Charles Hohensee

• University of Delaware

Quantitative reasoning (QR) is a key part of biology. Here, we apply the theoretical framework of student noticing to investigate into how students learn QR in an intro bio lab course. Using observations and interviews, we characterize what students notice when working with QR, their depth of noticing, and the factors that shape their noticing.

This study examined backward transfer , which we define as how students’ ways of reasoning about previously encountered concepts are modified when learning about new concepts. We examined the backward transfer produced when students learned about quadratic functions. We were specifically interested in how backward transfer may vary for students who...

Research has shown there are algebra concepts elementary teachers can introduce that help prepare elementary students for the eventual transition to algebra (e.g., the relational interpretation of the equal sign). Early algebra refers to the use of informal approaches to introduce such concepts to elementary students. Strip diagrams , a type of inf...

Early algebra can prepare elementary students for the transition they will need to make from arithmetic and algebra. Although teacher preparation programs emphasize the teaching of early algebra, research on how to prepare elementary pre-service teachers (PSTs) to teach early algebra is still scarce. The replication study reported in this article w...

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?” ar...

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. M...

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 re...

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...

spiepr Abs1 Every 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...

Backward transfer is defined as the influence that new learning has on individuals’ prior ways of reasoning. In this article, we report on an exploratory study that examined the influences that quadratic functions instruction in real classrooms had on students’ prior ways of reasoning about linear functions. Two algebra classes and their teachers a...

In this chapter, we trace the evolution of transfer research in the field of mathematics education (and associated domains, such as science education and computer science education), from the rejection of the transfer of learning construct to the recent flowering of research from progressive transfer perspectives. In the main body of the chapter, w...

In this chapter, I make a case for why the field of mathematics education is in need of theory development about a particular aspect of the transfer of learning called backward transfer. I begin the chapter by explaining my conceptualization of backward transfer and providing an illustrative example. Then, I present a three-part case. First, I prov...

The study reported in this article examined the ways in which new mathematics learning influences students’ prior ways of reasoning. We conceptualize this kind of influence as a form of transfer of learning called backward transfer. The focus of our study was on students’ covariational reasoning about linear functions before and after they particip...

This book provides a common language for and makes connections between transfer research in mathematics education and transfer research in related fields. It generates renewed excitement for and increased visibility of transfer research, by showcasing and aggregating leading-edge research from the transfer research community. This book also helps...

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 signifi...

(...) In this editorial, we discuss the first of the five overarching problems: defining and measuring learning opportunities precisely enough to study how to maximize the quality of the opportunities experienced by every student.

We concluded our November editorial (Cai et al., 2018b) with a promise to consider research paradigms that could bring us closer to the new world we have envisioned where research is intertwined with practice. We will call the paradigms we have in mind research pathways to avoid the range of complicated connotations often applied to the term paradi...

Cross-disciplinary self-studies and peer coaching have separately been shown to offer teacher educators meaningful professional development opportunities. However, the teacher educator literature has said little about combining cross-disciplinary self-studies with peer coaching. In this article, the authors report a two-year cross-disciplinary peer...

In this editorial, we elaborate our vision of the changing roles of researchers and teachers in a future world in which research has a much more direct and meaningful impact on practice (Cai et al., 2017). In previous editorials, we have described characteristics of this future world, including setting research agendas based on instructional proble...

In our May editorial (Cai et al., 2018a), we explored how collaborations among teacher-researcher partnerships could harness emerging technological resources to address the problem of isolation in the work of teachers and researchers. In particular, we described a professional knowledge base (Cai et al., 2018b) and a mechanism by which that knowled...

In our March editorial (Cai et al., 2018), we considered the problem of isolation in the work of teachers and researchers. In particular, we proposed ways to take advantage of emerging technological resources, such as online archives of student data linked to instructional activities and indexed by learning goals, to produce a professional knowledg...

In our November 2017 editorial (Cai et al., 2017), we presented a vision of a future in which research has a significant impact on practice. In the world we described, researchers and teachers work together, sharing similar goals and incentive structures. A critical feature of this brave new world is the existence of an online professional knowledg...

This editorial discusses the critical idea of replication in educational research.

