Concept maps: an essential tool for teaching and learning to learn science.

Abstract

This article is meant to offer math teachers a possibility to initiate their own study of concept maps, a powerful heuristic tool designed by Joseph Novak, on the ground that this tool can effectively help university students to face many difficulties for learning science and achieving a meaningful learning. This paper reflects the outcome of a research project undertaken at Universidad Nacional Experimental del Tachira (UNET), Venezuela, investigating different ways teacher and students may use concept mapping in physics. This paper hopes to engage educators on a discussion of this important issue and will focus on answering the following questions: What are Concept Maps? How are they constructed? What is the theory that supports Concept Maps? What are they used for? How can they be used with large groups of university students to facilitate the teaching-learning process? According to our experience, there are some possibilities to use Concept Maps in physics courses. Although we have faced difficulties in building individual concept maps with large groups of students, we are convinced that Concept Maps help to improve understanding of a given subject and facilitate building student's own knowledge, as long as the student has the opportunity to use, criticize, analyze, question or improve expert's maps or Concept Maps generated by his own peers.

Introduction

Many students at the Universidad Nacional del Tachira face difficulties in science learning. They find it hard to understand a whole body of information and to build their own knowledge about complex conceptual structures. They also find it difficult to link concepts and handle adequate representational techniques either to show or sum up information.

To help solve these problems we have found in Concept Maps a powerful help that facilitates students comprehension of physics at university level. Convinced of its advantages, we want to report our experiences hoping math teachers who have never worked before with Concept Maps may get involved and begin to use them with their students.

This paper is organized to answer, from our perspective, the following questions: What is a Concept Map? How is it constructed? What is its theoretical background? What are they used for? And finally, what strategies involving Concept Maps can be used with university students?

What is a Concept Map?

The first difficulty someone who attempts to comprehend a text faces is to understand what it is all about. That is, to grasp the global sense of the communication, understand its elements and the relationships among them. Imagine a student seeking information about frame of references finds the following text:

A Cartesian frame of reference is a set of two elements: a Cartesian,
linear coordinate system and a clock used to measure time. Cartesian
coordinates are rectilinear two-dimensional or three-dimensional
coordinates which are also called rectangular coordinates. The three
spatial axes of three-dimensional Cartesian coordinates conventionally
denoted the x, y, and z-axes, are chosen to be linear and mutually
perpendicular. Frames of references are used to describe and analyze
motion in one, two or three dimensions (using one two or three oriented
axes).

The student may understand some of the concepts involved in this definition. These concepts are linked by words forming whole sentences that seem to make sense. However, trying to understand the overall conceptual structure is more difficult. In our introductory physics courses at university level, it is a commonplace that there is no difference between a Cartesian frame of reference system and a Cartesian, linear coordinate system. Students that read carefully may notice that the two concepts are different. Moreover, one of them is included in the other.

Let the same information be presented in a different way (Figure 1).

It is probably easier for many students to grasp a whole sense of the concept frame of reference when faced with a graph like the one shown above. This is due to the powerful visual effect that a graph has in order to facilitate understanding of a concept or a conceptual structure.

[FIGURE 1 OMITTED]

This graphic is essentially a concept map. It is a map-like illustration that shows meaningful relationships between concepts (events, objects). Observe that this is a knowledge representation about a particular main idea (in this case: frame of reference), in the form of a graph comprised of boxes connected with labeled lines. Words or phrases that denote concepts (events, objects) are placed inside the boxes, and relationships between different concepts are specified on each line. Propositions (node-link-node triads) are a unique feature of concept maps, compared to other graphs. Propositions consist of two or more concept labels connected by a linking relationship that forms a semantic unit (Novak & Gowin, 1988).

And you may feel reading the above paragraph, if not familiar with concept maps, exactly as may have felt the student who saw for the first time the text about frame of reference: lost in the dark. That is, you may know some of the concepts mentioned before as separate entities but have no clear relationships among some of them. Obviously this makes difficult to understand the whole conceptual structure concept map. Let us then discuss some of the concepts involved in this definition: object, event, concept, proposition and meaningful relationships, to be able to understand what a concept map is.

Novak says the construction of new knowledge begins with the understanding of the terms event and object (Novak & Gowin, 1988). We represent event in Figure 2.

[FIGURE 2 OMITTED]

And also we may explain, following Novak's definition, what an object is in Figure 3.

