Structure recognition and generalization by second-graders in direct and inverse forms of a linear function

ABSTRACT

Within the context of the functional approach inherent in early algebra education, this paper reports the structures recognized and the generalizations shown by second-grade students (aged 7–8 years). Semi-structured interviews were conducted with 4 students selected from a cohort participating in a previous classroom teaching experiment. These students were tasked with solving word problems that required understanding of both the direct and inverse forms of linear functions. This paper details the structures recognized by the students both when working with specific cases and when attempting generalization. All students successfully identified the structures involved in both the direct and inverse forms of the function. Differences in the generalized structures were observed depending on the function and its form. A greater number of students was able to generalize structures for the inverse form than for the direct form of the functions examined. This study is presented as part of a broader research project at XXXX.

KEYWORDS:

• Algebraic thinking
• direct form of a function
• function
• functional thinking
• generalization
• inverse form of a function
• structure

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1. Question: Isabel pays for admission and (indeterminate quantity) rides. Explain how she knows how much she has to pay. Student response: 6 because there may be a 5 and 1 + 5 is 6.

2. 1. How many balls will come out of the machine if we put in 8 balls? ____ balls.

6. Now we are going to play a game in which the winner is the person who figures out how the machine works.

How can you tell how many balls will come out of the machine?

3. Número de paradas/personas = Number of stations/people.

Muchas paradas = A lot of stations

Additional information

Funding

The work was supported by the Spain’s National Research Agency (Spanish initials, AEI) and the European Regional Development Fund (ERDF).

Notes on contributors

María D. Torres González

María D. Torres González is Professor of Mathematics Education at the University of Granada (Spain). Her main line of research focuses on algebraic thinking in primary school students www.pensamientoalgebraico.es/en [1]).

María C. Cañadas

María C. Cañadas is Full Professor of Mathematics Education at the University of Granada (Spain) (https://produccioncientifica.ugr.es/investigadores/356190/detalle). Her main line of research focuses on algebraic thinking in early childhood and primary school students (www.pensamientoalgebraico.es/en [1]).

Antonio Moreno

Antonio Moreno is Professor of Mathematics Education at the University of Granada (Spain) (https://produccioncientifica.ugr.es/investigadores/352592/detalle). His main line of research focuses on algebraic thinking in primary school students (www.pensamientoalgebraico.es/en [1]).