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In discussing theories of teaching, we take the position that there is a two-way street between what we calltheory for teachingandteaching for theory. We articulate the linkages between these two dynamic processes through a particular conceptualization of professional knowledge for teaching carried by tangible artifacts. Within this context we have tried to answer a set of questions about theory and teaching: (1) What is a theory (of teaching)? (2) What should it contain and why? (3) Can such a theory accommodate differences across subject matters and student populations taught? If so, how? If not, why? (4) Do we already have a theory or theories on teaching? If so, which are they? (5) In the future, in what ways might it be possible, if at all, to create a (more comprehensive) theory of teaching? To answer these questions, we draw on the lens of Confucian learning as well as examples from Chinese and U.S. mathematics education to elaborate on understanding, assessing, and accumulating professional knowledge for teaching.

spiepr Abs1Every day people do research as they gather information to learn about something of interest. In the scientific world, however, research means something different than simply gathering information. Scientific research is characterized by its careful planning and observing, by its relentless efforts to understand and explain, and by its commitment to learn from everyone else seriously engaged in research. We call this kind of research scientific inquiry and define it as “formulating, testing, and revising hypotheses.” By “hypotheses” we do not mean the hypotheses you encounter in statistics courses. We mean predictions about what you expect to find and rationales for why you made these predictions. Throughout this and the remaining chapters we make clear that the process of scientific inquiry applies to all kinds of research studies and data, both qualitative and quantitative.

Every researcher wants their study to matter—to make a positive difference for their professional communities. To ensure your study matters, you can formulate clear hypotheses and choose methods that will test them well, as described in Chaps. 1, 2, 3 and 4. You can go further, however, by considering some of the terms commonly used to describe the importance of studies, terms like significance, contributions, and implications. As you clarify for yourself the meanings of these terms, you learn that whether your study matters depends on how convincingly you can argue for its importance. Perhaps most surprising is that convincing others of its importance rests with the case you make before the data are ever gathered. The importance of your hypotheses should be apparent before you test them. Are your predictions about things the profession cares about? Can you make them with a striking degree of precision? Are the rationales that support them compelling? You are answering the “So what?” question as you formulate hypotheses and design tests of them. This means you can control the answer. You do not need to cross your fingers and hope as you collect data.

If you have carefully worked through the ideas in the previous chapters, the many questions researchers often ask about what methods to use boil down to one central question: How can I best test my hypotheses? The answers to questions such as “Should I do an ethnography or an experiment?” and “Should I use qualitative data or quantitative data?” are quite clear if you make explicit predictions for what you will find and fully develop rationales for why you made these predictions. Then you need only worry about how to find out in what ways your predictions are right in what ways they are wrong. There is a lot to know about different research designs and methods because these provide the tools you can use to test your hypotheses. But as you learn these details, keep in mind they are means to an end, not an end in themselves.

Theoretical frameworks can be confounding. They are supposed to be very important, but it is not always clear what they are or why you need them. Using ideas from Chaps. 1 and 2, we describe them as local theories that are custom-designed for your study. Although they might use parts of larger well-known theories, they are created by individual researchers for particular studies. They are developed through the cyclic process of creating more precise and meaningful hypotheses. Building directly on constructs from the previous chapters, you can think of theoretical frameworks as equivalent to the most compelling, complete rationales you can develop for the predictions you make. Theoretical frameworks are important because they do lots of work for you. They incorporate the literature into your rationale, they explain why your study matters, they suggest how you can best test your predictions, and they help you interpret what you find. Your theoretical framework creates an essential coherence for your study and for the paper you are writing to report the study.

Building on the ideas in Chap. 1, we describe formulating, testing, and revising hypotheses as a continuing cycle of clarifying what you want to study, making predictions about what you might find together with developing your reasons for these predictions, imagining tests of these predictions, revising your predictions and rationales, and so on. Many resources feed this process, including reading what others have found about similar phenomena, talking with colleagues, conducting pilot studies, and writing drafts as you revise your thinking. Although you might think you cannot predict what you will find, it is always possible—with enough reading and conversations and pilot studies—to make some good guesses. And, once you guess what you will find and write out the reasons for these guesses you are on your way to scientific inquiry. As you refine your hypotheses, you can assess their research importance by asking how connected they are to problems your research community really wants to solve.

