hrvatski

Logarithms in Aequous Solutions

Mathematics is often viewed and taught as a subject almost completely unconnected to chemistry, particularly in the pre-university levels of education. Surely, there are some exceptions: the arithmetic techniques necessary for stoichiometry calculations and the use of graphs and mathematical functions. The last of the mentioned exceptions is a particularly problematic one, both from a mathematical and a chemical viewpoint. From the mathematical point, the functions used are often not understood well enough, sometimes even wrongly used, and usually they are reduced to buttons on a calculator1, both by chemistry teachers and pupils. From the chemical viewpoint, the mathematical functions are often an obstacle due to the mathematical formalism, becoming an opaque veil shading the real content they represent. As chemistry teachers often have very little mathematical training (and vice versa), it is hard to better integrate the teaching about mathematical functions and their application. One of the particularly important classes of mathematical functions for all levels of teaching chemistry are the logarithmic functions. The pupils first meet logarithms in chemistry courses when learning about the pH, and this is a topic studied on all levels of chemistry education. It has been often noticed2 that most of the main problems for pupils when learning about acids and bases are mathematical, because pupils usually do not know enough mathematics. Unfortunately, chemistry teachers often lack the sufficient knowledge about exponential functions and logarithms and are unable to explain them effectively to their pupils, thus being unable to help them. There have been several studies on how students learn exponential and logarithmic functions, e.g. by Weber3, and some (e. g. Strom4) on how science teacher think, reason and learn about them during their in-service training. Less practical help and instruction materials were offered to chemistry teachers to give them ideas how to reduce the problems they encounter when teaching about topics including logarithms. The aim of this contribution is to present an, as far the author knows new, approach to teaching logarithms for chemistry, adapted particularly for the use in the chemical classroom, at the same time being mathematically sound and not contradictory to the usual presentation of logarithms in the mathematics classroom. The idea is to develop the notion of logarithm starting from the chemical viewpoint, founded on experiments with measuring the pH of aqueous solutions, and to conclude about the basic properties of logarithms from the observed facts. Although it was not yet systematically tested, the teachers’ responses in informal interviews were very favourable. Also, by participant observation during and after the presentation of the proposed method at a recent meeting of the Educational Section of the Croatian Chemical Society it was possible to conclude that this suggestion could be effectively used in the classroom. Additionally, we plan to address the possible, unfortunately frequent, mistakes and misconceptions that arise from incorrect usage of mathematical formulas as a subject in their own right, without taking into account under what assumptions or approximations they were introduced. This kind of problems was discussed e.g. by Matsumoto et al5, and we shall address in more detail the problems connected to the interpretation of formulas containing logarithms and graphs that use a logarithmic scale (calculations of pH, titration curves etc.). In effect, we hope to show that there are simple ways to improve the understanding and correct usage of logarithms, and thus induce positive changes in the mathematical part of the chemistry teacher in-service training by introducing an “untraditional” method of developing mathematical notions from an applied context. References 1. D. J. Watters and J. J. Watters, Biochem. Mol. Biol. Educ. 34 (2006) 278–284 2. M. Drechsler and H.-J. Schmidt, Chem. Educ. Res. Pract. 6 (2005) 19–35 3. K. Weber, in Proceedings of the 24th Annual Meeting of the North American Chapter of Μathematics Εducation (2002), pp. 1019–1027. Retrieved October 14, 2010 from http://eric.ed.gov/PDFS/ED471763.pdf 4. A. D. Strom, in Proceedings of the 28th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (2006), pp. 624–630. Retrieved October 14, 2010 from http://www.pmena.org/2006/cd/book.pdf 5. P. S. Matsumoto, G. Tong, S. Lee and B. Kam, J. Chem. Edu. 86 (2009) 823–826

logarithms; pH; titration curves; errors due to incorrect applications of formulas

Radi se o dva srodna, ali ne sadržajno identična, priopćenja, jedno postersko i drugo usmeno. Pripadni cjelovit rad biti će objavljen u zborniku.

