction
It is important to note that there is a remarkable exception
to this point. A statement to be proven by mathematicalinduction is a universal proposition explicitly described as‘for all natural numbers’ using ordinary Japanese language.
This linguistic issue is also related to the epistemologicalnature of the Mathematical Theorem. We will illustrate thisby sharing a typical example from a Japanese textbook. Our
focus is not, however, on the details, but rather on what theexample reveals about how a statement using ordinary
words with an explicit universal quantification may matter,as well as the epistemological nature of the Mathematical
Theorem.
In Japan, proof by mathematical induction is taught inthe chapter ‘Sequences’. This is official content that isincluded in the upper secondary school subject ‘Mathematics B’ which is mostly studied in grade 11. Figure 1 andFigure 2 are excerpted from this chapter (Takahashi et al.,2012). Figure 2, which includes a proof by mathematicalinduction, is introduced in the final part of the chapter. One
can find similar descriptions elsewhere, for instance, inHarel (2002), Polya (1954), Tall et al. (2012), and probably
in many textbooks in other countries.
Figure 1 shows a description of the introduction of theformula:There is a statement (12 + 22 +...+ n2 = n(n +1)(2n +1) / 6)
and its proof, where a ready-made identity (k 3 − (k −1)3 =3k 2 − 3k +1) is applied for deducing the original statement.
This case is similar to what Tall et al. (2012) illustrate as analgebraic proof. On the other hand, one can notice that theequations, including the conventional notation ‘…’, are used
for representing ‘the sum of the first n terms’ and the opera-

Figure 1.

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A n excerpt from the chapter ‘Sequence’ in the
textbook. [4]tions. This notation in the proof involves some ambiguities
from the mathematical point of view as the operations
related to the dot notation, such as ‘adding to each side of theequation’ or transforming these equations into 12 + 22 +...+ n2= n(n +1)(2n +1) / 6 , are carried out without any explanation. Additionally, one may also notice that universal
quantification is not used for the formulations of the statement or proof in Figure 1.
In Figure 2, unlike in Figure 1, there is the statement
which includes a universal quantification formulated byusing the ordinary word ‘all’. As we have already mentioned, since universal quantification is rarely used inJapanese textbooks, even in natural language, this explicit
use is an exceptional case. In Figure 2, the statement isproven by mathematical induction, although the reference tothe Principle of Mathematical Induction (i.e., Peano’s fifth
axiom for the foundation of natural numbers) is still implicit.
In Figure 2, the notation ‘…’ is also used in a similar manneras in Figure 1, but it is only used to represent the equation
rather than to prove the statement or in operations on equations. The conventional use of dot notation can become more
rigorous if predicate logic and proof by mathematical induction are introduced. When comparing Figure 1 with Figure
2, one can identify an essential transition between them,which is related to the nature of theories in which the statement and proof take place. We will elaborate this transition
as a ‘reference epistemological model’ in the next section.
The necessity of a reference epistemologicalmodel
In the previous sections, we discussed some linguistic issues
related to proof that can be found in the Japanese curriculum. Although the illustrations we made are taken from theJapanese context, it is important to mention the theoreticaland methodological implications of the present study inorder to extend the discussion from the local context to theinternational context. Otherwise, it is difficult to exchange
our research results within the international community ofmathematics education. For this reason, we will pay special
attention to epistemological issues related to proof. The epistemological perspective is crucial for any researcher toFigure 2.

Proof by mathematical induction adapted from the
textbook.[5]FLM 38(1) - March 2018 - text_FLM 2018-02-10 2:13 PM Page 29

A proposition for an REM
Figure 3 represents the basic tenets of the REM we havedeveloped, based on some prior research results in the international context: the triplet which constitutes the
Mathematical Theorem and some additional input. In Figure
3, there are three layers according to the levels of theory,
which are adapted from the distinction between ‘small theory’ and ‘large theory’ (Hanna & Jahnke, 2002) and the ideaof local organization (Freudenthal, 1971, 1973). Statements
and proofs are interrelated and take place at each theoretical level, and the continuous evolution of each element may

depend on mathematical domains wherein the proof is carried out. We call these layers, respectively, ‘real-world
logic’, ‘local theory’, and ‘axiomatic theory’. The first levelis not the primary focus of study in secondary mathematics,
while it may be involved in the process of teaching mathematical proof. When someone accepts a geometric property
to be true by means of a physical experiment or a measurement of a real-world object, it can be interpreted that thetheory behind this argument is ‘real-world logic’.
We suggest that the developments of each element areusually implicit but essentially determine the teaching andlearning of mathematical proof. Even though mathematical
proof based on the axiomatic theory level is not explicitly
dealt with, its implicit nature might be involved in the mathematical proof of secondary school mathematics. This iswhat we have already discussed, relating to the distinction
between shōmei and ronshō described at the beginning of thepaper. For elaborating the two levels of the nature of theories
it is reasonable to call the two distinct levels of the natureof theories ‘the local theory’ and ‘the quasi-axiomatictheory’ respectively. We added the prefix ‘quasi’ here
because the axiomatization in secondary schools is not strict.
The terms axiom or postulate are not used, some properties
are introduced after observation, and some are admitted
implicitly. Therefore, the transition between local and quasiaxiomatic theory can be essential in developing indeveloping a curriculum in Japan. In this transition, thedevelopment in other elements, statements and proofs arealso crucial for the (local or axiomatic) nature of theory.
The two levels, local and quasi-axiomatic theory, may
allow us to understand the transition within the textbook
chapter which includes proof by mathematical induction(e.g., Figure 1 and 2). In doing so, we can become aware thatthe teaching of proof by mathematical induction can be seen
as ‘an exceptional case’ in the Japanese curriculum, because
one can find a statement with an explicit universal quantification using ordinary words as well as the evolution fromlocal theory to quasi-axiomatic theory. By showing such an
example, we intend to illustrate how the proposed REM may
work to describe and analyze the evolution that can be identified in the curricula of different countries.

