Professional learning opportunities for teachers of mathematics and science have increasingly focused on teachers’ content knowledge. Learning opportunities aimed at deepening teachers’ content knowledge often include a goal of helping teachers’ understanding of how knowledge is formally established in mathematics/science and the habits of working or thinking that characterize mathematics/science disciplines. Advice from experienced practitioners offers guidance for efforts to engage teachers in considering mathematics/science as ways of knowing. Insights provided by a group of expert practitioners with diverse backgrounds and experiences in working with teachers included the following ideas:
- Make it real—Provide teachers opportunities to experience what it means to “do” mathematics/science.
- Choose wisely—Select problems that are amenable to discussions of the nature of the discipline.
- Doing is not the only way to learn—Reading about how mathematicians/scientists generated knowledge can help teachers understand the nature of the disciplines.
- Be explicit—Focus teachers’ attention on what it means “to know” in mathematics or science.
- Sense-making is key—Debrief teachers’ learning experiences from the perspective of the nature of the discipline.
Practitioner Insights
Programs intended to deepen teacher content knowledge often have a goal of helping teachers understand mathematics and science as ways of knowing. These experiences entail deepening teachers’ understanding of how knowledge is formally established in mathematics/science and or the habits of working or thinking that characterize mathematics/science. Programs often included opportunities for teachers to experience what it means to “do” mathematics/science and focused explicit attention on disciplinary habits of mind.
When queried about strategies for engaging teachers with investigations of understanding mathematics/science as ways of knowing, experienced practitioners offered a number of insights, which are described below. After reviewing these insights, you will be provided with opportunities to share your own experiences about working teachers to develop their understanding of mathematics/science as ways of knowing. The information you provide will be analyzed along with the insights and examples from other practitioners as the website is periodically updated.
Make it real—Provide teachers opportunities to experience what it means to “do” mathematics/science.
There is little question that teachers need to understand mathematics and science disciplinary content, but as the primary representatives of the discipline for their students, teachers need more. Said an experienced program leader:
This is the part where the teachers’ understanding of the substantive content is not sufficient. If they don’t understand how the community of science works, how scientists put together questions, data, representations, and communications to reach consensus, how they argue about results, then teachers will have missed the opportunity to teach children about that and to facilitate kids’ learning in a way that is both effective and authentic to the nature of science.
Scientists and mathematicians are integrally involved in deepening teacher content knowledge in the MSPs. They bring with them not only a wealth of disciplinary content knowledge, but also a deep understanding of what it means to “do” mathematics and science. They want to ensure that students have the opportunity to experience these disciplines as dynamic bodies of knowledge, continually enriched by conjecture, investigation and analysis, and/or, reasoning, justification, and proof. As one MSP Co-PI shared:
Exploration is at the heart of mathematics. Mathematics is not just a powerful tool for understanding the world around us; it is also a powerful tool for discovering the world around us. When students learn mathematics through exploration, they experience a side of our subject that lies much deeper than the simple skills of algebraic manipulation and calculation.
Giving teachers opportunities to experience the discovery side of mathematics/science, program leaders suggest, increases the likelihood that they will be both willing and able to provide similar experiences for their students. A representative of an MSP that is working to help teachers gain a deeper understanding of the nature of mathematics described evidence of success the project has seen in promoting a “culture of exploration” in high school mathematics classrooms: “Our experience proves that the joys of exploration and discovery can be experienced by high school students and teachers in ways that are not all that different from what a seasoned mathematics researcher experiences.” Similarly, a science program leader commented:
I think unless we offer opportunities for teachers to investigate real questions, with real problems of things going wrong, the need for re-design of instruments or materials, questions about how to collect the data and why, problems in which the data don’t come out with a clear finding, and we have to go back to the drawing board again, and so forth, we haven’t really provided them with (1) an authentic view of what scientists encounter in their everyday work and (2) an authentic view of what’s going to happen with children’s work in classrooms, so that they can develop the reasoning abilities to help kids de-bug their investigations. I think many teachers avoid inquiry investigations with kids because they are scared of what to do if things “go wrong.” So, I think it’s critical that their own explorations encounter authentic problems, so that they know that’s what real science looks like, and have opportunities to discuss, as adult learners, how tricky science can be, and how rewarding it is to finally come up with a design that provides clear data, and enables one to make claims based on the evidence.
Choose wisely—Select problems that are amenable to discussions of the nature of the discipline.
