Mathematics as Problem Solving, Mathematics as Communication, Mathematics as Reasoning, and Mathematical Connections are critical items throughout the NCTM Standards. They appear at every level because they form the core of what it means to do mathematics.

"Wait a minute! How did they do that?" is a common thing to hear students say after they hear the story of the Hotel Infinity . Some students think that because it is logical, the events of the story are possible. Others will be certain that what the narrator describes cannot have occurred. Whether anyone changes their mind or not, the ensuing debate is a problem-solving session highly dependent on reasoning and communication.

Scrutinizing the foundation of our number system

Many of the Standards touch on the idea of numbers : number sense and numeration, concepts of whole number operations, whole number computation, number systems, and number theory.
Students deepen their understanding of numbers over a period of many years. But what, exactly, are numbers and why do they work?

Our number system is founded on the notion of counting. From counting and the idea of one-to-one correspondence come all of our ideas about the relationships between different quantities of objects. The familiar whole number operations of addition, subtraction, multiplication and division are all based on counting. All of the computations that we do, whether in our heads, with pencil and paper, or with the use of machines assume that counting is solid and reliable, that it makes sense in the world, and that it "works".

To "visit" the Hotel Infinity, we must examine what it is we mean by counting so that the assumptions which lead us to see counting as natural and normal do not lead to outlandish contradictions.