Sections:
Topics
Habits of Mind (Problem-Solving Strategies)
Technology
Mathematics Background
Duration
Keyword
Date

The list of topics was developed as a result of a detailed analysis of many current middle and high school curricula, both traditional and reform. In organizing the topics, our goal was to make it easy for you to find problems in ways that are already familiar and complement the textbooks you are using.

In addition to seven familiar areas—number, geometry and measurement, algebra and functions, statistics and probability, discrete math, trigonometry, and calculus—we included three that are less often identified as "areas" in the curriculum: topology; "math without math"; and logic, axiomatic systems, and foundations.

Three of the "joint" categories—geometry and measurement, algebra and functions, and statistics and probability—took a lot of debate. Even in grades 6 to 12, there are many situations that are best interpreted from a functions perspective and yet cannot be well represented with algebra, or at least with the algebra that the students have. Conversely, there are vitally important algebraic ideas—essential and standard parts of the curriculum—that have nothing to do with functions. The same kinds of distinctions can be made between geometry and measurement, and between probability and statistics. Why, then, should "x and y" be a single category? These combined categories are convenient both because they are commonly used, and because the problem sequences themselves often do contain elements of both and would have to appear twice if the topics were separated. We do list some problem sequences in more than one place—we quite deliberately sought to develop problems that interconnect two or more mathematical ideas—but we chose to create our topic categories in ways that did not escalate the multiple listings.

Two categories—logic, axiomatic systems, and foundations, and the thing we call "math without math"—were invented to hold important material that simply did not fit the conventional topic areas. The former is an aggregate of topics at the "foundations" of mathematics. The latter includes problems that require important mathematical methods but fit no particular mathematical topic. The best way to find out what these categories really mean is to scan the subheadings within them.

Mathematics is as much a set of ways of thinking—an evolving set of methods and habits of mind—as it is a body of results that have, over the centuries, been derived from those ways of thinking. Students who wish to advance in science or mathematics need a strong background in many of the results—the facts and procedures and topics that often form the tables of contents of our textbooks. But all students, those who may someday choose advanced mathematics and also those who will not, need the habits of mind that the study of mathematics hones and refines. They need to understand how mathematical results are created, and they need to be able to create results of their own. These ways of thinking must therefore be made an explicit part of mathematics curriculum, just as science curricula teach ideas about scientific method alongside the facts that scientific methods have uncovered.

However, the methods of mathematics are many and varied, and they don’t have standard names in wide popular use. The choices we made for this site were intended to be practical. We attempted to create a list of the habits of mind that would be most helpful to teachers looking for good problem sets to use with their classes. We hope that our choices are clear, useful, and reasonably comprehensive. Please let us know how well we did!

Mathematics background tells how much high school geometry and high school algebra a student should know before approaching a problem sequence. You can choose simultaneously from geometry (No Geometry, Some Geometry, or Completed High School Geometry) and algebra (No Algebra, Some Algebra, Advanced Algebra, or Calculus). For example, an advanced problem in algebra which does not use geometry at all will be classified as No Geometry and Advanced Algebra, even though students who are solving it usually know some geometry.

Mathematics background does not reflect a level of mathematical sophistication required for a problem. For example, a problem classified as No Geometry and No Algebra can require a high level of mathematical reasoning and be suitable for an advanced high school student. Please be sure to read the problem, and perhaps attempt solving it yourself, before deciding whether it is appropriate for your students.

You can search the database by the technology or manipulatives used in the problems, using the following categories.

• No technology The problem does not require any technology or manipulatives listed below. It can use general tools such as pencil, ruler, etc.

• Non-electronic tools and manipulatives The problem requires tools such as geoboards, counters, Cuisenaire rods, compass and straight edge, etc.

• Calculator/Graphing calculator Students must use calculators or graphing calculators to solve the problem.

• Geometry software The problem requires software such as The Geometer’s Sketchpad, Cabri Geometry, etc.

• Spreadsheets Students must use a spreadsheet in order to solve the problem.

• Computer Algebra System Students must use a Computer Algebra System, such as MathCad, Mathematica, LiveMath, etc., in order to solve the problem.

• Part of a Lesson, that is, from 5 to 30 minutes of work.
• Approximately One Day's Work, that is, about one class period time.
• More Than One Day's Work, that is, starting from a class period with a homework, up to multiple days project.