Fraction images: the case of six and a half

ABSTRACT

We explore prospective elementary-school teachers’ attempts to provide signs and symbols with mathematical meaning in a case involving non-integers as numerator or denominator. The data comprises 33 responses to a task inquiring about the existence of numbers between 1/6 and 1/7, in which the participants were asked to compose a hypothetical classroom dialogue addressing this issue, and provide explanations on the mathematics involved. The analysis utilised the theoretical frameworks of “semiotic representation” and “concept image”. The findings indicate a strong reliance on prevalent fraction images and difficulties in assigning meaning to “6½” when it appears in the numerator or denominator of a fraction representation. These difficulties are examined via three approaches demonstrated in the scripts, relating to the symbolic representation of fractions, visual representation of fractions, and permissible operations on fractions. We suggest teacher-education courses may put further emphasis on the set model of fractions and atypical semiotic representations.

KEYWORDS:

• Fractions
• semiotic representation
• concept image
• scripting tasks
• prospective teachers

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Ofer Marmur http://orcid.org/0000-0002-0195-4354

Xiaoheng Yan http://orcid.org/0000-0003-4860-5124

Rina Zazkis http://orcid.org/0000-0003-4617-4877

Notes

1 We note that in the literature, a variety of expressions are used interchangeably to describe ways of understanding fractions, among others including “conceptualizations of fractions”, “interpretations of fractions”, “models of fractions”, and “sub-constructs of fractions”.

2 We acknowledge that the bottom-right image in Figure 2 may be associated with different fractions depending on context. For example, if we speak about the ratio of males to females in the image, this would be associated with three quarters (which was our intention here); however, it could also be interpreted as the ratio of males (or females) to the total number of people, and respectively associated with three sevenths (or four sevenths).

3 We note that the participants used the representations 1/6.5, $\frac{1}{6.5}$, $\frac{1}{6\frac{1}{2}}$, “1/six and a half”, as well as other variations, interchangeably within a given assignment. In the presentation of the findings we try to follow the same representation provided by the discussed participant.

4 All handwritings and hand-drawn illustrations of the participants were recreated for purpose of anonymity (we made these using computer software, while staying true to the originals as much as possible).

5 Italicised here, not in original script.

6 Italicised here, not in original script.

7 Italicised here, not in original script.

8 Capitalised in original text.