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Ami Mamolo, Ph.D.

Tie a tie

Milk for milk

In mathematics, a word may be polysemous if its

mathematical meaning is different from its everyday,

familiar meaning (Durkin and Shire, 1991), or if it has

two related, but different, mathematical meanings

(Zazkis, 1998).

E.g., Continuity, function

E.g, Quotient, divisor

Tie a tie

Milk for milk

In a mathematical discourse, symbols such as +, =, and

1, may also be considered ‘words’

Gray and Tall (1994) advocated for flexible

interpretation of symbols such as 5+4 as processes or

concepts, i.e. procepts.

“This ambiguous use of symbolism is at the root of

powerful mathematical thinking” (p.125).

Byers (2007) suggested ambiguity in mathematics is

“an essential characteristic of the conceptual

development of the subject” and “opens the door to

new ideas, new insights, deeper understanding” (p.78).

Durkin and Shire (1991) suggested that enriched

learning may ensue from monitoring, confronting and

‘exploiting to advantage’ ambiguity

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 presentation will explore cases of ambiguity

connected to the context-dependent definitions of

symbols, that is, the polysemy of symbols. Specifically,

polysemy of ‘+’

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

Building on the idea of addition as a domain-dependent

binary operation, there are cases where extended meanings

of ‘a + b’ contribute to results that are inconsistent with the

‘familiar’.

Modular arithmetic with base 3

+3as addition over the set {0, 1, 2}

Transfinite arithmetic

+∞as addition over the class of cardinal numbers

The symbol +Nwill be used to represent addition over the set

of natural numbers, +Zas addition over the set of integers

Symbol Meaning in context of Z3

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

Dummit and Foote (1999) define the sum of

congruence classes by outlining its computation:

1+2 (modulo 3), is computed 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’.

E.g. 1 +

3

2

= (1 +

Z2) modulo 3

= (1 +

Z5) modulo 3

= (

-2 +Z -1) modulo 3

As 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.

Pimm (1987) notes that “the required mental shifts

involved [in extending meaning] 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.

Transfinite arithmetic may be thought of as an

extension of natural number arithmetic

its addends represent cardinalities of finite or infinite

sets

a sum is defined as the cardinality of the union of two

disjoint 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

(i.e. the transfinite numbers ℵ0, ℵ1, ℵ2, …).

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 itself” (p.107).

A generic example: the sum ℵ0+ 1

It’s the cardinality associated with the union set N{β},

where β is not in N.

The addends are elements of the (generalised) class of

cardinals, which includes transfinite cardinals.

Between the sets N{β} and N there exists a bijection, which,

in line with the definition (Cantor, 1915), guarantees that the

two sets have the same cardinality –that is, ℵ0+ 1 = ℵ0.

Symbol

Meaning in context of transfinite

arithmetic

1

Cardinality of the set with a single

element; class element

ℵ0

Cardinality of

N; transfinite number;

‘infinity’

1 + ℵ0

Cardinality

of the set Nβ; equal to ℵ0

+

Binary operation over the class of

transfinite numbers

ℵ0= ℵ0+ υ, for any υ ∈N, and ℵ0+ ℵ0= ℵ0.

Whereas with ‘+N’ adding two numbers always results

in a new (distinct) number, with ‘+∞’ there exist non-

unique sums.

A consequence: indeterminate differences.

Since ℵ0= ℵ0+ υ, for any υ ∈, then ℵ0-ℵ0has no unique

resolution.

The familiar notion that ‘anything minus itself is zero’

does not extend to transfinite subtraction.

Mason, Kniseley, and Kendall in research on literacy

suggest that knowledge of language includes “learning

a meaning of a word, learning more than one

meaning, and learning how to choose the contextually

supported meaning” (1979, p.64).

Similarly, 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.

Mason et al. (1979) suggest students “will choose a

common meaning [of a word], violating the context,

when they know one meaning very well” (p.63).

To what degree do analogous observations apply as

students begin to learn ‘+’ in new contexts?

Attending to the polysemy of symbols, as a learner, for

a learner, or as a researcher, may expose confusion or

inappropriate associations that could otherwise go

unresolved.

Echoing Pimm’s (1987) advice :

“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).

How to tease out influence of polysemy on student

(mis)understanding?

Could explicit attention help avoid difficulties?