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Editors' note: This column is the first in a regular series to appear in JOMA about reuse and interoperability of mathematical tools.]

Jeremy Roschelle and Chris DiGiano are at the Center for Teaching and Learning, home of the former ESCOT project. Victoria Hand was also a member of the ESCOT team.

In any academic discipline, one important role of journals is to ensure that knowledge is published in a form that others can use and build upon. A journal of interactive mathematical tools should also fulfill this role, adjusted to the reality that the field is now using and building upon tools and not just ideas. Where software is concerned, "using and building" thus translates into issues of reuse and interoperability.

As mathematics education does not have a long editorial tradition of examining software contributions for reuse and interoperability, we will use our column to begin exploring and articulating what these ideas mean for the audience of JOMA authors and readers. In this introductory column, we think especially about the "readers" -- what do mathematics teachers and instructors want from a collection of mathlets? First we consider the general question of what makes for a successful collection of digital resources. Then, we present five lines of triangulating data that help us understand what middle school math teachers want. (We explain later why our focus is on middle school rather than college.) Finally, we discuss the results and implications for the future contributions JOMA authors might make.

While finding and reusing applets on the Web is a fairly new practice in mathematics education, the general practice of organizing existing computational resources on a network for others to find and use has a reasonably long history, and that history has been the subject of studies in Computer Science. Specifically, the subfield of "reuse" in Computer Science has examined this question:

What makes one collection of computational resources better than another from the point of view of the consumer who would like to find an existing resource that they can adapt to their own specific need?

Research has progressed to a consensus answer on two points (Poulin, 1999, p. 98):

- Favored collections are "small libraries of greatly used, well-designed, domain-specific, high-quality components," on the order of 30 to 250 components.
- "The key to high levels of reuse comes from building a collection of components that all work together and that many applications will need."

Researchers summarize this as a "product line" approach -- consumers want a collection to be a well-conceived, organized, compatible collection of products that coherently address their needs. This is just common sense -- teachers will quickly become frustrated if they need to master a huge toolbox of thousands of mathlets, each of which is good for teaching a very small range of concepts. They will be especially frustrated if each slightly different tool in the toolbox has a slightly different way of controlling a common behavior, such as re-scaling a graph. In the end, teachers probably would be happier with a single or a few graphing tools, where the same tools are embedded in a wide variety of teaching contexts.

And herein lies a perplexing problem: Many JOMA mathlets will need a graph in them, but we can't expect all authors to converge on a single graphing tool immediately. How can JOMA be a positive force for moving from a realistic situation today (every author writes their own graphing tool) to an ideal situation (all authors extend a small family of graphing tools)?

Our position is simple: While JOMA mathlets may not share common code, they should contribute to a collective understanding of what a common, ideal tool would do. That is, at an absolute minimum we ought to encourage "Design Reuse" through publication of the best quality designs. Hence, an author seeking to build a "best of class" grapher should be able to read JOMA and figure out the set of features that ought to go into the best grapher, drawn from a long history of JOMA mathlets. And thus, part of the role of contributing to JOMA ought to be identifying what a particular mathlet does exceptionally well and how this innovation might contribute to the accumulated design wisdom of the JOMA community. The "product" of JOMA, thus, is not just a collection of individual mathlets, but also an increasingly refined body of design knowledge that can guide future products.

Jeremy Roschelle, Victoria Hand, Chris DiGiano, and the ESCOT Team, "Towards a Coherent On-line Collection of Tools - Collections of Tools," *Convergence* (November 2004)

Journal of Online Mathematics and its Applications