[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.

Successful Collections Are Organized as a Product

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):

1. Favored collections are "small libraries of greatly used, well-designed, domain-specific, high-quality components," on the order of 30 to 250 components.
2. "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.

The Product Line Can Emerge By Accumulating Best Features

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.

Published July, 2001

© 2001 by Jeremy Roschelle, Victoria Hand, and Chris DiGiano

So far, we have been talking in the abstract about a product line for math teachers and using a grapher as an example of a product within the line. What does the whole product line look like? What products do math teachers want? In the rest of this column, we seek to answer this question.

Because of resource limitations and existing project funding priorities, we have focussed our analysis on middle school mathematics. This is a simpler place to start than undergraduate mathematics, yet there is still a considerable variety of tools that appear desirable to middle school teachers. Computer technology has become a mainstay in middle-school mathematics and science classrooms. According to a recent report by Becker, Ravitz and Wong (1999), more than 80% of secondary school classrooms have a student-computer ratio of 1-to-4. While this statistic is promising, a deeper look into classroom practices indicates that only one third of the teachers assign computer work to students on a regular basis.

The nature of software use in the classroom is also limited. For example, Becker, et al. (1999) also found that middle school mathematics classrooms more often employ skills-practice games (popularly known as "drill and kill") over more innovative, conceptually - rich types of software. This may be due in part to ease of compatibility with traditional mathematics curricula found in many classrooms; however, it is also reasonable to suggest that many teachers are unaware of or confused about where to find quality interactive mathematics software appropriate for their classroom needs.

In sum, while teachers, educational technologists, and researchers acknowledge technology's potential to complement classroom learning in powerful ways, we have yet to see the fruits of this union. We believe that it is a good time to begin thinking about what kinds of tools middle school teachers ought to have available.

For our analysis, we considered five sources of data (see Table 1):

1. A survey posed to teachers who use the popular MathForum web site
2. An analysis of technology use and opportunities for use in four recent mathematics curricula, performed by our partners at the Show Me Center
3. A review of applets cataloged by the Educational Object Economy web site in the area of math education
4. A review of components in our ESCOT project, which we collected for use in middle school math Problems of the Week
5. A broad survey of technology use conducted by Becker and colleagues

Source	Type	Analysis	Sample
MathForum	On-line Questionnaire	Survey Questions: What mathematics text(s) are you currently using in your classes? Which mathematics software do you use in your classes? Which of the following mathematics tools would be most useful for you?	Math teachers and students. (N=1080 (527 M, 553 F)
Show Me Center	Curricula Analyses	Current uses of technology tools. Potential uses of technology tools.	Curricula: -Connected Mathematics Project (CMP) -Math in Context -MathScape -MathThematics
EOE	Applet Repository Review	Categorization of existing math applets
ESCOT	Math Components Review	Components used in the development of forty middle-school mathematics Problems-of-the-Week
Becker et al.	Technology Report

Table 1: Survey Data Sources

While our data sources are extensive and adequately represent the various perspectives in the field (e.g., teachers, developers, researchers, curriculum developers, etc.), we feel it important to mention that it does not capture the realm of technical and educational innovation. In other words, teachers' views on valuable resources are constrained by their potential to perceive innovative technical possibilities. (A good example of this is the possibility of integrating of web-based data collection tools with temperature or motion probes.) And, while technologists are generally on the cutting-edge of technical progress, they may not be familiar with reform movements in education that conceptualize learning and mathematical knowledge in new ways (e.g., collaborative learning, performance-based assessment). These two factors are cyclical in that each informs the other: technology can promote new ways of learning, and in turn, new concepts in education drive different computerized learning environments.

Given the data sources just discussed, we highlight some of the products teachers most want, in the hopes that future JOMA authors might consider submissions that address these needs:

Dynamic Geometry: The Geometer's Sketchpad^TM is the only math-specific piece of software that teachers nationwide identified in the Becker survey. It is the top software currently used in classrooms according to the Math Forum survey (25% of teachers found it useful). Dynamic geometry, in general, was important in every other data source, too (Show Me center survey: 734 total uses). Dynamic geometry on the web is reasonably well supported by Java Sketchpad and other Java-based tools, but clearly math teachers would appreciate if JOMA authors would extend the collection of great uses of dynamic geometry.

