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The Concept Map as anAid to Instruction inScience and Mathematics

John MaloneJohn Dekkers

All classroom teachers regularly en-counter students who are unable tounderstand science or mathematicsconcepts that have been carefullytaught to them. Research in bothscience and mathematics educationhas grappled with this problem overa considerable period of time. Therenow exists a substantial body ofknowledge that provides insights asto the reasons for this situation andalso provides the classroom practi-tioner with new teaching strategiesthat will enhance the understandingof those concepts which constitutethe foundation of both disciplines.AusubePs (1978) cognitive learn-

ing theory has been a guide to muchcurrent research on concept teach-ing in science and mathematics in-struction. He defines the notion of a"concept" in the following terms:

(<. . . . (any) objects, events, situationsor properties that possess common crit-ical attributes and are designated in anygiven culture by some accepted sign orsymbol" (Ausubel, 1978, P.105)

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The Concept Map 221

and he distinguishes between rote learning�simple memorization of def-initions and statements�and meaningful learning, relating new knowl-edge to knowledge previously learned via concepts. The key idea ofAusubeFs learning theory, that humans think with concepts, is sum-marized in Figure 1. Less rote learning takes place and becomes moremeaningful when instruction is organized in accordance with the proc-esses highlighted in the Figure 1. Studies concerned with meaningful

MEANINGFUL LEARNING: New knowledge is integrated into the existing network ofconcepts and propositions in the cognitive structure.

Subsumption: incorporation of new knowledge into specifically relevant existingconcepts or propositions,Integrative Reconciliation: new learning that results in explicit delineation of simi-larities and differences between related ideasSuperordinate Learning: elaboration and clarification of meanings of concepts orpropositions occurring over time as new subsumption, integrative reconciliation andor superordinate learning occursAdvance Organizer: a brief, meaningful learning task designed to help the learnerlink new specific knowledge to relevant concepts or propositions he/she alreadyknows.

ROTE LEARNING: Arbitrary verbatim incorporation of new information into cognitivestructures. (After Novak, 1981).

FIGURE 1Meaningful and Rote Learning Continuum

learning have attempted to "map" the concepts used by students. In con-cept mapping, both the concepts used and the links between concepts canbe identified. A variety of simplistic methods have been used to constructand study concept maps: by word association (Schaefer, 1979), by card-sorting and arrangement (Champagne et al. 1978) and free diagramsdrawn by students (Novak, 1977). However, it has been through themore recent work of Novak and co-workers that concept mapping hasemerged as an exemplary learning/teaching strategy in its own right.Novak (1981) describes concept mapping as:

a process that involves the identification of concepts in a body of study mate-rials and the organisation of those concepts into a hierarchical arrangementfrom the most general, most inclusive concept to the least general, most spe-cific concept" (Novak, 1981, p. 3).

The approach developed and used by Novak and co-workers isfounded on two of the key principles of teaching: to relate new knowl-edge first to the things that students already know, and, second, to thethings that they are currently learning in class.

This paper explains concept maps and presents a number of proce-dures for using and evaluating them.

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222 The Concept Map

Concept MappingThe concept map is a device to enable either the student or a teacher toexplicitly represent a number of concepts. The simplest concept mapwould consist of two concepts linked by "logical connectives" (Gardner,1980) as in the following example:

contains

The terms in the boxes are concepts and the verb or logical connectiveconstitutes a proposition,A map consisting of new concepts connected by propositions to pre-

viously learned concepts may be viewed as demonstrating the manner inwhich new knowledge has been integrated with an existing knowledgestructure. Furthermore, the nature of a person’s understanding of a con-cept changes as it is associated with a wider array of concepts and specificpropositions. Two examples of more complex maps are presented in Fig-ures 2 and 2A on the topic of oxidation and reduction, and Figure 2B onthe mathematics of the polygon. It is emphasised that the insertion of thepropositions linking various concepts is a most important component inthe use of this instructional tool. The prepositional statement providesan indication of what understanding exists and the depth of that under-standing.

Based on Ausubelian (1978) learning theory, it would be anticipatedthat concept maps demonstrating meaningful learning would possess anorganization of concept differentiation ranging from the most general,more inclusive concepts to more specific and less inclusive concepts.

