My research interests over the past few years have varied.  However, research and scholarly activity allows me to reflect upon how I teach as well as how students learn.  Primarily my research interests fall within the realm of mathematics/statistics education.  However, I have had some research interests that lie beyond mathematics/statistics education.  On this page, you will find some information about my current projects as well as some past projects.

Mathematics/Statistics Education

My current research in mathematics/statistics education entails multiple projects. 

My primary research (i.e. my dissertation) focuses on conceiving of the distribution of a random variable as the accumulation of outcomes of a random process with respect to the value of the random variable under study. Through my research I'm attempting to identify progress measures that I can use to inform a hypothetical learning trajectory for my proposed conception of the distribution of a random variable. Potential sites for progress measures include students' conceptions of the following: a trial/case, a random process, randomness and independence, a random variable, accumulation, and probability. As my research progresses, I will make as much available as I can here.

I made the following animation to show how I think about Statistics.

For the past five years, I've worked with Dr. Pat Thompson on Project Aspire (NSF funded grant) with several other colleagues. The aim of the project is to develop two instruments centered on secondary high school teachers' mathematical meanings for teaching. We intend for the first instrument (called the MMTsm) to allow insight into the nature of the mathematical meanings that teacher has for different mathematical concepts within the algebra/pre-caluclus secondary curriculum. The mathematical areas that we examine in the MMTsm include: co-variation (including variation and variables), functions (functions, functions as models, and function notation), frames of reference, rate of change, proportionality, reasoning with magnitudes, and representational equivalence (structure sense).  The second instrument is an extension of the Instructional Quality Assessment with the aim to help us capture mathematical meanings that a teacher presents during the course of instruction. While I work on many of the different aspects of the project, I spend a fair amount of time looking at teachers' meanings and use of syntactic structure within mathematics. (Also on the grant are Dr. Marilyn Carlson, Dr. Mark Wilson (Berkeley), Karen Draney (Berkeley), Dr. Stacy Musgrave (post-doc), and fellow graduate students Cameron Byerley, Hyunkyoung Yoon, and Surani Joshua.)

Within the context of Project Aspire, I'm involved in examining areas dealing with representational equivalence, functions (in particular function notation), and frames of reference. While these areas seem very different from my primary research area of statistics education, I find that paying attention to students' conceptions within each of these areas is of vital importance in statistics education. Given that statistics are functions from $\mathbb{R}^n\rightarrow\mathbb{R}$, how students understand and think about functions and function notation becomes central starting with descriptive statistics. Recognizing that the sample arithmetic mean is a function and not a value is often glossed over in many introductory statistics courses. Representational equivalence (an aspect of structure sense) plays roles in introductory statistics courses, upper-division and graduate-level calculus based courses. The two sides of representational equivalence, equality-preserving transformations and the substitution principle, assist student in parsing and building meaning for expressions as well as recognizing potential transformations (e.g. recognizing the kernel of a particular probability density function). Finally, how students' think within frames of reference plays an role in hypothesis testing and design of experiments. Any time use a test involving differences, students must commit to thinking within a frame of reference that way the values of the differences for the observations all have consistent and coherent meanings. Additionally, students must reconcile one-tailed hypothesis with the frame of reference they constructed when dealing with the data.

Consider the following example: A researcher collected $n$ observations on the same attribute for two populations, say $X$ and $Y$. The researcher has reason to believe that for some parameter of interest, say $\theta$, the value of this parameter will be greater for the $X$ population than the $Y$ population. Thus, the researcher tests $H_0:\theta_X =\theta_Y$ vs. $H_1: \theta_X>\theta_Y$. If the researcher thinks with the frame of reference where $Y-X$, then the researcher will end up doing a "lower-tail test". However, if the researcher thinks within an alternate frame of reference where $X-Y$, then the researcher does an "upper-tail test". The reconciliation between a hypothesis and how the student/researcher thinks in terms of a frame of reference becomes important area that is often left out of statistics courses.

Communication and Mathematics

I've found myself drawn to the field of Communication time and time again. This partially stems from my experiences in competing in forensics during my undergraduate career. I've dabbled with two lines of research dealing with communication and mathematics; reading/writing and discussion groups.

Reading and Writing

Early in my career as a mathematics education researcher, I found myself drawn to not only researching how we teach mathematics, but also how we teach mathematics communication skills.  (This is probably where my speech geek comes out to play.)  I believe that it is well-worth our time and effort as educators to study how we communicate as mathematicians and how we can get students of all levels to communicate mathematically and effectively.  We should not limit ourselves to only working on getting mathematics majors to improve their mathematical communication skills, but also work on getting all students to improve their mathematical communication skills.

To this end, I have began my research career by looking into the blending of mathematics pedagogy with reading instruction, writing instruction, and reading and writing instruction.  This gave birth to the research project entitled "Investigation into Mathematics Teaching Strategies Blending Reading and Writing" that I conducted in Spring 2010.  This study worked with developmental, undergraduate students at a medium-sized Midwest university.  I spoke on the theoretical underpinnings and some preliminary results stemming from this research at the 2010 Missouri Section meeting of MAA.  I then presented the final results from this project at the 2011 NCTM Annual Meeting and Exposition in Indianapolis, IN.

I learned a significant amount from this project; particularly about getting students to read mathematics effectively.  I have continued with this idea during the Fall 2010 semester to see what developed further.  This research project helped me discover that I am truly passionate about mathematics education research.  I enjoyed it.

Discussion Groups

In the spring of 2011, I began to study the effect of weekly discussion of the mathematics being presented to undergraduate students.  I was curious about the effect of the number of "expert presenters" that were present during the discussion.  Based off of an initial wave of research, I initiated a research project based off the concept of Literature Circles (as framed by Harvey Daniels).  I worked with a developmental mathematics class (Intermediate Algebra).  The research project looked at a control group as well as three discussion groups.  There was one discussion group that had two experts, one group that had three experts and one discussion group that had just one expert. Data collection finished in May 2011.  However, starting my Ph.D. program has put a hold on this research project.  I do intend to return to this project and conduct quantitative and qualitative analysis on the collected data.