Concept Mapping In Mathematics Pdf In Hindi

What We Know About Deepening Teachers' Content Knowledge: Engaging Teachers with Developing Conceptual Maps of Mathematics/Science Content

Practitioner Insights

Some professional development programs have teachers consider, and sometimes develop, conceptual maps of mathematics/science content to deepen their understanding of various mathematics/science concepts. "Concept mapping is a tool for representing the interrelationships between concepts in an integrated, hierarchical manner."† Maps typically depict concepts, in circles or boxes, and include labeled lines indicating the nature of the connections among the concepts.

When queried about this strategy for deepening teachers' content knowledge, experienced practitioners offered some insights, which are described below. After reviewing these insights, you will be provided with opportunities to share your own experiences with using this strategy for deepening teacher content knowledge. The information you provide will be analyzed along with the insights and examples from other practitioners as the website is periodically updated.

Show the flow—The process of concept mapping can help teachers understand how mathematics/science ideas connect.

Concept maps can be a powerful tool to illustrate a progression of ideas with in a unit of instruction or to demonstrate connections across the various content ideas within a unit of instruction, allowing participants to compare their own conceptual frames to others' frames. Said one program leader,

I think deep understanding is actually defined by having rich mental connections about a concept. So, if I understand something in multiple ways, if I can give a variety of examples (and non-examples) of a concept, if I can explain applications of the concept, if I can talk about simpler and more complex related concepts-then I really have good understanding. Sometimes making these connections explicit in professional development is helpful to teachers-and this may be done through concept mapping.

Professional development programs that utilize this strategy can work to deepen teachers' knowledge of the disciplinary content, including their understanding of distinctions among ideas as well as fundamental relationships in mathematics/science, and what their students are expected to learn. Often these programs emphasize how ideas can be represented in multiple ways. They may also focus on how a particular idea connects to other content ideas, either those that students had learned earlier or those that are being learned at the same grade level. Engaging teachers with concept maps that depict such "knowledge trajectories" in mathematics/science helps teachers understand where students have been and where they are going in their development of key concepts, facilitating engaging students in ways that will support their conceptual development now and in the future. One program, for instance, addressed how knowledge of place value is related to knowledge of operations, first focusing on increasingly large whole numbers, later on rational numbers, and still later on algebraic expressions.

Understanding is key—In concept mapping, focus on the key mathematics/science concepts and clarify their relationships to one another.

Although concept maps can be useful representations of mathematics/science content, experienced program leaders stress the need to focus on the deep conceptual ideas and the relationships among these concepts that the maps illustrate. Having teachers simply list topics and indicate connections without clarifying concepts and specifying the nature of relationships is unlikely to result in deepened understanding. Said one program leader: "The most difficult part for teachers was articulating the relationships. Frequent cajoling by the professional development providers was necessary to get relationships on the table."

Another experienced practitioner described how a program helped teachers develop conceptual maps, focusing on big ideas in science and their relationships.

Teachers in groups of three were given 3x5 cards with major concepts related to the topic printed on them along with some blank cards for teachers to use to add any concepts they thought would be useful. Teachers were asked to arrange the cards in ways that made sense to them on large sheets of paper. When consensus had been reached, teachers drew arrows connecting concepts, writing on the arrows the relationships between and among the concepts. (When groups could not reach consensus, another set of cards was provided so that the alternative representation could be displayed.) Groups then "presented" their maps and discussed differences and similarities in the maps.

The approach worked especially well when the topic was ecology. The relationships between energy conservation and flow of matter through ecosystems actually generated considerable expressions of "Ah ha" among the participants.

Insight in Action

A five-day institute for middle school science teachers focused on the flow of matter and energy in living systems. A central goal of the institute was to deepen teachers' content knowledge of a specific benchmark from Benchmarks for Science Literacy (American Association for the Advancement of Science, 1993),‡ which states:

Food provides molecules that serve as fuel and building materials for all organisms. Plants use the energy in light to make sugars out of carbon dioxide and water. This food [sugars] can be used immediately for fuel or materials, or it may be stored for later use. Organisms that eat plants break down the plant structures to produce the materials and energy they need to survive. Then they are consumed by other organisms. [5E(6-8)a, p. 120].

In working towards a deeper understanding of the ideas within this middle grades benchmark, facilitators addressed the content within the benchmark, and also related discussed common prior conception as well as later ideas about the content. Because this benchmark includes multiple abstract ideas, the institute was designed to address each specific component idea situated within the concrete notion of food. Daily focus questions led teachers through explorations of, and discussions about, these component ideas:

Daily Focus Questions
Day 1	What is food? Why do organisms need it?
Day 2	Where does food come from?
Day 3	How is food used for growth?
Day 4	How is food used for energy?
Day 5	What happens to matter/energy after death?

The process began with teachers developing a diagram (e.g., concept map, energy flow diagram, Venn diagram, or cartoon) depicting their current thinking in answer to these questions. Daily reflections on their content learning resulted in teachers modifying their existing diagrams and/or creating new diagrams. By the end of the week, the teachers had constructed a visual depiction of the entire benchmark, portraying understanding of each of the component ideas and the connections among them, conceptually similar to the diagram below:

food diagram

Tread lightly—When constructing concept maps, it is particularly important to build an atmosphere of respect and openness to ideas, since teachers' thinking is open to public examination.

Engaging teachers in the development of conceptual maps of mathematics/science content requires a culture in which teachers feel comfortable sharing ideas, exposing weaknesses in their understanding, collaborating with facilitators and peers, and openly reflecting on issues related to teaching and learning. This approach may be unfamiliar to many teachers, who may feel the appropriate course of action is to keep their weaknesses and concerns to themselves. Yet a safe, trusting professional development environment is essential if teachers are to critically and publicly examine their own knowledge.

The key, experienced program leaders suggest, is to create a learning environment that is simultaneously rigorous and respectful. An observer of a high-quality workshop for science teachers commented on the positive atmosphere, "The relationships among participants were congenial, collaborative, and professional; the discussions seemed open, candid, and reflective." A high quality session from another program was described as follows: "All the professional development offerings reflected the expectation that beliefs and assumptions need to be brought to light and discussed, and that all participating teachers have much to share and give in the way of dialogue and collaboration." An environment of respect and trust encourages teachers to share their thinking openly, including their own confusion and misconceptions about scientific or mathematical concepts.

It's fine to refine—Since incorrect content is likely to surface during concept mapping, be prepared to guide teachers to correct content understandings.

Concept maps are often used as a public representation of what teachers know and don't know about a particular set of ideas. Professional development providers need to be prepared to deal with instances when teachers get the content wrong, as teachers will inevitably have gaps in their understanding of some content areas. One program leader gave an example of how errors were handled respectfully and productively in a study group of second grade teachers working on a concept map for a unit on water.

Two teachers had placed "concepts" on the concept map that were in fact misconceptions. Other teachers gently pointed them to the resource materials we were using, and questioned their ideas. The whole group then spent some time investigating resource materials to make sure that what was on the concept map was accurate.

If you are interested in how these practitioner insights were collected and analyzed, a summary of the methodology can be found here.

† Mintzes, J. J., Wandersee, J. H., & J. D. Novak (Eds.). (2000). Assessing science understanding: A human constructivist view. San Diego, CA: Academic Press.

‡ American Association for the Advancement of Science/Project 2061. (1993). Benchmarks for science literacy. New York: Oxford University Press.