## Abstract

The aim of this research was to explore the mathematical connections that pre-university students make when they sketch the graph of a derivative function and an antiderivative function. Also, we tried to explain the origin of the mathematical connections identified. We assume mathematical connections as a cognitive process through which a person makes a true relationship between two or more mathematical ideas, concepts, definitions, theorems, or meanings with each other. Task-based interviews were used to collect data which included two graphical tasks that involved the derivative function and the antiderivative function. Through thematic analysis, we identified five types of intra-mathematical connections: procedural, different representations, part-whole, feature, and reversibility, which can serve as a preliminary theoretical framework to study mathematical connections in Calculus in future research; this is a contribution of this research. In addition, results indicated that Mexican students seldom used visualization to solve graphical tasks, so in future research, classroom intervention proposals should be developed to promote the use of visualization including the development of the ability to make mathematical connections, in order to improve their mathematical understanding.

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## Notes

- 1.
Pre-university level in Mexico includes the 10th, 11th, and 12th grades; it follows Junior High School that includes the 7th, 8th, and 9th grades and Elementary School from 1st to 6th grade. Pre-university level is equivalent to High School in the USA.

- 2.
We understand visualization as the ability to interpret, use of, and reflect on images on paper, with the purpose of depicting and communicating information (Arcavi 2003). Visualization can be a powerful tool to give meaning to mathematical concepts and identify the relationship (for us, make mathematical connections) between them (Rösken and Rolka 2006).

- 3.
In Mexico, the numerical scale of evaluation is from 0 to 10. In some pre-university schools (as in the school where this research was done), the minimum passing grade is 6.

- 4.
We assume that there is a primary belief when someone offers a proposition for which there is no other justification to support it (Green 1971).

- 5.
These mathematical connections are those that emerge inside mathematics and among mathematical entities (García-García and Dolores-Flores 2018).

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García-García, J., Dolores-Flores, C. Pre-university students’ mathematical connections when sketching the graph of derivative and antiderivative functions.
*Math Ed Res J* (2019). https://doi.org/10.1007/s13394-019-00286-x

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### Keywords

- Derivative and antiderivative functions
- Graphical representations
- Mathematical connections
- Pre-university level
- Task-based interviews