Uses of Concept Maps in Science and Mathematics Teaching

Concept maps have application in the teaching of the different sciences(e.g., biology, chemistry, physics) and mathematics (e.g., trigonometry,algebra, geometry) at all levels ranging from primary school to seniorhigh school. They can also be used with students who differ in intellec-tual ability. The following are examples of situations in which conceptmaps can be used:

1. Organizing information on a topic. Disorganized information is relatively useless.Useful knowledge must be organized so as to facilitate understanding and problem-solving ability. Students need to be taught how to construct hierarchical forms ofknowledge organization. The concept map organizes knowledge into categories andsub-categories so that it can be easily remembered and retrieved.

2. Motivate the study of a tjpic, Constructing a map has considerable motivationalvalue and challenge when introduced early in the study of a new topic. It can be

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The Concept Map

FIGURE 2A

223

ELECTRONTRANSFER ^

MOVEMENT OFIONS

ELECTRODES

OXIDATIONNUMBERS

ANODE

NIONS^E FROM

CATHODE

[CATIONSCOMEFROM

ANIONS CATIONS

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The Concept Map

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The Concept Map 225

looked upon as putting together in a puzzle. Maps made by students can be com-pared with other student maps and those of the teacher so that the students can"see" exactly what is known about the new topic at the outset.

3. Revision of a topic. The concept map can be constructed at the end of topics as aclass exercise or for homework. It is useful to compare student maps constructed as amotivational exercise with those constructed at the conclusion of the topic.

4. Generate discussion on topic. When using concept maps, students readily see that thestructure of the subject they are studying can be very complex and that any particu-lar concept can be related to many others.

5. Rank important ideas on a topic. While there is no one best way to represent knowl-edge of a topic, there tends to be agreement regarding the extent of exclusiveness(importance) and the inclusiveness (lesser importance) of concepts related to topic.Thus it is possible to determine from a number of maps the way concepts have beenranked.

6. Reinforce ideas about topic. The map can be used to demonstrate major ideas in atopic area and how these ideas can be related to other ideas.

All the above approaches to using concept maps in science and mathe-matics instruction can provide a teacher with valuable information re-garding the extent to which meaningful learning has taken place in class.

Teaching Concept MappingThe instruction and use of the concept map in science is now well docu-mented, but less comprehensively so in mathematics (see bibliography).Two main procedures can be identified. These procedures differ mainlyin the manner in which the hierarchical order of concepts is establishedon the map:

A Ranking Procedure

1. A suitable stand-alone topic is identified�for example, atomic theory, transforma-tion geometry, mechanics, sequences and series, genetics.

2. Concepts in the topics are then identified. This may be done by either the teacher"brainstorming" the subject, or by the students reading through a passage or chap-ter of the appropriate text, making a list of key concepts.

Often as many as twenty concepts can be identified in a single topic and these can bebroadly classified as

(a) entity concepts, i.e. objects, e.g. dog, triangle(b) relational concepts, e.g., proportion, between, less than(c) quantifying concepts, e.g., kinetic energy, volume, force.

Generally, only the entity and quantifying concepts are mapped. Should more than20 concepts be identified, it is preferable to reduce this number in order to reduce thecomplexity of the task to follow.

3. The concepts are then ranked from the "most important" or most general conceptto the "least important" or more specific concept. Individual students may at-tempt this task, or the class as a whole may work at achieving a consensus on thelist.

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226 The Concept Map

4. The map is then prepared. A sheet of paper, a pencil (and an eraser) are adequatematerials to use, although overhead transparency film and a water-soluble markerpen for each student are more convenient and also allow each student’s effort tobe projected for class discussion purposes.

The most general concepts are located at the top of the sheet or trans-parency, while the more specific concepts are placed at the bottom. Thelinkages (propositions) between the concepts are then written in.

An Association Pairing Procedure

1. A stand-alone topic is identified as in the previous procedure.

2. A previously prepared list of concepts is presented to students and they are asked toindicate the extent to which the concepts are related to one another. This is deter-mined by numerically scoring the relatedness between concepts from a classificationof very related (3 points) to not related (0 points).A summation is then made of the points for each concept (see Figure 3).

3. The concepts are then ranked in order and placed on the map. The most related orgeneral concept is placed near the top of the sheet and the least related, more specificconcept at the bottom. These key concepts are then connected by prepositional state-ments. This procedure is suitable for a map containing up to eight concepts. How-ever, when a greater number of concepts is to be mapped (say 15-30), then about halfof these, randomly selected from the total number should be mapped. The remainingconcepts are then linked into the structure by prepositional statements.

