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In this paper, we network five frameworks (cognitive demand, lesson cohesion, cognitive engagement, collective argumentation, and student contribution) for an analytic approach that allows us to present a more holistic picture of classrooms which engage students in justifying. We network these frameworks around the edges of the instructional triangle as a means to coordinate them to illustrate the observable relationships among teacher, students(s), and content. We illustrate the potential of integrating these frameworks via analysis of two lessons that, while sharing surface level similarities, are profoundly different when considering the complexities of a classroom focused on justifying. We found that this integrated comparison across all dimensions (rather than focusing on just one or two) was a useful way to compare lessons with respect to a classroom culture that is characterized by students engaging in justifying.




Classroom Testing Heaton Pdf


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The motivation for this work stems from the observation of two fourth-grade math lessons in the USA. The lessons shared many components: the nature of the content, students contributed frequently, and students had time to work on the mathematics. Yet, as researchers, we observed clear differences between the two lessons. There was a notable difference between classrooms in the quality and depth of the interactions in the classroom and the level of engagement with the content. This was particularly evident in terms of student contributions where justification appeared to be a strong aspect in one but not in the other lesson. We sought to make sense of various differences with regard to classroom culture characterized by students engaging in justifying in order to operationalize components of ambitious mathematics instruction.


Justifying provides essential tools in classrooms where students can come to make sense of important mathematical structures, ideas, and strategies. As such, we were looking for frameworks that emphasized a focus on deeply engaging with mathematical content and supporting students in justifying as a means for students to be positioned as contributors to mathematics. We describe each of the frameworks below and then illustrate how we see the frameworks connecting to one another. When viewed simultaneously and as interconnected, these frameworks helped us identify and operationalize how the classroom cultures of students engaging in justifying varied across the two lessons.


We describe each of these below and then coordinate them (Bikner-Ahsbahs & Prediger, 2010). In tandem, these frameworks helped us identify and operationalize how the classroom cultures of students engaging in justifying varied across the two lessons. For each of the analyses, we adjusted our unit of analysis to appropriately align with selected analytic frameworks adapting methods directly from framework authors when available. For example, when coding the coherence of the lesson, cognitive demand, and cognitive engagement activity, we segmented the lesson into chunks based on the focus of the segment/task (Smith & Stein, 1998), and then coded those segments. For collective argumentation and student contributions, we used the transcripts of the lesson to identify talk turns and then coded each talk turn (Conner et al., 2014). See Appendix Tables 3, 4, 5, 6, 7, 8, 9, and 10 for a full explication of these analytic methods.


Existing frameworks emphasize types of justifications (Sowder & Harel, 1998) and how justification develops in the classroom (Williams, 1993). In our analysis, we were interested in the distribution of student contributions and wanted a framework to describe the extent to which students engaged in the activities of justifying. Therefore, we used the Student Discourse Observation protocol (Melhuish et al., 2019) which parses student mathematical contributions into three categories: using procedures and facts, justification, or generalizations (see Fig. 1).


Through existing framings, such as the instructional triangle, researchers have argued that classroom instruction reflects complex and interdependent relationships between teacher, student(s), and content (e.g., Cohen, Raudenbush, & Ball, 2003; Hawkins, 2002; Lampert, 2001). The instructional triangle has been posited as a means to theorize high-quality instruction via attention to the complex relationships between teacher/teaching, students/learning, and content. Yet, when studying the effectiveness of mathematics classrooms, most analyses focused on a subset of the vertices or edges within the instructional triangle (Charalambous & Praetorius, 2018). Similarly, most observation tools have their key focus on the teacher (Praetorius & Charalambous, 2018).


By networking the five frameworks of lesson cohesion (Smith & Stein, 1998; Stein & Smith, 2011), cognitive demand (Smith & Stein, 1998), collective argumentation (Conner et al., 2014), student contributions related to argumentation (Melhuish et al., 2019), and cognitive engagement (Chi & Wylie, 2014) along the instructional triangle, we were able to explore the, at times, interdependent relationships between and among the teacher, student(s), and content in these two math lessons, enabling us to explore the relationships along the edges of the instructional triangle. Thus, we were able to meaningfully operationalize the relational arrows of the instructional triangle with an overarching lens for justification classroom culture through the use of multiple frameworks.


