Karl W. Kosko

Karl W. Kosko
Kent State University | KSU · School of Teaching, Learning & Curriculum Studies

Ph.D.

About

110
Publications
30,040
Reads
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913
Citations
Introduction
Karl W. Kosko is an associate professor in mathematics education at Kent State University. Dr. Kosko's research focuses on how mathematical meaning is conveyed.
Additional affiliations
August 2012 - present
Kent State University
Position
  • Professor (Assistant)
July 2010 - August 2012
University of Michigan
Position
  • PostDoc Position

Publications

Publications (110)
Article
Full-text available
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...
Article
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...
Article
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...
Conference Paper
Full-text available
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...
Conference Paper
Full-text available
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...
Conference Paper
Full-text available
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...
Conference Paper
Full-text available
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...
Article
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...
Chapter
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...
Article
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...
Article
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...
Conference Paper
Full-text available
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...
Conference Paper
Full-text available
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...
Conference Paper
Full-text available
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...
Article
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...
Article
Full-text available
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-...
Article
Use Cuisenaire™ Rods to emphasize the column-and-row structure in arrays for meaningful multiplication.
Article
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...
Article
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...
Chapter
Full-text available
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...
Article
Full-text available
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
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...
Preprint
Full-text available
This paper explores middle-grade students' conceptions of median. Describes, where and why they struggle and provides learning trajectory to improve their understanding.
Preprint
Full-text available
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...
Conference Paper
Full-text available
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...
Conference Paper
Full-text available
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.
Article
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...
Article
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...
Article
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...
Conference Paper
Full-text available
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...
Conference Paper
Full-text available
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...
Conference Paper
Full-text available
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-...
Article
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...
Article
Full-text available
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...
Article
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...
Article
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...
Technical Report
Full-text available
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...
Article
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...
Conference Paper
Full-text available
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...
Conference Paper
Full-text available
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...
Conference Paper
Full-text available
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...
Conference Paper
Full-text available
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...
Conference Paper
Full-text available
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...
Article
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...
Article
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...
Article
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...
Article
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...
Article
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...
Article
Full-text available
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...
Conference Paper
Full-text available
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...
Conference Paper
Full-text available
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...
Conference Paper
Full-text available
This paper explores middle-grade students' conceptions of median. Describes, where and why they struggle and provides learning trajectory to improve their understanding.
Conference Paper
Full-text available
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...
Conference Paper
Full-text available
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...
Technical Report
Full-text available
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...
Article
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...
Chapter
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...
Article
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...
Article
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...
Conference Paper
Full-text available
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...
Article
Full-text available
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...
Chapter
Elementary aged children need developmentally appropriate activities to engage them in argumentative writing in mathematics (math writing).
Article
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...
Conference Paper
Full-text available
The concept of equivalence is one of the first algebraic notions introduced to students in early childhood. The future success of students in mathematics, and related subjects, is believed to rely significantly on their understanding of this basic algebraic concept. However studies have found that the operational view of the equals sign is a major...
Conference Paper
Full-text available
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 use of the terms argumentation, justificat...
Article
Full-text available
This paper describes instruments designed to use multimedia to study at scale the instructional decisions that mathematics teachers make as well as teachers’ recognition of elements of the context of their work that might influence those decision. This methodological contribution shows how evidence of constructs like instructional norm and professi...
Article
Full-text available
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...
Conference Paper
Full-text available
As states, districts, schools, faculty, and other germane stakeholders prepare for full implementation of the Common Core State Standards for Mathematics (CCSSM), the number of resources designed to guide and support this transition continues to expand in both breadth and depth. Along with changes in mathematics content standards and their progress...
Article
[Draft version: http://deepblue.lib.umich.edu/bitstream/handle/2027.42/108487/JTE-FORUMANALYSIS_DB.pdf?sequence=1&isAllowed=y] It has been common to use video records of instruction in teacher professional development, but participants have rarely been encouraged to evaluate teachers and students’ actions in those records, allegedly because evaluat...