En meningsfull tilnærming til logaritmer En designstudie om introduksjon av logaritmer gjennom repetert divisjon
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This study is about finding a more effective method of introducing logarithms. As part of my study, I developed a new teaching material. The definition of logarithms in the new material is based on repeated division in the same way powers can be viewed as repeated multiplication. I have formulated the following research questions: 1) What are the characteristics of an effective method of teaching logarithms with regard to conceptual understanding? 2) Is the method where students are being introduced to logarithms through repeated division better than through the definition the Norwegian teaching materials to the greatest extent use? First, my literature review focuses on the approach Norwegian teaching materials use to introduce logarithms. A view of logarithms as part of algebra is then presented. In particular, I present literature describing student’s tendency to produce invalid algebraic transformations, so-called "visually salient rules" (Sleeman, 1986). These errors and other kinds of errors students make in logarithms are also described. Kilpatrick "intertwined strands of Mathematics" (Kilpatrick, 2001) is used to describe the term “conceptual understanding”. In addition, Anna Sfard’s description of mathematical concepts through the process-object duality (Sfard, 1991) is used. Together this constitutes my theoretical framework. As part of my Master’s thesis, I carried out a mixed-methods research, in which I designed, developed, and implemented an intervention on the teaching of logarithms in a Norwegian high school. I used interviews, observations, field notes, audio-recordings and tests (including a retention test three months later) to describe and evaluate the intervention. My findings indicate that repeated division is easier for students to relate to than the traditional approach, especially when it comes to the weaker students. The process-orientated approach seems to give logarithms more meaning, which makes it easier for them to reproduce the content of the definition. Students are better able to answer why lg1 = 0 and lg10 = 1. Students don’t have to memorize the rules as separate facts (as rules without reason). Instead, they can always reconstruct this knowledge using the repeated division-method. This reduces the amount of facts which must be memorized. These characteristics seem to be important features of an effective method of teaching logarithms. Inductive tasks were used in the experimental material. These tasks proved insufficient to bridge the gap between repeated division and logarithmic rules. "Visually salient rules" were found among students using the new materiel, in the same way they were found among students in the control group. The findings diverge about the effect of application of logarithms in the material. My study shows that students who were subject to the teaching approach score significantly better when asked to solve the equation lg(2x + 3) = 1 than the control group after 7 weeks. The new approach combined with the "cover-up"-method appears to have played a crucial role in this test result. Otherwise, I found no significant differences in test scores.
Masteroppgave matematikkdidaktikk- Universitetet i Agder, 2015