[FIGURE 3 OMITTED]

Now that we understand what objects and events are, we can define concepts as perceived regularities in events and objects, or records of events or objects, designated by a label (Novak & Gowin, 1988). Also, concepts are mental representation of objects or events with the following characteristics: a) correspondence among the concept and what it represents, b) absence of ambiguity and c) optimal use of the language involved. In concept mapping, concepts are usually written inside cells or boxes. Today you may design cells as you like. The important thing is that the graph highlights concepts visually in a clear and distinctive way.

But concepts are not isolated in a Concept Map. They are connected by labeled lines or arrows called links which consist of words, phrases or verbs that explain meaningful relationships between concepts by words or signs/symbols. Arrows, if used, designate the directionality of the relationship. Otherwise, the concepts must be arranged in a hierarchical way, from the most abstract and inclusive concepts on the top of the graph to the most concrete and specific, and it is assumed that the direction of the relationship is downward. This facilitates the reading of concepts and the links among them as whole sentences.

Relationships among concepts are diverse as seen in Figure 4.

[FIGURE 4 OMITTED]

Canas, Safayeni and Derbentseva (2004) classify them as static or dynamic relationships. Static relationships between concepts help to define, describe and organize knowledge for a given domain. Classifications and hierarchies are usually captured in relationships that indicate belongingness, composition, and categorization. These relationships comprise inclusion (a concept is part of another one), commonly membership (two concepts are part of another) and intersection (a concept is the meaning generated by crossing other two concepts). This intersection may be probabilistic (e.g., Polygons may be regular), or based on similarity between two concepts.

Dynamic relationships between two concepts show how changes of one concept cause change of the other concept in a proposition (e.g., Concept A leads to concept B). Dynamic relationships are those based on causality (e.g., volume is an inverse function of the density for a given mass), or those based on correlation/probability (e.g. Concept maps may help students to achieve a meaningful learning).

Scientific knowledge is based on both static and dynamic relationships among concepts. The teacher who is learning about Concept Maps for the first time must know that there are different types of relationships and should take them into consideration when he attempts concept mapping. Finally, a link may be simple (showing the connection between two concepts) or a crosslink (showing the relationships between ideas in different branches of the map).

Let us now talk about propositions. The basic unit of representation in Concept Maps is a proposition defined as two concepts plus a relationship, which is stated with a label on the link between concepts. Two or more concepts connected by a linking relationship forms a meaningful statement also called a semantic unit. Also it can be said that propositions are units of meaning constructed in the cognitive structure.

Each proposition is a sentence that has a unique standardized truth value (true or false) that is the basis for arguing whether the graph makes true or false assertions. Look at the following sentences: a) "Complex numbers include natural numbers," b) "Squares and rhombs are quadrilaterals," and c) "Mars has a unique moon." These sentences are propositions with a truth value (propositions a and b are true and c is false). And even if you do not know the veracity of any proposition, it is always possible to find an expert to judge it.

Instead, observe the following sentences: a) "Mathematics problems are easy to solve," b) "This sentence is false," c) "Goldbach conjecture will be solved one day." These sentences are not propositions as there is no way to know whether they are true or false. Some mathematics problems may be easy to solve ... but not all. Moreover, what is easy for one person may be difficult for someone else. The second sentence is a paradox, as it is impossible to decide its truth value. And finally, Goldbach conjecture may be really solved in the future ... or humankind becomes extinguished before finding the answer. Scientific knowledge, what we are trying to reflect in concept maps, deals with the truth that is knowledge widely accepted as true by a given scientific community. That is the reason why Novak insists on constructing maps with real meaningful propositions.

Novak (2004) says that a map is always built as an answer to a focus question (e.g., what is a polygon?). The main idea (the polygon) is the supraordinate concept, a subsumer, the most general and more inclusive concept. From this onwards, subordinate (less important) concepts related to it begin to emerge and defining appropriate relationships of this main concept with subordinate concepts makes possible the construction of propositions. Similarly collateral structures begin to appear from the connections among propositions.

Let us finish this section summing up in Figure 5 our first answer to the focus question: What is a concept map?

[FIGURE 5 OMITTED]

How to Build a Concept Map

The most appropriate way to learn to swim is swimming by your own means. In the same way you will only learn to construct maps by doing them. There are some basic rules you may follow of course, but the only way to understand the whole process involved in concept mapping is by facing the difficulty of building a conceptual structure all by yourself.

How to Build a Concept Map about a Topic One Knows

A concept map will be the final product of an effort to answer specific questions about a concept, situation or event that is attempted to be understood. It is helpful to select a limited domain of knowledge the first time you attempt concept mapping. Let us explain how to construct a concept map about a topic one knows, for example area of a triangle at a basic level.

