This report articulates some of the recent research on different methods of optimization. Topics discussed include an implicit differentiation process in which the primary substitution method is not used and a relationship among variables method is introduced. In addition, a finding extrema without limits method is explored. Also included is a discussion on the depth of optimization taught in secondary schools and the different methods and levels of instruction on this topic. / text

There is an overemphasis on procedures and manipulation of symbols in calculus and not enough emphasis on conceptual understanding of the subject. Specifically, students struggle to understand and correctly apply concepts in calculus such as the chain rule, implicit differentiation, and related rates. Students can learn mathematics more deeply when they make connections between different mathematical ideas. I have hypothesized that students can make powerful connections between the chain rule, implicit differentiation, and related rates through the mathematical concept of nested multivariation. Based on this hypothesis, I created a hypothetical learning trajectory (HLT) rooted in nested multivariation for students to develop an understanding of these three concepts. In this study, I explore my HLT through a small-scale teaching experiment with individual first-semester calculus students using tasks based on the HLT.Based on the teaching experiment, nested multivariational reasoning proved to be critical in understanding how the variables within a function composition change together and in developing intuition and understanding for the multiplicative nature of the chain rule. Later, nested multivariational reasoning was mostly important in recognizing the existence of a nested relationship and the need to use the chain rule in differentiation. Overall, through the HLT, students gained a connected and conceptual understanding for the chain rule, implicit differentiation, and related rates. I also discuss how the HLT might be adjusted and improved for future use.