Cronin et al. (1982) have found that once students are familiar withthis procedure, a map containing between 15-20 concepts can be con-structed within forty minutes without reference to a textbook. However,the technique has been used in homework revision tasks with studentsbeing encouraged to use their textbooks. In these circumstances studentstend to take a significantly longer time to complete the map.

Evaluation of Concept Maps

Concept maps can be used by the classroom teacher to determine the ex-tent meaningful learning has taken place for a particular topic. Maps willvary from individual to individual in that some will be more completethan others, thus it is useful to be able to evaluate or assess differentmaps. Table 1 presents a scheme devised for scoring concept maps basedon criteria that are easily recognizable/identifiable (Cronin et al., 1982).Furthermore, the scheme has a scoring system that can differentiate be-tween different levels of concept development.The evaluation scheme is based on Ausubelian learning principles. The

propositions that link concepts are seen as a measure of the degree of dif-ferentiation of constituent concepts. Hierarchical differentiation pro-vides an indication of a student’s ability to distinguish between exclusive-ness and inclusiveness of particular concepts. Figure 2B shows hier-archical differentiation for types of polygons and the characteristics of

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The Concept Map 111

polygons.Grouping of concepts on a map is a further way of demonstrating rela-

tionships and interrelationships between concepts and can be seen as ameasure of integrative reconciliation of meanings. Grouping scores canbe effectively used to identify the extent of the concept of growth inlearning that has occurred. Figure 2A has, for example, a closed group-ing for the concepts permanganate titration, oxidation, reduction andoxidation number and an open grouping for concepts associated withelectrodes.

TABLE lEvaluation of a Concept Map

ProfileScoreDefinition Scoring ProcedureCriteria

ConceptRecognition

Grouping

Concepts are objects, events,situations or properties ofthings that are designated bya label or symbol.Groupings are the ways con-cepts can be linked or joinedtogether. Three types ofgroupings are:

Point grouping: a number ofsingle concepts emanate fromone concept.

Open grouping: Three ormore concepts are linked in asingle chain.

Count all concepts that areconnected to other conceptsby propositions. Score onpoint for each concept.Scoring of groupings

Point grouping: 1 point foreach concept in the group

Open grouping: 2 points foreach concept in the group

Closed grouping: Concepts Closed groupings: 3 pointsfrom a closed system, for each concept in the

group.

Hierarchy Concepts on a map can berepresented as a hierarchicalstructure in which the moregeneral, more inclusive con-cepts are at the top of themap; the specific and exclu-sive concepts are at the lowerend of the map.

Only First Column conceptsare scored for the degree ofhierarchy in the map. This isbased upon the extent con-cepts are present in assignedlevels. Four points are givento each concept correctly as-signed to a level; 2 points foreach concept on level re-moved from an assigned lev-el; no score for concepts thatare two levels removed.

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228TheConcept Map

TABLE 1�ContinuedEvaluation of a Concept Map

ProfileScoreCriteria Definition Scoring Procedure

Branching Branching of concepts refersto the level differentiationamong concepts, that is, theextent the more specific con-cepts are connected to moregeneral concepts.

Proposition Concepts acquire meaningthrough the relationship be-tween concepts. The relation-ships are represented by con-necting word(s) phrases writ-ten on the line joining anytwo concepts.Simple Proposition is a sim-ple English word or phrase.

Scientific Proposition is aphrase or statement that iscomposed of technical or sci-entific word(s).

Score one point for eachbranching point which has atleast two statement lines.

Simple Proposition: scoreone point for each word orphrase; give half a point forrepeated use of simple propo-sitions.Scientific Propositions:score two points for eachproposition. Give one pointfor repeated use of scientificproposition.

FIGURE 3Year 11 Oxidation-Reduction Revision

A CONCEPT MAP is to be prepared from the oxidation-reduction concepts listed in theFIRST and SECOND COLUMN

FIRST COLUMN SECOND COLUMNelectron transfer cationsoxidation reduction process anionscorrosion redox titrationpermanganate-titration oxidationelectrochemical reductionmovement of ions electrolyte

equilibriumanodecathode

FIRST TASKThe KEY IDEAS from the first column are listed in pairs. Score each pair as follows:

They are not related ==0 They are slightly related =1They are quite related = 2 They are very related = 3

oxidation-reduction process ..... permanganate titration = ____electron transfer ..... movement of ions = ____

movement of ions ..... oxidation-reduction process = ____

oxidation-reduction process ..... corrosion = ____

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The Concept Map 229

FIGURE 3�Continued

corrosion

permanganate titrationpermanganate titrationelectrochemical celloxidation-reduction processcorrosionelectrochemical cellcorrosionpermanganate titrationelectrochemical cellpermanganate titrationelectrochemical cellelectrochemical cell

electron transfermovement of ions

corrosionelectron transfer

movement of ionsoxidation-reduction process

electron transfermovement of ionselectron transfer

SECOND TASKNow add up the individual scores, crossing out each KEY IDEA as you go.