1. Write a focus question and the main idea. The focus question could be: what is the area of a triangle and how can it be measured? Begin by writing down on the top of a sheet of paper the main concept (e.g., area of a triangle) and write all the other concepts you are thinking may be related to this one (without any order). You are already thinking of many concepts associated with this one: area, triangle, surface, sides, base, height, vertex, length, internal region, and so on. Obviously, according to your own expertise on the subject, you may have thought about other concepts related to the main idea. Let us focus on the ones mentioned above.

2. Highlight concepts and work out relationships among them. Enclose each concept inside any shape form you like (e.g., squares). Now, try to work out the relationships between them. Any two concepts related must be linked in the graph with a line and a label explaining the relationship among the selected concept (Remember that the main point is to work out meaningful propositions). In your map you may have linked concepts forming propositions like these: "Area is the size of a surface." "The triangle area measures the size of the region enclosed by the triangle." "Area is usually expressed in terms of some square units such as square meters." "Base (b) stands for the measure of the selected length side."

3. Organize and rearrange items. Create groups and subgroups of related items. Try to group items to emphasize hierarchies. You may rearrange items and introduce new items omitted initially until you are satisfied you can easily read whole sentences that are meaningful propositions. You may notice that some propositions have to do with the definition of area and others attempt to answer how it is measured. These give birth to different branches in the map. Take care that the different branches are logically interconnected and point out different aspects of the main idea.

In Figure 6 you have our map of the concept triangle area.

[FIGURE 6 OMITTED]

Yours may look different. Concepts may be arranged in a different way or links may be expressed rather differently from the ones shown above. Do not worry if that happens. Variations are possible, but not the essence of what is to be presented by a Concept Map. The reason is that everyone has a different cognitive structure and maps are somehow a reflection of our way of seeing things. The important point is that the student who is building the map achieves a clear understanding of the topic, its elements and the relationships among concepts.

There is no such thing as "the right map." You may express this conceptual structure on a Concept Map in a better way than your students--of course you know the subject! But two maps may look slightly different being both of them correct, as long as the propositions built are meaningful and there is logical coherence in the whole body of ideas expressed.

A Concept Map is a very appropriate heuristic tool to help teachers and students to answer questions in science such as: What is it? What is it used for? How is it constructed? And so on, leading toward the significance of a given topic. Many conceptual structures in mathematics and physics could be understood in an easier way if they were introduced by means of concept maps. The important thing is the process that occurs in the mind of anyone making a Map while the designer attempts to clarify to himself the conceptual structure. Thus, the process is worthwhile and may help students to understand a particular topic in a meaningful way.

Therefore, think about the possibility of building your own maps. Concept Maps will aid your students and perhaps more important will help you clarify many ideas, concepts or relationships among them that never deserved from you more than a casual analysis because they seamed to be too elementary or obvious.

How to Build a Concept Map from a Written Text about a Topic We Want to Learn?

Sometimes we need to learn about a topic or understand a concept or conceptual structure, and we find the necessary information in a written text. Let us share with the reader the way we proceed in order to build a concept map. Suppose you want to learn about polygons and find just the following text. (You must assume you are not an expert and it is the first time you study the topic)

  A polygon (from the Greek poly, for "many", and gonos, for "angle") is
a closed planar path composed of a finite number of sequential straight
line segments. The straight line segments that make up the polygon are
called its sides or edges and the points where the sides meet are the
polygon's vertices. A polygon has inner angles (formed internally
between two adjacent sides), and external angles (formed externally
between two adjacent sides). A polygon has equal number of sides,
vertices and inner angles.
  The closed planar path is also called boundary and separates the plane
surface in two regions (internal and external). The measure of the inner
region is called area of the polygon. The length of the path of the
polygon is defined as perimter.
  Polygons are named according to the number of sides, combining a Greek
root with the suffix-gon, e.g., pentagon, dodecagon. The triangle and
quadrilateral are exceptions.
  A polygon is called simple if it is described by a single,
non-intersecting boundary; otherwise it is called complex. A simple
polygon is called convex if it has no internal angles greater than
180[degrees], otherwise it is called concave.

To build a Concept Map from this information we recommend following these steps:

1. Construct a good focus question. You must read the whole text trying to work out what it is all about and which question may be answered by this information. We decide the question is: What is a polygon?

