China
Africa
Explore chapters and articles related to this topic
Implicit Functions and Optimization
Published in John Srdjan Petrovic, Advanced Calculus, 2020
Example 12.1.3 can be considered from a geometric viewpoint. The graph of x - y2 = 0 is a parabola. The reason why this equation does not define a function is that its graph does not pass the “vertical line test”, i.e., a vertical line intersects the parabola at 2 points. There is a way around this predicament. We will illustrate this on the point P1 = (4, 2) on the parabola. We will use a rectangle [3,5] × [1,3] that contains P1. If we zoom in on this rectangle and ignore anything outside of it, then the graph of x - y2 = 0 passes the vertical line test, so we obtain a function y = √x.
Eliciting the coordination of preservice secondary mathematics teachers’ definitions and concept images of function
Published in International Journal of Mathematical Education in Science and Technology, 2022
Allison W. McCulloch, Jennifer N. Lovett, Lara Dick, Milan Sherman, Cyndi Edgington, Michael Meagher
Crucially, teachers’ understanding of the function concept has been shown to impact the pedagogical choices they make during instruction. In a study of 152 PSMTs, Even (1993) found PSMTs could not justify the need for univalence and did not know why it was important to distinguish between functions and non-functions. Owing to this lack of content knowledge, the PSMTs limited the exposure of their students to various function representations and emphasized procedures such as the vertical line test in identifying functions. Building on Even’s work, Hatisaru and Erbas (2017) found when a practicing teacher had a robust concept of function their students, in turn, developed a high level of content knowledge of functions and when the teacher had limitations and constraints in knowledge their students exhibited those same limitations and constraints. In addition to having a rich understanding of the definition, Bannister (2014) suggests that PSMTs who are adept at translating between algebraic and graphical representations of functions, may be better prepared to understand diverse student conceptions when they encounter them during instruction.
Challenges of maintaining cognitive demand during the limit lessons: understanding one mathematician’s class practices
Published in International Journal of Mathematical Education in Science and Technology, 2019
Although the first graph by the student was incorrect, Dr A still invited her students to evaluate the claim which gives students the idea that there is something to be learned from incorrect claim. As the discussion progressed, Dr A tried to invite her students to think and probe mathematical situations and Dr A needed to provide different images (second and third pictures) to address the claims made by students. Thus, while getting to the desired conclusion, Dr A assessed her students’ prior knowledge about various mathematical topics – there is an asymptotes, it fails the vertical line test, and it crosses the x-axis – to determine that it is impossible to have a function that satisfies the initial condition. Compared to the previous episodes, we were able to observe students’ voice and work. The main difference in this episode is students also had opportunities to evaluate other students’ work. Furthermore, during the discussion, we can see that Dr A’s role was not necessarily to give lectures but to facilitate the class discussion because she was encouraging, revoicing and pressing students for answers by asking novel (probing) questions. This episode was unique because this was the only case that they used students’ thinking and responses during discussions. To increase cognitive demand further, students can be asked to provide more graphs and evaluate each graph but students are not given those opportunities. At the end, the intermediate value theorem was the only calculus topic that Dr A and her students were able to increase cognitive demand.
Enhancing pre-service mathematics teachers’ understanding of function ideas
Published in International Journal of Mathematical Education in Science and Technology, 2022
Hande Gülbağci Dede, Zuhal Yilmaz, Hatice Akkoç, David Tall
Concept images can be used to express a personal concept definition that can reflect different function ideas. Besides, these ideas may or may not be expressed in individuals’ personal concept definitions (Ayalon et al., 2017; Tall & Vinner, 1981). For instance, a PMT can define functions as ‘a formula’. Although this definition reflects the idea of algebra and rule, other personal concept definitions may not reflect any function idea. For instance, in Even’s (1990) study, some teachers used the ‘vertical line test’ rule in their function definition. In such a definition, the rule is used to decide whether a given graph represents a function rather than explicitly emphasize function ideas.