Geometric Functions
With Technologically Embodied Geometric Functions, students develop conceptual metaphors that directly relate computer-based sensory motor experiences of abstract function concepts. This approach relies on four foundations:
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The articles and papers listed here are related to geometric functions, by which we mean the use of geometric transformations to provide students with sensorimotor experiences that develop their understanding of function concepts by creating and manipulating the variables and the function rules while attending to relative rate of change and other important function-related concepts.
- A Geometric Path to the Concept of Function.This article from Mathematics Teaching in the Middle School describes how transformations using dynamic software can provide a unique perspective on a common topic.
- Why Students Should Begin the Study of Function Using Geometric Points, Not Numbers
Abstract: By beginning the formal study of functions through geometric transformations that take a point as input and produce a point as output, students can form stronger and clearer concepts of independent and dependent variables, domain and range, relative rate of change, function notation, composition of functions, and inverse functions. This paper describes cognitive, kinesthetic, visual, and structural advantages of a geometric pathway to function concepts, and concludes by describing connections between geometric (ℝ2 → ℝ2) functions and numeric (ℝ → ℝ) functions that can facilitate students’ ability to transfer these function concepts between the two realms.
Download the paper. -
Using Multiple Representations to Teach
Composition of Functions. This article from the November 2012 Mathematics Teacher describes how "Experience with multiple representations fosters students’ robust understanding of what functions are, how they behave, and how they can be composed.
Access the article. Visit the support page. - Connecting Functions in Geometry and Algebra. This article, the cover story of the February 2016 issue of Mathematics Teacher, was selected as the Editorial Panel's favorite article of the 2015-2016 year.
Abstract: Students use geometric transformations to create a generalized linear function and construct its Cartesian graph. They dilate and then translate a point, restrict these points to number lines, and ultimately observe that in the algebraic equation y = mx + b, multiplying x by m corresponds to dilating x, and adding b corresponds to translating the result.
Support: The article is supported by two units of activities based on Web Sketchpad. The activities are freely available through a Creative Commons license, and are available here:
Activities for Introducing Functions Activities Activities for Connecting Geometry and Algebra Through Functions
Update History:
05 November 2015: Created this page.
This web site is based in part upon work supported by the National Science Foundation under NCSU IUSE award 1712280 (July 2017 through June 2019) and KCP Technologies DRK-12 award ID 0918733 (September 2009 through August 2013). Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.
We are deeply grateful to McGraw-Hill Education for making Web Sketchpad available for these activities.
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