RDP - Final Statement of Interest

At the beginning of the semester, I described my three related areas of interest using the following research questions:

• How do students think about mathematics concepts when engaging with dynamic representations and manipulatives?
• How might real-time feedback on curricular adaptations affect the instructional planning decisions of elementary-school mathematics teachers?
• How does integration of computational thinking concepts into mathematics change the way teachers and students approach mathematics tasks?

To organize my thinking and work over the course of the semester, I used a Venn diagram to document articles related to each area of interest, key search terms, important scholars, journals, and organizations, and research that I felt represented intersections between my interests. The resulting diagram is shown below. At the top left is content related to my first question above, at the top right is content related to my second question, and at the bottom is content related to my third question.

In this ending statement of interest, I will organize my discussion around my three original areas of interest, describing how each has changed and become more focused as a result of reading related literature and discussing it with colleagues and advisors. I will conclude with a brief discussion of a couple of interesting areas of overlap among the three topics I discovered this semester and how and to what extent I plan to manage pursuit of all three areas during my years of graduate study and later in my career.

Changes and New Foci in Each Area of Interest

Students' Use of Digital Tools. I first became interested in this area as part of the curriculum development work I did prior to coming to graduate school. As part of this work, I developed physical/print and digital versions of the same curriculum resources, and became very interested in the ways that the experience of each would be different for students. When I first starting exploring this idea in the research literature, I found it rather strange that there were not more experimental or quasi-experimental studies pitting physical versus digital manipulatives against each other in terms of aiding student learning or any other outcome of interest. It seemed like low-hanging fruit. A meta-analysis by Moyer-Packenham and Westenskow (2013) helped me understand why this might be. Moyer-Packenham and Westenskow looked not only at effects of virtual manipulatives over physical versions, but also at the particular affordances of the virtual manipulatives that seemed to be linked to positive learning outcomes. They identified five particular affordances that seemed to be beneficial: focused constraint, creativity, linked representations, precise representations, and motivation. I will note that I had difficulty thinking of creativity and motivation as affordances of manipulatives, but even so, this study helped me realize that comparisons between physical and digital manipulatives may be hard to interpret given the wide variation in the particular affordances of each.

Along the same lines, a study by Anderson-Pence and Moyer-Packenham (2016) also aided my thinking about studies of digital manipulatives. Anderson-Pence and Moyer-Packenham studied how students' use of three types of virtual manipulatives (linked, pictorial, and tutorial) related to the kinds of discourse that occurred in small groups, with particular attention to generalization, justification, and collaboration. The kinds of discourse differed by manipulative type. This study helped me to realize that specificity in outcome might be just as important as specificity in affordances of the manipulatives when designing a study.

Thus, while my original research area for this question was quite general ("How do students think about mathematics concepts when engaging with dynamic representations and manipulatives?"), Moyer-Packenham and Westenskow's (2013) study and Anderson-Pence and Moyer-Packenham's (2016) study helped me realize that one way to focus this question would be to choose particular affordances of the manipulatives (more specific than just the fact that they were digital) and perhaps also to choose particular outcomes (more specific than just increased learning). My new question to investigate further in this area would be something like the following:

• How does students' use of manipulatives with linked digital representations support them in making mathematical generalizations?

Teachers' Use of Digital Mathematics Curriculum Materials. My original statement of the question related to teachers' use of curriculum materials ("How might real-time feedback on curricular adaptations affect the instructional planning decisions of elementary-school mathematics teachers?") reflected my background as a designer of such materials. It was framed as an exploration of the effect on teachers of a feature -- real-time feedback -- that a designer might incorporate into a curriculum resource. As I've read more literature related to curriculum enactment, however, it's become clear to me that conceptualizing teacher behavior as a dependent variable is problematic. In particular, a literature review article by Remillard (2005), which outlines a framework for the study of curriculum enactment, emphasizes that the relationship between teachers and curriculum materials is bi-directional. Teachers use curriculum resources to guide their instruction, but teacher characteristics also influence the way the resources are read and interpreted, and therefore the effects the materials have on students.

