Trying on a Ethonograpy and Grounded Theory Lense

My focus is continaully evolving by reading the statistical literature. At one point I want to dive into student understanding of particular concepts in statistics. Other times I want to look more at the general aspect of developing reasoning and the thought process of being a statisitician. When I look at the ethnography lense, I can see how the second might surmise in an ethnographic study very easy. If I were to attempt to use an ethonograpy to understand my classroom, there wouldn’t be a clear question. I would hope by being involved in the class enviornment some of the methodologies and interactions would demonstrate this understanding and relationship. Students would perhaps reflect on the opportunities to learn and how these efforts were either successful or not. The grounded theory approach would probably work very well for my first question. How does a student develop or understand a particular concept in statistics. By posing a question to a group of students and analyzing responses, I could understand the misconceptions and previous knowledge of students in order to determine how to best make more questions of the topic I am addressing. I could resample the same group or a different group of students after reformulating my questions to see if this new methods provides more insights. It seems as if a final product for this would be an understanding of how a student comes to know a particular concept in statistics and perhaps even meaningful tasks to use and develop student understanding.

It seems that with each type of qualitative study, the large difference is what is aquired at the end. In order to determine the type of study that you will use, the researcher should make clear what their intentions are for their research. This relates very much to the generalizability found in quantiative literature and the transferability in the qualitative. How would you like to transfer the information in your reserach for use in others.


Formative Assessment

Though evaluation and assessment are many times used interchangeably, there is a subtle difference. Evaluations are many times viewed as summative, a final product that demonstrates student conception or understanding of a concept. Assessments can be used as a means to evaluate students, but assessments should do more. Assessments should move students forward in their understanding of a concept. Assessments geared toward increasing understanding are commonly referred to as formative assessments. Petit, Zawojewski, and Lobato (2010) describe formative assessment as “a way of thinking about gathering, interpreting, and taking action on evidence of learning by both the teachers and students as learning occurs. (p.68)”
According to Petit, Zawojewski, and Lobato (2010), formative assessments should improve instruction by clarifying and sharing learning intentions and criteria for success, providing feedback that moves learners forward, and activating students as instructional resources for one another. Making expectations and objectives clear can help students attempt to achieve satisfactory results as well as demonstrating how students can achieve these expectations (Petit, Zawojewski, & Lobato, 2010). Teachers who provided feedback to student assessments encourage self-reflection; however, identifying special actions a student can do that can support improvement are even stronger examples of teacher feedback (Petit, Zawojewski, & Lobato, 2010). Mathematical practice standards within the Common Core State Standards for Mathematics (NGA & CCSO, 2010) and the Alabama College and Career Readiness Standards (ASDE, 2010) require teachers to have students critique the reasoning of others, one of Petit, and Zawojewski’s (2010) elements to effective formative assessments. Though these three elements provide an excellent starting point for a teacher to use formative assessments to guide instruction, students can be more active in this process.

Petit, Zawojewski, and Lobato (2010) clarify that formative assessments should activate students as the owners of their learning and engineer effective classroom discussions, questions, and learning tasks that elicit evidence of learning. Activating students as owners of their own learning provide openness (NCTM, 1995) within assessment that is not offered in all mathematical classrooms. Students who complete self-assessment tasks may be asked to create an individual work plan, create problem-solving activities to explore, provide opportunities to evaluate their own performance against their work plan or against jointly created criteria for performance, identify specific problems they are having (Petit, Zawojewski, & Lobato, 2010). The use of assessment portfolios is a good example of this type of formative assessment (Garfield & Chance, 2000). Analyzing discourse within the classroom can provide strong evidence of what is valued within the classroom. Orchestrating productive mathematical discussions is an art that not only provides a means for students to build mathematical power, but also continually assess student understanding in the present(Stein & Smith, 2011). Through monitoring, teachers can provide sequenced explanations that build on each student strengths and help build connections between mathematical concepts (Stein & Smith, 2011).
After incorporating formative assessment in the classroom, it is imperative to know what to look for using the assessment triangle (Petit, Zawojewski, & Lobato, 2010). The assessment triangle’s vertices consist of observation, interpretation, and cognition (Petit, Zawojewski, & Lobato, 2010). The cognition vertex refers to the ways students understand, misrepresent, misunderstand, or use prior knowledge that influences this understanding of a particular concept (Petit, Zawojewski, & Lobato, 2010). The observation vertex refers to the descriptions from research that produces specific responses (Petit, Zawojewski, & Lobato, 2010). Interpretation in formative assessment or in the assessment triangle is the tools and methods used to reason from the evidence (Petit, Zawojewski, & Lobato, 2010). “To develop effective assessments that lead to sound inferences, each corner of the assessment triangle must connect to the other in a significant way regardless of the level of assessment. (Petit, Zawojewski, & Lobato, 2010, p. 70-71)”
Formative assessment is one of the most robust tools teachers can use to improve student learning within the classroom (Petit, Zawojewski, & Lobato, 2010). Though it may seem like more work for a teacher who has not incorporated these strategies in the past, students engaging in self-reflection and critique actually take the burden away from teacher grading and move toward teacher facilitation. Good instruction uses formative assessment in ways that when summative assessment takes place, there is no surprise. It is imperative that teachers use these tools appropriately to improve student understanding within the classroom.

