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This article explores instances of symbol polysemy within mathematics as it manifests in different areas within the mathematics register. In particular, it illustrates how even basic symbols, such as ‘+’ and ‘1’, may carry with them meaning in ‘new’ contexts that is inconsistent with their use in ‘familiar’ contexts. This article illustrates that knowledge of mathematics includes learning a meaning of a symbol, learning more than one meaning, and learning how to choose the contextually supported meaning of that symbol.
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TMME, vol7, nos.2&3, p.247
Polysemy of symbols: Signs of ambiguity
Ami Mamolo1
Department of Mathematics and Statistics
Queen’s University
Abstract. This article explores instances of symbol polysemy within mathematics as it manifests
in different areas within the mathematics register. In particular, it illustrates how even basic
symbols, such as ‘+’ and ‘1’, may carry with them meaning in ‘new’ contexts that is inconsistent
with their use in ‘familiar’ contexts. This article illustrates that knowledge of mathematics
includes learning a meaning of a symbol, learning more than one meaning, and learning how to
choose the contextually supported meaning of that symbol.
1 Queen’s University
Department of Mathematics and Statistics
Kingston, ON
K7L 3N6
Email: ami@mast.queensu.ca
The Montana Mathematics Enthusiast, ISSN 1551-3440, Vol. 7, nos.2&3, pp.247- 262
2010©Montana Council of Teachers of Mathematics & Information Age Publishing
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Ambiguity in mathematics is recognized as “an essential characteristic of the conceptual
development of the subject” (Byers, 2007, p.77) and as a feature which “opens the door to new
ideas, new insights, deeper understanding” (p.78). Gray and Tall (1994) first alerted readers to
the inherent ambiguity of symbols, such as 5 + 4, which may be understood both as processes
and concepts, which they termed procepts. They advocated for the importance of flexibly
interpreting procepts, and suggested that “This ambiguous use of symbolism is at the root of
powerful mathematical thinking” (Gray and Tall, 1994, p.125). A flexible interpretation of a
symbol can go beyond process-concept duality to include other ambiguities relating to the
diverse meanings of that symbol, which in turn may also be the source of powerful mathematical
thinking and learning. This article considers cases of ambiguity connected to the context-
dependent definitions of symbols, that is, the polysemy of symbols.
A polysemous word can be defined as a word which has two or more different, but
related, meanings. For example, the English word ‘milk’ is polysemous, and its intended
meaning can be determined by the context in which it is used. Mason, Kniseley, and Kendall
(1979) observed that word polysemy in elementary school reading tasks was a source of
difficulty – students demonstrated a tendency to identify the common meaning of words, despite
being presented contexts in which an alternative meaning was relevant. Durkin and Shire (1991)
discussed several instances of polysemous words within the mathematics classroom. They noted
confusion in children’s’ understanding of expressions that had both mathematical and familiar
‘everyday’ meanings. In resonance with Mason, Kniseley, and Kendall (1979), Durkin and Shire
found that “when children misidentified the meaning of an ambiguous word in a mathematical
sentence, the sense they chose was often the everyday sense” (1991, p.75).
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In addition to potential confusion between a word’s ‘everyday’ meaning and its
specialized meaning within mathematics, learners are also often faced with polysemous terms
within the mathematics register. Zazkis (1998) discussed two examples of polysemy in the
mathematics register: the words ‘divisor’ and ‘quotient’. These words were problematic for a
group of prospective teachers when confusion about their meanings could not be resolved by
considering context – both meanings arose within the same context. In the case of ‘divisor’,
attention to subtle changes in grammatical form was necessary to resolve the confusion. In the
case of ‘quotient’, a conflict between familiar use and precise mathematical definition needed to
be acknowledged and then resolved. Zazkis relates to the mathematics register Durkin and
Shire’s (1991) suggestion that enriched learning may ensue from monitoring, confronting and
‘exploiting to advantage’ ambiguity.
I would like to continue the conversation on polysemy within the mathematics register,
and extend its scope to consider the polysemy of mathematical symbols. This article examines
the polysemy of the ‘+’ symbol as it manifests in different areas within the mathematics register.
The article begins with a reminder of the ‘familiar’ – addition and addends in the case of natural
numbers – as well as a brief look at an example where meanings of symbols are extended within
the sub-register of elementary school mathematics. Following that, I focus on two instances
where meanings of familiar symbols are extended further: the first involves modular arithmetic,
while the second involves transfinite arithmetic. I chose to focus on these cases for two reasons:
(i) the extended meanings of symbols such as ‘a+b’ contribute to results that are inconsistent
with the ‘familiar’, and (ii) they are items in pre-service teacher mathematics education.
This article presents an argument that suggests that the challenges learners face when
dealing with polysemous terms (both within and outside mathematics) are also at hand when
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dealing with mathematical symbols by starting with ‘obvious’ and well-known illustrations of
symbol polysemy in order to prepare the background to analogous but not-so-obvious
observations. It focuses on cases where acknowledging the ambiguity in symbolism and
explicitly identifying the precise, context-specific, meaning of that symbolism go hand-in-hand
with understanding the ideas involved.
Building on the familiar: from natural to rational
The main goal of this section is to establish some common ground with respect to ‘familiar’
meanings of symbols of addition and addends. In the subsequent sections, the meanings of these
symbols will be extended in different ways, dependent on context. Their extensions will be
explored so as to highlight ambiguity in meanings which can be problematic for learners should
it go unacknowledged.
