Programs for mathematics and science teachers have increasingly focused on deepening teachers’ understanding of disciplinary content. One strategy used to deepen teachers’ knowledge of mathematics and science concepts is engaging teachers in developing conceptual maps. Developing conceptual maps involves working with a particular concept or set of concepts and identifying the relevant ideas that make up the concept(s) as well as the relationships among those ideas. When engaged in this intervention, teachers may be asked to identify both the ideas and their relationships, or they may be provided with the ideas and asked to focus only on the relationships among them. Advice from experienced practitioners offers guidance for efforts to engage teachers with developing conceptual maps of mathematics and science content as a strategy for deepening their content knowledge. Insights provided by a group of expert practitioners with diverse backgrounds included the following ideas:
- Show the flow—The process of concept mapping can help teachers understand how mathematics/science ideas connect.
- Understanding is key—In concept mapping, focus on the key mathematics/science concepts and clarify their relationships to one another.
- 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.
- It’s fine to refine—Since incorrect content is likely to surface during concept mapping, be prepared to guide teachers to correct content understandings.
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 3×5 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:
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.
Teacher Content Knowledge Matters
Empirical evidence demonstrates that teachers’ mathematics/science content knowledge makes a difference in their instructional practice and their students’ achievement. Consistent findings across studies include:
- Teachers’ mathematics/science content knowledge influences their professional practice.
- Teachers’ mathematics/science content knowledge is related to their students’ learning.
Learn more about research on why teachers’ mathematics/science content knowledge matters.
Research on Engaging Teachers with Developing Conceptual Maps of Mathematics/Science Content
The literature search identified one research study of a professional development program that engaged teachers in developing conceptual maps of mathematics content, as one of several strategies. The study provided preliminary evidence of positive effects on teachers’ mathematics content knowledge (Clark & Schorr, 2000). This study focused on the middle grades, with teacher participants ranging from grade 6 to grade 8. Topics in number and operations were addressed. Although the study did not investigate the unique contribution of the strategy of engaging teachers in developing conceptual maps of content, the positive results suggest that developing content maps is a potentially useful professional development strategy and a fruitful area for future research.
Research on Engaging Teachers with Developing Conceptual Maps of Mathematics Content
Programs for mathematics teachers can work to deepen teachers’ content knowledge by guiding teachers in development of conceptual maps of student-level content ideas and their representations. These maps focus attention on how a particular idea connects to other content ideas, either those that were learned earlier or those that are being learned at the same grade level. One research study (Clark & Schorr, 2000) identified in a review of research on deepening teacher content knowledge investigated a professional development program that included this strategy for mathematics teachers.
Findings from Research
The Clark and Schorr (2000) study of a professional learning experience that included engaging teachers with developing conceptual maps provided positive results on participating teachers’ content knowledge. Although the study did not investigate the unique contribution of the strategy of engaging teachers with developing conceptual maps, the positive results suggest that the use of such maps may be an effective professional development strategy and a fruitful area for future research.
Teacher participants in the study ranged from grade 6 to grade 8. Topics in number and operations were addressed. The experience for teachers was structured as a course with meetings over 14 weeks, during which they worked through problems for middle grades students and developed curriculum maps to visually demonstrate their understanding of the interconnectedness among the skills and concepts in the problems. They then implemented these same problems in their classrooms and documented their students’ thinking, revisiting the curriculum maps in the process.
Teachers participated in the study on a voluntary basis, so generalizability of the findings must be considered in this light. Although the study traced changes in teachers’ thinking over time, it did not use a comparison group of teachers who did not participate in the professional development program. It is possible that participating teachers might perform better as time went by simply because they learned to focus on the ideas the researchers were assessing. As is often the case in published research studies, very little detail was provided about the intervention. Similarly, little information was provided on how the measures were developed and validated for the purpose of assessing growth in teachers’ content knowledge.
For the research on engaging teachers with developing conceptual maps of mathematics content bibliography, click here. [PDF 7K]
The study described above was part of a more inclusive review of research on experiences intended to deepen teachers’ mathematics content knowledge. For more information, you are invited to read a summary of research on experiences intended to deepen teachers’ mathematics content knowledge, click here. [PDF 120K]
The literature search surfaced one research study of a professional development program that included engaging teachers with developing conceptual maps of science content (Shymansky et al., 1993). The study focused on teachers of grades 4 through 9, with various physical, life, and earth science concepts targeted among different groups of teachers. Although the study was not designed to systematically study the particular strategy of using concept maps for deepening teacher content knowledge, it offers some support regarding its effectiveness in deepening teachers’ science content knowledge, suggesting that additional research in this area is warranted.
Research on Engaging Teachers with Developing Conceptual Maps of Science Content
Professional learning opportunities for teachers of science have increasingly focused on deepening teachers’ content knowledge. One strategy sometimes used to deepen teachers’ content knowledge is engaging them in developing conceptual maps of content targeting specific science concepts. Developing conceptual maps involves working with a scientific concept or concepts and identifying relevant ideas that make up the concept(s), and the relationships among those ideas. When engaged in this task, teachers may be asked to identify both the ideas and their relationships, or they may be provided with the ideas and asked to focus on just the relationships. One research study (Shymansky et al., 1993) identified in a review of research on deepening teacher content knowledge investigated a professional development program that included this strategy for science teachers.
Findings from Research
The Shymansky and colleagues (1993) study engaged teachers with the development of conceptual maps of content and reported at least some positive results on participating teachers’ content knowledge. Although the study was not designed to systematically investigate this particular strategy for deepening teacher content knowledge, there was some support regarding its effectiveness in deepening teachers’ science content knowledge. Some of the participating teachers’ incorrect prior ideas changed over the course of the intervention, but some participants expressed new misconceptions at the end of the intervention.
Shymansky and colleagues (1993) analyzed teachers’ concept maps to assess changes in understanding of 42 4th through 9th grade teachers at three points in time over six months: at a spring orientation meeting before a one-week summer workshop, immediately after the summer workshop, and again after follow-up activities during the fall. The teachers divided themselves among 10 topic teams, ranging from astronomy to electricity to ecology.
The teachers’ maps were rated by advanced doctoral students in science departments. To limit biases, raters were not aware of the time each content map was produced nor which teacher produced each one. The raters identified valid and invalid propositions depicted on the maps. The study found that “all postfall valid proposition mean scores exceeded the spring mean scores, but weighted invalid mean scores in 8 of 10 topic areas did not change or even increased. These data suggest that the teachers successfully incorporated new valid propositions in their conceptual frameworks, but did not necessarily discard or discount all previously held misconceptions” (p. 747-748). Only three of the ten topic teams (heredity, substance abuse, and ecology) evidenced consistent growth across the three time points. Valid proposition means in the other seven areas did not change significantly from spring to summer or from summer to fall. It is unclear whether the differential outcomes are related to the specific topics, whether the small group of teachers who participated in those three topics differed from the other teachers, or whether the experts in charge of the various topics were differentially effective.
Analysis of the conceptual maps was a novel way to assess teacher content knowledge. Although the inter-rater reliability process was adequately described, validation processes were not described in detail, which is a particular concern when using a new method of assessment. In addition, teachers participated in the study on a voluntary basis, so generalizability of the findings must be considered in this light.
For the research on engaging teachers with developing conceptual maps of science content bibliography, click here. [PDF 7K]
The study described above was part of a more inclusive review of research on experiences intended to deepen teachers’ science content knowledge. For more information, you are invited to read a summary of research on experiences intended to deepen teachers’ science content knowledge, click here. [PDF 134K]