Experienced program leaders had advice for the nature of mathematics and science problems teachers should address if the goal is to develop an understanding of disciplinary habits of mind. A scientist in one MSP suggested that a relatively simple experiment that is likely to give highly reproducible results with little variation is a good entry point for understanding the nature of science as a discipline.
If participants are still developing their confidence/understanding in science as a way of knowing, you don’t want a data set that requires complex statistics to confirm that the results are significant. You want one where the variables can be controlled – but also easily identified. You may want one where there are new variables that can be easily manipulated for further exploration to add additional clarity, to demonstrate how knowledge leads to new questions that can be further investigated, and to confirm their understanding of variables and controls.
In mathematics, the recommendation was to engage teachers in investigations that involve inductive reasoning to formulate conjectures, and deductive reasoning to prove or disprove their conjectures, and to include explicit discussions of both types of processes. In both mathematics and science, program leaders noted that some investigations lend themselves to discussions of the nature of the discipline and others do not. One of the primary justifications for professional development at times addressing content that might not be directly relevant to the teachers’ classrooms is its helpfulness for exploring the nature of science or mathematics.
With the exception of “accessible, unsolved problems,” program leaders cautioned that it is not realistic to expect teachers to work at the cutting edge of the disciplines. Said one:
In mathematics, you can expose teachers to the types of questions mathematicians address, but most will not have the sophistication to be able to address them themselves, except for well-designed investigations. Mathematicians operate at a very abstract level that is well beyond most teachers.
Other program leaders agreed. One MSP CO-PI shared:
It’s unlikely that non-specialists could understand the questions that are at the frontier of the discipline – it takes years of background building. But the wonderful thing about our discipline is that, while the questions at the frontier are often not tractable to non-experts, the methods used by researchers are quite accessible to everyone – teachers and students. It’s these habits of mind that need to be at the center of content immersion, and, rather than trying to find fancy topics, one needs to look for low-threshold, high ceiling areas (like number theory or geometry) that allow easy entry but that give a genuine experience of mathematics research.
Another program leader suggested that problems from the history of mathematics can provide a genuine way for teachers to approximate how mathematicians actually work.
The development of ideas over the centuries often is recapitulated by the development of ideas within individuals – and this can be very interesting for teachers to see. For example, working with ancient numeration systems helps us appreciate the power of zero – and helps us to recognize why understanding our place-value number system can be so challenging. Recognizing that negative numbers – and irrational numbers – were not always understood and appreciated, even by mathematicians of times past, helps teachers see why these numbers are hard for students to comprehend.
Experienced program leaders recommend that teachers have an opportunity to investigate ideas from more than one perspective, noting that encountering “opportunities to propose, challenge, debate, try out, evaluate, and discuss different procedures for answering the same question” is particularly helpful. Said one program leader:
Ultimately, you’ll want a variety of experimental designs to demonstrate that direct investigation is not always possible (e.g. astronomy), that not all experiments are “hypothesis driven” (e.g., natural history), and break out of the very traditional notion of THE scientific method… . To get a clear picture of science as a way of knowing and to avoid the “recipe” for the scientific method, multiple experiences really are necessary. Participants must learn that knowledge is generated in a variety of experimental (or theoretical) designs and each is valid.
Doing is not the only way to learn—Reading about how mathematicians/scientists generated knowledge can help teachers understand the nature of the disciplines.
Some experienced program leaders noted that having teachers do investigations is not the only, nor perhaps the best way, of engaging teachers when the goal is for teachers to gain a deeper understanding of the nature of science. Said one program leader:
I think to treat “nature of the discipline” solely through investigations is not an accurate representation of the discipline. The organization of concepts within the discipline, what is known and not known in the discipline, and what is considered law/theory or fact/opinion in the discipline are all aspects of the nature of the discipline. I think that investigations help illustrate some aspects of the nature of the discipline that cannot be experienced through other kinds of learning activities. BUT other learning activities can also address the ideas related to the nature of the discipline.
“If the goal is to have teachers learn about the nature of science then have them read about how scientists do their work and the kinds of questions that scientists investigate,” said one program leader. Another program leader suggested using case studies or viewing videos based on the real work of scientists: “A facilitated conversation about the reading/video can provide teachers with a structured opportunity to view the problem through the scientist’s eyes and learn something about the scientist’s perspective on the problem.”
Be explicit—Focus teachers’ attention on what it means “to know” in mathematics or science.