Since experiences with symbols in mathematics often start with the natural numbers, it
seems fitting that this paper should start there as well. Natural numbers may be identified with
cardinalities2, or ‘sizes’, of finite sets – where ‘1’ is the symbol for the cardinality of a set with a
single element, ‘2’ the symbol for the cardinality of a set with two elements, and so on. With
such a definition, addition over the set of natural numbers may be defined as the operation which
determines the cardinality of the union of two disjoint sets (Hrbacek and Jech, 1999; Levy,
1979). As noted earlier, a symbol such as ‘1+2’ can be considered a procept, and as such may be
viewed as both the process of adding two numbers and also the concept of the sum of two
numbers. For the purposes of this paper, it is enough to restrict attention to the concept of ‘1+2’
(and hereafter all other arithmetic expressions), though the process of ‘1+2’ is no less
polysemous.
2 Natural numbers may also be identified with ordinals; however addition of ordinals is not commutative (Hrbacek
and Jech, 1999), and thus in doing so one loses a fundamental property of natural number arithmetic.
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A more formal definition of addition over the set of natural numbers,Գ, can be written as
the following:
xif A and B are two disjoint sets with cardinalities a, b in Գ, then the sum a + b is
equal to the cardinality of the union set of A and B, that is, the set (AB).
Table 1 below summarizes the meanings of the symbols ‘1’, ‘2’, and ‘1+2’ when
considered within the context of natural number addition:
Symbol Meaning in context of natural numbers
1 Cardinality of a set containing a single element
2 Cardinality of a set containing exactly two elements
1+2 Cardinality of the union set
Table 1: Summary of familiar meaning in Գ
Sensitivity towards various meanings attributed to arithmetic symbols is endorsed by teacher
preparation guides and texts, such as Van de Walle and Folk’s Elementary and Middle School
Mathematics, which notes that “each of the [arithmetic] operations has many different meanings”
and that “Care must be taken to help students see that the same symbol can have multiple
meanings” (2005, p.116). Van de Walle and Folk highlight as an example the ‘minus sign’,
which they observe has a broader meaning than ‘take away’. However, they seem to take for
granted that their readers are familiar with exact mathematical meaning of arithmetic symbols.
For instance, they introduce addition as a ‘big idea’ which “names the whole in terms of the
parts” (p.115), but without explicitly defining addition over the natural numbers, nor
distinguishing conceptually natural number addition from, say, rational number addition. Rather,
they recommend that “the same ideas developed for operations with whole numbers should apply
to operations with fractions. Operations with fractions should begin by applying these same ideas
to fractional parts” (p.244). This advice has dubious implications both conceptually and
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pedagogically when we consider the definition of natural (and whole) numbers as cardinalities of
sets. Rational numbers do not have an analogous definition as cardinalities, and indeed, the idea
that a set might contain ½ or ¼ of an element is not meaningful. Instead, rational numbers may
be described as numbers that can be represented as a ratio v:w, where v and w are integers.
Campbell (2006) warns against conflating whole number and rational number arithmetic,
and suggests that merging the two ideas may be the root of both conceptual and procedural
difficulties during an individual’s transition from arithmetic to algebra. Campbell identifies a
source for this confusion as the
“relatively recent development in the history of mathematics that has logically
subsumed whole (and integer) numbers as a formal subset of rational (and real)
numbers. This development appears to have motivated and encouraged some serious
pedagogical mismatches between the historical, psychological, and formal
development of mathematical understanding” (2006, p.34)
Campbell asserts that the set of natural (and whole) numbers are not a subset of the set of rational
numbers, but rather are isomorphic to a subset of the rational numbers. As such, this distinction
is significant as it carries with it separate definitions for the set of natural numbers (and its
corresponding arithmetic operations) and the subset of the rational numbers to which it is
isomorphic. In particular, although the symbols appear the same, their meaning in this new
context is different, as illustrated in Table 2.
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Symbol Meaning in context of rational numbers
1 A ratio of integers equivalent to 1:1
2 A ratio of integers equivalent to 2:1
1+2 A ratio of integers equivalent to 3:1
Table 2: Summary of extended meaning in Է
Campbell suggests that although the
“standard view… is to claim that young children are simply not developed or
experienced enough to grasp the various abstract distinctions and relations to be
made between whole number and rational number arithmetic… it may be the case
that the cognitive difficulties in children’s understanding of basic arithmetic is a
result of selling short their cognitive abilities” (2006, p.34).
Thus, although it may seem cumbersome to distinguish between 1 א Գ and 1 (or 1.0) א Է, where
Է symbolizes the set of rational numbers, it is conceptually important. In a broad context, the
operation of addition may be considered as a binary function, and as such, its definition depends
on the domain to which it applies. Recalling Table 1, we may add another row:
Symbol Meaning in context of natural numbers
1 Cardinality of a set containing a single element
2 Cardinality of a set containing exactly two elements
1+2 Cardinality of the union set
+ Binary operation over the set of natural numbers
Table 1B: Summary of familiar meaning in Գ
It is useful for purposes of clarity in this paper to distinguish between different definitions of the
addition symbol as they apply to different domains. The symbol +N will be used to represent
addition over the set of natural numbers, +Z to represent addition over the set of integers, and +Q
to represent addition over the set of rational numbers. +N and +Q have, to apply Zazkis’s (1998)
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phrase, the ‘luxury of consistency’ – despite the different definitions, 1 +N 2 = 3 and 1 +Q 2 = 3.
