Inspiration
Map 1
Inspiration
Map 2
Inspiration
Map 3
Inspiration
Map 4
Keywords:mathematical function, concept attainment (math, fun, con)
I.Objectives.These objectives are partly based upon NCTM Standard 6:(1997)[1] Functions, for grades 9-12.Students are to be able to –
¨represent and analyze relationships using tables, verbal rules, equations, and graphs;
¨translate among tabular, symbolic, and graphical representations of functions;
¨recognize that a variety of problem situations can be modeled by the same type of function.
As a result of this lesson, students will be able to –
A.differentiate between functions and other more general relations between variables;
B.identify and apply function notation;
C.recognize various types of functions as functions;
D.attend to some of their own thinking processes.
Prerequisite to this lesson is a knowledge of and ability to manipulate algebraic expressions.Also, the teacher and the students will have had previous experience using the Inspiration® software package.
II.Procedures.This lesson utilizes the Concept Attainment model[2] to develop a definition for the mathematical concept of “function.”This definition is formed as the class differentiates the attributes of functions from examples and non-examples.The objectives are assessed as the students sort out further examples and non-examples on their own.
Materials required include a computer lab equipped to run the Inspiration® package with enough stations for individual use.If there are not enough computers, the students may be assessed in groups at each station.Each student individually should then be able to explain the choices made.A means of projecting the instructor’s screen must be employed during the presentation phase of the lesson.
This lesson should be completable
in one class meeting, although it may be extended or elaborated upon as
needed to ensure that objectives are attained.Once
attained, the concept of functions may subsequently be developed further
by differentiating between types of functions and their applications.
The lesson proceeds in the following steps:
The preparation phase.
1.Select and define the concept.
A function is a mathematical relation between at least two variable quantities where one of them is dependent upon the others for its values, and has a defined value for any of those considered.
2.Select the attributes.A function…
omay be expressed as an equation;
ohas at least two variables;
ocan be solved for at least one of its variables unambiguously so that it can "depend" for its values on the values of the other “independent” variables.In other words,
§The "dependent" variable has a value that is defined for every value of the others being considered.
§The "dependent" variable has only one value for every value of the others being considered.
3.Develop positive and negative examples.
These examples appear in the “space” around the list of positive and negative examples and are to be added to this list during…
The presentation phase.
4.Introduce the process to the students.
The objectives and the steps involved are introduced.Also it may be explained that this procedure reflects a natural cognitive process that people use to understand their world.
5.Present the examples and list the attributes.
The empty concept maps for examples and attributes are displayed for the class from a prepared Inspiration® document.These are surrounded by ‘free-floating’ examples that will be linked to the positive/negative list while attributes are keyed into the other.During this process, text boxes or “bubbles” may be replaced via the software’s editing capabilities by crossed out versions that retain the text.Attributes common to positive examples are retained, while negative examples highlight the non-essential ones.
6.Develop a concept definition.
When sufficient positive attributes are accumulated, class members formulate a definition in their own words.
7.Give additional examples.
At this point, understanding is checked and reinforced using further examples.
8.Discuss the process with the class.
This bit of meta-cognition should aid the students in acquiring further concepts by becoming more aware of the processes involved.
9.Evaluate.
The degree of attainment of the objectives is to be assessed at this point when the students turn to their own stations and use another prepared Inspiration® document adding examples to their own lists as was similarly done in step number five.
III.Evaluation.
Evidence will be apparent that students are able to –
A.differentiate between functions and other more general relations between variables when they sort examples and non-examples;
B.identify and apply function notation; and to
C.recognize various types of functions as functions when they sort a variety of these;
D.attend to some of their own thinking processes when they demonstrate their ability to perform some of these steps on their own.
The concept of functions is sometimes difficult even for students having a background in algebra.They are used to symbolizing numbers with letters, having attained that level of abstraction, but may stumble at a notation that represents by a single letter all the operations performed on a quantity together with, to them, a non-standard use of parentheses.I chose the Concept Attainment model to introduce functions because it does so independently of notation.A definition is attained naturally modulo details, and these details, including notation, may be categorized later when the concept is developed.
The Inspiration® software package specializes in concept mapping.The graphical representation of the structure of a concept being attained aids in the construction of the concept internally.When used as a mindtool[3] together with computers, this semantic networking tool provides an engaging experience in thinking critically about the structure of an idea.