Many programs engage teachers in content-based exploration as a means of deepening their mathematics/science content knowledge, modeling the processes they intend teachers to use in their classrooms. Some programs go further, focusing explicit attention on how knowledge is generated, including “ways of doing and thinking about” these disciplines, and what it means “to know” in mathematics or science.
A number of MSP projects intentionally designed professional development opportunities to engage teachers in doing mathematics or science to enable them to develop a deeper understanding of the discipline. Said one PI:
The idea is that teachers should understand mathematics on some level the same way mathematicians do; at least teacher leaders, content leaders should have some view of mathematics that’s connected to the way a mathematician sees it. Basic things like, that there should be more questions than answers; that mathematics has a tradition and a history and a culture; that mathematics is something that one can experience, that it’s real.
Another program leader described a similar set of key ideas that teachers need to understand about science as a way of knowing:
Understanding how scientists use evidence to develop explanations based on logic that may eventually become a scientific theory in a discipline is at the heart of understanding scientific inquiry. Learning about the role skepticism plays in advancing science is another aspect of understanding the nature of science. How scientists judge their own work and the work of others helps us have a greater appreciation for the years of work that sometimes lead to a new “discovery” in science.
A science program leader explained the particular importance of focusing on the nature of science knowledge given the current debate about the theory of evolution:
In the age of creationism and claims about its viability as a theory, the role of evidence in supporting scientific theories and persuading practitioners in the field of their usefulness in explaining phenomena is so important for teachers to understand. There may be conflicting evidence and arguments about theories and evidence in science, but that process is still the way progress is made, and an accepted process across disciplines. If teachers don’t understand this, what are the chances that they will encourage arguments about evidence in their classrooms, and be able to explain to children that this is what grown up scientists do, when they disagree? Scientists fall back on the evidence and the way it was collected and whether they all agree that that’s the way to get good results.
Scientists and science educators bristle at the idea that there is a prescribed set of steps to follow in generating scientific knowledge, noting that “cookbook” laboratory activities that take teachers (or students) through a specific sequence of steps misrepresent the nature of science. One program leader suggested that:
On the scale of explorations from confirmatory laboratory exercises to open inquiry, explorations in the guided to open realm are most productive for providing teachers with the opportunity to discuss and develop an understanding of the nature of the discipline.
Insight in Action
A leader in an MSP project preparing high school mathematics teachers to work with their peers described how they try to avoid the “follow-the-dots” approach to solving mathematics problems, giving teachers a better idea of what is entailed in doing mathematics:
We are in the midst of our second four-week-long summer institute. For our mathematics content strand we decided to follow the approach that the Park City Mathematics Institute has taken for some years: Provide teachers with problem sets in an area of mathematics that they typically do not teach deeply; and allow them to mess with the mathematics rather than approach it from an algorithmic perspective to solve application problems. These problem sets contain items that are doable by all, some that are difficult and others that are interesting extensions. We expected some resistance. However, we based our focus on the idea that as teachers of mathematics we should actually do mathematics ourselves.
Sense-making is key—Debrief teachers’ learning experiences from the perspective of the nature of the discipline.
Just like sense-making is important to help solidify understanding of mathematics/science concepts developed during investigations, teachers need opportunities to consider what they are learning about mathematics and science habits of mind. A mathematics program leader shared:
When teachers engage first in explorations and then discuss the process, they have a better chance of truly understanding the nature of mathematics as a discipline because they will be building their own understanding on personal experiences. They are learning about the nature of mathematics through actually doing mathematics.
Another program leader described engaging teachers in discussions that are more advanced than their students will have, for the purposes of helping them be aware of their own reasoning.
For example, I ask them questions they will find difficult, and have them work on it for a while, and then stop them and ask them to reflect on the sorts of things they are doing. We get a list of things like “trying simpler cases,” “trying extreme cases,” “trying to mathematize the question,” “look for connections to other ideas,” and so on.
And since the ultimate goal of deepening teacher content knowledge is improved teaching and learning in the K–12 mathematics and science classroom, it is important that teachers are able to apply what they are learning about habits of mind in mathematics and science to their instruction. Said one program leader:
By engaging [teachers] in inquiry around topics that they find challenging, they can experience for themselves the sorts of uncertainties and frustrations and dead-ends that their students will experience with topics that the teachers once found challenging but may not remember what it was like. They can explore these topics, and then the professional development can step back and have them reflect on what it is they were doing, how they got started, what strategies they had when they were stuck, etc., and those things can be the content they will tie to their teaching- how they will help their students learn to do those things that they do, as learners/problem solvers, faced with difficult material.