2. Write down the main idea (polygon) and extract the most important ideas from each paragraph to construct propositions. Begin reading the first paragraph. Notice all terms and concepts associated with the topic of interest. At first you may feel you are just copying down sentences expressed in the paragraph. Focus on the concepts and begin to make clear relationships among them.

Let us analyze the first paragraph:

  A polygon (from the Greek poly, for "many", and gonos, for "angle") is
a closed planar path composed of a finite number of sequential straight
line segments. The straight line segments that make up the polygon are
called its sides or edges and the points where the sides meet are the
polygon's vertices. A polygon has inner angles (formed internally
between two adjacent sides), and external angles (formed externally
between two adjacent sides). A polygon has equal number of sides,
vertices and inner angles.

As a result you may come up with something as seen in Figure 7.

[FIGURE 7 OMITTED]

Now carry on with the analysis of the next paragraph. It says:

  The closed planar path is also called boundary and separates the plane
surface in two regions (internal and external). The measure of the inner
region is called area of the polygon. The length of the path of the
polygon is defined as perimeter.

Extract the important ideas and make new propositions. It is a completely different idea but can be added to the map (subsumed) from the concept closed planar path as in Figure 8.

[FIGURE 8 OMITTED]

Now carry on with the third paragraph:

  Polygons are named according to the number of sides, combining a Greek
root with the suffix -gon, e.g., pentagon, dodecagon. The triangle and
quadrilateral are exceptions.

Rearrange or relocate concepts, propositions, and branches at any time. You may need to rearrange sections to emphasize organization and appearance. In our map we added a new branch derived directly from the main idea. To do so it was necessary to modify the layout. Our concept map now looks like the map in Figure 9.

[FIGURE 9 OMITTED]

Finally we read the last paragraph:

  A polygon is called simple if it is described by a single,
non-intersecting boundary; otherwise it is called complex. A simple
polygon is called convex if it has no internal angles greater than
180[degrees], otherwise it is called concave.

This obliges the designer to open a new branch classifying polygons as simple and complex and the simple ones as convex and concave. In our map we solved this problem by adding examples with drawings that illustrate what is meant by each one of the four terms (Remember, an image is worth a thousand words).

Make final refinements. We had to relocate some concepts, rearrange branches and change the selected shape to a way we found more suitable. Be creative in a constructive way through the use of colors, fonts, shapes. The only limit is your imagination. Finally, after some modifications, the map may look like this one in Figure 10.

[FIGURE 10 OMITTED]

Observe that examples are not framed (to differentiate them from concepts). One last remark: you may argue that as a beginner you have no idea of the meaning of some concepts. For example, you may not know what a closed planar path means. Well, first you locate this concept in the map according with the propositions that emerge from the text. Later on, you will look for the meaning of that new concept somewhere else (go to a dictionary, a math book, the web, consult an expert, etc). This is what it is all about. That is the way you are building more and more complex structures in your mind all the time. If the new concept needs to be clarified, then you may add a new branch to the map later on.

How to Build a Concept Map for a Topic We Want to Learn?

1. Construct a good focus question.

2. Gather information from different sources. Select from each one the most important ideas. Highlight main concepts and try to understand relationships among them.

3. Proceed to outline the main points which may help to answer the focus question. You may come out with written propositions, paragraphs you have extracted from different sources, important sentences you may have synthesized or rephrased. Begin to make a draft ordering the main ideas as a written text or initiate your concept map.

4. Write down the main idea, organize hierarchies. You will come out with different branches related to different aspects of the main concept. Feel free to rearrange things at any time.

5. Add more information (if needed). Once you have the basic ideas, add new ones and proceed as we explained before until you have a version of the map you are satisfied with. Do not expect your layout to look exactly like that of anyone else.

Remember there is nothing like the right map. But you can always ask your colleagues for advice about your design. They will tell you the weaknesses or incongruence they may see on the conceptual structure, wrong propositions, or obscure relationships among concepts. Follow advice. If there is something your peers do not understand there may be a problem. It if is just a matter of form, it is your choice to decide whether you change it or not.

There are some practical problems when creating a graphical representation like a concept map by means of paper and pencil. It is not easy to determine in advance the space requirements and the relative positions of the different branches of the map and it is a time consuming task. Yet there are several suitable drawing programs that enable the authors to pay most of their attention to the conceptual content of the subject matter (Inspiration 7.5; Cmaps, Vision, etc.) including a freely available drawing tool called Cmap Tools (Canas, et al., 2004). You may begin with Inspiration or Cmap Tools. They are easy to handle and you will be surprised with the results.

