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Introduction
Publications
Publications (80)
We report on a written measure of multiplicative double counting (mDC), a form of children’s multiplicative reasoning. Our aim is to apply theory of students’ cognition to validate a written measure for grade 3–8 students as a proxy of interviewing. The 4item measure consists of hintfree and hintcontaining wordproblems, following the finegrain...
This study investigates the relationship between children’s subitizing activity and their construction of arithmetic units. In particular, the study hypothesizes a positive association between children’s construction of subitized units and their construction of arithmetic units, and hypothesizes that children who can subitize larger units, such as...
We examine a hypothesis implied by Steffe’s constructivist model of children’s numerical reasoning: a child’s spontaneous additive strategy may relate to a foundational form of multiplicative reasoning, termed multiplicative double counting (mDC). To this end, we mix quantitative and qualitative analyses of 31 fourth graders’ responses during clini...
Preorder link to Corwin:
https://us.corwin.com/enus/nam/numeracyforalllearners/book268322

This book – Numeracy for All Learners: Teaching Mathematics to Students with Special Needs – is a significant and important addition to the current Mathematics Recovery series. The eight books in this series address the teaching of early number, whol...
In a previous study, we validated the fractions schemes and operations trajectory (Norton and Wilkins 2012; Steffe in J Math Behav 20(3): 267–307, 2002; Steffe and Olive Children’s fractional knowledge. Springer, New York, 2010; Wilkins and Norton 2011) with PreK8 prospective teachers and explored PreK8 prospective teachers’ understanding of frac...
Number sequences are defined in terms of children’s abilities to construct and transform units. Children operating with the initial number sequence (INS) construct units of 1. They construct other numbers as strings of 1s and can count on, by 1s, from one number to a subsequent number. A critical benchmark in children’s further numerical developmen...
Units coordinating has emerged as an important construct for understanding students’ mathematical thinking, particularly their concepts of multiplication and fractions. We conducted an elevensession constructivist teaching experiment with a pair of sixthgrade students to investigate how children coordinate whole number and fractional units across...
This opening chapter provides an introduction to the book. It also introduces a theme that integrates many of the contributions from the remaining chapters: we adopt Kant’s perspective for merging rationalist and empiricist philosophies on the construction of knowledge. In particular, we focus attention on ways that biologically based abilities and...
The overarching theme of this book can be simply stated: Building on a foundation of biologically based abilities, children construct number via sensorimotor and mental activity. In this chapter, we return to this theme, and we connect it to three additional themes that emerge across chapters: comparing competing models for conceptual change; consi...
Psychological studies of early numerical development fill a void in mathematics education research. However, conflations between magnitude awareness and number, and overattributions of researcher conceptions to children, have led to psychological models that are at odds with findings from mathematics educators on later numerical development. In th...
The book synergizes research on number across two disciplines—mathematics education and psychology. The underlying problem the book addresses is how the brain constructs number. The opening chapter frames the problem in terms of children’s activity, including mental and physical actions. Subsequent chapters are organized into sections that address...
Problem solving lies at the core of engineering and remains central in school mathematics. Word problems are a traditional instructional mechanism for learning how to apply mathematics to solving problems. Word problems are formulated so that a student is able to identify data relevant to the question asked, and choose a set of mathematical operati...
This commentary addresses the role of theoretical frameworks in building models of students’ mathematics. Specifically, it compares ways that the Learning Through Activity framework (LTA) and scheme theory explain and predict students’ mathematical activity. Both frameworks rely on Piagetian constructs—especially reflective abstraction—to build exp...
We report on a teaching experiment intended to foster a concept of multiplication that would
both subsume students’ multiplegroups concept of whole number multiplication and provide a
conceptual basis for understanding multiplication of fractions. The teaching experiment, which
used a rigorous singlesubject methodology, began with an attempt to b...
This study 1 provides statistical analysis that corroborates a prediction implied by Les Steffe's model: the strength of children's conception of number as a composite unit predicts their ability to reason multiplicatively. In individual clinical interviews, 33 fourth graders (age ~10) correctly solved a 1digit addition word problem (8+7). Student...
