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• Author or Editor: Eileen Fernández
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## Sharing Teaching Ideas: Understanding Functions without Using the Vertical Line Test

Since I started teaching precalculus eighteen years ago, I have struggled with how to convey a function's definition so that its importance and usefulness comes across to students. This is especially the case with the Vertical Line Test (VLT), a mechanism for testing whether the graph of a relation is a function. (Vertical lines are drawn through the graph. If every vertical line intersects the graph in exactly one point, the relation is a function.) In this era of “learning mathematics with understanding” (NCTM 2000, p. 20), the Vertical (and Horizontal) Line Tests present a challenge. I find that once students are introduced to these devices, they tend to disregard the concepts that the devices are testing and instead mechanically apply the tests. This rote application limits not only the students' understanding of the concepts but also their ability to work within and across representations to solve problems related to identifying functions.

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## Early Childhood Corner: Taking Advantage of Everyday Activities to Practice Math Every Day

When my son started kindergarten, I discovered that many everyday activities at home allow us to practice the mathematics he was learning (and would learn) in school and encourage mathematics use in his life. Because “children learn through exploring their world … everyday activities are natural vehicles for developing mathematical thinking” (NCTM 2000, p. 74).

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## Flower Power – Sunflowers as a Model for Logistic Growth

Logistic growth displays an interesting pattern: It starts fast, exhibiting the rapid growth characteristic of exponential models. As time passes, it slows in response to constraints such as limited resources or reallocation of energy (see fig. 1). The growth continues to slow until it reaches a limit, called capacity. When the growth describes a population, capacity is defined as “the maximum population that the environment is capable of sustaining in the long run” (Stewart 2008, p. 628).

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## 3D Designing for Mathematical Learning

Describes an approach using 3D printing technologies to produce a new physical tool for mathematical learning called Thirty6.

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## Extending Mathematical Practices to Online Teaching

The NCTM Process Standards provide a window to analyze two lesson excerpts.