We contend that one of the most powerful insights from this coordination of frameworks into the reconceived instructional triangle with measurable components is that it allowed us to parse apart the complexity of classroom cultures focusing on students engaging in justifying. Each framework individually told us something about the lessons. For example, when comparing the two lessons (Fig. 3), we see that each of the component frameworks along the arrows of the triangle differ across the two lessons. Notably, given our focus on eliciting student reasoning, the difference in the level of student justification from lesson 1 to lesson 2 was a statistically significant increase.


Yet, these frameworks do not work disjointedly, rather, they work in relationship with each other. All frameworks capture a component of the classroom that may reflect or contribute to a culture of justification, and thus, a change in one part of the framework has the potential to affect changes in the others. For example, the difference in teacher questioning toward requests for ideas and elaborations appeared to support students in lesson 2 in deeper engagement with the content and each other, which may account for the increase in justification contributions. It was by examining both the individual components and then the whole of the lessons that we gained a more complete picture of the actions and interactions that supported more high-quality mathematics instruction in the second lesson than the first.


It must be noted that we intentionally selected the component frameworks for this analysis based on extant literature documenting elements of high-quality mathematics instruction with a focus on a classroom culture that is characterized by students engaging in justifying (e.g., Ball, 1993; Jacobs & Spangler, 2017; Nasir & Cobb, 2006; Schoenfeld, 2011; Turner et al., 2013). We found these frameworks to be fruitful in carefully analyzing these two case study lessons. However, analyzing different classrooms with different characteristics could provide more insight into the complexities of the teacher-student(s)-content instructional relationships and, particularly, how these varying frameworks may work in concert with one another and in tandem with the instructional triangle. Moreover, we recognize that these five frameworks are not the only elements of high-quality mathematics instruction, and that there may be other frameworks that could be networked using the instructional triangle to explore other relationships in the teacher, student(s), and content connections.


Transferring existing neuropsychological instruments to new settings, particularly resource-limited settings, is not a simple task. In addition to cultural differences in skill sets, differences in and even a total lack of formal education can be factors especially in rural settings where lack of familiarity with writing instruments, much less computers, place limitations on assessments. Familiar stimuli in Western cultures such as subways, escalators, and even certain foods and animals, are unfamiliar and inappropriate for use in testing people in other cultures. Visual, auditory, and other stimuli therefore, may need to be redesigned for use in a particular setting. In settings where other alphabets are standard for example, the Trail Making Test has been replaced with the Color Trails Test which removed the English alphabet from the instrument. Investigators at the University of Miami altered word lists and phrases in the instructions and certain passages of six verbal learning, memory, and fluency instruments in an attempt to make these instruments valid for use in Hispanic populations (Wilkie et al. 2004).


One of the most challenging terms for professional educators is 'test.' Even seasoned instructors may not always feel at ease with putting a grade or a mark on a student's final paper. If an entire class does well, the instructor feels proud that work has been accomplished; however, if a large number of students do not perform well, instructors are disappointed and sometimes need to reevaluate the objectives of the entire course. Certainly, students show signs of stress and anxiety before exam periods. Most of us may recall the hollow feeling in our own stomachs the minute just before a test was distributed as well as the silence in the classroom when instructors handed back the corrected papers.


Instructors and curriculum designers today seem to be convinced that a more learner-centered, creative and flexible teaching system motivates students. They also see the necessity to adapt testing methods to the revised curricula and methodologies. Peer correction, self- and portfolio evaluation are becoming common in even the most traditional university settings. Instructors who emphasize a communicative type of testing may promote a more efficient learning environment. They certainly contribute to making tests less traumatic. Nevertheless, it seems that the instructor's testing methods do have a lasting effect on the learning experience, the students' attitude as well as the teacher's enthusiasm (Schmidt & McCutcheon, 1994:118.) Traditional testing is still a critical aspect of education; research in North America has shown that students who take frequent instructor-developed assessments scored higher on national tests (Linking Effective Teaching to Test Scores, 2001). This may be the case not only because of the value of testing, but because the tests are well thought out and allow students to apply the process of what was learned in class as well as the content of the instruction. Instructors must not overlook the importance of student motivation to do well as one important factor in the success of testing. In one survey, students themselves requested numerous quizzes and tests--testimony to the critical role testing plays in a university setting (Kfouri, 2003).


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