This quantitative study corroborates a conceptual linkage implied by Tzur et al.'s (2013) model of progression in schemes for multiplicative reasoning. We demonstrate that the Same Unit Coordination (SUC) scheme serves as a conceptual screener for the Mixed Unit Coordination (MUC) schemeand hence for base10, place value (PVB10) reasoning. Soluti...
Background
Fractions continue to pose a critical challenge for students and their teachers alike. Mathematics education research indicates that the challenge with fractions may stem from the limitations of partwhole concepts of fractions, which is the central focus of fractions curriculum and instruction in the USA. Students’ development of more s...
In an effort to expand our knowledge base pertaining to preK8 prospective teachers’ understanding of fractions, the present study was designed to extend the work on fractions schemes and operations to this population. One purpose of our study was to validate the fractions schemes and operations hierarchy with the preK8 prospective teacher popul...
Through their work on the Fractions Project, Steffe and Olive (2010) identified a progression of fraction schemes that describes students' development toward more and more sophisticated ways of operating with fractions. Although several quantitative studies have affirmed this progression, the question has remained open as to whether it is specific...
The main goal of this paper was to examine middle school students’ game engagement and its effect on math performance. For the game, we developed [Math App], an educational video game intended to support students’ understanding of fractions. Using [Math App] in a quasiexperimental research setting, we collected data on game engagement, game featur...
Units coordination has emerged as an important construct for understanding students’ mathematical thinking, particularly their concepts of multiplication and fractions. To explore students’ units coordination development, we conducted an elevensession constructivist teaching experiment with a pair of sixthgrade students, investigating how they co...
A growing body of research implicates students' ability to coordinate multiple levels of numerical units as an important aspect of their mathematical development. In this paper, we consider relationships between the ways students coordinate units with whole numbers (their multiplicative concepts) and the ways students coordinate units with fraction...
A new app helps students learn to construct fractions and develop algebra readiness, to take some of the challenge out of both areas.
In considering mathematical development across multiple domains, researchers have implicated the critical role of an individual's ability to produce and coordinate units. Here, we describe a theoretically grounded instructional approach for promoting growth in units coordination. Our approach is informed by neuroscience, as well as existing researc...
Students’ ability to coordinate multiple levels of units constitutes a cognitive core in their mathematical development across several domains, including counting, whole number multiplication, integer addition, fractions concepts, and algebraic reasoning. Identifying a progression of students’ units coordination activity would help educators levera...
Units coordination refers to students’ abilities to create units and maintain their relationships with other units that they contain or constitute. In recent research, units coordination has arisen as a key construct that mediates opportunities for student learning across several domains of mathematics, including fractions knowledge and algebraic r...
Units coordination refers to students’ abilities to create units and maintain their relationships with other units that they contain or constitute. In recent research, units coordination has arisen as a key construct that mediates opportunities for student learning across several domains of mathematics, including fractions knowledge and algebraic r...
In an effort to maximizing success in mathematics, our research team implemented an educational video game in fifth grade mathematics classrooms in five schools in the Eastern US. The educational game was developed by our multidisciplinary research team to achieve a hypothetical learning trajectory of mathematical thinking of 5th grade students. I...
This study examined the effects of a learning game, [The Math App] on the
mathematics proficiency of middle school students. For the study, researchers
recruited 306 students, Grades 6–8, from two schools in rural southwest Virginia.
Over a nineweek period [The Math App] was deployed as an intervention
for investigation. Students were assigned to...
“Serious digital games” for education are presumed to be engaging, but little is known about whether engagement is ubiquitous, whether it persists over time, whether it is found for all students across the full range of prior gaming experience, and whether it is actually associated with gamebased learning outcomes. To address these gaps, student b...
> Context . This paper outlines how radical constructivist theory has led to a particular methodological technique, developing secondorder models of student thinking, that has helped mathematics educators to be more effective teachers of their students. > Problem . The paper addresses the problem of how radical constructivist theory has been used...
