# The Language of “Rate of Change” in Mathematics

^{*}

*Eur. J. Investig. Health Psychol. Educ.*

**2021**,

*11*(4), 1599-1609; https://doi.org/10.3390/ejihpe11040113

## Abstract

**:**

## 1. Introduction

## 2. The Role of Language in Mathematics Education

## 3. Method

## 4. Results

#### 4.1. Rate of Change Aspects

#### 4.2. Concepts Related to “Rate of Change”

#### 4.2.1. Slope

#### 4.2.2. Tangent

#### 4.2.3. Derivative

#### 4.2.4. Speed-Velocity

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Riccomini, P.J.; Smith, G.W.; Hughes, E.; Fries, K.M. The Language of Mathematics: The Importance of Teaching and Learning Mathematical Vocabulary. Read. Writ. Q.
**2015**, 31, 235–252. [Google Scholar] [CrossRef] - Schleppegrell, M.J. The Linguistic Challenges of Mathematics Teaching and Learning: A Research Review. Read. Writ. Q.
**2007**, 23, 139–159. [Google Scholar] [CrossRef] - Barwell, R. Ambiguity in the Mathematics Classroom. Lang. Educ.
**2005**, 19, 117–125. [Google Scholar] [CrossRef] - Orton, A. Students’ understanding of differentiation. Educ. Stud. Math.
**1983**, 14, 235–250. [Google Scholar] [CrossRef] - Orton, A. Understanding rate of change. Math. Sch.
**1984**, 13, 23–26. [Google Scholar] - Teuscher, D.; Reys, R.E. Connecting Research to Teaching: Slope, Rate of Change, and Steepness: Do Students Understand these Concepts? Math. Teach.
**2010**, 103, 519–524. [Google Scholar] [CrossRef] - Ubuz, B. Interpreting a graph and constructing its derivative graph: Stability and change in students’ conceptions. Int. J. Math. Educ. Sci. Technol.
**2007**, 38, 609–637. [Google Scholar] [CrossRef] - Ärlebäck, J.B.; Doerr, H.M.; O’Neil, A.H. A Modeling Perspective on Interpreting Rates of Change in Context. Math. Think. Learn.
**2013**, 15, 314–336. [Google Scholar] [CrossRef] - Tall, D.; Vinner, S. Concept image and concept definition in mathematics with particular reference to limits and continuity. Educ. Stud. Math.
**1981**, 12, 151–169. [Google Scholar] [CrossRef] - Vinner, S. The Role of Definitions in the Teaching and Learning of Mathematics. Available online: https://link.springer.com/chapter/10.1007/0-306-47203-1_5 (accessed on 30 August 2021).
- Bezuidenhout, J. First-year university students’ understanding of rate of change. Int. J. Math. Educ. Sci. Technol.
**1998**, 29, 389–399. [Google Scholar] [CrossRef] - Cornu, B. Advanced Mathematical Thinking. Available online: https://link.springer.com/book/10.1007/0-306-47203-1 (accessed on 30 August 2021).
- Weber, E.; Dorko, A. Students’ concept images of average rate of change. In Proceedings of the 35th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, Chicago, IL, USA, 14–17 November 2013; pp. 386–393. [Google Scholar]
- Tall, D. Inconsistencies in the learning of calculus and analysis. Focus Learn. Probl. Math.
**1990**, 12, 49–63. [Google Scholar] - Rubenstein, R.N.; Thompson, D.R. Understanding and Supporting Children’s Mathematical Vocabulary Development. Teach. Child. Math.
**2002**, 9, 107–112. [Google Scholar] [CrossRef] - Gagatsis, A.; Patronis, T. Using geometrical models in a process of reflective thinking in learning and teaching mathematics. Educ. Stud. Math.
**1990**, 21, 29–54. [Google Scholar] [CrossRef] - Dunn, P.K.; Carey, M.D.; Richardson, A.M.; Mcdonald, C. Learning the Language of Statistics: Challenges and Teaching Approaches. Stat. Educ. Res. J.
**2016**, 15, 8–27. [Google Scholar] [CrossRef] - Kaplan, J.J.; Fisher, D.G.; Rogness, N.T. Lexical Ambiguity in Statistics: What do Students Know about the Words Association, Average, Confidence, Random and Spread? J. Stat. Educ.
