## About

130

Publications

43,304

Reads

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1,473

Citations

Introduction

Karl W. Kosko is an professor in mathematics education at Kent State University. Dr. Kosko's research focuses on how mathematical meaning is conveyed.

Additional affiliations

August 2012 - December 2023

July 2010 - August 2012

## Publications

Publications (130)

Mathematical knowledge for teaching (MKT) is a professional knowledge crucial for effective mathematics instruction, encompassing both content knowledge (CK) and pedagogical content knowledge (PCK). The purpose of this study is to investigate the hierarchical structure of PCK for fractions by seeking further evidence for the PCK-Fractions assessmen...

Quantitative measures in mathematics education have informed policies and practices for over a century. Thus, it is critical that such measures in mathematics education have sufficient validity evidence to improve mathematics experiences for students. This article provides a systematic review of the validity evidence related to measures used in ele...

Teacher education has begun to embrace the use of 360 videos to improve preservice teachers' (PSTs) engagement and immersion. While recent research on such use is promising overall, there are specific questions that have been left unanswered about the construct of presence in 360 videos. More specifically, research has yet to fully explore how vide...

This study used traditional and holographic video, along with eye-tracking technology, to examine how preservice teachers' physical act of looking is associated with how they attend to and assess students' fraction reasoning. Findings revealed that, although viewing of holograms may have influenced more focus on students' work area, there was littl...

There is emerging evidence that professional noticing is embodied. Yet, there is still a need to better under embodied noticing at a fundamental level, especially from the preservice teachers. This study used traditional and holographic video, along with eye-tracking technology, to examine how preservice teachers' physical act of looking interacts...

Manipulatives are considered essential tools to scaffold student learning of mathematical topics. Despite the importance of using such tools, the acceptability and affordances of different manipulatives and teachers' unfamiliarity with them constrain their use. In this chapter, we provide two different maker tools (i.e., 3D printers and Cricut) for...

Professional teacher noticing is a key skillset for mathematics teaching that is informed by various psychological and physiological factors. The present study investigated the described about students after viewing a 360 video of a primary grades mathematics lesson. Results indicated a statistically significant

In recent years, teacher noticing has gained prominence as a theoretical construct in mathematics education, highlighting the dynamic, situational aspects of teaching that underlie instructional decisions and actions. This research forum explores innovative research approaches to teacher noticing in mathematics education, focusing on four key areas...

A growing body of research has supported the implementation of innovative and immersive video for teaching and learning across the lifespan. Immersive video, delivered through eXtended Reality (XR) tools like 360 video, provides users with new ways to see real or created environments. Unfortunately, most of the existing research has highlighted imm...

This study sought to explore math and science teacher educators' use of various media to represent practice within methods courses. There is little understanding of why certain media is used over other representations and the rationale for these choices. Specifically, the study focused on the prevalence and familiarity of teacher educators with com...

This study discusses the difference in a preservice teachers’ (PSTs) mathematical noticings and the relationship between where they attend in a 360 video. Using a convergent mixed methods approach, our evidence provides support that PSTs who are more explicit in their written noticing of mathematical references are more focused in their viewing of...

Comparing expert and novice teachers' professional noticing via eye-tracking in a 360 video.

This study presents an extension of the validity argument for the PCK-Fractions measure. PCK-Fractions is designed to assess the effectiveness of professional experiences in facilitating teachers’ pedagogical content knowledge (PCK) for children’s fraction reasoning in grades 3-5. We examined data across 101 participants from two Midwest universiti...

Immersive 360 videos are increasingly being used in pre-service teachers (PST) education. There is preliminary evidence that this technology may benefit future educators' focus and attention to classroom settings and events. However, more analytical efforts are needed to better understand its potential impact on reported focus of attention (RFA) am...

If there was a bright side to the COVID-19 pandemic, particularly
related to education, it was the massive and rapid
introduction of educational technologies to scaffold teaching
and learning. Most notably, within teacher education, this
included extended reality (XR) technologies to supplement
or replace face-to-face field experiences. With the pa...

This research report provides an overview of validity evidence, assessment items, and scoring table for the PCK-Fractions measure.