However, if we consider summing non-integer rational numbers, there are pedagogical
consequences for neglecting the distinction between natural number addition and rational
number addition. In particular with respect to motivating and justifying the specific algorithms
applicable to computations with fractions, and also with respect to interpreting student error. A
classic error such as
may be seen as a reasonable interpretation of Van de Walle and
Folk’s (2005) advice of applying whole number operations to fractional parts. Without
distinction,
is, for a learner, equivalent to
. This latter expression is logically
problematic: as a binary function, +N is applicable only to elements in its domain – the set of natural
numbers – in which the fractions

are not.
may be viewed as an algorithm that
restricts the function +N to elements of its domain (the two numerators, and the two denominators).
Adequate knowledge of addition as an operation whose properties depend upon the domain to which it
applies, offers teachers a powerful tool to address the inappropriateness of this improvised algorithm.
The following sections build on the idea of addition as a domain-dependent binary
operation. They explore examples of two domains for which a ‘luxury’ of consistency is absent:
(i) the set {0, 1, 2} and (ii) the class of (generalised) cardinal numbers. When clarification is
necessary, the notation +3 will be used to represent addition over the set {0, 1, 2} (i.e. modular
arithmetic with base 3), and + will be used to represent addition over the class of cardinal
numbers (i.e. transfinite arithmetic). The sections take a close look at familiar and not-so-familiar
examples of domains for which an understanding develops hand-in-hand with an understanding
of the associated arithmetic operations.
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TMME, vol7, nos.2&3, p.255
Extending the familiar: an example in modular arithmetic
Modular arithmetic is one of the threads of number theory that weaves its way through
elementary school to university mathematics to teachers’ professional development programs – it
is introduced to children in ‘clock arithmetic’, it is fundamental to concepts in group theory, and
it is a concept that has helped teachers develop both their mathematical and pedagogical content
knowledge. This section considers the context of group theory. It takes as a generic example the
group Ժ3 – the group of elements {0, 1, 2} with the associated operation of addition modulo 3.
Within group theory the meanings of symbols such as 0, 1, 2, +, and 1+2 are extended
from the familiar in several ways. As an element of Ժ3, the symbol 0 is short-hand notation for
the congruence class of 0 modulo 3. That is, it is taken to mean the set consisting of all the
integral multiples of 3: {… -6, -3, 0, 3, 6, …}. Similarly, the symbol 1 represents the congruence
class of 1 modulo 3, which consists of the integers which differ from 1 by an integral multiple of
3, and 2 represents the congruence class of 2 modulo 3, which consists of the integers which
differ from 2 by an integral multiple of 3. The symbol ‘+’ also carries with it a new meaning in
this context: it is defined as addition modulo 3. As Dummit and Foote (1999) caution:
“we shall frequently denote the elements of Ժ/nԺ [or Ժn] simply by {0, 1, … n-1}
where addition and multiplication are reduced mod [modulo] n. It is important to
remember, however, that the elements of Ժ/nԺ are not integers, but rather collections
of usual integers, and the arithmetic is quite different” (p.10, emphasis in original)
Pausing for a moment on the symbol ‘1+2’, we might explore just how different the meaning of
addition modulo 3 is from the ‘usual integer’ addition. Since the symbols ‘1’ and ‘2’ (in this
context) represent the congruence classes {… -5, -2, 1, 4, 7, …} and {… -4, -1, 2, 5, 8,…},
respectively, the sum ‘1+2’ must also be a congruence class. Dummit and Foote (1999) define
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the sum of congruence classes by outlining its computation. In the case of 1+2 (modulo 3), we
may compute the sum by taking any representative integer in the set {… -5, -2, 1, 4, 7, …} and
any representative integer in the set {… -4, -1, 2, 5, 8,…}, and summing them in the ‘usual
integer way’ (i.e. with the operation +Z). Having completed this, the next step is to determine the
final result: the congruence class containing the integral sum of the two representative integers.
Defined in this way, addition modulo 3 does not depend on the choice of representatives taken
for ‘1’ and ‘2’. Thus, recalling the notation introduced in the previous section, sample
computations to satisfy this definition include:
1 +3 2 = (1 +Z 2) modulo 3
= (1 +Z5) modulo 3
= (-2 +Z-1) modulo 3
all of which are equal to the congruence class 0.
Laden with new meaning, these symbols pose a challenge for students who must quickly
adjust to a context where the complexity of such compact notation is taken for granted, and
where inconsistencies arise between the symbols’ specialized meaning and their ‘familiar’,
‘usual’ meaning. Table 3 below summarizes the meanings of the symbols ‘1’, ‘2’, and ‘1+2’, and
‘+’ when considered within the context of Ժ3:
Symbol Meaning in context of Ժ3
1 Congruence class of 1 modulo 3: {… -5, -2, 1, 4, 7, …}
2 Congruence class of 2 modulo 3: {… -4, -1, 2, 5, 8,…}
1+2 Congruence class of (1+2) modulo 3: {…, -3, 0, 3, …}
+ Binary operation over set {0, 1, 2}; addition modulo 3
Table 3: Summary of extended meaning in Ժ3
The process of adding congruence classes by adding their representatives is a special case
of the more general group theoretic construction of a quotient and quotient group – central ideas
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in algebra, and ones which have been acknowledged as problematic for learners (e.g. Asiala et
al., 1997; Dubinsky et al., 1994).These concepts are challenging and abstract, and are made no
less accessible by opaque symbolism. As in the case with words, the extended meaning of a
symbol can be interpreted as a metaphoric use of the symbol, and thus may evoke prior
knowledge or experience that is incompatible with the broadened use. In a related discussion of
the challenges learners face when the meaning of a term is extended from everyday language to
the mathematics register, Pimm (1987) notes that “the required mental shifts involved can be
extreme, and are often accompanied by great distress, particularly if pupils are unaware that the
difficulties they are experiencing are not an inherent problem with the idea itself” (p.107) but
instead are a consequence of inappropriately carrying over meaning from one register to the
other. A similar situation arises as learners must stretch and revise their understanding of a
symbol within the mathematics register – an important mental shift that is taken for granted
when clarification of symbol polysemy remains tacit.
