- Using this site


CHALLENGES - the applets

The applets that this site will lead you to are primarily designed for teachers of mathematics who want to deepen their understanding of the mathematics they teach and that their students are expected to learn.

In many cases, different approaches are used with familiar topics.
  •         linear & quadratic functions  in parameter space as well as in the Cartesian plane
  •         solving the equation  f(x) = g(x)  by transforming graphs rather than symbols
  •         UNsolving equations and inequalities
  •         formulating measures as an essential first step in building mathematical models
  •         generalizing geometric constructs
  •         geometry from an algebraic perspective
  •         algebra from a geometric perspective

a caution & an attempt to help

Many teachers will approach these applets with the idea of using the applets in their classes with their students. Please think carefully about doing so. The applets were designed for use in small groups by people who already know and understand the mathematical topic in question. They are designed for you, the teacher, to help provide you with a wider and deeper perspective. In as many instances as possible I have included the question

What questions could/would you pose to your students based on this applet ?     Read The Parable of ADD THE EGG !

That having been said, you will no doubt find that there are occasions when an applet in one of the collections is a 'perfect fit' for a student or students who are somewhat more advanced or who might think differently about the material you are teaching. Toward that end we urge you to read the essay on this web site entitled "Functions - objects & actions". That essay provides some of the background thinking that went into the structure of this site and includes a mapping, albeit a crude one, from a more traditional organization of the mathematical content to the organization used in this web site.

We find the applets to be most effective when teachers, in groups of two or three, explore the applets together and discuss them.

go to CHALLENGES - the applets
THEMES among the applets

These are smaller collections of applets, all drawn from the challenge archive, that are organized according to thinking strategies that cut across traditional mathematics subject matter.

Click here to go to THEMES among the applets

This section of the site contains several essays that reflect some of what the author and his colleagues have learned over the years about the teaching (and learning) of mathematics. Topics include
  • developing 'function sense'
  • the power of multiple representations
  • the role of conjecture in teaching, learning and doing mathematics
  • the multiple meanings of the equal sign
  • the two faces of the function - object and action
  • the arithmetic of quantity (as opposed to number).
  • qualitative graphing
  • number lines, quantity lines, and the graphing of functions of quantity
  • representing word problems graphically
  • models, simulations & exploratory environments
  • on the nature of making mathematical models
Click here to go to ESSAYS on teaching mathematics



Always resist the temptation to pose problems that have only one correct answer.

Judah L. Schwartz
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mathMINDhabits by Judah L. Schwartz is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.