In this concluding chapter, we summarize and consider the consequences of various conceptions of argumentation, justification, and proof, as revealed by the chapter authors’ analyses and syntheses. In discussing these consequences, we note productive ways the field can move forward, while also noting the importance of attending to both research and...

This chapter provides an overview of data analyzed by authors focusing on the constructs of argumentation, justification, and proof. Data comes from a US-based second-grade (ages 7–8) lesson focusing on the concept of equivalence. Evidence from whole class discussions, as well as written work and use of classroom tools incorporated are described in...

This preliminary study explored how many representations of standard videos, animations/comics, and 360 videos are being used in mathematics methods courses to teach future teachers. Drawing on knowledge from prior studies on standard videos, this study aimed to address the gaps in literature to encompass other representations that are being utiliz...

Professional noticing is an essential skill for teachers that is enacted by teachers via their embodied senses (sight, sound, etc.). To better understand the nature of teacher noticing, 44 preservice teachers (PSTs) viewed a 360 video of an elementary mathematics lesson while wearing virtual reality headsets. PSTs writings of what they noticed and...

Preservice teachers (PSTs) often demonstrate difficulty learning to attend to content-specific student actions in-the-moment. However, machine learning algorithms applied to PSTs' viewing of 360 videos provides a potentially useful tool for teacher educators. In this paper, we describe the initial development of such a tool and the implications for...

Attending to students' actions and mathematical thinking is an important aspect of professional teacher noticing. In this paper, we used 360 videos as a medium to examine the relationship between preservice teachers'(PSTs) observed attending behaviors and their written noticing. Findings suggest that PSTs focusing on students, instead of the teache...

Argumentation, justification, and proof are conceptualized in many ways in extant mathematics education literature. At times, the descriptions of these objects and processes are compatible or complementary; at other times, they are inconsistent and even contradictory. Regardless of the descriptions of these processes, however, given the importance...

Teacher noticing is a crucial facet of math and science teacher education, with one goal being to shift preservice teachers' (PSTs) noticing from teacher-centered to student-centered. In this study, we used 360 videos to examine PSTs' choices of where to look in a classroom. We discuss differences in attending behavior of those PSTs who focused on...

The quality of teachers’ mathematical knowledge for teaching (MKT) is critical for effective teaching and mathematical learning of students. However, most efforts on measuring MKT tend to focus on teachers’ content knowledge (CK), with less attention to teachers’ pedagogical content knowledge for teaching (PCK). This study reports on our initial ef...

Controlling and standardizing experiments is imperative for quantitative research methods. With the increase in the availability and quantity of low-cost eye-tracking devices, gaze data are considered as an important user input for quantitative analysis in many social science research areas, especially incorporating with virtual reality (VR) and au...

The use of technology affords opportunities for prospective teachers to engage in actions that are proximal to the work of teaching. The authors designed a task in which prospective teachers (n = 95) at four institutions created an animation or depiction of a classroom scenario using one of two technology platforms: GoAnimate or LessonSketch. They...

The use of video is commonplace for professional preparation in education and other fields. Research has provided evidence that the use of video in these contexts can lead to increased noticing and reflection. However, educators now have access to evolving forms of video such as 360 video. The purpose of this study was to adapt and validate an inst...

Teachers' pedagogical content knowledge (PCK) influences their instruction and, by consequence, their children's opportunities to learn better. Within the domain of fractions, items assessing PCK are nested within larger assessments, with little explicit focus on the PCK domain. We report on the development and initial validity argument for a PCK f...

Multiplication and division are vital topics in upper level elementary school. A teacher's pedagogical content knowledge (PCK) influences both instruction and students' learning. However, there is currently little research examining teachers' PCK within this domain, particularly regarding professional education of future teachers. To help address t...

There is growing evidence to suggest that construction of knowledge is an embodied activity (Alibali & Nathan, 2012; Barsalou, 1999; Gallese & Lakoff, 2005). We conjecture that embodied cognition is a useful theoretical lens for explaining teachers' professional noticing. Evidence from eye-tracking studies suggest that experienced secondary teacher...