Now you are just at the starting point. To become a good swimmer you must go inside the pool. You will become an expert in concept mapping only through practice.

What is the Theoretical Foundation of Concept Maps?

Ausubel's meaningful learning and constructivist epistemology (Ausubel, Novak & Hanesian, 1978), provide the basis for concept mapping. The key idea in Ausubel's theory is the concept of meaningful learning. It is a process controlled by the learner in which new information is related to an existing relevant aspect of the learner's knowledge structure. It happens when the learner connects the new piece of information to information already known. Ausubel talks about assimilation of new information into existing knowledge frameworks. The principles of the Assimilation Theory of Meaningful Learning provide the basis to the developing of Concept Maps.

Subsumption: derivative and correlative 

Existing concepts are subsumers of new concepts, that is, the existing concept provides a base for linkage between the new information and previously acquired knowledge.

Derivative subsumption

A student learns through the process of derivative subsumption when a new concept (information) is an example of a concept that has already been learned. For example, a learner has acquired a basic concept such as "polygon" and knows that a polygon is a closed planar path with sides, angles and vertices. Next, he learns about a kind of polygon he has never seen before (e.g. a heptagon) that conforms to his previous understanding of polygon. The learner's new knowledge of a heptagon is attached to his concept of polygon.

Correlative subsumption

Suppose a student has learned about polygons and examples shown to him are convex polygons. Now he encounters a new kind of polygon that is concave. In order to accommodate this new information, he has to alter or extend his concept of polygon to include the possibility of polygons with one inner angle > 180[degrees]. He has learned about this new kind of polygon through the process of correlative subsumption which enriches the higher-level concept.

Progressive differentiation

As meaningful learning happens, development and elaboration of subsuming concepts naturally occur. There is a process of refinement of the concept meanings in cognitive structure, which adds more precision and specificity to those concepts. Ausubel believes concept development occurs best when the most general, most inclusive concepts are introduced first and then these concepts are progressively differentiated. For example, polygon is an inclusive concept with classes (simple, complex) and subclasses (e.g., simple may be convex or concave).

Integrative reconciliation

Integrative reconciliation is a process by which cross linkages are formed and new interrelationships are established between concepts in the cognitive structure. For example, a student may have learned that squares are regular polygons having four equal sides and right angles. On the other hand, he may have developed an understanding of a cube as a parallelepiped (a prism whose faces are all parallelograms) with six congruent faces. The learner establishes a clear relationship among both concept structures (square and cube) when he comes up with a proposition such as: "Squares constitute the faces of a cube."

Superordinate learning

Superordinate learning occurs when new concepts are constructed and integrate large domains of knowledge that were not previously recognized as intimately related. For example, a physics student could be acquainted with the concepts "affine function," "potential function" and "exponential function" without knowing they were examples of mathematical models in Physics. In this case, the learner already knew a lot of examples of the concept, but did not know the concept itself until it was taught to him. This is superordinate learning.

Novak (2004) says that a Concept Map is a tool for organizing and representing knowledge. A good concept map is built according to the Ausubelian principles mentioned above. The author emphasizes the importance of hierarchical structures in concept mapping. Concepts are represented in a hierarchical fashion with the most inclusive, most general concepts at the top of the map and the more specific, less general concepts arranged hierarchically below. On the other hand, cross-links help us to see how some domains of knowledge represented on the map are related to each other through the process of integrative reconciliation. In the creation of new knowledge, cross-links often represent creative leaps on the part of the knowledge producer.

What are Concept Maps Used For?

Concept Maps are used around the world by educators and researchers alike. Many journals, such as The Journal of Research in Science Teaching, have published numerous articles on Concept Maps. Although they were originally proposed by Novak as learning tools, they have demonstrated to be more powerful. They are used for project planning, evaluation and management, curriculum development, human resource management, product and business development and so on (Proceedings of the First International Congress on Concept Mapping, 2004). Let us focus on the most important uses in the teaching and learning process: instruction, learning, and evaluation.

[FIGURE 11 OMITTED]

Concept Maps to Aid Learning Process

We have already explained how concept mapping may be used by students as a tool for organizing and representing knowledge to achieve meaningful learning. In the prior section you practiced building maps on your own about a topic explained in a written text (polygon). In the same way, students may use concept maps to construct their own knowledge about a subject, the sources of information being very diverse: conferences, classes, written texts or their own experience. We also encourage you (and students) to use Concept Maps to take notes on a class or conference. We call them "Instant Concept Maps." That is, maps you build when listening to a conference. You will be amazed to learn how it helps you (or the student) to focus on the main ideas exposed, the concepts explained and the relationships among them. Concept Maps can also be built in small groups to reach common shared knowledge about a subject. In this way it is possible to stimulate cooperative learning.