Inspired by Remillard's (2005) piece, I read and synthesized several more articles that explored the teacher-curriculum relationship, specifically studies of teachers' use of digital curriculum resources. Interestingly, the synthesis revealed that teachers' perception of their role in the design of a digital curriculum resource may influence whether or not particular features of the resources are used in instruction. In Hansen, Mavrikis, and Geraniou's (2016) study of teachers' use of a fractions manipulative they co-designed with researchers, teachers reported that the dynamic representation embedded in the manipulative significantly influenced their instruction. Similarly, in Hoyles, Noss, Vahey, and Roschelle's (2013) study of teachers' use of a curriculum deliberately designed to be flexible, teachers chose instructional strategies that took advantage of the technology features. By contrast, in Drijvers, Tacoma, Besamusca, Doorman, and Boon's (2013) study of teachers' use of a digital curriculum resource, teachers' instructional strategies did not change from those they used with print resources. A key difference in these studies is that in the teachers in the former two studies understood themselves to be partially responsible for the design of the resource. In Drijvers et al.'s (2013) study, they did not.

This contrast has piqued my interest in processes of co-design of digital curriculum resources involving teachers and curriculum developers or researchers. As such, I would rephrase my original question about this topic as follows:

• How does teachers' involvement in the design of digital curriculum resources relate to the ways in which they enact those resources?

Integration of Computational Thinking into Elementary Mathematics. My original focus in this area was rather general. After doing some literature searches combining computer science and mathematics as keywords, however, I happened upon a line of work that helped me identify a more specific area of inquiry. Orit Hazzan (Hazzan & Zazkis, 2005) is a scholar of computer science education who has done a number of studies of how students reduce the abstraction level of problems as a coping strategy when they encounter problems they don't understand. For example, students will tend to think about just one case instead of a whole set of cases, presumably because single cases are more concrete. They will also restate an unfamiliar task in familiar terms (sometimes erroneously) in order to make the task tractable. Finally, they will sometimes rely on canonical procedures without attending to the conceptual basis of those procedures or of the problems they are solving. Hazzan began her exploration of this phenomenon with undergraduate computer science students, but then extended her reducing-abstraction framework to study undergraduate mathematics and later school mathematics (mostly middle and high school). She found that students use the same techniques of reducing abstraction to cope with abstract problems in both disciplines.

I was struck by this pattern of reducing abstraction, because I see it as a way to interpret several lines of research in the elementary mathematics space. There is research, for example, showing that elementary students tend to apply whole-number reasoning to problems with fractions, which seems like a case of reformulating unfamiliar problems in familiar ways. Similarly, there is a long line of research showing that students solve word problems simply by picking out the numbers and applying a computation algorithm, without attending to the meaning of the problem. This seems like a case of relying on canonical procedures.

If these are, indeed, examples of reducing abstraction as explained by Hazzan, it seems to me that the pedagogical frameworks computer science researchers have developed to promote abstract thinking (e.g., Muller & Haberman, 2008; Statter & Armoni, 2016) have relevance to addressing these issues. I'm interested in exploring how the frameworks could be adapted for use in elementary school classrooms. In short, Hazzan's line of work on reducing abstraction, and the connections I see with some common and enduring challenges in mathematics, have led me to change my focus in this area to questions such as this:

• How can pedagogical frameworks designed to promote development of abstraction be leveraged to help elementary students productively engage with challenging mathematics topics?

Interesting Intersections

As they were originally conceived, the two top circles on my Venn diagram had two major differences:

• The left circle focused on students, and the right circle on teachers.
• The left circle focused on digital tools and the right circle on comprehensive curriculum materials in a digital format.

Based on my reading of the literature, I've come to realize that the while the first distinction is meaningful from a research perspective, the second distinction may not be. While there are studies that examine teachers' use of comprehensive digital curriculum materials, there are other studies, similar in structure, that examine teachers' use of individual tools. For example, Moyer-Packenham, Salkind, and Bolyard (2008) studied teachers' decisions about when to you physical and digital manipulatives. This realization has led me to believe that research questions related to digital tools could have a teacher component, and the definition of "digital curriculum materials" can include stand-alone tools as well as comprehensive programs. In this way, the lines between the research interests represented in the top two circles of the diagram have become blurred.

