There have been ongoing discussions amongst educational researchers concerning how teachers can support students in making connections between mathematics topics (The National Council of Teachers of Mathematics, 2000). Conventional instruction, with its a sequential presentation of materials in textbooks and the rote completion of problem sets, often fails to help students develop a deep understanding. This is particularly true in regard to the interconnections amongst mathematical concepts, which often come across to students as completely separate topics (Hiebert, 1984).
In response, working closely with a school math teacher, we co-designed (Penuel, et al., 2007) a curriculum to engage several small groups of students working in parallel as they “tagged” a common set of math problems. In so doing, a collaborative visualization emerged as the curriculum synthesized the combined tags from all groups. A set of thirty problems developed by the teacher belonged to one or more of four category groups: Algebra & Polynomials, Functions & Relations, Trigonometry, and Graphing Functions. The basic goal of this activity was to help students understand the relationships between these four aspects of mathematics by having them visualize the association of math problems with multiple categories.
Within our S3 classroom, students were automatically grouped and placed at one of the room’s visualization displays, and usinglaptops were asked to “tag” (label) a total of 30 questions. Each group’s display showed a graphical visualization of their collective responses. Students were then asked to collaboratively solve their tagged questions and vote and comment on the validity of other groups’ tags. A central display showed a larger real-time aggregate of the all groups’ tags as a collective association of links. As students voted on these tags, agreements resulted in thicker link lines than those that fostered disagreement.
Preliminary findings, while representing only a small number of participants, showed an upward trend of increasing accuracy and structuredness for the experimental condition. The improved accuracy from the pre-test to the curriculum activity and post-test suggests the importance of how we ask students to make connections to problems, with greater accuracy derived from a collaborative design which shares responsibility. The structuredness, which measured students’ recognition of the connections, shows increasing willingness to characterize math problems from different perspectives.
Overall, students found the visualizations useful in showing different mathematical themes from which a problem could be approached. One student indicated that the visualization was helpful when he could not solve a problem. Students also stated that, over time and with more contributors, the system would become increasingly valuable for studying purposes.
Students also commented that they became more cognizant of the connections amongst mathematics ideas and themes. It is noteworthy that students gained awareness that one could discuss properties of math problems and their relevant themes rather than simply answer them.
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