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## Using Concept Maps to Assess Conceptual Knowledge of Function

In this study I examine the value of concept maps as instruments for assessment of conceptual understanding, using the maps to compare the knowledge of function that students enrolled in university calculus classes hold. Twenty-eight students, half from nontraditional sections and half from traditional sections, participated in the study. Eight professors with PhDs in mathematics also completed concept maps. These expert maps are compared with the student maps. Qualitative analysis of the maps reveals differences between the student and expert groups as well as between the 2 student groups. Concept maps proved to be a useful device for assessing conceptual understanding.

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## Functions

One of the most interesting and important unifying concepts in school mathematics is the function concept. One often hears such expressions as “the function of the brain is to enable a person to make rational decisions” or “the function of a lamp is to give light.” However, the mathematical use of the word “function” is linked to the historical use of the word by mathematicians and scientists. Historically, mathematicians and scientists have used the word “function” to illustrate how one condition or “state” affects another. For example, they use such phrases as “distance is a function of time,” “water pressure is a function of depth,” and “the area of a circle is a function of its radius.” You will note that these expressions suggest pairs—distance and time, pressure and depth, area and radius. In the following discussion we will first review the mathematical definition of function, and then present some practical uses of functions. Part 2 of this article give examples of functions that can be used to describe some real life situations.

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## Functions*

Some of the recent elementary school arithmetic textbooks introduce functions, a topic formerly appearing no earlier than in high school. The University of Illinois Arithmetic Project has long used functions (called “jumping rules” by the Project) in classes for elementary school children.

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## The Effects of a Graphing-Approach Intermediate Algebra Curriculum on Students' Understanding of Function

In this study, we extended O'Callaghan's computer-intensive algebra study by using his component competencies and the process-object framework to investigate the effects of a graphingapproach curriculum employing the TI-82 graphing calculator. We found that students in the graphing-approach classes demonstrated significantly better understanding of functions on all 4 subcomponents of O'Callaghan's Function Test, including the reification component, than did students in the traditional-approach classes. Additionally, no significant differences were found between the graphing-approach and traditional classes either on a final examination of traditional algebra skills or on an assessment of mathematics attitude.

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## The Slippery Road From Actions on Objects to Functions and Variables

Functions are all around us, disguised as actions on concrete objects. Composition of functions, too, is all around us, because these actions can be performed in succession, the output of one serving as the input for the next. In terms of Gray and Tall's (2001) “embodied objects” or Lakoff and Núñez's (2000) “mathematical idea analysis,” this “embodied scheme” of action on objects may serve as intuitive grounding for the function concept. However, as Gray, Tall, and their colleagues have shown, such embodied schemes can also lead to serious “epistemological obstacles” in later stages of concept development. In the same vein, our own data show that the intuitions about change and invariance entailed by the action-on-objects scheme, although helpful in earlier stages of learning functions, may later come to clash with the formal concepts of function and of composition of functions.

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## Supporting Innovation: The Impact of a Teacher's Conceptions of Functions on His Implementation of a Reform Curriculum

In this study we investigate the content conceptions of an experienced high school mathematics teacher and link those conceptions to their role in the teacher's first implementation of reform-oriented curricular materials during a 6-week unit on functions. The teacher communicated deep and integrated conceptions of functions, dominated by graphical representations and covariation notions. These themes played crucial roles in the teacher's practice when he emphasized the use of multiple representations to understand dependence patterns in data. The teacher's well-articulated ideas about features of a variety of relationships in different representations supported meaningful discussions with students during the implementation of an unfamiliar classroom approach to functions.

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## Is This Vending Machine FUNCTIONing Correctly?

will be released? As a matter of fact, vending machines are programed with functions that assign to each button pressed a particular item to dispense. Given that vending machines are a common experience for students in today's world, they provide a

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## Interesting Functions

Contexts from the fields of geography and history emphasize functions as situations in which each input has exactly one output.

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## Hyperbolic Functions

The development of hyperbolic functions in the traditional trigonometry courses (if this is ever reached during a one-semester instruction) is usually confined to purely algebraic methods. However effective the latter procedures may be, it is doubtful that a student realizes the import of the properties of hyperbolic functions. The student is never offered the opportunity to realize the fact that, essentially, the properties of hyperbolic functions are analogous to the properties of circular functions. It is possible, however, to develop the properties of hyperbolic functions in a manner which is analogous to the processes which are employed in the development of circular functions. Thus, it is proposed to examine and to develop hyperbolic functions by means of a geometric approach.

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## An Alternative Approach for Defining a Quadratic Function

; Schoenfeld & Herrmann, 1982 ; Silver, 1997 ; Star & Rittle-Johnson, 2008 ). CONCLUSION Defining a quadratic function by its slopes of secant/tangent lines is not a common approach in school mathematics. The legend that linear functions