Category Archives: Education

Blogs related to education.

Pilot Test

So I have finally decided to go with a qualitative narrative for my pilot study on how assessment and instruction interact to shape student attitudes. I found a class to sample this summer. I am planning to just watch the class in a few hours to get an idea of how the class operates as far as student and teacher interaction. I will later meet with the teacher to get an idea of how he believes assessment and instruction interact in his class. I will also see how he believes these two things shape student attitudes. At the end of the class, I plan to introduce myself and my research question. I’m hoping for volunteers who will be willing to tell me about a 30 minute story regarding this interaction and their experience in this and other courses. I suppose if I constrain them to this class, I will have a type of case study; however, I hate to refrain students from sharing how their attitudes have been shaped. For this reason, I’m not going to refrain students from discussing other classes.


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?

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.

Mixed Ability Grouping – Reading Research

The national cry for charter and magnet schools is becoming larger and quite overwhelming in many areas. The background for this public support is largely based on what is seen as strong public education for students. The introduction of these type schools however often tends to create many equity issues and frequently aggregate students according to ability in specific geographic regions. Research by Lincheuvski and Kutscher (2002) focused on the impact of aggregating students according to ability within schools largely from a quantitative standpoint. This research found that not only does mixed ability grouping increase low and middle level ability students, but does little to support the achievement of higher ability students (Lincheuvski and Kutscher, 2002).
Lincheuvski and Kutscher (2002) focused largely on three studies that pertain to student achievement and teacher attitudes toward mixed ability grouping. Evidence before this time of the study suggested that grouping students according to ability had significant effects on increasing the differences found between students before heterogeneous grouping (Lincheuvski and Kutscher, 2002). Research prior to Lincheuvski and Kutscher (2002) also suggested that mixed ability grouping. The expansion by Lincheuviski and Kutscher (2002) focused on whether this gap was created by the increase of high ability students or the decrease in low ability students and whether this gap can be avoided by mixed ability grouping. Lincheuviski and Kutscher (2002) also explained the relationship between prior research on low expectations of students in low ability groups and low quality teaching. This relationship lead Lincheuviski and Kutscher (2002) to their third study based on the relationship of teacher beliefs of mixed ability grouping, professional development, and ability to teach mixed ability groups affectively.
Evidence from the two studies by Lincheuviski and Kutscher (2002) on student achievement suggested that mixed ability grouping did little to affect achievement of high ability students. In fact, only two schools of twelve had any significant affect by placement of students in homogenous settings when prior ability was adjusted (Lincheuvski and Kutscher, 2002). High ability students from homogenous settings scored slightly higher than their counterparts in heterogeneous settings, but not significantly different (Lincheuvski and Kutscher, 2002). Larger light was shed on the positive impacts of middle and low level abilities students’ effects on heterogeneous grouping (Lincheuvski and Kutscher, 2002). Students in these mixed ability group settings scored significantly higher than their homogenous counterparts (Lincheuvski and Kutscher, 2002). These findings suggest that ability differences found in previous studies are more related to a decrease in ability of low and middle ability students than in the increase of high ability students (Lincheuvski and Kutscher, 2002). These findings should carry heavy weight for those who choose to continue on the path of tracking and streamlining students. School policies such as these tend to increase ability gaps not by increasing your best students’ knowledge, but by decreasing or lessening opportunities for lower ability students.
Previous research by Dar (as cited in Lincheuviski and Kutscher, 2002) on teachers’ attitudes show that prior experience with heterogeneous settings improves teacher’s attitude of effectiveness and impact of students. Using surveys, Lincheuviski and Kutscher (2002), attempted to measure support for heterogenetic classroom settings and workshop contributions to attitude and practice. Results from their study showed that the impact of workshops on the teaching of heterogeneous classrooms positively affected teachers’ attitudes (Lincheuvski and Kutscher, 2002). In fact, the more teachers were involved in the program the larger the impact of their attitude and effect of teaching mixed ability classrooms (Lincheuvski and Kutscher, 2002). Teachers in the study also expressed interest as well as concern for the continuing chances to develop and discuss the teaching of mixed ability classrooms because of fundamental changes that are required in instructional methods (Lincheuvski and Kutscher, 2002). Professional development for mixed ability classrooms provided strong tools and discussions that teachers would not have had opportunities to acquire, foster, and develop otherwise (Lincheuvski and Kutscher, 2002).
Implications for school systems from this research are imperative to today’s classrooms. We can no longer hold to the idea that we are intending to improve education for all by tracking our students into separate modified classrooms. Silver (1990) discussed that the relationship between research, policy, and practice should be bidirectional. Research has shown us that tracking and streamlining students fosters ability gaps between students. Not only this, but these classrooms do little to improve the achievement of our best students and devastate the achievement of lower and middle achieving students (Lincheuvski and Kutscher, 2002). Our service to the community in which we serve as teachers should be to develop each child to his or her full potential. Affective mixed ability settings with guided mentorship during the teaching of these classes fosters a community of successful students and not cluster of mathematically proficient school and community population. It is now our turn as a practitioners to direct questions to researchers about or incorporation of sound research on mixed ability classrooms.