If you are interested in how these practitioner insights were collected and analyzed, a summary of the methodology can be found here.
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 in Considering Mathematics/Science as Ways of Knowing
One research study that engaged teachers with activities that targeted their understanding of mathematics as a way of knowing was identified in a search of the published literature. This study provided evidence of positive effects on teachers’ mathematics problem solving (Stecher & Mitchell, 1995). Teacher participants in the study taught grade 4. Topics from various mathematics strands were addressed. Although the study did not investigate the unique contribution of the strategy of engaging teachers in considering mathematics as a way of knowing, the positive results suggest that it may be an effective professional development strategy and a fruitful area for future research.
Research on Engaging Teachers in Considering Mathematics as a Way of Knowing
Professional learning opportunities for teachers of mathematics have increasingly focused on deepening teachers’ content knowledge. One method of addressing teachers’ knowledge is to engage them in considering mathematics as a way of knowing. This strategy may involve focusing teachers on the nature of mathematics content embedded in activities they use with their students, or engaging teachers with investigations designed specifically for adult learners of the content. A common feature of such investigations is a focus on the specific content of an investigation as teachers experience firsthand the nature of the work done in the discipline of mathematics. One research study identified in this review investigated a professional development program that included this strategy.
Findings from Research
The Stecher and Mitchell (1995) study of a professional learning experience that included investigations into mathematics problem solving provided positive results on participating teachers’ knowledge of mathematics problem solving. Although the study did not investigate the unique contribution of the strategy of engaging teachers with investigations into mathematics as a way of knowing, the positive results suggest that it may be an effective professional development strategy and a fruitful area for future research.
Teacher participants in the study all taught grade 4 using portfolios in their mathematics instruction for at least one year and up to four years. All attended summer and academic year workshops provided by the state on using portfolios in mathematics instruction during the past two years. Topics from various mathematics strands were addressed in the workshops.
In addition to the use of the strategy of engaging teachers in investigations of mathematics problem solving, the professional development also addressed mathematics task selection for student portfolios, tying teachers’ learning specifically to their classroom instruction.
Although teachers participated in the workshops on a voluntary basis and at variable levels, the use of portfolios in mathematics instruction and appropriate training to do so was an expectation in the state. Also, teachers were selected through stratified random sampling for the study, so generalizability of the findings certainly extends to the population of grade 4 teachers in Vermont at the time of the study. However, the unique circumstances of the state’s mathematics instructional program at the time may limit generalizability to other contexts.
For the research on engaging teachers in considering mathematics as a way of knowing bibliography, click here. [PDF 10K]
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 eight research studies of professional development programs that included engaging teachers with activities that targeted their understanding science as a way of knowing. None of the studies was designed to measure the unique influence of focusing teachers’ learning on science as a way of knowing. Still, they each reported at least some positive results regarding the effectiveness of the programs in deepening teachers’ science content knowledge (Chun & Oliver, 2000; Lord & Peard, 1995; Nehm & Schonfeld, 2007; Niaz, 2008; Niaz, 2009; Odom, 2001; Radford, 1998; Tuan & Chin, 1999; Wang, 2001).
Research on Engaging Teachers in Considering Science as a Way of Knowing
Professional learning opportunities for teachers of science have increasingly focused on deepening teachers’ content knowledge. One method of addressing teachers’ knowledge is to engage them in considering science as a way of knowing. This strategy may involve focusing teachers on the nature of science content embedded in activities they use with their students, or engaging teachers with investigations designed specifically for adult learners of the content. A common feature of such investigations is a focus on the specific content of an investigation as teachers experience firsthand the nature of the work done in a science discipline. Eight research studies identified in this review investigated professional development programs that included this strategy.
Findings from Research
Each of the eight studies that included engaging teachers in considering science as a way of knowing reported at least some positive results on participating teachers’ content knowledge. None of the studies was designed to systematically examine only the strategies for focusing teachers’ learning on science as a way of knowing; rather, the studies looked at programs comprising multiple interventions with multiple goals, without isolating the influence of any one strategy. In this sense, the studies are more akin to program evaluations than systematic research on specific interventions. Although none of the studies investigated the unique contribution of strategies for engaging teachers in considering science as a way of knowing, generally positive results across programs support claims regarding the effectiveness of the professional development approaches that addressed this goal.