The purpose of this paper is to provide a theoretical stance on integrating implementation and research. Implementation should be made integral to research because viewing research and implementation as an integrated whole is a more useful perspective for educational research and practice. A consequence of this position is a marked blurring of the boundaries between researchers and practitioners and an expanding of the vision of the nature of the products of research and how they are shared. From this position, we conceptualize four characteristics of research in which implementation is integral, and we illustrate them with the example of a case study of a teacher learning to teach linear functions through problem posing. In particular, we highlight the notion of an artifact that embodies and provides space for the dynamic intertwining between implementation and research. This intertwining represents what we mean by implementation being integral to research, and it has implications that ultimately blur the boundaries between the roles of researchers and teachers. We conclude by raising several pressing questions about the conduct and communication of this kind of research in education.

Problem solving and problem posing have long been of interest to the mathematics education community. In this survey paper we first look at some of the seminal moments in the history of research on the important topics. We then use this history to position the state-of-the-art research being done in both problem solving and problem posing, before introducing the presented state-of-the-art developments in problem solving and problem posing. We then use this work as a backdrop against which to introduce the 16 empirical papers that make up this special issue. Together these 16 papers add nuance to what is already known about problem solving and problem posing; this nuance is the result of attending to very specific contexts and purposes in which these activities are embedded. We end the paper by discussing the future directions these fields can take.

This study examined how expert and novice (preservice) teachers solved mathematical modelling tasks as well as how they noticed written artifacts of student thinking that were in response to the mathematical modelling tasks. Some teachers in both groups were aware of the openness and underdetermination of the modelling tasks and that these characteristics implied that underlying assumptions needed to be made in order to solve the tasks. Indeed, nearly all of the expert teachers addressed the need to make assumptions for the Seashell Task. When examining student work, both expert and preservice teachers interpreted positive aspects in the students’ solutions and provided feedback to students. However, almost all of the expert teachers responded by asking questions, whereas around a third of preservice teachers directly corrected students’ mistakes and another third pointed out mistakes without correcting them. This study provides a new angle to study teachers’ mathematical competencies, the assessment of modelling competencies, and considerations for the development of teachers’ modelling competencies.

Research has shown that students’ beliefs related to mathematics are connected to their affect and motivation as well as their mathematical thinking and activity. Moreover, high parental expectations have been shown to play a role in students’ positive beliefs and self-efficacy. This paper reports on two studies investigating parental expectations related to students’ beliefs and their perceived achievement. These studies highlighted an unusual, but confirmed finding that students who do not perceive that their parents have specific mathematics achievement expectations show more positive mathematics-related beliefs and higher perceived mathematics achievement than students who believe their parents do have specific expectations. This paper explores possible reasons for this finding related to parents’ methods of conveying their expectations.

This study investigated 45 undergraduate students’ mathematical problem posing based on four problem situations. This study was designed to better understand the kinds of problems they pose, and the kinds of mathematical ideas exhibited when problem posing. The findings of this study support that underprepared college students can pose mathematical problems and were largely capable of engaging with the critical mathematical ideas embedded in problem-posing tasks. Problem-posing tasks, then, appear to be a context within which underprepared undergraduate students can successfully engage with key mathematical ideas. The main implication of these findings is that problem posing appears to be amenable to being harnessed as a pedagogical tool for developmental mathematics courses.

Fostering conceptual understanding in mathematics classrooms is an important goal in mathematics education. To support this goal, we need to be able to diagnose and assess the extent to which students have conceptual understanding. In this study we employed a problem-posing task and a problem-solving task in order to diagnose and assess preservice teachers’ mathematical understanding of fraction division. The results of the study show that although over 99% of the preservice teachers could correctly perform fraction division, they were much less likely to exhibit conceptual understanding of fraction division either through a problem-solving task involving graphical representation or through a problem-posing task. However, engaging in problem posing appears to more likely than making a graphical representation to elicit preservice teachers’ conceptual understanding. Moreover, a conceptual cue about the meaning of fraction division appears to greatly increase the likelihood that preservice teachers will exhibit conceptual understanding of fraction division in their posed problems and generate conceptually based diagrams of fraction division. The findings highlight the usefulness of problem-posing tasks for diagnosing and assessing preservice teachers’ mathematical understanding.

In this study we aimed to understand teaching mathematics through problem posing based on an analysis of 22 teaching cases. Teaching mathematics through problem posing starts with problem-posing tasks. This study provides not only specific examples of problem-posing tasks used in classrooms but also related task variables to consider when developing problem-posing tasks. This study also contributes to our understanding of how teachers can deal with student-posed problems in the classroom. In these 22 teaching cases, there was a typical pattern to how teachers dealt with the students’ posed problems in the classroom according to the instructional goals. For future research, we need to accumulate additional teaching cases and explore possible discourse patterns concerning how teachers handle students’ posed problems, as well as identify the most effective discourse patterns when teaching mathematics through problem posing.

