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The reviewer, Judy Holdener, is an Associate Professor of Mathematics at Kenyon College.

Tom Leathrum's collection, *Mathlets: Java Applets for Math Explorations*, provides a set of interactive learning tools for Precalculus, Calculus, and beyond. For explorations in Precalculus, the collection includes interactive graphing tools for lines, parabolas, sine curves, roots of polynomials, exponential functions, and conic sections. The user can change coefficients and parameters and examine the output to better understand elementary functions and the significance of certain coefficients. For Calculus, Leathrum designed *Mathlets* to explore limits, concavity, sequences and series, Taylor polynomials, Newton's Method, parametric surfaces, the Jacobian, differential equations, and many other topics.

Although the collection is clearly a work in progress – many sections are under construction – Leathrum's website is easy to navigate, and users can learn how to use the materials quickly. Each mathlet is accompanied with instructions describing how to use it (sample), along with several characteristic examples (sample). The *Limits* mathlet, for example, demonstrates the limit of a function *f*(*x*) at *x* = *c* by generating tables of values of *f*(*x*) for values of *x* near *c*. To accompany the mathlet, Leathrum includes examples illustrating

- the limit of an indeterminate form,
- the limit of a difference quotient,
- the limit at a jump discontinuity, and
- the limit at a vertical asymptote.

With some of the applets, Leathrum also includes "other notes" (sample), aimed at students and instructors alike, to provide mathematical context for the applet, describe limitations of the applet, and summarize the main points of the concept under exploration.

For my own use in teaching Calculus, I imagine that Leathrum's *Mathlets* would be best used as an exploratory engine for students to answer a series of questions designed for guided discovery. It would likely take five or ten minutes getting the class up to speed on the use of the mathlet, and then I would set the class free to answer the questions. With carefully designed questions – yes, finding the right questions is the difficult part! – Leathrum's collection would be an effective learning tool. Although similar goals can be accomplished with computer algebra systems such as *Maple* or *Mathematica*, mathlets are perhaps more effective because students are less likely to be distracted by the technology. Although the user must learn how to represent functions and mathematical expressions in a form acceptable to the input text fields (and Leathrum includes a clear description of the acceptable expressions), *Mathlets* require very little knowledge of additional syntax.

Leathrum illustrates how he uses his collection in an exploration, Exponential Functions and their Derivatives, published in the first issue of JOMA. This exploration uses a small subset of the applets in the *Mathlets* collection, embedded within a series of web pages, to examine exponential functions and their derivatives as one cohesive story. I would have preferred that there be student exercises included within the web pages – I have found that some students are still developing their skills as active readers – but the mathlets themselves are interactive. The user can change parameters, examine the output, and analyze the results with very little trouble. It is not hard to imagine a future where mathematics texts are equipped with embedded mathlets allowing for greater interaction between the reader and the material!

Judy Holdener, "Leathrum's Mathlets Site," *Loci* (November 2004)

Journal of Online Mathematics and its Applications