Beyond the familiar: an example in transfinite arithmetic
Transfinite arithmetic may be thought of as an extension of natural number arithmetic – its
addends (transfinite numbers) represent cardinalities of finite or infinite sets. Transfinite
arithmetic poses many challenges for learners, not the least of which involves appreciating the
idea of ‘infinity’ in terms of cardinalities of sets. Before one may talk meaningfully about
polysemy and ambiguity in transfinite arithmetic, it is important to first develop some ideas
about ‘infinity as cardinality’, which is where this section will begin.
Infinity is an example of a term which is polysemous both across and within registers.
The familiar association of infinity with endlessness is extended into the mathematics register in
areas such as calculus where the idea of potential infinity is indispensible. Potential infinity may
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be thought of as an inexhaustible process – one for which each step is finite, but which continues
indefinitely. In calculus for example, the idea of limits which ‘tend to’ infinity relates the notion
of an on-going process that is never completed. This extension across registers preserves some of
the meaning connected to the colloquial use of the term ‘infinity’, however it is distinct from
intuitions which, say, connect infinity to endless time or to the all-encompassing (see Mamolo
and Zazkis, 2008). Within the mathematics register, the term ‘infinity’ is extended further to the
idea of actual infinity, which is prevalent in the field of set theory. Actual infinity is thought of as
a completed and existing entity, one that encompasses the potentially infinite. The set of natural
numbers is an example of an actually infinite entity – it contains infinitely many elements and, as
a set, exists despite the impossibility of enumerating all of its elements. The cardinality of the set
of natural numbers is another instance of actual infinity; it is also the smallest transfinite number.
Transfinite numbers are generalised natural numbers which describe the cardinalities of
infinite sets. As implied, infinite sets may be of different cardinality: the set of natural numbers,
for example, has a different cardinality than the set of real numbers, though both contain
infinitely many elements. Cardinalities of two infinite sets are compared by the existence or non-
existence of a one-to-one correspondence between the sets. Two sets share the same cardinality
if and only if every element in the first set may be ‘coupled’ with exactly one element in the
second set, and vice versa. This is a useful approach, and I will return to it when illustrating
properties of transfinite arithmetic. The point I am trying to make here is that the concept of a
transfinite number, which intuitively may be thought of as an ‘infinite number’, requires
extending beyond the familiar idea of infinity as endless (and thus unsurpassable). Also, in
resonance with Pimm’s (1987) observation regarding negative and complex numbers, the
concept of a transfinite number “involves a metaphoric broadening of the notion of number
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itself” (p.107). In this case, the broadening includes accommodating some arithmetic properties
which are both unfamiliar and unintuitive.
As in the case with arithmetic over the set of natural numbers, transfinite arithmetic
involves determining the cardinality of the union of two disjoint sets. The crucial distinction is of
course that at least one of these sets must have infinite magnitude – its cardinality must be equal
to a transfinite number. To illustrate some of the distinctive properties of transfinite arithmetic
consider, without loss of generality, the cardinality of the set of natural numbers, denoted by the
symbol Յ0. Imagine adding to the set of natural numbers,Գ, a new element, say ȕ. This union set
Գ {ȕ} has cardinality equal to Յ0 + 1 – there is nothing new here. However, each element in Գ
can be ‘coupled’ with exactly one element in Գ {ȕ}, and vice versa. By definition, two infinite
sets have the same cardinality if and only if they may be put in one-to-one correspondence, thus
the cardinality of Գ is equal to the cardinality of Գ {ȕ}. As such, Յ0 = Յ0 + 1. Similarly, it is
possible to add an arbitrary natural number of elements to the set of natural numbers and not
increase its cardinality, that is Յ0 = Յ0 + ȣ, for any ȣא Գ,and further Յ0 + Յ0 = Յ0.
This ‘tutorial’ in transfinite arithmetic is relevant to the discussion on polysemy as it
illustrates how the symbol ‘+’ in this context is quite distinct in meaning from addition over the
set of natural numbers. Whereas with ‘+N’ adding two numbers always results in a new (distinct)
number, with ‘+’ there exist non-unique sums. Further, since the concept of a set of numbers
must be extended to the more general ‘class’ of transfinite numbers, the symbol ‘1’ in the
expression ‘Յ0 + 1’ also takes on a slightly new meaning since it must be considered more
generally as a class (rather than set) element3. Extended meanings connected to transfinite
arithmetic are summarized in Table 4:
3 For distinction between set and class, see Levy (1979).
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Symbol Meaning in context of transfinite arithmetic
1 Cardinality of the set with a single element; class element
Յ0Cardinality of Գ; transfinite number; ‘infinity’
Յ0 + 1 Cardinality of the set Գȕ; equal to Յ0
+ Binary operation over the class of transfinite numbers
Table 4: Summary of extended meaning in transfinite arithmetic
A specific challenge related to the polysemy of + in this context derives from the
existence of non-unique sums, a consequence of which is indeterminate differences. Explicitly,
since Յ0 = Յ0 + ȣ, for any ȣא Գ, then Յ0 - Յ0 has no unique resolution. As such, the familiar
experience that ‘anything minus itself is zero’ does not extend to transfinite subtraction. This
property is in fact part and parcel to the concept of transfinite numbers. Identifying precisely the
context-specific meaning of these symbols (‘+’ and ‘’) can help solidify the concept of
transfinite numbers, while also deflecting naïve conceptions of infinity as simply a ‘big unknown
number’ by emphasizing that transfinite numbers are different from ‘big numbers’ since they
have different properties and are operated upon (arithmetically) in different ways.
