Theoretical Framework

Technology possesses increasing sophistication and "multirepresentational capability" (Heid & Blume, 2008, p. 58).  It can affect students' opportunities to engage in conceptualizing, representing, generalizing, working with symbols, and modeling (Heid & Blume, 2008). One of the theoretical lenses through which the impact of technology on mathematics education can be examined is representation. Technology provides visualization, speed, and the opportunity to move seamlessly between different representations (Hennessy, Fung, & Scanlon, 2001). It can also provide an external reference object to facilitate student discourse, helping students make their thinking explicit and encouraging them to clarify their ideas (Hennessy et al., 2001).

Hsieh and Lin (2008) noted that technology helped students build an understanding of the ideas verbal problems were communicating, allowing an internal change in representation from words to symbols or diagrams. Students were able to "internalize prior experience" with the technology and integrate it with existing ideas to create new representations which helped them solve problems (Hsieh & Lin, 2008, p. 227). Such results indicate an emphasis on students' internal representations of mathematics – those mathematical forms which exist within the students' minds (Goldin, 2003). Goldin (2003) noted that research into students' internal representations relies on observations of their interactions with and production of external representations. Smith (2003) noted that inferences can be made about students' abilities to represent and understand mathematics by examining their language about their mathematical representations as well as their attitudes.

Importance of the Work

Technological representations must be considered from a framework of ideas related to internal representations. Duval (2006) noted that understanding the nature of the difficulties that students have with mathematics is particularly important where the goal is to prepare students to face a technological environment of increasing complexity.  Smith (2003) believed that "research that attempts to understand the 'internal representational systems' of children is necessary to understand how representations enable and constrain learning" (p. 273).

Insufficient internal representation can lead to manipulation of external representations without attached meaning (Saul, 2001). Evidence exists that such a mindless view of mathematics as mere manipulation is still prevalent (Roberts & Tayeh, 2011). Multiple representations which model various aspects of a mathematical situation can, however, provide the user with a new, more accurate internal representation (Van Dooren, De Bock, Hessels, Janssens, & Verschaffel, 2004). Technology can provide a dynamic link to multiple representations. Since technology does have the potential to help build sufficient internal representations, then it is well worth the research community's time to carefully examine how related research is being done, what constructs are arising from it, how the work can be advanced, and how it can be translated into practical instructional interventions.

The Work of the Session

Four constructs related to a study linking technology and internal representations will be discussed upon which it is hoped further research and suggestions for practical instructional interventions can be built. Three of the constructs have been found helpful in examining the impact of technology: valid internal representations, useful internal representations, and enduring internal representations. Table 1 provides definitions of those terms.           

Table 1

Interpretive framework for ideas related to internal representations

Internal representations are
Valid if they	Useful if they	Enduring if they
Accurately reflect the mathematics they seek to represent, are flexible enough to allow additional mathematical ideas to be built upon them, are accompanied by sound mathematical habits of mind.	Are accessible for reasoning and sense-making, communicating new mathematical ideas, and building new understanding.	Remain with the student in various situations apart from the environment in which they were initially developed and are carried forward, built upon, and refined over a period of time.

Indicative movements are gestures made along paper or technological representations that provide insight into a student's internal representations of mathematics that are not otherwise evident. The idea was built upon Campbell's (2003) use of dynamic tracking to gain insight into cognitive processes from detailed examinations of students' interactions with technology, including descriptions of mouse movements which helped clarify student thinking. During the first hour of this work session, participants will be introduced to these ideas, provided with examples from the research, and asked to consider the following questions: 

·         How do the ideas of valid, useful, and enduring internal representations relate to other representational ideas with which you are familiar? How might they be built upon to further examine the use of technology?

·         How does the idea of indicative movements relate to your research and classroom experiences? How can it be built upon to further examine the use of technology?

During the final 30 minutes of the session, the participants will be asked to consider the impact of these ideas on classroom practice, particularly on instructional interventions. They will be divided into 3 groups, each of which will consider one of the following sets of questions for 10-15 minutes. Each group will then be permitted 5 - 7 minutes to present their ideas.

·         How can such research into internal representations, which often requires in-depth study of a small number of subjects, be translated into effective classroom practices? How could the constructs described above be made useful for classroom practitioners?

·         How can the use of technology to examine internal representations be implemented in a practical way that allows teachers to gain insight into otherwise hidden misunderstandings? How could such use lead to beneficial instructional interventions?

·         Technology can record students' interactions with software as well as what they say. Campbell (2003) noted that his work “demonstrated that dynamic tracking in the service of . . . investigations [into studying and enhancing teaching] can be carried out in both classroom and clinical settings” (p. 81). How do you envision such work taking place in the future?

References

Campbell, S. R. (2003). Dynamic tracking of elementary preservice teachers' experiences with computer-based mathematics learning environments. Mathematics Education Research Journal, 15(1), 70-82.

Duval, R. (2006). A cognitive analysis of problems of comprehension in a learning of mathematics. Educational Studies in Mathematics, 61(1/2), 103-131.

Goldin, G. A. (2003). Representation in school mathematics: A unifying research perspective. In J. Kilpatrick, W. G. Martin & D. Schifter (Eds.), A research companion to principles and standards for school mathematics (pp. 275-285). Reston, VA: National Council of Teachers of Mathematics.

Heid, M. K., & Blume, G. W. (2008). Algebra and function development. In M. K. Heid & G. W. Blume (Eds.), Research on technology and the teaching and learning of mathematics: Volume 1. Research Syntheses (Vol. 1, pp. 55-108). Charlotte, NC: Information Age Publishing.

Hennessy, S., Fung, P., & Scanlon, E. (2001). The role of the graphic calculator in mediating graphing activity. International Journal of Mathematical Education in Science and Technology, 32(2), 267-290.

Hsieh, C., & Lin, S. (2008). Dynamic visual computer design for factors and multiples word problem learning. International Journal of Mathematical Education in Science & Technology, 39(2), 215-232.

Roberts, S. K., & Tayeh, C. (2011). All These Rays! What's the Point? Mathematics Teaching in the Middle School, 16(7), 408-413.

Saul, M. (2001). Algebra: What are we teaching? . In A. A. Cuoco & F. R. Curcio (Eds.), The roles of representation in school mathematics: 2001 yearbook (pp. 35-43). Reston, VA: National Council of Teachers of Mathematics.

Smith, S. P. (2003). Representation in school mathematics: Children's representations of problems. In J. Kilpatrick, W. G. Martin & D. Schifter (Eds.), A Research Companion to Principles and Standards for School Mathematics (pp. 263-274). Reston, VA: National Council of Teachers of Mathematics.

Van Dooren, W., De Bock, D., Hessels, A., Janssens, D., & Verschaffel, L. (2004). Remedying secondary school students' illusion of linearity: a teaching experiment aiming at conceptual change. Learning & Instruction, 14(5), 485-501.