KEY IDEAS INDIVIDUAL SCORES TOTAL

electrochemical cells

corrosion

oxidation-reduction process

electron transfer

movement of ions

permanganate titration

THIRD TASKNow write the KEY IDEAS in order of relatedness from highest score to lowest. (If somehave the same score put them on the same line.)

FOURTH TASKTake the sheet of printed labels of KEY IDEAS and stick them LIGHTLY on the CON-CEPT MAP (Green Paper) starting with the highest score near the top of the sheet. Leavetwo or three level lines free at the bottom. Later when you have finished the whole CON-CEPT MAP, you can press down the labels firmly.FIFTH TASKNow join up the KEY IDEAS and write SENTENCE WORD(s) on each connecting line be-tween the KEY IDEAS. Try to cross link as much as possible.SIXTH TASKGO BACK TO THE TOP OF THE PAGE. THE SECOND COLUMN on the map usingyour sheet of printed labels. TRY and place all these as the most important ones on thehigher levels and others on the lower levels. Then try to join all the key ideas up with SEN-TENCE LINES. (Remember to put SENTENCE WORDS on each line!).

YOU HAVE NOW COMPLETED YOUR MAP

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LABELS LEVELS GROUPING BRANCHING SIMPLESCIENTIFICPROPOSI-PROPOSI-TIONTION

FIGURE 4Student Concept Map Profile Scores

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The Concept Map 231

Conclusion

In most instances, the score of any on? map is presented as a total value.We have experimented with the idea of presenting a learning profilewhich plots out the scores of the various attributes of the map (e.g., hier-archial structure, prepositional statements) in a graphical format (seeFigure 4). The advantage of this approach is that differences betweenindividuals can be pinpointed to specific weaknesses, thereby identifyingthe need for remedial action.Concept maps have been called the "windows to the mind" of the stu-

dents we teach: for seeing in (by the teacher and other students), for see-ing out (by the student) and for reflecting on one’s own perceptions (byeverybody). These maps facilitate a sharing of meaning unhampered byany lack of verbal skills. Consequently both teacher and students are ableto judge with some degree of clarity how well they themselves havegrasped a particular concept and how well their colleagues or classmateshave done so.

References

1. Ausubel, D. P., J. D. Novak, and H. Hanesean. Educational Psychology: A Cogni-tive View (2nd Ed.). New York: Holt, Rinehart & Winston, 1978.

2. Champagne, A. G., L. E. Klopfer, A. T. De Sena and D. A. Squires. Content Struc-ture in Science Instructional Materials and Knowledge Structure in Student Memories.Learning Research and Development Centre, University of Pittsburgh, 1978.

3. Cronin, P. J., J. Dekkers, J. G. Dunn. A Procedure for Using and Evaluating Con-cept Maps. Research in Science Education, 1982 (in press).

4. Gardner, P. L. Logical Connectives in Science. A Summary of the Findings Researchin Science Education. Research in Science Education, 1977, 7, 9-24.

5. Moreira, M. A. Concept Maps as Tools for Teaching. Journal of College ScienceTeaching, 1979, 5(5), 283-286.

6. Novak, J. D. A Theory of Education. Ithaca, New York: Cornell University Press,1977.

7. Novak, J. D. The Use of Concept Mapping and Gowin’s "V" Mapping InstructionalStrategies in Junior High School Science. Report of the Cornell University "LearningHow to Learn" Project, Ithaca, New York, 1981.

8. Shaefer, G. Concept Formation in Biology: The Concept of Growth. European Jour-nal of Science Education, 1979,7,87-101.

9. Stewart, J., J. Vankirk, and J. Rowell. Concept maps: A Tool for Use in BiologyTeaching. The American Biology Teacher, 1979, 41 (3), 171-175.

10. Williams, C. K. The Use of Concept Maps in a Personalised Learning Programme inBiology. Unpublished report, Wanneroo Senior High School, Perth, 1981.

John Malone and John DekkersWestern Australian Institute of TechnologyScience and Mathematics Education CentreKent Street, Bentley, W.A. 6102

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