Problem posing, the process of formulating problems based on a given situation, is an essential practice in mathematics and other disciplines. Although this is acknowledged in policy documents, problem posing is neither substantively nor consistently included in the school mathematics curriculum. In this paper, we first comment on the state of problem posing in school mathematics and discuss three recommendations to improve its integration into curriculum materials and classroom practice. These recommendations present a low barrier to entry for teachers and students and require only minor changes to common mathematics classroom activities and curriculum materials. Based on the three recommendations, as well as the features of effective teacher professional development, a program was created to investigate longitudinally the impact of problem-posing professional development on teachers’ conceptions of problem posing and their design of lessons to teach mathematics using problem posing, as well as on students’ learning. Initial findings are presented, including the significant changes in teachers’ conceptions of problem posing and teachers’ design of lessons to teach mathematics using problem posing, as well as the impact on students’ learning.

This study explored 1634 Chinese eighth-grade students’ domain- and task-specific self-efficacy and its relationship to their problem-posing performance. In particular, the linear regression model, generalized additive model (GAM), and piecewise regression model (PRM) were used to detail the linear and non-linear relationships between these variables. The findings indicate that most (92.5%) of the students could pose mathematical problems in all tasks, but the effect of their domain-specific self-efficacy on their problem-posing performance was lower than the effect of their task-specific self-efficacy. Students’ problem-posing performance and their task-specific self-efficacy were not always matched when the requirements of the problem they posed varied in difficulty. As the level of difficulty increased, the correlation coefficient between task-specific self-efficacy and problem posing declined from 0.22 to 0.06. Furthermore, PRM confirmed that there were significant changes of the slope around the cut-point of the relationship between task-specific self-efficacy and students’ problem-posing performance. Moreover, the relationship between task-specific self-efficacy and posing performance was different for easy and difficult problems, as the cut-point and slopes before and after the point varied. The findings of this study contribute both to understanding self-efficacy as well as advancing understanding about the characteristics of problem posing from a non-cognitive perspective.

Any effort to integrate problem-posing instruction in school mathematics must attend to teachers’ beliefs about the advantages of teaching through problem posing and especially their beliefs about the challenges of teaching in this way. This study investigated teachers who were learning how to teach mathematics through problem posing. The primary foci were the teachers’ beliefs about teaching through problem posing and their problem-posing performance. The findings demonstrate that the participating teachers were able to engage successfully in problem-posing tasks and that they held a number of different beliefs about the advantages and challenges of teaching through problem posing. After participating in three workshops, a sample of teachers showed gains in problem-posing performance and in the scope and specificity of their beliefs about teaching through problem posing.

This study serves two purposes. The first purpose is the development and validation of a Mathematics Teachers’ Beliefs Scale (MTBS). In addition to providing a valid and reliable MTBS instrument, this study provides information on the process of developing and validating the instrument, which shall be useful for the development and validation of similar instruments. The second purpose of this study is to explore the relationship between teachers’ beliefs and their teaching experience. Using a validated MTBS, this study assessed the beliefs of four groups of Chinese high school mathematics teachers with varying years of professional teaching experience: 1–5 years (n = 25), 6–10 years (n = 70), 11–20 years (n = 48), and 21+ years (n = 28). The MTBS is composed of 26 items distributed across five subscales spanning beliefs about mathematics, learning, teaching, students, and teachers. Among the four groups, the results of this study indicate that there were no significant differences in scores on beliefs about mathematics, mathematics learning, mathematics teaching, students, and teachers. Further item analysis revealed that longer teaching experience does not necessarily mean that a teacher holds reform-oriented views. However, teachers with 6–10 years of experience were most likely to hold a more dialectic view of mathematics.

Significant research in science and mathematics education should advance the field’s knowledge and understanding of the teaching and learning of science and mathematics. How, then, should the significance of a research question in science and mathematics education be assessed? And, when disseminating the findings of research, how should the significance of the research questions be communicated? In this study, we analyzed peer reviews to answer these questions. Our analysis revealed the main issues peer reviewers identify about research questions and the ways they are communicated during the dissemination of research. The findings provide insights for new and experienced researchers about communicating the significance of research questions, and they also illustrate how reviewer comments in peer-reviewed journals can provide a window into the field’s frontiers.

Although often asked tactfully, a frequent question posed to authors by JRME reviewers is “So what?” Through this simple and well-known question, reviewers are asking: What difference do your findings make? How do your results advance the field? “So what?” is the most basic of questions, often perceived by novice researchers as the most difficult question to answer. Indeed, addressing the “so what” question continues to challenge even experienced researchers. All researchers wrestle with articulating a convincing argument about the importance of their own work. When we try to shape this argument, it can be easy to fall into the trap of making claims about the implications of our findings that reach beyond the data.