Linchevski, L. & Kutscher, B. (2002). Tell me with whom you’re learning, and I’ll tell you how much you’ve learned: Mixed-ability versus same-ability grouping in mathematics. In J. Sowder & B. Schappelle (Eds.), Lessons learned from research (pp. 68-82). Reston, VA: National Council of Teachers of Mathematics.
Silver, E. A. (1990). Contributions of research to practice: Applying findings, methods, and perspectives. In T. J. Cooney & C. R. Hirsch (Eds.), Teaching and learning mathematic in the 1990’s (pp 1-11). Reston, VA: National Council of Teachers of Mathematics.

Moving Beyond Analyzing to Engineering Change in Mathematics Education

Discussions throughout education, government, and media often discuss the achievement gap between white, black, and Lationo/a students.  Many times these discussions focus around what is wrong with education, but do little to offer suggestions for closing the achievement gaps between these groups of students (Rousseau & Tate, 2007).  In order to engineer change in mathematics, the first objective for a district, school system, school, or teacher is to believe that all students are capable of learning, doing, and understanding advanced mathematics.  This shift in thought can be found within the mathematics education community and nation through documents from the National Council for Teachers of Mathematics standards documents and the passing of the No Child Left Behind Act of 2001 (Rousseau & Tate, 2007).

Trends in mathematical achievement for minority and white students have increased every year from 1973 to 1999 which is very promising; however, significant differences between student achievements still exist.  Along with these trends of achievement come trends of how the United States population is changing at the same time (Rousseau & Tate, 2007).  Where white students were once a large majority, 78% in 1972, of the population this number has shifted to a majority by 58% (Rousseau & Tate, 2007).  This shift comes largely in light of an increasing immigrant population and a decreasing white population (Rousseau & Tate, 2007).  Though this analysis of the current state of the nation is where much effort and discussion stops, it is important for the state of the nation, these increasing populations, and mathematics as a whole to engineer change in students learning of mathematics.

In order for districts to adequately address factors that may contribute to student achievement such as race, ethnicity, socio-economic status, etc. there needs to be a shift in student progress monitoring (Rousseau & Tate, 2007).  Many districts rely solely on standardized testing results that are many times disseminated after policies and decisions have been carried out by school systems (Rousseau & Tate, 2007).  District leaders should also be able to aggregate data in order to implement and monitor changes within their districts which is extremely overlooked throughout the nation (Rousseau & Tate, 2007).  Aggregation of student data is important to understand if differences exist and if program implementations foster closing these differences.

Policy decisions made at district levels also do much to control the opportunity for students to learn mathematics, such as quality, time, and design (Rousseau & Tate, 2007).  Tracking policies within school systems are many times justified as a means to promote growth of students who are behind and accelerate those that are ahead; however, research has showed and is showing that these practices do little to foster either (Rousseau & Tate, 2007).  Decisions such as this jeopardize the quality of mathematics instruction that is being given to different students (Rousseau & Tate, 2007). 

Districts serving large minority populations often have large teacher turnover rates (Rousseau & Tate, 2007).   Teachers serving in these schools are many times emergency hires who lack teacher certification or new teachers with very little experience (Rousseau & Tate, 2007).  Teachers who teach in these areas for extended times often seek transfers to more affluent areas in their districts starting the process over for schools serving minority students in essence jeopardizing the quality of instruction within these schools (Rousseau & Tate, 2007).  With these issues in mind, schools serving large percentages of minority students are encouraged to create curriculum materials that are easy to implement, understand, and rigorous while at the same time providing resources and professional development that seamlessly integrate with the curriculum (Rousseau & Tate, 2007).  Providing this for new teachers and under qualified teachers can promote better teaching with less planning time (Rousseau & Tate, 2007). Constructs such as these highlight opportunity to learn (OTL) variables such as content exposure and emphasis variables (Rousseau & Tate, 2007).  Districts and states should look more deeply into ways to retain teachers in minority school districts to promote OTL variables such as quality of instructional delivery (Rousseau & Tate, 2007).

The quality of teachers in a school provides correlates strongly with student achievement (Rousseau & Tate, 2007).  High performing schools spend more time in mathematical discourse, provide ongoing professional development, design rigorous curriculums with high expectations of all students, and address underachievement of low performing students (Rousseau & Tate, 2007).  Ongoing, quality professional development is key to helping teachers undergo fundamental changes in their pedagogy (Rousseau & Tate, 2007).  The creation of curriculum materials and opportunities for collaboration between teachers, researchers, and master teachers that focuses on student understanding like the project IMPACT, MARS, and QUASAR can have significant increases in student achievement and teacher quality (Rousseau & Tate, 2007). 

Though increases have been seen in relationship to student achievement across racial groups, there is still much room for improvement (Rousseau & Tate, 2007).  Shifting focus from what is the problem to what can be done to fix the problem can be paramount for those students traditionally underserved.  I believe the achievement gaps prevalent in today’s literature such as controlled for by race, ethnicity, SES, parental achievement, etc. are just easy categories that researchers use to aggregate students test scores.  I believe if an in-depth qualitative analysis of students’ culture could control for practically all of these typically aggregated fields.  I believe this link would center around what is valued and expected in particular households and classrooms.

Tate, W. & Rousseau, C. (2007).  Engineering change in mathematics education. In F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 1209-1246). Charlotte, NC: Information Age.