In this section, to address issues of polysemy of symbols, it was necessary to first glance
at the polysemy of the term infinity. It is a complex concept that can encompass different
connotations across and within different registers. Within mathematics, it is difficult to think of
infinity – even in the context of transfinite numbers – without imagining that well-known symbol
’. Informally, the symbol ‘Յ0 + 1’ might be thought of as ‘ + 1’. This informal symbolism
suggests the idea of adding 1 to a ‘concept’ rather than a ‘set number’, of adding 1 to
endlessness. Notwithstanding the formal use of ‘Յ0’, an intuition of ‘’ may persist (if only
tacitly), carrying with it all sorts of inappropriate associations.
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Concluding Remarks
This article examined instances of symbol polysemy within mathematics. The intent was to
illustrate how even basic symbols, such as ‘+’ and ‘1’, may carry with them meaning that is
inconsistent with their use in ‘familiar’ contexts. It focused on cases where acknowledging the
ambiguity in symbolism and explicitly identifying the precise (extended) meaning of that
symbolism go hand-in-hand with developing an understanding of the ideas involved. While this
article focused on particular examples of distinguishing among the symbolic notation for
arithmetic over the set of natural numbers, rational numbers, equivalence classes, and transfinite
cardinals is fundamental to appreciating the subtle (and not-so-subtle) differences among the
elements of those sets, this argument has broader application. I suggest that the challenges
learners face when dealing with polysemous terms (both within and outside mathematics) are
also at hand when dealing with polysemous symbols. Just as knowledge of languages such as
English include “learning a meaning of a word, learning more than one meaning, and learning
how to choose the contextually supported meaning” (Mason et al., 1979, p.64), knowledge of
mathematics includes learning a meaning of a symbol, learning more than one meaning, and
learning how to choose the contextually supported meaning of that symbol. Further, echoing
Pimm’s (1987) advice and extending its scope to include mathematical symbols:
“If … certain conceptual extensions in mathematics [are] not made abundantly clear
to pupils, then specific meanings and observations about the original setting, whether
intuitive or consciously formulated, will be carried over to the new setting where
they are often inappropriate or incorrect” (p.107).
Sfard (2001) suggests that symbols – such as the ones discussed here, but also in a more
general sense – are not “mere auxiliary means that come to provide expression to pre-existing,
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pre-formed thought” but rather are “part and parcel of the act of communication and thus of
cognition” (p.29). As such, attending to the polysemy of symbols, either as a learner, for a
learner, or as a researcher, may expose confusion or inappropriate associations that could
otherwise go unresolved. Research in literacy suggests that students “may rely on context when a
word does not have a strong primary meaning to them but will choose a common meaning,
violating the context, when they know one meaning very well” (Mason et al., 1979, p.63).
Further research in mathematics education is needed to establish to what degree analogous
observations apply as students begin to learn ‘+’ in new contexts.
Acknowledgements
I would like to thank Rina Zazkis for her insightful comments and encouragement on an early
draft of this paper.
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... This may be depicted in contexts of problem-solving and multiple representations revealing levels of proficiency in individuals (Kaput & Shaffer, 2002) though there are expected challenges. These challenges may include "Polysemy symbols" -symbols with many meanings (Mamolo, 2010), different symbols representing the same concepts, the 'procept' nature of symbols-symbols representing both concepts and process (Gray & Tall, 1994), and contextual meaning (Mamolo, 2010). ...
... This may be depicted in contexts of problem-solving and multiple representations revealing levels of proficiency in individuals (Kaput & Shaffer, 2002) though there are expected challenges. These challenges may include "Polysemy symbols" -symbols with many meanings (Mamolo, 2010), different symbols representing the same concepts, the 'procept' nature of symbols-symbols representing both concepts and process (Gray & Tall, 1994), and contextual meaning (Mamolo, 2010). ...
... Such interpretation of singular meanings ought to have not been the case with symbols such as the equal (=) sign and plus (+) sign that learners have met on several occasions. This contrasted with the findings from a research study by Mamolo (2010) that a symbol meaning revealed in some contexts may not be in agreement with other usual contexts. In the Kenyan context, a research study done by Mulwa (2015) has it that there were difficulties in the use of mathematical terms and concepts. ...
Article
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This study examined the students’ proficiency in using mathematical symbols. The need to carry out the study was prompted by the Kenya National Examinations Council (KNEC) report that revealed dismal performance in mathematics. Therefore, the study objectives explored how symbols affect learning of mathematics, students’ perception of the role of symbols in mathematics learning and students’ use of mathematical symbols. The basis of the study relied on a conceptual framework of epistemological approach to notations and supportive and problematic conceptions as a lens that helped in dissecting the kind of symbol sense that exist amongst students. The study targeted mathematics teachers and form four students and was therefore conducted in a public secondary school in Rarieda Sub-County, Siaya County, Kenya. A qualitative approach with a case study research design was employed with sampling techniques such as convenience, purposive, and stratified sampling used to locate the research site and recruit participants. Data collection instruments included interview guides and document analysis protocol. Thematic analysis was used. The findings of the study showed that symbols influenced the learning of mathematics in terms of prior knowledge and symbol meanings at hand, thereby posing challenges in the learning of mathematics. Also, the findings revealed that students had a perception of the role of mathematical symbols in giving easy time in understanding concepts due to their precise and succinct nature, conserved time and assisted in the solution of mathematics problems and that use of symbols is profound in the linkage of concepts across topics, multiple representations and problem-solving. It may be recommended that prominence ought to be put on various ways of symbol representation to enable comprehension of symbols and meanings; better instructional techniques ought to be used to reduce the symbol cognitive load on students.
