Mathematics is not only a body of content knowledge developed over the course of many millennia, but also a way of thinking. A good mathematics student is a skillful thinker who can flexibly and creatively connect pieces of content knowledge to produce new mathematical results. This kind of thinking is important not only because of its value in further work in mathematics and science, but also because it can be applied in a variety of situations, many of which are not ostensibly "mathematical."

Successful students of mathematics call on "habits of mind," or processes they use as they do mathematics. The MathScape curriculum aims to make students proficient in the following processes:

• Seeking patterns and relationships
  Whether students are trying to find an algorithm to describe a situation or discover a shortcut for a long calculation, students can become skilled at analyzing situations of change to discover underlying patterns.
• Inventing
  Students can flexibly think about problems and invent their own ways to represent situations along with notation necessary for the representation. Whether they are working with an invented representation or not, students can be in the practice of coming up with their own, unique ways of approaching problems and should be encouraged to use their creativity as they do their mathematics work.
• Experimenting
  Just as mathematicians tinker, so, too, can students learn to play around with problem situations to get a feel for the underlying mathematics. Once a specific problem has been solved, students can make it a habit to test the generality and limits of ideas involved.
• Trying alternative representations and strategies
  Students can represent problem situations by various means: words, diagrams, tables, expressions and equations, graphs, and other invented ways. In particular, they can be skilled in visualizing problem situations, whether the visualizations are direct representations of problem situations or more abstract ones that get at some of the mathematics behind problems. Furthermore, students can get in the habit of linking the representations to one another so as to come to a deeper understanding of the underlying mathematics. Finally, they can think flexibly about different strategies for solutions within and across these various representations.
• Abstracting essential aspects of situations
  Students can learn to generalize about relationships they find among particular numbers or objects in specific problems; additionally, students can become skilled at making connections across problem situations.
• Wondering, proving, and conjecturing
  Students can be in the habit of questioning why their results hold and wondering how they might be justified. Even though some convincing arguments might be out of reach for some students, students can at least recognize where a justification is lacking and wonder about the proof. When reasonable, students can provide justification for their thinking. Additionally, students can be curious enough that they begin to make their own conjectures and wonder about the validity of those conjectures.
• Communicating mathematically
  Students can be in the practice of describing their thinking in both oral and written forms. They can incorporate mathematical vocabulary and symbols into their language in order to communicate with precision.
• Reflecting upon their own thinking
  Students can engage in meta-thinking so that they are aware not only of the mathematical content they are learning, but also the ways of thinking in which they are engaging.