**2009**, 17, 1–19. [Google Scholar] [CrossRef] - Adu, E.O.; Olaoye, O. An Investigation into Some Lexical Ambiguities in Algebra: South African Experience. Mediterr. J. Soc. Sci.
**2014**, 5, 1243. [Google Scholar] [CrossRef] - Olaoye, O.; Adu, E.O.; Moyo, G. Lexical Ambiguity in Algebra, Method of Instruction as Determinant of Grade 9 Students’ Academic Performance in East London District. Mediterr. J. Soc. Sci.
**2014**, 5, 897. [Google Scholar] [CrossRef] - Amit, M.; Vinner, S. Some misconceptions in calculus anecdotes or the tip of an iceberg? In Proceedings of the 14th Conference of the International Group for the Psychology of Mathematics Education, Mexico City, Mexico, 15–20 July 1990; Volume 1, pp. 3–10. Available online: https://files.eric.ed.gov/fulltext/ED411137.pdf (accessed on 30 August 2021).
- Adams, T.L. Reading mathematics: More than words can say. Read. Teach.
**2003**, 56, 786–795. [Google Scholar] - Abedi, J.; Lord, C. The language factor in mathematics tests. Appl. Meas. Educ.
**2001**, 14, 219–234. [Google Scholar] [CrossRef] - Klymchuk, S.; Zverkova, T.; Gruenwald, N.; Sauerbier, G. University students’ difficulties in solving application problems in calculus: Student perspectives. Math. Educ. Res. J.
**2010**, 22, 81–91. [Google Scholar] [CrossRef] [Green Version] - Kajander, A.; Lovric, M. Mathematics textbooks and their potential role in supporting misconceptions. Int. J. Math. Educ. Sci. Technol.
**2009**, 40, 173–181. [Google Scholar] [CrossRef] - Moschkovich, J.N. “I Went by Twos, He Went by One”: Multiple Interpretations of Inscriptions as Resources for Mathematical Discussions. J. Learn. Sci.
**2008**, 17, 551–587. [Google Scholar] [CrossRef] - Remoundou, D.; Avgerinos, E. Rate of change: Necessity, epistemological obstacles and proposals for teaching. In Proceedings of the First Congress of Greek Mathematicians (FCGM) 2018, Athens, Greece, 25–30 June 2018; pp. 201–215, ISBN 978-960-7341-41-9. [Google Scholar]
- Thompson, P.W.; Carlson, M.P. Variation, covariation, and functions: Foundational ways of thinking mathematically. In Compendium for Research in Mathematics Education; Cai, J., Ed.; National Council of Teachers of Mathematics: Reston, VA, USA, 2017; pp. 421–456. [Google Scholar]
- Thompson, P.W. The development of the concept of speed and its relationship to concepts of rate. In The Development of Multiplicative Reasoning in the Learning of Mathematics; Harel, G., Confrey, J., Eds.; State University of New York Press: Albany, NY, USA, 1994; pp. 179–234. [Google Scholar]
- Park, J. Is the derivative a function? If so, how do students talk about it? Int. J. Math. Educ. Sci. Technol.
**2013**, 44, 624–640. [Google Scholar] [CrossRef] - Gough, J. Conceptual complexity and apparent contradictions in mathematics language. Aust. Math. Teach.
**2007**, 63, 8. [Google Scholar] - Doerr, H.M.; Bergman Ärlebäck, J.; O’Neil, A.H. Interpreting and communicating about phenomenon with negative rate of change. In Proceedings of the 120th ASEE (American Society for Engineering Education) Annual Conference & Exposition, Atlanta GA, USA, 23–26 June 2013. [Google Scholar] [CrossRef]
- Zandieh, M.J.; Knapp, J. Exploring the role of metonymy in mathematical understanding and reasoning: The concept of derivative as an example. J. Math. Behav.