Because of their focus on psychological structures and operations, neoPiagetian approaches to learning lend themselves to neurological hypotheses. Recent advances in neural imaging and educational technology now make it possible to test some of these claims. Here, we take a neoPiagetian approach to mathematical learning in order to frame two stud...
The [Math App], developed for the iOS platform (primarily targeting iPads), is an educational video game (EVG) intended to support students' understanding of fractions and thus promote algebrareadiness. In working with fractions, most fifth grade students in the U.S. rely on partwhole conceptions alone, inhibiting development toward algebra in hi...
Explore a new app that allows students to develop a more sophisticated understanding of fractions.
With the emergence of free Massive Open Online Courses (MOOCs), online education has been in the headlines in recent years. Many universities are offering online courses, some free and some for pay tied to a degree program. However, the lack of sufficient quality in existing online courses is undeniable. In particular, teaching Modeling and Simulat...
The purpose of this paper is to illustrate cognitive challenges introduced by Common Core State Standards for Mathematics (2010) with regard to conceptualizing fractions. We focus on a strand of standards that appear across grades three through five, which is best represented in grade four, by standard 4.NF.4a: “[Students should] understand a fract...
Previous research has demonstrated the effectiveness of particular instructional practices that support students’ constructions of the partitive unit fraction scheme and measurement concepts for fractions. Another body of research has demonstrated the power of a particular mental operation—the splitting operation—in supporting students’ development...
Recent research suggests that video games and social media may influence youths’ lives in ways that deserve attention from psychologists, mathematics educators, and learning scientists. For example, positive effects on engagement, which can increase probability of mathematics proficiency, have been reported in the literature. We examine this issue...
Piagetian theory describes mathematical development as the construction and organization of mental operation within psychological structures. Research on student learning has identified the vital roles 2 particular operations—splitting and units coordination—play in students' development of advanced fractions knowledge. This article demonstrates th...
The current body of literature suggests an interactive relationship between several of the process standards advocated by National Council of Teachers of Mathematics. Verbal and written mathematical communication has often been described as an alternative to typical mathematical representations (e.g., charts and graphs). Therefore, the relationship...
In this article six mathematics teacher educators describe a collaborative selfstudy that examined personal beliefs about mathematics teacher education. We were striving to understand more fully our beliefs and belief structures, including how these beliefs influence our instructional practices. We describe four beliefs about mathematics teacher e...
We have used letter writing as a means for preservice teachers (PSTs) to develop ability to design effective tasks, in terms of eliciting high levels of cognitive activity from students. Studies on studentdependent task analyses, by assessing the levels of cognitive demand indicated in students’ responses, have demonstrated significant growth amon...
The Candy Factory, an app developed for the iOS platform targeting iPads, is an educational game intended to prepare middle school students for algebrareadiness. The Candy Factory differs from existing educational games along three dimensions: 1) the app is designed following evidencebased theories of cognition and engagement; 2) the app scaffold...
In order to evaluate the effectiveness of an experimental elementary mathematics field experience course, we have designed
a new assessment instrument. These videobased prediction assessments engage prospective teachers in a video analysis of a
child solving mathematical tasks. The prospective teachers build a model of that child’s mathematics and...
Researchers have hypothesized that children's construction of splitting operations is crucial to their construction of more advanced fractions concepts (Steffe, 2002). The authors propose that splitting constitutes a psychological structure similar to that of a mathematical group (Piaget, 1970): a structure that introduces mutual reversibility of s...
In building models of students’ fractions knowledge, two prominent frameworks have arisen: Kieren's rational number subconstructs, and Steffe's fractions schemes. The purpose of this paper is to clarify and reconcile aspects of those frameworks through a quantitative analysis. In particular, we focus on the measurement subconstruct and the partitiv...
Directly or indirectly, The Fractions Project has launched several research programs in the area of students’ operational development. Research has not been restricted to fractions, but has branched out to proportional reasoning (e.g., Nabors 2003), multiplicative reasoning in general (e.g., Thompson and Saldanha 2003), and the development of early...