Research and practitioner articles advocate the use of visual representations in scaffolding elementary students’ learning of multiplication and division. Prior research suggests students use different strategies when provided with different visualized representations of multiplication and division. However, there is relatively little study examini...

Research has provided evidence of the value of producing multiple representations of content for learners (e.g., verbal, visual, etc.). However, much of the research has acknowledged changes in visual technologies while not recognizing or utilizing related audio innovations. For instance, teacher education students who were once taught through two-...

Use Cuisenaire™ Rods to emphasize the column-and-row structure in arrays for meaningful multiplication.

Researchers have documented that high engagement in learning is linked to high academic achievement, and that low engagement is linked to low academic achievement, student alienation, and withdrawing from school. While research provides insights into student engagement and instructional practices, this research is typically conducted in traditional...

Use of video as a representation of practice in teacher education is commonplace. The current study explored the use of a new format (360 video) in the context of preservice teachers’ professional noticing. Findings suggest that preservice teachers viewing 360 videos attended to more student actions than their peers viewing standard video. In addit...

This paper presents Empeiría, which uses cutting-edge technologies and novel virtual reality systems to enhance future in-class education training. Empeiría incorporates JavaScript, WebGL, WebVR, and powerful web graphics engines like Babylon.js to create immersive training experiences. It demonstrates a useful application of computer science conce...

The global COVID-19 pandemic has disrupted normal face-to-face classes across institutions. This has significantly im-pacted methods courses where preservice teachers (PSTs) practice pedagogy in the field (e.g., in the PreK-12 classroom). In this paper, we describe efforts to adapt an assignment originally situated in a face-to-face school placemen...

Article is available here: https://rdcu.be/b4LgZ
Although it is a relatively new innovation, 360 videos have gained commercial and public popularity. Given their novelty, there is a relative dearth of studies on their effectiveness. Most of the existing research is published as technical articles dealing with standards and specifications, as theo...

This paper explores middle-grade students' conceptions of median. Describes, where and why they struggle and provides learning trajectory to improve their understanding.

This paper explores two preservice mathematics teachers' understanding of mode. Participants' initial understanding and understanding following use of an interactive virtual manipulative is examined. Findings suggest that participants initially operated with less-effective definitions of mode. Preservice teachers' developed understanding following...

Videos of mathematics instruction are commonplace in mathematics teacher education. However, videos' effectiveness in facilitating mathematics preservice teachers' attention is often contingent upon their decomposition and/or approximation of practice. This pilot study examined whether 360 video technology affected the specificity of mathematics at...

This paper reports on one mathematics teacher educator’s self-study of implementing 360 videos to facilitate preservice teacher education. Three themes emerged that guided implementation: accessibility of the medium, the perceptual capacity of the recorded scenarios, and how/whether preservice teachers’ attending to mathematics was facilitated.

In recent years, teacher professional noticing has become more prevalent internationally. One common definition to distinguish professional noticing from everyday noticing includes the three interrelated skills of attending, interpreting and deciding to respond. We used qualitative analysis to explore prospective teachers' approximations of practic...

Visual representations and manipulatives are a highly advocated mathematical tool for the teaching and learning of multiplication and division. Although there is some prior research on elementary teachers’ general use of manipulatives and visual representations, there is little to no specific focus on use of such representations on a specific mathe...

Multiplicative reasoning is critical for the development of more advanced mathematical concepts such as fractions and algebra, and therefore is a critical process for teachers and researchers to assess in an effective and efficient manner. Unfortunately, the vast majority of available assessments examine students’ skill with algorithms or memorized...

Evidence suggests that visual models used to represent multiplication are not always interpreted multiplicatively. This study examined elementary students' strategies across different models. Findings suggest students will count by 1s when a model illustrates all ones. By contrast, models illustrating composite units elicited a wider variety of str...

Multiplication and division are important topics that are formally introduced in third grade. Visual models and representations are key scaffolds for facilitating the teaching and learning of these topics, but there is little research on the prevalence of these models in classrooms. The present study examined six U.S. textbooks and found that three...