Concept Maps to Aid Teaching Process

Teachers may use Concept Maps mainly for Evaluation, Instructional Design and Teaching.

Evaluation

Concept Maps are very useful to evaluate what students have learned about a topic. You can check partial or total comprehension of a concept by asking them to build a map in the middle or at the end of a specific teaching learning process. Concept Maps are very powerful tools to access prior knowledge and find out misconceptions. Concept maps are effective in identifying both valid and invalid ideas held by students. On the other hand, a concept map might be used in assessing declarative and procedural knowledge, both of which have a place in the science classroom. There is a wide range of methods for constructing and scoring maps. However, certain questions about the reliability and validity of Concept Maps as an assessment remain unanswered.

In our experience it has been difficult to use Concept Maps as an assessment tool with large groups of students as concept mapping is a time consuming task even though you will find references in the literature arguing it is possible to do so with the help of specially designed software.

Instructional design

An instructor may use concept mapping to help him design new courses, and its modules. It helps teachers to see where themes and continuities exist, and to notice where redundancies and omissions occur. It is invaluable for curriculum development and implementation.

Somehow Concept Maps have substituted the stage of task analysis in instructional design. That is, instead of breaking down objectives to analyze what should be done, with Concept Maps the analysis must start from a main idea (for the whole course). Then, through a brainstorm process, the designer may write down all the topics that must be covered in a course and begin to relate them until he builds an initial Concept Map. The main topics in this map will give rise to other sub maps. That is, this is the starting point from which the designer initiates the construction of more specific maps for each topic to organize the different modules of a course.

The maps built for each module will orientate instructional design. Maps are a wonderful aid in a web published material to serve as the menus from which a student begins to open sub menus seeking more and more information about a subject.

Teaching

Concept Maps may be used to introduce a topic, orientate a process, recapitulate main ideas, or summarize information. In this article we used maps in different ways: we presented the whole idea of a concept map with an example (Figure 1). We introduced two new concepts (event and object) in a simple way (Figures 2 and 3). We summarized the whole idea about Concept Maps at the end of the first section with a Concept Map (Figure 5). We used Figures 6, 7, 8 and 9 to exemplify and allow you to check partial comprehension of the process of constructing a concept map. Again we used a map (Figure 1) as an advanced organizer at the beginning of this section to explain uses of Concept Maps in teaching and learning. Finally in the last section we will present some strategies we used in the teaching-learning process of the introductory physics course at the university level.

How Can Concept Maps be Used in Physics Introductory Courses?

Introductory Physics at the UNET is a course lasting 16 weeks (six hours per week). The general strategy include lectures, workshops, practical demonstrations, problem solving and the use of concept maps. Obviously, the teachers had to be trained in concept mapping prior to applying diverse strategies with their students. This was attained by engaging teachers in activities about concept maps specifically designed for them (Ramirez de M & Sanabria, 2004).

For actual course work we have included Concept Maps with two purposes in mind. The first is to help students understand what is a concept map, how it is built and actually allows them to generate their own concept maps as we are convinced of the advantages of them as a learning tool for any subject. The second purpose is to facilitate the learning process. We use Concept Maps to help students to understand each conceptual structure, explore and develop concepts; we show them how to build a Concept Map from certain information given or how to use it as an advance organizer. The use of Concept Maps as a help to find "the right function or formula" is also encouraged. We have constructed Maps for each main topic and obviously we may use them in different ways. Let us present one way we use them from the beginning of the physics course.

[FIGURE 12 OMITTED]

Introducing Concept Maps

1. Generate a discussion about the words concepts, sentences, propositions and its elements. Exemplify and ask for other examples.

2. Ask for a proposition. Write it down as a Concept Map showing students how concepts may be highlighted with any shape (circle, square, etc.). Explain the purpose of the links (lines and labels).

3. Show a simple Concept Map with one or two propositions. Ask students to explain in their own words what a concept map is.

4. Now select a simple Topic (e.g., "a tree"). Ask them to build a Concept Map including possible answers to the questions. What is it? What is it good for? How is it formed? Etc. Tell them to link concepts by relating each other with a line and a label explaining the type of relationship among them.