... Several researchers have characterized polysemous and homonymous words (e.g., Zazkis, 1998) and symbols (Mamolo, 2010;Zazkis & Kontorovich, 2016) in mathematics education. For example, Zazkis (1998) described the word 'divisor' as a polysemous word within mathematics as it can refer to multiple different but related ideas; a divisor can be used to either refer to the number that a dividend is being divided by or a number that divides another number without a remainder (e.g., 3 is a divisor of 12). ...
... For example, Zazkis (1998) described the word 'divisor' as a polysemous word within mathematics as it can refer to multiple different but related ideas; a divisor can be used to either refer to the number that a dividend is being divided by or a number that divides another number without a remainder (e.g., 3 is a divisor of 12). Mamolo (2010) extended the use of polysemous words in mathematics education to include polysemous symbols, characterizing the different ways individuals use the '+' symbol in mathematics to denote an operation among elements of different sets (e.g., , ℤ 5 ). ...
... We add to the previous literature characterizing polysemous and homonymous symbols in mathematics (Mamolo, 2010;Zazkis & Kontorovich, 2016). We characterized the extent to which both we, as researchers, and students understood various inequality symbols and statements as polysemous (our conceiving the multiple ways in which inequality symbols can be used to represent quantitative relationships), homonymous (the students only assimilating statements with inequality symbols as representing comparative inequalities), and synonymous (the students' meanings for "U = Bigger" and "U > I"). ...
Article
Inequalities are an important topic in school mathematics, yet the body of research exploring students’ meanings for inequalities largely points to difficulties they experience. Thus, there is a need to further explore students’ meanings for inequalities. Addressing this need, we conducted an exploratory teaching experiment with two seventh-grade students to investigate their developing meanings for inequalities. We distinguish between two types of inequalities in student thinking: comparative and restrictive inequalities. Whereas a student reasoning about a comparative inequality compares two quantities’ values or magnitudes, reasoning about a restrictive inequality entails reasoning about a range of one quantity’s magnitudes or values. We realized a complexity arose in our interactions with students due to our conceiving the use of inequality symbols across the two types of inequalities as polysemous, whereas the students did not. Attending to these two types of inequalities has important implications for the teaching and learning of inequality.
... Lexical ambiguities may occur only within the mathematics register (e.g., Mamolo, 2010;Zazkis, 1998;Zazkis & Kontorovich, 2016). For instance, Zazkis (1998) examined the polysemy of the school mathematics terms "divisor" and "quotient." ...
... She stressed that the quotient refers to the result of division in the context of rational numbers (e.g., 13 ÷ 5 = 2.6), while it refers to the integral part of the result of the division algorithm in the context of whole numbers (e.g., 2 is the quotient and 3 is the remainder in 13 ÷ 5). Mamolo (2010) extended the notion of polysemy from mathematical concepts to mathematical symbols by focusing on the different but related meanings of the addition sign ( +) in the advanced mathematical contexts involving modular and transfinite arithmetic. She explained that the notation + 3 can be used to represent addition over the set {0, 1, 2} (i.e., modular arithmetic with base 3), and that + ∞ can be used to represent addition over the class of cardinal numbers (i.e., transfinite arithmetic). ...
Article
The purpose of the current study was to explore pre-service middle school mathematics teachers’ personal concept definitions of a trapezoid, parallelogram, rectangle, rhombus, square, and kite. The data were collected by a self-report instrument through which the pre-service teachers provided their answers in written form. The participants’ definitions were coded by using Zazkis and Leikin’s (Educational Studies in Mathematics, 69(2), 131–148, 2008) framework, which includes the following four main categories for determining mathematical correctness: necessary and sufficient, necessary but not sufficient, sufficient but not necessary, and neither necessary nor sufficient. The findings revealed that about half of the all definitions were correct. More specifically, the participants generated considerably higher proportion of correct definitions for a parallelogram and rhombus, while they displayed a very low performance in defining a kite. The possible reasons of the participants’ varied performance levels in defining the six basic quadrilaterals are discussed based on the linguistic (syntactic, semantic, and lexical) structure of the Turkish names given to the these quadrilaterals. The current study may provide some feedback to teacher education programs regarding the extent of knowledge that should be possessed by the pre-service teachers about definitions of special quadrilaterals before they leave their programs. Such feedback may also help mathematics teacher educators ponder on more fruitful approaches that may promote the development of pre-service teachers’ knowledge and understanding of definitions of special quadrilaterals.
... Zazkis (1998) exemplified the ambiguity of "divisor" with the exercise 12 ÷ 5 = 2.4 where in the domain of rational numbers, the number 5 can be addressed as a divisor, since it is defined as the denominator of a fraction. However, if the exercise is considered in the domain of integers, 5 is not a divisor of 12 because there exist no integer that when multiplied by 5 equals 12. Mamolo (2010) focused on the polysemy of symbols '+' and '1'. Her analysis accounted for the changes in the definitions and, consequently, in symbols' meanings in the contexts of modular arithmetic, transfinite mathematics, et cetera. ...
... Her analysis accounted for the changes in the definitions and, consequently, in symbols' meanings in the contexts of modular arithmetic, transfinite mathematics, et cetera. Based on their analyses, Mamolo (2010) and Zazkis (1998) argued that polysemy in mathematics is a potential source of struggle for learners. This study can be considered as an examination of their argument. ...
