What We Know About Deepening Teachers' Content Knowledge: Considering Mathematics/Science as Ways of Knowing

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.

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.

Now It's Your Turn...

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