5. Working in couple, ask each student to read his peer's map. Are their maps clear? Are the concepts highlighted in a way to distinguish them from links? Can propositions be read correctly? Are they meaningful? Does the whole structure make sense?

6. Ask them to make the necessary arrangements. Serve as a referee, explain and look for agreements about the general idea.

7. Make students now build a Concept Map for a concept from Physics Science (e.g., Frame of Reference shown in Figure 1). Discuss their maps and present the map made by an expert (yourself). Make clear that there are different ways of presenting the same information. The layout may look different but the propositions must be correctly stated. Discuss advantages of Concept Map.

Concept Maps as Advance Organizers 

We can think of Concept Maps as advance organizers. These are simply devices or mental learning aids to help us "get a grip" on the new information. The first two weeks we study Motion in one, two or three dimensions. We use two maps: one about motion and the other related to linear motion with constant speed to help students understand each conceptual structure and explore new concepts (Figures 13 and 14).

Motion

1. Show a Concept map of the new conceptual structure to be learned. In this case we begin with a map, in Figure 13, over Motion in two dimensions that is in a plane.

[FIGURE 13 OMITTED]

2. Ask students to study the map and express in their own words concepts they may have understood. Explain the meaning of the graphics. Give explanations or examples if required.

3. Ask them to recall, for instance, what average velocity is. Ask them to compare it with what is expressed in the map and its mathematical expression.

4. Demand them to compose a definition of any concept appearing in the map to check existing correspondence with their prior knowledge. For example, what is acceleration?

5. Solve problems using the map as a support for the application of concepts.

Linear motion with constant speed

1. Ask students to define some basic concepts of linear motion based on their own prior experience. For example, what is uniform linear motion? What is accelerated motion? (This will give the teacher the possibility of identifying some students' basic alternative conceptions).

2. Show the map, Figure 14, about Linear Motion with constant speed constructed by the teacher.

[FIGURE 14 OMITTED]

3. Read the map in detail with the students. Ask them to recall the definition they made of a given concept (e.g., accelerated motion.) Make them read the map and find out any correspondence among their definition and the similar one expressed in the map.

4. Give them time to analyze the map looking for the different concepts involved in the conceptual structure of linear motion.

5. Discuss with students examples shown in the map to allow them to understand the relationships between each concept, its mathematical expression, and graphical representations X(t) and V(t).

6. Problems requiring the use of the concepts implied in this map and their mathematical expression are now solved. Each time a concept is needed, ask students to find it in the map and look carefully at its definition and mathematical expression.

7. Encourage use of Concept Maps as a summary and secure support in problem solving.

Concept Maps to Synthesize Information

1. Explain accelerated motion as usual (without the aid of the corresponding Concept Map).

[FIGURE 15 OMITTED]

2. Define [x.sub.(t)], [v.sub.(t)] [a.sub.(t)]. Explain the difference between accelerated motions with a > 0 and a < 0.

3. Give examples and teach students how to build a graph from the mathematical expressions of position, velocity and acceleration. Show them the reciprocal process (how to construct mathematical expressions from graphs).

4. Show them the map about accelerated motion (Figure 15) as a synthesis of the information given.

5. Begin to solve problems using the map as a support.

Concept Maps to Integrate Information From Related Topics

1. Explain projectile motion. According to the Galilean Independence of Movements it can be interpreted as a combination of two movements that occur simultaneously: one in a horizontal axis with constant speed and the other in a vertical one under the influence of gravity (accelerated motion with g = 9.8m/[s.sup.2]).

2. From this information ask students, with the aid of the corresponding maps used before for uniform motion or accelerated motion, to analyze what happens in each movement. Make them write down mathematical expressions for both of them (Ask them to use x for the horizontal axis and y for the vertical one).

3. They will come out with separate equations in x and y that describe projectile's movement. For example positions in both axes will be expressed as: x = [x.sub.0] + [v.sub.0x]t and y = [y.sub.0] + [y.sub.0y]t + 1/2 g[t.sup.2].

4. Give an example to calculate x and y for a certain instant of time.

5. Draw a vector to show position of a projectile in a certain time. Ask them to look at the Concept Maps of motion (Figure 13) and find out how to describe the position of a particle moving in a plane, just like the projectile you are analyzing. They will have to combine the two values as the position of a projectile in the plane defined by an x-y coordinate system is given by a set of two values (x, y) for the same instant of time (e.g. r=4i + 5j).

6. Make them notice that the same will happen if they are asked for the velocity of the projectile in a certain time. Perpendicular components of vectors are independent of each other. They must calculate the two components and put them together, according to Concept Maps of Motion, to express the vector velocity (e.g. v=8i + 7j).