Conference Paper
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This article is concerned with cognitive aspects of students' struggles in situations in which familiar concepts are reconsidered in a new mathematical domain. Examples of such cross-curricular concepts are divisibility in the domain of integers and in the domain of polynomials, multiplication in the domain of numbers and in the domain of vectors. The article introduces a polysemous approach for structuring students' concept images in these situations. Post-exchanges from an online forum were analyzed for illustrating the potential of the approach for indicating possible sources of students' misconceptions and meta-ways of thinking that might make students aware of their mistakes.
... Zazkis (1998) exemplified the ambiguity of Bdivisor^with the exercise 12 ÷ 5 = 2.4 where in the domain of rational numbers, the number 5 can be addressed as a divisor, since it is defined as the denominator of a fraction. However, if the exercise is considered in the domain of integers, 5 is not a divisor of 12 because there exists no integer that when multiplied by 5 equals 12. Mamolo (2010) focused on the polysemy of symbols B+^and B1.^Her analysis accounted for the changes in the definitions and, consequently, in symbols' meanings in the contexts of modular arithmetic, transfinite mathematics, et cetera. Based on their analyses, Mamolo (2010) and Zazkis (1998) argued that polysemy in mathematics is a potential source of struggle for learners. ...
... However, if the exercise is considered in the domain of integers, 5 is not a divisor of 12 because there exists no integer that when multiplied by 5 equals 12. Mamolo (2010) focused on the polysemy of symbols B+^and B1.^Her analysis accounted for the changes in the definitions and, consequently, in symbols' meanings in the contexts of modular arithmetic, transfinite mathematics, et cetera. Based on their analyses, Mamolo (2010) and Zazkis (1998) argued that polysemy in mathematics is a potential source of struggle for learners. This study can be considered as an examination of their argument. ...
Article
Full-text available
This article is concerned with cognitive aspects of students’ struggles in situations in which familiar concepts are reconsidered in a new mathematical domain. Examples of such cross-curricular concepts are divisibility in the domain of integers and in the domain of polynomials, multiplication in the domain of numbers and in the domain of vectors, and roots in the domain of reals and in the domain of complex numbers. The article introduces a polysemous approach for structuring students’ concept images in these situations. Post-exchanges from an online forum and excerpts from an interview were analyzed for illustrating the potential of the approach for indicating possible sources of students’ misconceptions and meta-ways of thinking that might make students aware of their mistakes.
... We argue that this is indeed a very good description of how mathematics works at any level of sophistication. The phenomenon has been studied in mathematics education research under the concept polysemy (see Mamolo, 2010 for an overview). While some may see polysemy as a problem of ambiguity, we see polysemy as an important feature of mathematical conceptualization and of mathematics itself. ...
Conference Paper
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Iszák and Beckmann recently argued for a coherent approach to multiplication, based on a specific model for thinking about multiplication. We argue theoretically against this variant of coherence. We base our arguments on an exemplification of how mathematical thinking at four different cognitive levels involves an element of polysemy and exemplify how an innovative instructional approach not building on coherence might be designed by referring to an ongoing intervention project.
... Symbolic ambiguity, according to Foster (2011), appears when the same symbol is used to represent different ideas or concepts. For example, Foster noted that the × sign could be used for multiplication of real numbers or the cross product of vectors; Zazkis and Kontorovich (2016) focused on the phenomenon of the symbol □ −1 denoting both an inverse function and a reciprocal of a number; and Mamolo (2010) illustrated how in certain contexts, even basic symbols such as "1" and " + " may carry meaning that is inconsistent with their use in familiar contexts. ...
Article
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We explore the responses of 26 prospective elementary-school teachers to the claim “1/6.5 is not a fraction” asserted by a hypothetical classroom student. The data comprise scripted dialogues that depict how the participants envisioned a classroom discussion of this claim to evolve, as well as an accompanying commentary where they described their personal understanding of the notion of a fraction. The analysis is presented from the perspective of productive ambiguity, where different types of ambiguity highlight the prospective teachers’ mathematical interpretations and pedagogical choices. In particular, we focus on the ambiguity inherent in the aforementioned unconventional representation and how the teachers reconciled it by invoking various models and interpretations of a fraction. We conclude with a description of how the perspective of productive ambiguity can enrich teacher education and classroom discourse.
... In mathematics education, the role of polysemy cannot be overstressed as it can be found in many mathematical terms (e.g., Presmeg 1992; Zazkis 1998; Kontorovich 2018a) and symbols (e.g., Kontorovich 2016Kontorovich , 2018bKontorovich , 2018cMamolo 2010). Two types of mathematical polysemy can be distinguished. ...
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This study is concerned with the reasoning that undergraduates apply when deciding whether a prompt is an example or non-example of the subspace concept. A qualitative analysis of written responses of 438 students revealed five unconventional tacit models that govern their reasoning. The models account for whether a prompt is a subset of a vector space, whether the zero vector is included, the structure of vectors, their number in the formula for the general solution to the system of linear equations, and the corresponding coefficient matrix. Furthermore, a conception was identified in students’ responses, according to which the algebraic structure of a vector space passes from a ‘parent’ space to its subset, turning automatically it into a subspace. For many students this conception of an inheriting structure was instrumental for identifying and reasoning around subspaces. Polysemy of the prefix ‘sub’ and students’ prior experiences in identifying concept examples are used for offering explanations for the emergence of the conception.