Figure 3.

Final remarks
Since each country has its own curriculum, we cannot ignore
the influences of curricular contents and sequencing on students’ constructions of mathematical proof (Hoyles, 1997).
In this paper, we illustrated how cultural aspects (linguistic
issues in particular) may potentially affect curriculum development of mathematical proof in the Japanese educational
context. As a result, we discussed how the Japanese language
may influence the formulation of universal quantification as
well as different technical meanings of proof in the Japanese
language by comparing with other languages such as English
and French. This issue is a challenge in curriculum development in that there may be obstacles in applying priorinternational research results to the Japanese context. This
also implies that it is important to consider cultural differences
when examining mathematical proof in different countries.
On the other hand, we also mentioned the necessity of an
REM to extend the discussion from the local context to theunderstand the object of the study undertaken (Balacheff,
2008; Reid, 2015). This perspective may correspond to what
Bosch and Gascón (2006, 2014) called the ‘reference epistemological model’ (REM), which constitutes the basic
theoretical lens for researchers to analyze differing mathematical knowledge among different institutions (Chevallard,
1985/1991). Such an REM is also needed for the development of curricula because this model can help to clarify what
constitutes proof in the curriculum of a given country.
There is another purpose of an REM. It is to detach ourselves, as researchers, from educational institutions where thepractices of mathematics education occur, such as the development of curricula, textbooks, teaching materials, etc.
Japanese researchers, however, cannot easily detach themselves from educational institutions. This is probably due to theuniversity researcher’s close relationship to the community ofmathematics education; for example, he or she may be a textbook author, a member of the advisory board of the Ministryof Education, a teacher educator or an adviser to in-service
teachers in schools. In fact, research on mathematics education in Japan often aims at developing a curriculum that may
have some impact on the national curriculum or mathematics
textbooks. The theoretical and methodological reflection that
we will address is important because, as Bosch and Gascón(2006) stated, “we need to elaborate our own ‘reference’ modelof the corresponding body of mathematical knowledge” (p. 57).
To conceptualize such an REM we need to be able to free ourselves from our educational institution. We think that thisdetachment is a crucial point, especially for Japan.

A n image of the proposed REM.

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international context. Regarding the proposed REM, weconceptualized that the statements and proofs are interrelated according to the nature of theory and illustrated how
this model helps to describe the evolution of each element.
This model may help us to understand the gaps in the evolution of the formulation of a statement and how different
meanings of proof relate to the distinction between local and(quasi-) axiomatic levels of theory. Consequently, it is meaningful for Japanese researchers (and hopefully forinternational researchers as well) to understand why teaching of proof by mathematical induction is ‘an exceptional
case’ in the Japanese curriculum.
Further research is needed to examine how the linguistic
issues we identified influence actual classroom lessons andhow the constructed REM can analyze such classroom phenomena. Although we have focused on specific linguistic
issues related to proof, it is also important to investigate different cultural constraints in the educational contexts ofJapan and other countries. Since what constitutes ‘proof’varies from institution to institution and country to country,
further international comparative studies are needed todevelop a deeper understanding of each country’s curriculum and classroom teaching. The model we have outlined
may be elaborated upon by future empirical studies.
Notes[1] It should be noted that these three characters have several other meanings, since the same character can have both different meanings anddifferent pronunciations.
[2] In fact, at the most recent conference held in 2016, there were elevenpresentations within the category shōmei (including ronshō and setsumei).
Four of those were categorized as shōmei, two studies were categorized asronshō, three studies were categorized as setsumei (explanation) and two
studies were on other topics (logical thinking and justification).
[3] We do not intend to claim that this feature of the Japanese language necessarily affects how universal quantifications are spoken and written. A
reviewer suggested to us that the Russian language also does not use articles, but many Russian textbooks do use universal predicates. In
advanced-level Japanese textbooks, there are many universal quantifiers
based on predicate logic. What we would like to point out here is the factthat Japanese secondary textbooks rarely use universal quantification formulated in ordinary Japanese language.
[4] The original of Figure 1 can be found in a Japanese mathematics textbook (Takahashi, 2012, p. 23). The statement and proof were translated by
the authors.
[5] The original of Figure 2 can be found in a Japanese mathematics textbook (Takahashi, 2012, p. 40). The statement and proof were translated andrearranged by the authors. Although there are no essential modifications,
the words ‘base step’ and ‘inductive step’ have been added to the proof text.

Acknowledgements
We are very grateful to the editors and reviewers for helpful comments on
earlier versions of this article. Special thanks go to Marianna Bosch andYves Chevallard for valuable discussions, and Hiroaki Hamanaka for useful comments on earlier drafts. This research is supported by JSPS
KAKENHI (JP24330245 and JP15H03501).

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