Alabama State Department of Education. (2010). Alabama course of study mathematics: Building mathematical foundations of college and career readiness. Montgomery, AL: Author.
Garfield, J. & Chance, B. (2000). Assessment in statistics education: Issues and challenges. Mathematics Teaching and Learning, 2, pp. 99-125.
National Council of Teachers of Mathematics. (1995). Assessment standards for school mathematics. Reston, VA: Author
National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Common Core State Standards for Mathematics. Washington, DC: Authors.
Petit, M.M., Zawojewski, J.S., & Lobato, J. (2010). Formative assessment in secondary school mathematics classrooms. In J. Lobato (Ed.), Teaching and learning mathematics: Translating research for secondary school teachers (pp. 67-75), Reston, VA: NCTM.
Stein, M. & Smith, M. K. (2011). 5 practices for orchestrating productive mathematical discussions. Reston, VA: NCTM.

Trying on a Phenomenology and Narrative Lens

My focus the last week has been on reading and understanding phenomenology and narrative qualitative research. When I first read about narrative research, I was under the impression that respondents provided data through a narrative and the researcher reported results using a narrative. After reading different studies, this is not always the case. This makes the use of a narrative in my research much more “doable”. Possibly because I’m not much of a wordsmith, but even more largely affected by my personal preference to having results just given and not hidden in interpretation. I always came away after reading a book with a different idea than what my instructor wanted in my college literature courses. Through my current reading and writing for my literature review and dissertation question, I am going to try on two types of questions. The qualitative narrative and phenomenological approach. Through my recent readings of dissertations on IASE and journal articles on JSE, there seems to be little emphasis on how teaching statistics develops students statistical reasoning. Authors have seemingly neglected the role of student connections from their courses to projects in determining how informal and formal reasoning in statistics develop.
Overarching Question: How do pedagogical techniques influence students ability to reason statistically and attitudes?
Possible Narrative Question:
Tell me what you did in class through the year that changed your understanding of statistics.

Possible phenomenological Question:
Explain to me your experience of answering a statistical question.
When have you made a deduction or inference in statistics?

The Beginning of a New Research Era (Qualitative Research)

Those of you who are familiar with me will know that I love numbers. I feel like the world of numbers can bring insights into phenomena in our lives that we can not see or understand. I do feel however that this is not the only process to understand our world of course. Qualitative inquiry and research is an emerging discipline to conduct educational research based largely from anthropologists. I’m really trying to wrap my head around this idea and its implications for my research. How to control for bias within the researcher seems to be much harder to understand even though this bias is articulated in almost all qualitative studies. In most statistical studies this bias is controlled for through instruments that measure quantities; however, surveys have been seen to have bias in their wording.
Moving from generalizability to transferability is definitely a different concept than what I have experienced. From my past experiences I would have said that this is useless, but I’m beginning to feel differently. Even though a study may be of a small sample size or sample a population that doesn’t relate to my life, there could be implications that transfer to my own. I’m looking forward to this approach to research. Looking through qualitative Inquiry and Research Design by John Creswell, I am definitely a postpostitivist and similarly see the quality in phenomenological studies. I’m interested see how my views change as I progress through my PhD in math education.

Is there a difference in it Happening and Making it Happen?

If you’ve noticed, I’ve been writting rather extensively about my geometry class in a the private section of my blog. I’ve really been focusing on fostering reasoning and sense making in the class by promoting and looking at classroom discussion. I teach two of these classes and their classroom discussion dynamics are slightly different. This has really made me think about the difference in the two classes and the ways that I have to really encourage certain aspects of a lesson in one class and the other flows naturally. Today’s lesson one finding the relationship between volume of prisms and pyramids with identical bases and heights went really well; however, it seemed like I really forced some of the same aspects in my second class that flowed naturally in my first. I wonder if the lesson would have been just as good for the kids if I wouldn’t have forced some of the same insights that were found in my first class. Did the insights that they saw really play a critical role in the understanding of the concepts of volume? (No). I believe if I would have allowed students to develop the main concepts more adequately the second go around, this would have helped develop a better environment for student discussion. Students would have began to respect their own ideas and look to one another for clarification and justification rather than myself. Small things like this really play a crictical role in setting up your classroom for good student discourse.

Protected: Persistence

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Protected: Student Statistical Interest

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