STATISTICAL KNOWLEDGE FOR TEACHING: ELEMENTARY PRESERVICE TEACHERS

INTRODUCTION What is the statistical knowledge needed for teaching in grades K-8? Burgess (2009) notes, "There is an extensive research literature on teacher knowledge needed for teaching mathematics. Statistics education research has a much shorter history and its literature pertaining to teacher knowledge is relatively scarce" (p. 18). Further, Sanchez, Silva and Coutinho (2011) believe there is an urgent need for studies of teachers' professional knowledge for teaching variation. Building upon recent work with middle school students in exploring measures of center and given the American Statistical Association's (ASA) endorsement of the Guidelines for Assessment and Instruction in Statistics Education (GAISE) report FRAMEWORKS In his paper conceptualizing statistical knowledge for teaching (SKT), Groth (2007) indicated there is sufficient overlap from the work describing the mathematical knowledge for teaching to inform the study of SKT. With this in mind, Groth (2012) presented a model for investigating SKT based on Ball, Thames, and Phelps' (2008) model of Mathematical Knowledge for Teaching. Our primary concern has focused on the aspects of the model that pertain to subject matter knowledge. Formulate questions II. Collect data III. Analyze data IV. Interpret results V. Nature of variability VI. Focus on variability The levels A, B, and C are intended to show what appropriate grade-level understandings are for K-5, 6-8, and 9-12 respectively. However, as in the van Hiele levels for geometry (van Hiele, 1959), it is quite often the case that students do not exhibit understandings appropriate to their age group due to a lack of experience with statistics. Thus, when reflecting on our elementary PSTs' performance at the outset of the class we found a non-trivial portion of our classes performing at level A. Our PSTs have some statistical background, but it is chiefly based on a few high school course units related to statistics embedded within other mathematics courses, such as algebra. It is not typical for our elementary PSTs to have successfully completed an AP statistics course in their high school program. Teaching probability and statistics to elementary and middle school children requires all three aspects of subject matter knowledge which suggests, given HCK, that PSTs should have a minimal requirement of performing within level C, preferably surpassing it. We endeavored to design tasks to move the PSTs' understanding towards that level. CONTEXT The context of our study was the course, Probability and Statistics for Elementary/Middle School Teachers, a content course designed for future elementary/middle school teachers. It is one of three required mathematics content courses for all undergraduates in elementary education at a doctoral-granting university in the U.S. Midwest. The class of 12 students was taught during the seven-week Summer term in 2013, with it meeting four days a week for 100 minutes. The technology environment in which the class was held included several interactive whiteboards (IWBs) that were utilized throughout the term. Students were expected to bring their laptop computers and graphing calculators to each session. The Web 2.0 tool Google Documents was used daily. The instructor (one of the co-authors) would post the content focus of each session, course announcements, and indicate which student would write up a summary of key ideas for that day's session on these documents. PSTs would use the same Google document to raise homework questions with their classmates. PSTs were encouraged to respond to the posted questions at the beginning of class on the Google document itself; those that were left with no response were used as initial class discussion points. PSTs were assigned to write daily session summaries and these were posted on a class Wikispaces page. The initial redesign of current tasks, the incorporation of new tasks, and the sequencing of topics were conducted in order to study what tasks may facilitate the development of the PSTs' statistical knowledge for teaching, and to begin to study the learning trajectories of major statistical concepts. Some of the newly created tasks evolved around the use of TinkerPlots® ICOTS9 (2014) Invited Paper -Refereed Browning, Goss & Smith -2 - METHODS The work described here is a part of a larger design experiment conducted throughout the semester consisting of cycles and iterations of developing rich tasks that made strategic use of digital tools to engage the PSTs in statistical and probabilistic reasoning. The intent of the design experiment To help understand the impact of the designed tasks on student understanding, we collected four different types of data on the thinking of the PTSs. First were the instructor's lesson notes created in preparation for class and then modified afterwards. The second data source included classroom observations where field notes were taken during the implementation of the tasks. The third was the class work that PSTs generated consisting of assignments, quizzes, exams, and projects. Much of the time these work items had a technological tool component. This digital work was collected through Dropbox, software that allows one to store data and provide access to many people at one time. The final pieces of data on the PSTs' thinking were student interviews. The PSTs (who volunteered) participated in a 20-30 minute interview with one of the researchers shortly following the completion of the course. The purpose behind collection of the interview data was to provide PSTs with the opportunity to explain their thinking on particular tasks and to comment on affordances and constraints of the tasks and technological tools utilized in the course. With regard to data analysis, this paper presents preliminary findings as we are only in the beginning stages of analysis. The qualitative analysis of the data will include in-depth descriptions of coursework, analysis of themes within individual PSTs' work, analysis of themes across their work, and interpretation of these themes. A constant comparative method, where multiple researchers read, code critical points, discuss, and refine codes, will be used to reduce threats to validity BACKGROUND AND DESCRIPTION OF TASKS PSTs had been involved in several small in-class "statistical investigations" that made use of the first four process components outlined in the GAISE document As an initial step in one of these cycles, PSTs explored data plots initially using the divider tools in TinkerPlots®. The use of the divider tools was not prescribed, but was a student-generated idea originating through exploration within TinkerPlots®. Divider tools allowed the PSTs to divide the data set with visually shaded regions, either by equal count or equal width. The equal count option set a stage for discussing a box plot. While discussing features of the box plot, PSTs were presented with two sets of data where the mean, median, and mode of both sets were identical, yet the data had very different distributions. Box plots of the two data sets nicely presented these different distributions by accentuating some clear visual differences. The idea of the median was shown to be a key feature in a box plot, with PSTs finding the median of a data set and then upper and lower "medians" for the upper and lower halves of the data. While discussing how the data related to the median in the box plot, PSTs were asked if perhaps another measure of spread other than the median could be used to provide "differences from a center". They chose the mean and so we began to investigate the idea of the mean absolute deviation