... verbs, nouns and adjectives, to texts and stories and even to symbolic constructions such as symbols (Taylor, 2002). Polysemy is a phenomenon that is believed to result from an economy of wordsor symbolsaiming at preserving lexical storage space (Mamolo, 2010), at speeding lexical access (Crossley, Salsbury, & McNamara, 2010;Eddington & Tokowicz, 2014) and at adding to the discourse nuances related to a given situation (Benjafield, 2012). In fact, Riggs (1987) points out that the tolerance for ambiguity, which stems from a frequent use of polysemic terms, is intrinsic in social interaction and helps deal with delicate situations. ...
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This study examines approaches to infinity of two groups of university students with different mathematical background: undergraduate students in Liberal Arts Programmes and graduate students in a Mathematics Education Master's Programme. Our data are drawn from students’ engagement with two well-known paradoxes – Hilbert's Grand Hotel and the Ping-Pong Ball Conundrum – before, during, and after instruction. While graduate students found the resolution of Hilbert's Grand Hotel paradox unproblematic, responses of students in both groups to the Ping-Pong Ball Conundrum were surprisingly similar. Consistent with prior research, the work of participants in our study revealed that they perceive infinity as an ongoing process, rather than a completed one, and fail to notice conflicting ideas. Our contribution is in describing specific challenging features of these paradoxes that might influence students’ understanding of infinity, as well as the persuasive factors in students’ reasoning, that have not been unveiled by other means.
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The research reported in this paper explores the nature of student knowledge about group theory, and how an individual may develop an understanding of certain topics in this domain. As part of a long-term research and development project in learning and teaching undergraduate mathematics, this report is one of a series of papers on the abstract algebra component of that project.The observations discussed here were collected during a six-week summer workshop where 24 high school teachers took a course in Abstract Algebra as part of their work. By comparing written samples, and student interviews with our own theoretical analysis, we attempt to describe ways in which these individuals seemed to be approaching the concepts of group, subgroup, coset, normality, and quotient group. The general pattern of learning that we infer here illustrates an action-process-object-schema framework for addressing these specific group theory issues. We make here only some quite general observations about learning these specific topics, the complex nature of understanding, and the role of errors and misconceptions in light of an action-process-schema framework. Seen as research questions for further exploration, we expect these observations to inform our continuing investigations and those of other researchers.We end the paper with a brief discussion of some pedagogical suggestions arising out of our considerations. We defer, however, a full consideration of instructional strategies and their effects on learning these topics to some future time when more extensive research can provide a more solid foundation for the design of specific pedagogies.
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Traditional approaches to research into mathematical thinking, such as the study of misconceptions and tacit models, have brought significant insight into the teaching and learning of mathematics, but have also left many important problems unresolved. In this paper, after taking a close look at two episodes that give rise to a number of difficult questions, I propose to base research on a metaphor of thinking-as-communicating.This conceptualization entails viewing learning mathematics as an initiation to a certain well defined discourse. Mathematical discourse is made special by two main factors: first, by its exceptional reliance on symbolic artifacts as its communication-mediating tools, and second, by the particular meta-rules that regulate this type of communication. The meta-rules are the observer’s construct and they usually remain tacit for the participants of the discourse. In this paper I argue that by eliciting these special elements of mathematical communication, one has a better chance of accounting for at least some of the still puzzling phenomena. To show how it works, I revisit the episodes presented at the beginning of the paper, reformulate the ensuing questions in the language of thinking-as-communication, and re-address the old quandaries with the help of special analytic tools that help in combining analysis of mathematical content of classroom interaction with attention to meta-level concerns of the participants.
Article
This paper reports on a continuing development of an abstract algebra course that was first implemented in the summer of 1990. This course was designed to address discrepancies between how students learn and how they were traditionally being taught. Based on results from the first implementation, pedagogical changes were made, including increased computer programming activities and other exercises which were designed to give the students the opportunity to build experience to draw on in order to construct understanding of the topics in class. A second experimental course was run. To assess the impact of these methods, and to continue to better understand how students go about learning, test results from the latter class and interviews with students from both experimental courses and a lecture-based class were analyzed. The students in the second experimental course demonstrated a deep understanding of the title concepts, especially cosets and normality. We discuss the details of the revised experimental course; the epistemological theory behind its design; and the framework used to analyze the results. We demonstrate through examples from interviews and test results the applicability of this analysis to the data, and the strides made by the students in comparison with the students from the lecture-based course, and with the students from the first experimental course. We hope to illustrate difficulties students face in learning abstract algebra, and to discuss instructional strategies to help students overcome these difficulties.
Article
This book is a methods book to help teachers guide children to develop ideas and relationships about mathematics. The methods and activities are designed to get children mentally involved in the construction of those ideas and relationships. Chapter 1 discusses what it means to teach mathematics and suggests some of the important variables shaping mathematics education. Chapter 2 describes the general philosophy behind the subtitle, "Teaching Developmentally." In chapter 3, meaningful learning of mathematics is shown to be a problem-solving process regardless of the particular content. Chapters 4 through 19 each address a different part of the elementary mathematics curriculum. Activities (stressed as the most important feature of these chapters), problems for discussion and exploration, and suggested reading references are provided for each of the chapters. Chapter 20 describes the role of calculators and computers in mathematics. Chapter 21 discusses planning lessons, classroom use of materials, cooperative learning groups, homework, and the role of the basal textbooks. Chapter 22 is concerned with assessment with an emphasis on diagnosis and how to listen to children. The last chapter explores the special considerations that should be given to children with special needs. The appendices contain summaries of the NCTM Standards for grades K-8, guides for mathematics learning activities, and masters and construction tips. (YP)