Numerous different recommendations for the order that multiplication should be taught exist. Woodward (2006) suggested introducing multiplication using 0s, 1s, squares, 5s, 9s and 10s, followed by other operands. Researchers in neuroscience suggest that multiplication with operands 2-5 is simpler and uses different neural processes than operands 6-...

We examine how instructors at different grade levels proposed mathematical tasks to support students’ engagement in constructing viable arguments and critiquing others’ reasoning. During small group interviews with pairs of practicing elementary and collegiate mathematics instructors, they created tasks intended to support students’ ability to crea...

Elementary preservice teachers at six universities engaged in a task that provided them opportunities to articulate their professional noticing within video representations, written decompositions, and animated approximations of practice. The preservice teachers’ written accounts indicated that a majority attended to students or student thinking; h...

Multiplicative reasoning is a key concept in elementary school mathematics. Item
statistics reported by the National Assessment of Educational Progress (NAEP) assessment provide the best current indicator for how well elementary students across the U.S. understand this, and other concepts. However, beyond expert reviews and statistical analysis, th...

Multiplicative reasoning is essential for students’ engagement with various mathematical concepts. Although the field’s understanding of children’s multiplicative concepts has grown over the past 30 years, relatively few studies have examined the development of multiplicative concepts with whole numbers, and even fewer have studied this phenomenon...

The first mathematics education faculty member at Kent State (Dr. James W. Heddens) arrived on campus in 1961. In the more than 50 years that followed, the program was witness to the development of a doctoral program in the 1960s, the first mathematics specialist graduate program in the U.S., the second mathematics laboratory in the U.S., the devel...

In this mixed methods study, we used a convergent paral- lel design to create and analyze a task that was implement- ed across four US institutions with 99 preservice teachers (PSTs) using one of two multi-media platforms: LessonS- ketch or GoAnimate. We analyzed PSTs’ comic-based depic- tions or animations to examine teacher questioning and ut- te...

This paper explores two preservice mathematics teachers' understanding of mode. Participants' initial understanding and understanding following use of an interactive virtual manipulative is examined. Findings suggest that participants initially operated with less-effective definitions of mode. Preservice teachers' developed understanding following...

Research suggests that students' engagement in writing in mathematics has a positive effect on mathematics achievement (e.g., Cross, 2009). Recently, evidence has emerged suggesting a relationship between elementary children's mathematical argumentative writing and their early algebraic reasoning (Kosko & Singh, 2016). To explore the phenomenon fur...

This study is focused on the structure of equivalence problem to probe the evolution from operational to relational view of students' understanding of equals sign. We propose a modified construct map which incorporates the intermediate levels in such a transition which were previously ignored. Our findings suggest that the structure of number sente...

It is generally accepted that a relational conception of equivalence (sameness or interchangeability of expressions), is critical for elementary students' success in latter mathematics (Blanton & Kaput, 2005). By contrast, an operational view of the equals sign (i.e. producing the answer) hinders their success. Researchers often use different types...

Argumentation, justification, and proof are conceptualized in many ways in extant mathematics education literature. At times, the descriptions of these objects and processes are compatible or complementary; at other times, they are inconsistent and even contradictory. The inconsistencies in definitions and usages of the terms argumentation, justifi...

Multiplicative reasoning as articulated through scheme theory suggests that students improve the sophistication of their reasoning as they are able to coordinate additional levels of units, often described through students’ use of schemes and operation at particular multiplicative concepts. Theoretical descriptions of the second and third multiplic...

Recent educational policy documents have encouraged engaging students in mathematical argumentation via discussion and writing. Most recently in the U.S., the Common Core State Standards recommend that children construct viable arguments and critique the reasoning of others. One often advocated means of engaging students in this mathematical practi...

Mathematical communication has been an important feature of standards documents since National Council of Teachers of Mathematics’ (NCTM) (1989) Curriculum and Evaluation Standards. Such an emphasis has influenced content standards of states from then to present. This study examined how effective the prevalence of various forms of mathematical comm...

In an effort to elicit elementary preservice teachers' mathematical noticing, mathematics teacher educators at 6 universities designed and implemented a 3-step task that used video, writing, and animation. The intent of the task was to elicit preservice teachers' mathematical noticing—that is, noticing specific to mathematics content and how studen...