We began our editorials in 2017 seeking answers to one complex but important question: How can we improve the impact of research on practice? In our first editorial, we suggested that a first step would be to better define the problem by developing a better understanding of the fundamental reasons for the divide between research and practice (Cai e...

An understanding of partitive division is foundational for numerous other mathematics topics, including unit rate, slope, and probability. However, research has shown that learners tend to have a limited understanding of partitive division when the divisor is a proper fraction. To extend research on conceptions of partitive division in this study,...

In our May editorial (Cai et al., 2017), we argued that a promising way of closing the gap between research and practice is for researchers to develop and test sequences of learning opportunities, at a grain size useful to teachers, that help students move toward well-defined learning goals. We wish to take this argument one step further. If resear...

Researchers have argued that integrating early algebra into elementary grades will better prepare students for algebra. However, currently little research exists to guide teacher preparation programs on how to prepare prospective elementary teachers to teach early algebra. This study examines the insights and challenges that prospective teachers ex...

In our last editorial, we considered the impact of research on students' learning. In clarifying our perspective, we answered the question of “impact of research on what” to include both cognitive and noncognitive outcomes in students as well as long-term impact on students that goes well beyond short-term cognitive impact. A natural next step is t...

In our first editorial (Cai et al., 2017), we highlighted the long-standing, critical issue of improving the impact of educational research on practice. We took a broad view of impact, defining it as research having an effect on how students learn mathematics by informing how practitioners, policymakers, other researchers, and the public think abou...

How can research have a larger impact on educational practice? What kinds of research can have the greatest impact on educational practice? These are perennially thought-provoking questions for mathematics education researchers (e.g., Battista et al., 2007; Boerst et al., 2010; Heck et al., 2012; Heid et al., 2006; Herbel-Eisenmann et al., 2016; La...

This study examines the degree to which individual and social aspects of student noticing in classroom settings during new learning influence students’ ways of reasoning about previously-encountered concepts. Seventh- and eighth-grade students (N = 7) participated in an instructional unit on quadratic functions (the new concept) and clinical pre- a...

In this study, I examined the degree to which experienced teachers were aware of the relationship between prior knowledge and new learning. Interviews with teachers revealed that they were explicitly aware of when students made connections between prior knowledge and new learning, when they applied their prior knowledge to new contexts, and when th...

The purpose of this study was to examine and elaborate upon elementary prospective teachers’ (PSTs) conceptions of partitive division with fractions. We examined the degree to which PSTs’ conceptions were connected (i.e., capable of translating between representations correctly; aware that partitive division generates a unit rate for its quotient)...

Even in the resource-rich, more ideal conditions of many design-based classroom interventions, unexpected events can lead to disappointing results in student learning. However, if later iterations in a design research study are more successful, the previous failures can provide opportunities for comparisons to reveal subtle differences in instructi...

The transfer of learning has been the subject of much scientific inquiry in the social sciences. However, mathematics education research has given little attention to a subclass called backward transfer, which is when learning about new concepts influences learners’ ways of reasoning about previously encountered concepts. This study examined when a...

Despite recent research interest in student-created diagrams, little research has systematically investigated students' diagram-construction processes, meaning the order and manner in which students create markings as they physically generate diagrams. In this study, we characterize the various processes students use to create diagrams that represe...

Even in simple mathematical situations, there is an array of different mathematical features that students can attend to or notice. What students notice mathematically has consequences for their subsequent reasoning. By adapting work from both cognitive science and applied linguistics anthropology, we present a focusing framework, which treats noti...

As transfer researchers have begun to investigate a broader range of phenomena, they have correspondingly put forward new processes to provide explanatory accounts for the occurrence of transfer. This move coincides with a call to acknowledge the contribution of social interactions, language, cultural artifacts, and normed practices to the generali...

Despite the proliferation of mathematics standards internationally and despite general agreement on the importance of teaching for conceptual understanding, conceptual learning goals for many K-12 mathematics topics have not been well-articulated. This article presents a coherent set of five conceptual learning goals for a complex mathematical doma...

This two-phase study investigated high school student difficulties with graphing and understanding piecewise functions, with a focus on how students thought about multiple domain statements for a single function. The report on the first phase details the essential aspects of student thinking and highlights underlying reasons that begin to account f...