This colorful illustration of a primary component of modern cryptography—the DiffieHellman key exchange—draws students into the secret world of message encoding and decoding.
Mathematical letter writing can be a mutually beneficial partnership between high schools and universities.
Recognizing schemes, which are different from strategies, can help teachers understand their students' thinking about fractions.
Teaching experiments with pairs of children have generated several hypotheses about students’ construction of fractions. For example, Steffe (2004) hypothesized that robust conceptions of improper fractions depends on the development of a splitting operation. Results from teaching experiments that rely on scheme theory and Steffe's hierarchy of fra...
After Peano gave an arithmetic construction, Hilbert developed a geometric construction for spacefilling curves. This paper describes the key idea of Hilbert's construction, here called 'the nestedsquares criterion,' implicit in Hilbert's writing but, once explicated, generalizes to a whole class of spacefilling curves that correspond to a speci...
An account is given of the importance and role of trust in motivating teachers to understand students and develop instructional approaches. Three examples of student actions, teacher learning, and instructional change that resulted from understanding students' goals are presented, as well as guidelines for teacher reflection. (Contains 3 figures.)
We report on quantitative methods applied to a pair of hypotheses formed through teaching experiments. In particular, our research affirms conceptual distinctions between partwhole and partitive reasoning with fractions, as theorized in previous literature (e.g., Steffe, 2002). These distinctions include a developmental hurdle in moving from parti...
The challenge that we address concerns teachers’ shifts toward studentcentered instruction. We report on a yearlong professional
development study in which two United States elementary school teachers engaged in a teaching experiment, as described by
Steffe and Thompson (in: Lesh and Kelly (eds) Research on design in mathematics and science educat...
Each year, more teachers learn about the successful intervention program known as Math Recovery (USMRC 2008; Wright 2003). The program uses Steffe's wholenumber schemes to model, understand, and support children's development of wholenumber reasoning. Readers are probably less familiar with Steffe's fraction schemes, which have proven similarly u...
This article reports on students' learning through conjecturing, by drawing on a semesterlong teaching experiment with 6 sixthgrade students. It focuses on 1 of the students, Josh, who developed especially powerful ways of operating over the course of the teaching experiment. Through a finegrained analysis of Josh's actions, this article integra...
The goal of this article is to examine students' mathematical development that occurs as a teacher works within each of 2 zones of learning: students' zones of proximal development (ZPD) and students' zones of potential construction (ZPC). ZPD, proposed by Vygotsky, is grounded in a social constructivist perspective on learning, whereas ZPC, propos...
A report on a sample of fourth graders' responses to the 2003 NAEP exam. Readers are invited to analyze the responses to assess students' concepts of fractions. The article has four examples of student work.
This article provides detailed analysis, from a radical constructivist perspective, of a sequence of letterwriting exchanges between a preservice secondary mathematics teacher and a high school student. This analysis shows the ways in which the preservice teacher gained understanding of the high school student's mathematics and attempted to pose t...
This paper is a reflection on activities designed for computerenhanced inservice training of high school mathematics teachers. The goal of these activities is twofold: to promote advanced mathematical thinking, and to introduce new uses of existing technology tools. The authors suggest using jointly a computerbased graphing calculator, a dynami...
This paper introduces technologybased teaching ideas that facilitate the development of qualitative reasoning techniques in the context of quadratic equations with parameters. It reflects on activities designed for prospective secondary mathematics teachers in accord with standards for teaching and recommendations for teachers in North America. Th...
The mathematics education community fosters discourse on a wide variety of personal and social factors influencing mathematical development in the individual and in the mathematics community. Some authors have focused on the issues of race and gender in mathematical learning (e.g. Moody, 1998; Fennema, 1990). Others have focused on the issue of soc...
This article confronts the issue of why secondary and postsecondary students resist accepting the equality of 0.999… and 1, even after they have seen and understood logical arguments for the equality. In some sense, we might say that the equality holds by definition of 0.999…, but this definition depends upon accepting properties of the real numbe...