**2006**, 25, 1–17. [Google Scholar] [CrossRef] - Lobato, J.; Thanheiser, E. Re-thinking slope from quantitative and phenomenological perspectives. In Proceedings of the 21st Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, Cuernavaca, Mexico, 23–26 October 1999; Volume 1, pp. 291–297. [Google Scholar]
- Nagle, C.; Moore-Russo, D.; Viglietti, J.; Martin, K. Calculus Students’ and Instructors’ Conceptualizations of Slope: A Comparison Across Academic Levels. Int. J. Sci. Math. Educ.
**2013**, 11, 1491–1515. [Google Scholar] [CrossRef] - Confrey, J.; Smith, E. Exponential Functions, Rates of Change, and the Multiplicative Unit. Learning Mathematics; Springer: Dordrecht, The Netherlands, 1994; pp. 31–60. [Google Scholar]
- Avgerinos, E.; Remoundou, D. Epistemological obstacles for the concept of “slope of a line” in secondary education (in Greek). In Proceedings of the 34th Panhellenic Conference of Mathematical Education, Athens, Greece, 7–9 December 2018; pp. 93–102. [Google Scholar]
- Ömer, D.; Kabael, T. Students’ Mathematization Process of the Concept of Slope within the Realistic Mathematics Education. Hacet. Univ. J. Educ.
**2016**, 32, 1–20. [Google Scholar] [CrossRef] - Stump, S.L. High School Precalculus Students’ Understanding of Slope as Measure. Sch. Sci. Math.
**2001**, 101, 81–89. [Google Scholar] [CrossRef] - Stump, S.L. Developing preservice teachers’ pedagogical content knowledge of slope. J. Math. Behav.
**2001**, 20, 207–227. [Google Scholar] [CrossRef] - Coe, E. Modeling Teachers’ Thinking about Rate of Change. PhD Thesis, Arizona State University, Phoenix, AZ, USA, 2007. [Google Scholar]
- Biza, I.; Christou, C.; Zachariades, T. Student perspectives on the relationship between a curve and its tangent in the transition from Euclidean Geometry to Analysis. Res. Math. Educ.
**2008**, 10, 53–70. [Google Scholar] [CrossRef] [Green Version] - Ricardo, H.J.; Schwartzman, S. The Words of Mathematics: An Etymological Dictionary of Mathematical Terms Used in English. Am. Math. Mon.
**1995**, 102, 563. [Google Scholar] [CrossRef] - Vincent, B.; LaRue, R.; Sealey, V.; Engelke, N. Calculus students’ early concept images of tangent lines. Int. J. Math. Educ. Sci. Technol.
**2015**, 46, 641–657. [Google Scholar] [CrossRef] - Vincent, B.; Sealey, V. Students’ concept image of tangent line compared to their understanding of the definition of the derivative. In Proceedings of the 19th Annual Conference on Research in Undergraduate Mathematics Education, Pittsburgh, PA, USA, 25–27 February 2016; pp. 1360–1366. [Google Scholar]
- Bingölbali, E.; Monaghan, J. Concept image revisited. Educ. Stud. Math.
**2008**, 68, 19–35. [Google Scholar] [CrossRef] - Thompson, P.W. Images of Rate and Operational Understanding of the Fundamental Theorem of Calculus. In Learning Mathematics; Cobb, P., Ed.; Springer: Dordrecht, The Netherlands, 1994; pp. 125–170. ISBN 978-94-017-2057-1. [Google Scholar]
- Polya, G. How to Solve It: A New Aspect of Mathematical Method; Princeton University Press: Princeton, NJ, USA, 2004. [Google Scholar]

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Avgerinos, E.; Remoundou, D.
The Language of “Rate of Change” in Mathematics. *Eur. J. Investig. Health Psychol. Educ.* **2021**, *11*, 1599-1609.
https://doi.org/10.3390/ejihpe11040113

**AMA Style**

Avgerinos E, Remoundou D.
The Language of “Rate of Change” in Mathematics. *European Journal of Investigation in Health, Psychology and Education*. 2021; 11(4):1599-1609.
https://doi.org/10.3390/ejihpe11040113

**Chicago/Turabian Style**

Avgerinos, Evgenios, and Dimitra Remoundou.
2021. "The Language of “Rate of Change” in Mathematics" *European Journal of Investigation in Health, Psychology and Education* 11, no. 4: 1599-1609.
https://doi.org/10.3390/ejihpe11040113