Cartesian Graphing: Cartesian graphing of algebraic functions and data sets came up in nearly all the sources we considered as well (Math Forum: 23% of teachers in the survey found it useful; Show Me center survey: 193 current uses/143 potential). It was also one of the most common tools available on the EOE site. The popularity of graphing calculators points to the importance of this capability. In our experience with the EOE site, for example, many online graphing calculators offer only partial features relative to popular Texas Instruments devices. As we argued above, JOMA could serve an important role by pushing for integration of powerful graphing features into a common graphing tool that many lesson authors could use.

Calculators: It was not surprising to find that both teachers and students regarded graphing calculators as the most useful classroom tool (Math Forum survey, 25% of teachers; 13% of students). Similarly, four-function and scientific calculators were the tools most often directly called for in the four curricula that the Show Me center evaluated, and many simple calculators are available on line through the EOE site and others. Our experience tells us, however, that students and teachers find the numbers of buttons on physical calculators confusing for teachers and students, so an interesting JOMA contribution could focus on a calculator with a more refined interface, which is more easily tuned to the computations a student is likely to do in a particular chapter or course.

Probability Simulators and Stats Plots: Probability tools were also one of the top tool requests made by teachers in our Math Forum survey. Both the Show Me curricular analysis and development of ESCOT components helped to specify important features: the ability to generate large numbers of random experimental trials appears to be an important function. Moreover, the survey suggested that a rather small collection of simulated devices could cover most needs: a collection of spinners, cards, dice, coins, dominos, and marbles in a bag would cover a lot of ground. Of course, to go with these simulators, math teachers need a collection of statistics plots, such as histograms, box & whiskers plots, scatter plots, etc. This area is fertile for component-based designs that allow mixing and matching random generators and plot types (as in the topnotch commercial offering, Fathom, from KCP Technologies). While probability applets were found on the EOE site, many of them were gambling-related and therefore not necessarily appropriate for middle-school students.

Spreadsheets, Tables and Lists: These tools emerged across our surveys as representations and tools teachers frequently use, and that fill many curricular needs in middle school. For example, 75% of all ESCOT PoW's incorporated some form of a data table or spreadsheet. Yet most noncommercial web-based mathlets we have seen have fairly primitive and ad hoc data organizers. Commercial spreadsheets, on the other hand, appear to be overly complicated for most K-12 students to master. This category strikes us as fertile ground for JOMA authors to investigate and innovate in.

Computer Algebra Processing: A subset of graphing tools, computer algebra processing tools also have widespread uses on their own. The Show Me center data found 50% more potential uses for this type of technology than currently are called for in reform mathematics textbooks. Although many educational applications could benefit from including computer algebra processing, computer algebra processing code is hard to write and is commonly available only in proprietary systems. This makes computer algebra a great candidate for a reusable library that many mathlets could draw upon. Perhaps JOMA authors would step up to the challenging of an open source computer algebra system for education.

This list is by no means complete, but it covers the kinds of tools that came up most frequently across the data sources we considered. Other needs that came up less frequently include number lines, charting tools, visualizations of fractions, visualizations of fractals, turtle graphics, pattern explorations (e.g. tessellations), number sense manipulatives, and Venn diagrams.

Conclusion

The results of this study hold several implications for the development, collection and distribution of technology tools for math learning for middle school classrooms and beyond. Most important from the perspective of the electronic marketplace, there appears to be some consensus among the various stakeholders in mathematics education regarding the "product line" of mathlets that should be made available for classroom use. In addition, because this list is targeted and primed for reuse, development efforts are estimable and can be focused. Finally, we hope that the "product line" presented here will become a catalyst for grounded discussions of design features, reuse issues, and engineering approaches among the JOMA community.

JOMA will be successful only if it becomes more that a catalogue of the best of "what's out there" in mathlets. Like any good journal, it should encourage authors to contribute to cumulative, reusable, increasingly integrated outcomes. In this case, those outcomes should be a product line of mathlets that meets teachers' needs. In this column, we articulated some of these needs, as well as the minimal form a contribution ought to take, a reusable design. In future columns, we'll talk more specifically about how authors can structure their contributions to build on others' work and allow others to build on their work.

References

Poulin, J.S. (1999). Reuse: Been There. Done That. Communications of the ACM, 42, pp. 98-100.

Becker, H. and Reil, M. (1999). Teacher and teacher-directed students' use of computers and software. Center for Research on Information Technology and Organizations. University of California, Irvine, CA, 3.