Multiplicative reasoning is essential for developing meaningful understandings of fractions, rational number, and algebraic relationships. Several mobile applications are available for parents and teachers to download and supplement elementary students’ learning experiences with multiplication and division, and some studies support the use of vario...

This study set out to examine two important aspects of the use of eWriters by early elementary students. First, it explored the impact of eWriters on literacy motivation and self-efficacy of students in Pre-Kindergarten, Kindergarten, and First grade. Second, it explored if and how the technology implementation would affect parent and teacher relat...

Detailing is a linguistic tool for mathematical argumentation in which given mathematical information is operationalized through one's warrants to support a claim. Recent literature suggests that students' detailing is related to their early algebraization. This study examined 168 elementary students' use of detailing in two mathematical argumentat...

This paper explores the use of animations as an approximation of practice to provide a transformational technology experience for elementary mathematics preservice teachers. Preservice teachers in mathematics methods courses at six universities (n=126) engaged in a practice of decomposing and approximating components of a fraction lesson. Data anal...

This paper explores middle-grade students' conceptions of median. Describes, where and why they struggle and provides learning trajectory to improve their understanding.

Argumentation, justification, and proof are conceptualized in many ways in extant mathematics education literature. At times, the descriptions of these objects and processes are compatible or complementary; at other times, they are inconsistent and even contradictory. The inconsistencies in definitions and usages of the terms argumentation, justifi...

Preservice elementary mathematics teachers' choice of questioning sequence in a hypothetical representation of practice was examined in comparison with their choice of follow-up task. Findings indicate that preservice teachers' task choice, and their rationales for task choice, has a statistically significant influence on their choice of questionin...

Facilitating mathematical discussions has consistently been identified as beneficial to students’ mathematical learning, with teachers’ use of questioning a primary identifier of appropriate facilitation. Although many teachers report familiarity with appropriate questioning techniques, we hypothesized that some teachers may not work in contexts wh...

Research has documented the potential positive impact of mobile devices and apps across multiple formal and informal learning environments. Less is known about the role of mobile technologies for the use of engaging learners with special needs or disabilities. The purpose of this report is two-fold: 1) to describe the findings from a review that wa...

Although writing in mathematics has been advocated in practitioner journals, policy documents, and various research, surveys of mathematics teachers between 1988 and 2008 in various English-speaking countries have reported relatively infrequent incorporation into mathematics lessons. Further, much of prior survey research has yielded only anecdotal...

This chapter focuses on how digital technologies provide affordances for Practice-Based Mathematics
Teacher Education (PBMTE) and maps needed research on technologically mediated teacher education.
The chapter elaborates on Grossman’s pedagogies of practice, describing how they can be enacted in
digital environments. Using LessonSketch as a prototy...

This paper explores the use of animations as an approximation of practice to provide a transformational technology experience for elementary mathematics preservice teachers. Preservice teachers in mathematics methods courses at six universities (n=126) engaged in a practice of decomposing and approximating components of a fraction lesson. Data anal...

The nature of mathematical argumentative writing (MAW) is relatively understudied, particularly in the context of early childhood. While much of the literature on MAW, and mathematical argumentation in general, has focused on the manner in which procedures and warrants are sequenced, the present study focused on the grammatical resource of mathemat...

Children’s development of mathematical argument has long been identified as facilitated by early algebraization. A fundamental feature of argumentation is colligation, or the ability to synthesize one’s warrants towards a claim. In this study, we examined the presence of colligation across 168 second and third grade students’ written responses to m...

The present study explored whether primary grades teachers chose probing questions, given two hypothetical mathematics lesson scenarios. After responding to the mathematics lesson scenarios, participating teachers completed the Problems in Schools survey assessing dispositions to support student autonomy, and the Mathematical Knowledge for Teaching...

Elementary aged children need developmentally appropriate activities to engage them in argumentative writing in mathematics (math writing).

Prevalent characteristics of effective mathematical discourse in the literature include student autonomy, a positive social environment, and a degree of perceived mathematical competence. These characteristics are also found in Self-Determination Theory (SDT), which posits that actions are self-regulated when students’ autonomy, competence, and soc...