However, not all of the studies reported uniformly positive results. Two studies indicated that pre-intervention measures of teachers’ knowledge indicated that they already showed understanding of at least some aspects of science as a way of knowing (Chun & Oliver, 2000; Wang, 2001). In two of the studies, teachers’ understanding of science as a way of knowing progressed, but not for all ideas that were addressed or not to the level of understanding the intervention targeted (Odom, 2001; Wang, 2001).
Teacher participants in the eight studies ranged from grades Kindergarten through 12, with more studies of teachers in the middle and secondary grades than elementary grades. Particular studies focused on topics in life or physical science, while others addressed science topics from various disciplines.
The experiences for teachers in the eight studies varied in intensity and duration Several studies focused on semester courses that lasted one semester or a full year (Nehm & Schonfeld, 2007; Niaz, 2008, 2009; Tuan and Chin, 1999; Wang, 2001). Others examined intensive summer workshops, one of three weeks in duration (Lord and Peard, 1995), and another of a series of three years of summer workshops (Chun & Oliver, 2000). The other two studies investigated impacts of in-service professional development programs, one pairing teachers working with scientists on science research projects (Odom, 2001), and the other engaging teachers in inquiry learning experiences using their student instructional materials and through independent science research projects (Radford, 1998).
In addition to a wide range in the duration of interventions, the studies also used several different measures to assess gains in teachers’ understanding of science as a way of knowing. Some selected existing survey-type assessments with scale scores (Chun & Oliver, 2000; Tuan & Chin, 1999; Wang, 2001). One of these studies (Wang, 2001) also collected lesson plans and classroom artifacts, and included interviews with teachers to triangulate data on teachers’ understanding of science as a way of knowing. Other studies drew on written or performance assessments used within the institute or course in which the teachers participated (Lord and Peard, 1995; Nehm & Schonfeld, 2007), or as a pre- and post-intervention measure (Odom, 2001). Measurement of teachers’ understanding of science as a way of knowing in two studies was accomplished through examination of multiple sources of data, such as course/workshop assignments, participant presentations, and observations of the course or workshop (Niaz, 2008, 2009; Radford, 1998).
Although unknown for one study (Chun & Oliver, 2000), teachers participated in the experiences investigated in all of the other studies on a voluntary basis. In addition, the populations that the participating teachers represent are limited to those willing and able to commit to extensive interventions, so generalizability of the findings from the studies must be considered in this light. The small sample size of several studies also raises questions about how appropriate it is to generalize the results.
All eight studies used either a pre-post design to measure changes in teachers’ content knowledge or traced changes over the time they were involved in the intervention. It is possible that participating teachers might perform better on a post-intervention measure of knowledge of science as a way of knowing simply because they had completed it previously, in one case (Lord & Peard, 1995) only a few weeks before. Only one study (Radford, 1998) involved a comparison group of teachers who did not participate in the intervention, but data regarding understanding of science as a way of knowing was not part of what was collected from comparison group teachers. Several of the studies used measures of teacher content knowledge that were either unidentified or not well described (Lord & Peard, 1995, Nehm & Schonfeld, 2007; Niaz, 2008, 2009; Odom, 2001; Radford, 1998), and researchers did not generally include evidence of reliability and validity of their measures.
Additional limitations were noted regarding some of these studies. Three of the studies did not describe the intervention in enough detail to support interpretations linking teachers’ experience with the results (Chun & Oliver, 2000; Lord & Peard, 1995; Odom, 2001). Potential biases were also an issue for some of the studies. In at least three of the studies, the researchers designed or implemented the intervention, leading to a possible investigator bias (Lord & Peard, 1995; Niaz, 2008, 2009; Radford, 1998). There are also questions of potential bias in seven of the studies for which the samples were unique in ways that might have interacted with the treatments (Chun & Oliver, 2000; Lord & Peard, 1995; Nehm & Schonfeld, 2007; Niaz, 2008, 2009; Odom, 2001; Radford, 1998; Tuan & Chin, 1999). In most cases, these studies provided adequate descriptions of the samples, although there was no investigation of how characteristics of the sample may have influenced outcomes. In one study (Odom, 2001) some of the participants did not complete the treatment and in another (Nehm & Schonfeld, 2007) some participants did not complete post-treatment measures, but no follow-up procedures to assess potential biases were described in either case.
For the research on engaging teachers in considering science as a way of knowing bibliography, click here. [PDF 119K]