Concept Worksheet: An Important Tool for LearningAuthor(s): Charalampos ToumasisSource: The Mathematics Teacher, Vol. 88, No. 2 (FEBRUARY 1995), pp. 98-100Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/27969225 .

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Concept Worksheet: An Important Tool for Learning

A

The

approach is beneficial

for all students

student's ability to learn mathematics is directly related to his or her understanding of mathemati cal concepts and principles (Cooney 1977). Experi enced teachers know that concepts are the basic

building blocks of thinking, particularly higher level thinking in mathematics. This article explores the concept worksheet, which can help students

identify and analyze important mathematical con

cepts and is a simple assignment that can be com

pleted by students overnight. The following four

types of exercises are part of the worksheet.

Definition

Knowing the definition of a concept is important, since students will need to communicate about the

concept and eventually be able to interpret and name it. Many students tend to repeat the textbook

definition, so they should have experience using their own words and vocabulary to describe critical attributes that are present in every example to dis

tinguish it from others. Asking students to express the definition in their own words enhances the pre cision of their terms. This approach helps them to

analyze the definition by comparing attributes and

by giving examples and nonexamples.

Web of attributes

Concepts have attributes that describe and charac terize them. These attributes are implied by the de finition and establish conditions for an object to be an example of a concept. Teachers should emphasize the various ways that these properties are connect ed and help students identify properties of objects that exist because of several types of relationships.

Charts, diagrams, webs, and pictures can be

employed as visual examples of abstract concepts (Klausmeier 1980). A semantic web presents a visu al image of the characteristics and relationships generated by the core idea of the concept. To help students construct a web for each concept, suggest that they follow a four-step procedure:

1. Create the core, which is the focus of the web. This core will be the name of the concept.

2. Draw strands branching out from the core, which are the critical attributes. Give each a name.

3. Draw line segments connecting the various strands to show how the attributes and dimen sions differentiate.

4. Tie the various strands together to show rela

tionships among the attributes.

Examples

Asking students to show examples for various mathematical concepts helps them to grasp the

concepts that exhibit greater abstraction or more

relationships and ideas (Henderson 1970). Most students can give examples of some mathematical

concepts if they have previously discussed what constitutes an example of a concept. Students must realize that examples must satisfy all the conditions related to the defining attributes of the concept.

The teacher must be certain that students know the meaning of the words they are using to define the new concept. A teacher may need to stop, exam ine each phrase, discuss its meaning, and demon strate or refer to an example. Students are required to give two examples of the specific concept in the worksheet and to supply reasons why each is an

example of this concept. Having students cite

examples of mathematical concepts can be a good diagnostic tool for instructors. Sometimes students form wrong impressions about a mathematical con

cept because irrelevant properties inherent in sim

ple examples may become part of the students' con

cept formation. When such misunderstanding occurs, various carefully planned examples can

help rectify these wrong impressions.

Nonexamples

Asking students to list nonexamples enables them to recognize the insufficiency of an incomplete con

cept and creates a clearer grasp of the concept. The teacher should force attention to nonexamples to

Charalampos Toumasis works at the Centre for In-Service Education, Maizonos 19, 26223, Patras, Greece. He is interested in alternative teaching strategies and curricu lum development.

98 THE MATHEMATICS TEACHER

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Vol. 88, No. 2 ? February 1995 99

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emphasize that a nonexample is some object that satisfies fewer than all the conditions related to the

defining attributes of the concept. Showing stu dents several nonexamples enables them to focus on concept-relevant properties by noting their absence and to recognize the insufficiency of an

incomplete concept. The teacher may help students find various nonexamples by suggesting that their choices closely resemble examples or contain some

attributes of the concept. Using a triangle as a

jii ? ? nonexample of a square would not reinforce a stu

dent's concept of a square as well as would a non

foTT?ldl square rhombus. Students are required to give two CCM be used examples of the specific concept in the worksheet

., and to explain what attribute is missing in each Wlin

nonexample. various Asking students to complete the concept work

topiCS sheet for each important mathematical concept re

quires time and effort for both teacher and student. But this opportunity for students to clarify and

analyze various mathematical concepts is a won

derful way to bridge gaps in their knowledge and

help them verbalize their understanding. Giving the concept-worksheet assignment, of course, does not automatically generate effective learning. The teacher must respond to student writings. It is use

ful to select the most exhaustive webs or the most

original and novel examples or nonexamples and

place them on the overhead projector for classroom

viewing. Encourage discussion and reactions, which

helps focus students' thinking and encourages them to compare attributes of the examples and non

examples. The discussion allows students to ex

change ideas and to develop an idea in greater depth. The students' answers to the concept worksheet

often contain oversights, misunderstandings, or

ambiguous points. After the need for precise descriptions has been explained through an evalua

tion, students may be asked to rewrite their work sheet. Encourage students to repeat the exercise another day. This approach helps them understand the mathematical concept in depth and offers an excellent learning opportunity for students to reconstruct more correct answers. Furthermore, the revision process allows students to communicate with the teacher and exchange ideas and views. The satisfaction of having created a useful learning tool seems to be a natural and more appropriate reward for students instead of applying a letter

grade to the project. A complete concept worksheet

appears in figure 1.

BENEFITS AND FINAL COMMENTS This activity has many benefits and advantages. First, it requires students to identify and analyze important mathematical concepts and to think about the meaning of these concepts by integrating visual and verbal understanding. The weaker stu

dent may acquire the basis for understanding the mathematical concept. The stronger student experi ences an in-depth, intellectually inspiring conceptu al view rather than a surface, uninspiring mechani cal process. Second, the concept worksheet gives the teacher an easy way to identify students' mis

conceptions and misunderstandings. Responses can uncover some thinking by students of which the teacher is not aware, thus allowing feedback about the grasp of the concept. Third, it gives students an

efficient tool for reviewing material before the final examination. Fourth, the process of writing about a

mathematical concept increases the students' abili

ty to think critically and organize their thoughts (Johnson 1983). Finally, this experience allows both the teacher and the student to communicate during the revision phase and to exhange useful ideas, thoughts, and views.

The author uses the concept worksheet assign ment continually in his classes and, with some minor

changes and modifications, with students of all lev els and grades. However, teachers can use the cen

tral ideas of this technique in their own special ways,

according to the needs of the curriculum and their mathematics classes. Each class is different; what works for one may not work for another. This type of activity can help develop students' critical think

ing and intellectual growth, and opportunities for these experiences should be given in each classroom.

REFERENCES Cooney, Thomas J. "Organizing Instruction: Logical Considerations." In Organizing for Mathematics Instruction, 1977 Yearbook of the National Council of Teachers of Mathematics, edited by F. Joe Cross white. Reston, Va.: The Council, 1977.

Henderson, Kenneth B. "Concepts." In The Teaching of Secondary School Mathematics. Thirty-Third Yearbook of the National Council of Teachers of Mathematics, edited by Myron F. Rosskopf. Reston, Va.: The Council, 1970.

Johnson, Marvin L. "Writing in Mathematics Classes: A Valuable Tool for Learning." Mathematics Teacher 76 (February 1983):117-19.

Klausmeier, Herbert. Learning and Teaching Con

cepts. New York: Academic Press, 1980. H?

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