International Portal of Teacher Education

Source: Journal of Mathematics Teacher Education, Volume 16, No. 2, April 2013, p. 85-103.
(Reviewed by the Portal Team)

This article examines prospective elementary teachers’ difficulties and growth with language for defining the whole.
The study investigates the following research questions:
• What fraction understandings do prospective elementary teachers have with respect to defining the whole?
• How does prospective teachers’ understanding of defining the whole develop?

Methodology
Thirty-three prospective elementary teachers participated in a semester-long classroom teaching experiment conducted at a large metropolitan university in the southeastern part of the United States.
The study was conducted in a content course focusing on mathematics for teaching elementary school.
All participants were women, in at least their sophomore year of college, and either majoring in elementary or exceptional education.

Rational numbers constituted nine class days.
It was the second topic taught in the course following a unit focusing on whole number concepts and operations.
Language for fractions was introduced on the first and second day and continued throughout the duration of instruction.

Data collection
The data collected included video recordings and transcripts of whole-class discussions, and student work from in-class activities, two homework assignments, and an end-of-unit examination.
Research team field notes and reflective journals were also collected for each class session.

The results of this study indicate that three mathematical ideas became taken-as-shared as prospective elementary teachers developed an understanding of language use for defining the whole.
The first was that fractional solutions depend on a group or whole.
The second included defining an of what.
The third was developing language in terms of what the denominator represents.
Difficulties prospective teachers had conceptualizing language included distinguishing among the phrases of a, of one, of the, and of each.

Other difficulties the prospective teachers have are distinguishing between the questions how much and how many.
When the question how much is asked toward the end of the discussion of sharing four pizzas among five people, the class understands the solution to be 4/5.

Though this study incorporated a familiar context, prospective teachers struggled with understanding the language for defining a whole.
Thus, for the prospective teachers, a familiar context did not seem to be enough to aid in their understanding.
Their understanding also relied on the language used in the question.

The results provide insight into the types of language understandings prospective teachers bring to mathematics teacher education programs and documents how language understanding develops.
The results have several implications for teacher education programs and future research studies focusing on mathematics content courses.
By knowing when an idea shifted position in an argument and/or was no longer questioned, an analysis could be done which illustrated when prospective teachers developed fraction language and how.
In this study, the mathematical ideas intertwined to the extent that at one point during the first day all three mathematical ideas were emerging before any one idea became taken-as-shared.

Language learning is complex and the tasks used in this study were successful in eliciting valuable and informative conversations.
By discussing how to group pieces students were able to formulate the idea of defining and describing an of what.
This then provided them with a foundation to start developing the idea that fractions depend on a referent whole and lead to an understanding of what is meant by the phrases of a, of one, of the, and of each.

The findings also indicate that when prospective teachers develop an understanding of language for fractions less than one, this does not signify their understanding of language for fractions greater than one.
In this study, language for fractions less than one was developed six class days prior to language for fractions greater than one.
This gap may have resulted from the fact that all of the tasks used in this study incorporated fractions less than one, but not every task included fractions greater than one.

With the need for teachers to have a deep understanding of mathematics, they also need to be able to communicate that knowledge effectively.
By fostering language, education programs can simultaneously support prospective teachers’ development of and communication skills with the mathematics they are to teach.