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 informal diagram, can be used for early algebra. We present an instructional sequence mathematics teacher educators could use to introduce elementary preservice teachers to or reacquaint them with strip diagrams. We also present strategies elementary pre-service teachers used to solve strip diagram problems before and after participating in our instructional sequence. We conclude with a discussion about the strategies and implications for teaching early algebra to elementary preservice teachers.

Quantitative reasoning (QR) is a key skill for undergraduate biology education. Despite this, many students struggle with QR. Here, we use the theoretical framework of student noticing to investigate why some students struggle with QR in introductory biology labs. Under this framework, what students notice when given new information and data influences how they process this information and connect it with other events to form new conceptions. Students must mentally isolate given features, create mental records of those features, and identify features or objects that they connect to existing knowledge. Identifying these features or objects is thus critical since they form the foundation upon which learning takes place. We conducted observations of groups in introductory biology labs involving QR, which informed follow-up interviews to examine what students notice, the level/relevance of their noticing, and factors that shape student noticing. We find that some students are noticing more perceptual features, often focusing on less relevant trends and features, with others noticing deeper, more relevant patterns that facilitate conceptual sensemaking. In addition, we find multiple factors, including students' expectations and their attitude toward QR and biology, that shape student noticing. We conclude with implications for instructors and the biology education research community.

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 whose incoming conceptions about linear functions were at different levels of development. Our study comprised a two-week quadratic functions instructional unit emphasizing covariational reasoning bracketed by pre- and postassessments and interviews. Our analysis focused on four students with incoming linear functions conceptions at different levels of development. Findings revealed that students experienced different kinds of backward transfer. This study generated new insights into backward transfer in the context of mathematics education.

This book is about scientific inquiry, providing a practical manual for conducting and communicating quality research in (mathematics) education.

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 was a conceptual replication study designed to examine Iranian PSTs' reasoning about pre-symbolic early algebra by looking at what was more, somewhat, and less challenging. The aims of the replication study aligned with the original study (Hohensee, 2017). Results from the replication study show that participating PSTs (N = 15) found the early algebra approach to variables and functions more challenging, indeterminable unknowns somewhat challenging, and equivalence and equations less challenging. We make comparisons with the original study, as well as offer implications and suggestions for preparing PSTs to teach early algebra.

Backward transfer, defined as the influence that learning about new concepts has on individuals' reasoning about previously learned concepts, is a relatively new topic in mathematics education. Our exploratory study compared the backward transfer produced in a mathematics enrichment programme that emphasised covariational reasoning to that produced in real algebra classrooms that did not emphasise a particular type of reasoning. Quadratic and linear functions were the new concepts being learned about and the concepts previously learned about, respectively. One theme that emerged from our comparisons was that quadratic instruction that emphasised covariational reasoning had less, but more consistent, backward transfer than quadratic instruction that did not have a strong emphasis on a particular type of reasoning. This and other findings led us to new insights about backward transfer, as well as tentative new ideas about backward transfer more generally, that can be tested in future studies..

For serious games on education, understanding the effectiveness of different learning methods in influencing cognitive processes remains a significant challenge. This study investigates the impact of serious games on graph structure learning. For this, we compared our in-house game-based learning (GBL) and video-based learning (VBL) methodologies by evaluating their effectiveness on cognitive processes by oxygenated hemoglobin levels using functional near-infrared spectroscopy (fNIRS). We conducted a 2 x 1 between subjects preliminary study with twelve participants, involving two conditions: game and video. Both groups received equivalent content related to the basic structure of a graph, with comparable session lengths. The game group interacted with a quiz-based game, while the video group watched a pre-recorded video. The fNIRS was employed to capture cerebral signals from the prefrontal cortex, and participants completed pre- and post- questionnaires capturing user experience and knowledge gain. In our study, we noted that the mean levels of oxygenated hemoglobin were higher in the GBL group, suggesting the potential enhanced cognitive involvement. Our results show that the lateral prefrontal cortex (LPFC) has greater hemodynamic activity during the learning period. Moreover, knowledge gain analysis showed an increase in mean score in the GBL group compared to the VBL group. Although we did not observe statistically significant changes due to participant variability and sample size, this preliminary work contributes to understanding how GBL and VBL impact cognitive processes, providing insights for enhanced instructional design and educational game development. Additionally, it emphasizes the necessity for further investigation into the impact of GBL on cognitive engagement and learning outcomes.

AbstractEvery 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.

AbstractBuilding 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.

AbstractTheoretical 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.

AbstractIf 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.

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 at two comprehensive high schools served as the participants. Both schools drew from low-socioeconomic urban populations. The study involved paper-and-pencil assessments about linear functions that were administered before and after a four- to five-week instructional unit on quadratic functions. The teachers were instructed to teach the quadratic functions unit using their regular approach. Qualitative analysis revealed three kinds of backward transfer influences and each influence was related to a shift in how the students reasoned about functions in terms of an action or process view of functions. Additionally, features of the instruction in each class provided plausible explanations for the similarities and differences in backward transfer effects across the two classrooms. These results offer insights into backward transfer, the relationship between prior knowledge and new learning, aspects of reasoning about linear functions, and instructional approaches to teaching functions.

Abstractspiepr 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.

Pre-symbolic algebra has been advocated for as a mathematics topic elementary students should experience to better prepare them for middle and high school algebra. However, most elementary pre-service teachers have little to no experience with pre-symbolic algebra. The study reported here analysed the struggles that ten elementary pre-service teachers experienced when learning about pre-symbolic algebra in a mathematics content course. Three types of struggles emerged, struggles with changes in the artifacts of algebra, the objects of algebra, and pre-service teachers’ role while doing algebra. This study could inform efforts to better support elementary PSTs in preparing to teach pre-symbolic algebra.

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, we present the key features of six progressive perspectives on the transfer of learning, using examples of their recent use in STEM education research. Finally, we end with a discussion of the motivation for and organization of this book.

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 provide an argument for why the field needs more theory development generally, and why this especially applies to theory about backward transfer. Second, I present the current state of research and theory about backward transfer in the context of mathematics education and related fields. Finally, I outline five aspects of theory about backward transfer in the context of mathematics education for which there is a pressing need for development.

For five decades, JRME has sought to publish high-quality mathematics education research that advances the field's knowledge and has a positive impact on the teaching and learning of mathematics in the classroom. The journal's 50th anniversary represents an opportune time for the research community to take a step back, assess what progress has been made on the major problems of the field, and consider the most important problems that could orient research in the future. As we look across educational scholarship, we find that among the most robust findings from research on teaching and learning is that students'

Over the past several decades, educators have become increasingly intent on using data to inform decision-making at all levels of the educational system (Cho & Wayman, 2014; Mandinach, 2012; Means et al., 2010). The underlying reasoning is sound: Better decisions can be made with relevant data. Policymakers have reasoned that instructional decisions made by teachers that are based on data relevant to the classroom will help students to learn and achieve more. Indeed, from the introduction of No Child Left Behind (NCLB) to the current policies of the Every Student Succeeds Act (ESSA), the