engleski

Logarithms in Aequous Solutions

Mathematics is often viewed and taught as a subject almost completely unconnected to chemistry, particularly in the pre-university levels of education. Surely, there are some exceptions: the arithmetic techniques necessary for stoichiometry calculations and the use of graphs and mathematical functions. The last of the mentioned exceptions is a particularly problematic one, both from a mathematical and a chemical viewpoint. From the mathematical point, the functions used are often not understood well enough, sometimes even wrongly used, and usually they are reduced to buttons on a calculator1, both by chemistry teachers and pupils. From the chemical viewpoint, the mathematical functions are often an obstacle due to the mathematical formalism, becoming an opaque veil shading the real content they represent. As chemistry teachers often have very little mathematical training (and vice versa), it is hard to better integrate the teaching about mathematical functions and their application. One of the particularly important classes of mathematical functions for all levels of teaching chemistry are the logarithmic functions. The pupils first meet logarithms in chemistry courses when learning about the pH, and this is a topic studied on all levels of chemistry education. It has been often noticed2 that most of the main problems for pupils when learning about acids and bases are mathematical, because pupils usually do not know enough mathematics. Unfortunately, chemistry teachers often lack the sufficient knowledge about exponential functions and logarithms and are unable to explain them effectively to their pupils, thus being unable to help them. There have been several studies on how students learn exponential and logarithmic functions, e.g. by Weber3, and some (e. g. Strom4) on how science teacher think, reason and learn about them during their in-service training. Less practical help and instruction materials were offered to chemistry teachers to give them ideas how to reduce the problems they encounter when teaching about topics including logarithms. The aim of this contribution is to present an, as far the author knows new, approach to teaching logarithms for chemistry, adapted particularly for the use in the chemical classroom, at the same time being mathematically sound and not contradictory to the usual presentation of logarithms in the mathematics classroom. The idea is to develop the notion of logarithm starting from the chemical viewpoint, founded on experiments with measuring the pH of aqueous solutions, and to conclude about the basic properties of logarithms from the observed facts. Although it was not yet systematically tested, the teachers’ responses in informal interviews were very favourable. Also, by participant observation during and after the presentation of the proposed method at a recent meeting of the Educational Section of the Croatian Chemical Society it was possible to conclude that this suggestion could be effectively used in the classroom. Additionally, we plan to address the possible, unfortunately frequent, mistakes and misconceptions that arise from incorrect usage of mathematical formulas as a subject in their own right, without taking into account under what assumptions or approximations they were introduced. This kind of problems was discussed e.g. by Matsumoto et al5, and we shall address in more detail the problems connected to the interpretation of formulas containing logarithms and graphs that use a logarithmic scale (calculations of pH, titration curves etc.). In effect, we hope to show that there are simple ways to improve the understanding and correct usage of logarithms, and thus induce positive changes in the mathematical part of the chemistry teacher in-service training by introducing an “untraditional” method of developing mathematical notions from an applied context. References 1. D. J. Watters and J. J. Watters, Biochem. Mol. Biol. Educ. 34 (2006) 278–284 2. M. Drechsler and H.-J. Schmidt, Chem. Educ. Res. Pract. 6 (2005) 19–35 3. K. Weber, in Proceedings of the 24th Annual Meeting of the North American Chapter of Μathematics Εducation (2002), pp. 1019–1027. Retrieved October 14, 2010 from http://eric.ed.gov/PDFS/ED471763.pdf 4. A. D. Strom, in Proceedings of the 28th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (2006), pp. 624–630. Retrieved October 14, 2010 from http://www.pmena.org/2006/cd/book.pdf 5. P. S. Matsumoto, G. Tong, S. Lee and B. Kam, J. Chem. Edu. 86 (2009) 823–826

logarithms; pH; titration curves; errors due to incorrect applications of formulas

Radi se o dva srodna, ali ne sadržajno identična, priopćenja, jedno postersko i drugo usmeno. Pripadni cjelovit rad biti će objavljen u zborniku.