Coming to understand the overlap between my interests in computational thinking and in teaching mathematics with technology was a somewhat different process. In the case of the two math-focused circles, understanding the overlap came from reading the work of others. In the case of computational thinking (CT), there is a small amount existing work that examines potential relationships between mathematics and CT, with Hazzan's work on reducing abstraction as the primary example. Even this line of work, however, does not directly relate the use of technology in mathematics education to CT.

To make such a connection, I ended up re-reading some of the mathematics education literature with a specific eye toward abstraction, which Hazzan's work suggests is a strong connection between the disciplines. Interestingly, Moyer-Packenham and Westenskow's (2013) meta-analysis of research on digital manipulatives showed that some of the affordances of digital manipulatives seem to promote mathematical abstraction skills. Thus, explorations of digital manipulatives and abstraction may be a key area of research overlap between use of technology in mathematics education and computational thinking.

While I am a long way from being able to articulate a single, focused research question that sits comfortably in the middle of my Venn diagram, my RDP has helped me articulate a set of research questions that do seem to reside there when considered in concert:

• How do digital manipulatives assist students in developing abstraction skills?
• How does the development of mathematical abstraction skills relate to readiness for computer science -- which relies on abstraction?
• How do teachers consider these ideas when making decisions about what digital mathematical tools to use in the classroom?

I am interested in all the questions I've articulated in this statement: those in the sections discussing my three interests individually and those relating more directly to the overlap between my interests. I hope to explore them all in some capacity during my research career. For the purposes of graduate school, however, I am currently contemplating the prospect of focusing my practicum and dissertation work on the questions that more directly address the ways in which my interests overlap. In this manner, I hope to position myself to be able to explore a wider range of interests later in my career.

References

Anderson-Pence, K., & Moyer-Packenham, P. (2016). The Influence of Different Virtual Manipulative Types of Student-Led Techno-Mathematical Discourse. Journal of Computers in Mathematics and Science Teaching, 35(1), 5-31.

Drijvers, P., Tacoma, S., Besamusca, T., Doorman, M., & Boon, P. (2013). Digital Resources Inviting Changes in Mid-Adopting Teachers' Practices and Orchestrations. ZDM - International Journal on Mathematics Education, 45(7), 987-1001.

Hansen, A., Mavrikis, M., & Geraniou, E. (2016). Supporting teachers' technological pedagogical content knowledge of fractions through co-designing a virtual manipulative. Journal of Mathematics Teacher Education, 19(2-3), 205-226.

Hazzan, O., & Zazkis, R. (2005). Reducing Abstraction: The Case of School Mathematics. Educational Studies in Mathematics, 58(1), 101-119.

Hoyles, C., Noss, R., Vahey, P., & Roschelle, J. (2013). Cornerstone Mathematics: Designing digital technology for teacher adaptation and scaling. ZDM - International Journal on Mathematics Education, 45(7), 1057-1070.

Moyer-Packenham, P., Salkind, G., & Bolyard, J. (2008). Virtual Manipulatives Used by Teachers for Mathematics Instruction: Considering Mathematical, Cognitive, and Pedagogical Fidelity. Contemporary Issues in Technology and Teacher Education, 8(3), 202-218.

Moyer-Packenham, P., & Westenskow, A. (2013). Effects of Virtual Manipulatives on Student Achievement and Mathematics Learning. International Journal of Virtual and Personal Learning Environments, 4(3), 35-50.

Muller, O., & Haberman, B. (2008). Supporting Abstraction Processes in Problem Solving through Pattern-Oriented Instruction. Computer Science Education, 18(3), 187-212.

Remillard, J. (2005). Examining key concepts in research on teachers' use of mathematics curricula. Review of Educational Research, 75(2), 211-246.

Statter, D., & Armoni, M. (2016). Teaching Abstract Thinking in Introduction to Computer Science for 7th Graders. In Proceedings of the 11th Workshop in Primary and Secondary Computing Education - WiPSCE '17 (pp. 80-83). New York: ACM.