7. Show them the Concept Map, Figure 16, for parabolic projectile motion.

8. Notice that the left branch explains uniform motion and the right one explains an accelerated motion (with g = 9.8m/[s.sup.2]).

9. The central branch shows the mathematical expressions to describe position, velocity and acceleration for a projectile.

10. In this way students may crosslink the information they had about linear uniform motion with that of accelerated motion which is what happens in the movement of a projectile.

11. Make questions to reinforce concepts and to allow them to understand that all they have learned about the two single movements can be applied in the movement of a projectile.

[FIGURE 16 OMITTED]

12. Begin to solve problems using the Concept Map as a support.

Final advice: students need time to get used to working with Concept Maps. We allow them to read carefully the maps, generate discussion and let them suggest changes to improve content or layout of the maps. The important thing is that they understand concepts, their relationships and can effectively solve problems.

Building Maps to Learn From Texts

1. Having acquired elementary knowledge about Concept Maps, students are occasionally asked to build their own concept maps about certain topics. They are given short written texts and asked to hand them in as homework.

2. In any case we always discuss some of the maps with students. They are also given a Concept Map made by an expert (the teacher) so that, without being "the correct one" it may serve as a reference to compare with their Concept Maps.

3. By the end of the course students are asked to design Concept Maps for whole conceptual structures.

4. Along the course students are encouraged to use Concept Maps as an aid to solve problems. We insist Concept Maps not only give students the right formulae to apply, but allow the students to see the whole context, the principles and theories that frame some phenomena. And obviously that is an invaluable help in problem solving.

Finally, we must admit we have faced difficulties in building individual concept maps with large groups of students as it is a time demanding task. However, we are convinced that Concept Maps help to improve understanding of a given subject and facilitate building student's own knowledge, as long as the student has the opportunity to use, criticize, analyze, question or improve expert's maps or maps generated by his own peers.

In conclusion, we have presented Concept Maps, their theoretical background, how to build them and the variety of uses focusing on applications for teaching and learning processes. In addition, we have suggested from our research and experience, possible ways a math teacher may be initiated in concept mapping. Finally, we have shown some strategies we use in the teaching-learning processes of elementary physics that involve the use of Concept Maps, with two goals in mind: to get students to acquire basic knowledge and use of Concept Maps as a tool for their own learning process and to help them to understand in a simpler way complex conceptual structures.

We are convinced that our experience with physics teachers would be applicable to math teachers, too. Novak's tool has been there for more than twenty years and like a good wine has done nothing but improve with time. Concept Maps are waiting for you to take advantage of concept mapping in teaching mathematics. Thus we hope you, as a teacher, begin building your own knowledge about Concept Maps and find the best way to use them with your students. That is the challenge. It is up to you.

References

Ausubel, D. P., Novak, J. D., & Hanesian, H. (1978). Educational psychology: A cognitive view (2nd ed.). New York: Holt, Rinehart and Winston.

Canas, A. J., Safayeni, F., & Derventseva, N. (2004). Concept Maps: A theoretical note on concepts and the need for cyclic concept maps. Retrieved December 10, 2004, from University of West Florida, Institute for Human and Machine Cognition Web site: http://cmap.ihmc.us/Publications/ResearchPapers/Cyclic%20Concept%20Maps.pdf

First International Congress on Concept Mapping (CMC). (2004). Proceedings of the CMC. Retrieved January 9, 2005, from University of West Florida, Institute of Human and Machine Cognition Web site: http://cmc.ihmc.us/CMC2004Programa.html

Novak, J. D. (2004). The theory underlying concept maps and how to construct them. Retrieved January 21, 2004, from University of West Florida, Institute for Human and Machine Cognition Web site: http://cmap.coginst.uwf.edu/info/

Novak, J. D., & Gowin, D. B. (1988). Aprendiendo a aprender. Barcelona: Martinez Roca.

Ramirez de M, M., Sanabria, I. (2004). El Mapa Conceptual como Elemento Fundamental en el Proceso de Ensenanza-Aprendizaje de la Fisica a Nivel Universitario. Retrieved December 3, 2004, from University of West Florida, Institute for Human and Machine Cognition Web site: http://cmc.ihmc.us/papers/cmc2004-086.pdf.

Maria S. Ramirez de M., Mario Aspee S., Irma Sanabria., Decanato de Investigacio Universidad Nacional Experimental del Tachira, San Cristobal, Venezuela