Karl W. Kosko

Mathematical Knowledge for Teaching and its Specificity to High School Geometry Instruction

This chapter documents efforts to develop an instrument to measure mathematical knowledge for teaching high school geometry (MKT-G). We report on the process of developing and piloting questions that purported to measure various domains of MKT-G. Scores on a piloted set of items had no statistical relationship with total years of experience teaching, but all domain scores were found to have statistically significant correlations with years of experience teaching high school geometry. Other interesting relationships regarding teachers’ MKT-G scores are also reported. We use these results to propose a way of conceptualizing how instruction-specific considerations might matter in the design of MKT items. In particular, we propose that the instructional situations that are customary to a course of studies can be seen as units that organize much of the mathematical knowledge for teaching such a course.

An Analysis of Evaluative Comments in Teachers' Online Discussions of Representations of Practice.

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 evaluation deters from the development of a professional discourse. In this study, we inspected teachers’ online discussions of animations of classroom episodes realized with cartoon characters, looking at the difference in the content of conversation turns when members made evaluative comments and when they did not make evaluative comments. We were interested in finding out whether making evaluative comments correlated with participants’ reflection on their professional practice and proposal of alternative teaching actions; for that purpose we used systemic functional linguistics (SFL) to develop a coding scheme that attended to evaluation, alternatives, and reflection in forum discussions. We found statistically significant evidence that the more the participants actively evaluated the teaching in the animations, the more they proposed alternative teaching actions and reflected on instructional practice. We relate these findings to the notion of social presence in online discussions.

The Effect of Student Discussion Frequency on Fifth-Grade Students’ Mathematics Achievement in U.S. Schools

Student discussion about mathematics has been said to improve the mathematical understanding of students. Yet, some studies suggest that this may not be the case (i.e., Shouse, 2001). Therefore, the authors used hierarchical linear modeling to investigate the effect frequent discussion has on math achievement. Findings showed no statistically significant mean difference in math achievement between weekly and less than weekly discussion in fifth-grade mathematics classrooms, but significant variability in the effect of discussion on achievement was found across classrooms and schools. This indicates that weekly discussion can have large positive effects in some classrooms/schools and negative effects in others. These results and their implications are discussed in context with current literature.

Students’ Quality of Mathematical Discussion and Their Self-Determination in Mathematics

Abstract Mathematical discussion allows for students to reflect upon math concepts and understand such concepts at a deeper level. This process of reflection requires a certain amount of internalization on the part of the student. This internalization is facilitated by meeting the needs of autonomy, competence, and relatedness as advocated by Self-Determination Theory. The current study provides evidence of a relationship between fulfillment of these psychological needs and the quality of mathematical discussion students report they engage in. Correlational analyses and structural equation modeling of data from 176 high school Geometry students were conducted to examine this relationship. Results support the claims of a connection between fulfillment of students’ autonomy, competence, and relatedness and their reported engagement in mathematical discussion.

Communicating Quantitative Literacy: An Examination of Open-Ended Assessment Items in TIMSS, NALS, IALS, and PISA

Quantitative Literacy (QL) has been described as the skill set an individual uses when interacting with the world in a quantitative manner. A necessary component of this interaction is communication. To this end, assessments of QL have included open-ended items as a means of including communicative aspects of QL. The present study sought to examine whether such open-ended items typically measured aspects of quantitative communication, as compared to mathematical communication, or mathematical skills. We focused on public-released items and rubrics from four of the most widely referenced assessments: the Third International Mathematics and Science Study (TIMSS-95): the National Adult Literacy Survey (NALS; now the National Assessment of Adult Literacy, NAAL) in 1985 and 1992, the International Adult Literacy Skills (IALS) beginning in 1994; and the Program for International Student Assessment (PISA) beginning in 2000. We found that open-ended item rubrics in these QL assessments showed a strong tendency to assess answeronly responses. Therefore, while some open-ended items may have required certain levels of quantitative reasoning to find a solution, it is the solution rather than the reasoning that was often assessed.

Mathematical Communication in State Standards Before the Common Core

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 communication in 2009 state standards documents was in regard to National Assessment of Educational Progress (NAEP) 2009 achievement scores for Grade 4. Analysis suggests mixed results with potential implications as states move toward fully implementing the Common Core State Standards in Mathematics. Specifically, although including language requiring mathematical descriptions from students had a positive effect on Grade 4 NAEP 2009 achievement scores, including language requiring rationales and justifications was not found to have a statistically significant effect.

Emergence of argument in children’s mathematical writing

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 practice is mathematical writing. This requires students to develop mathematical writing that demonstrates careful analysis, a command of sequence, and a level of detail considered fundamental for constructing effective argumentative, persuasive and informative mathematical explanations. However, there is currently little to no research examining how mathematical writing develops in elementary grades. The present study examined K-3 students’ mathematical writing using modified Piagetian tasks. Incorporating elements of Toulmin’s argumentation scheme, a set of classifications for mathematical writing emerged from K-3 student samples. Further, these classifications are sequential, with strong statistical correlations associated with children’s grade levels. The findings indicate a potentially useful set of classification schemes for identifying children’s writing and examining how such writing develops in early grades.

Going beyond Numbers to Complicate the Given Information

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