The application format for mathematics concept formation through Rigorous Mathematical Thinking (RMT) involves three factors – topic, grade level, and time of application. Although the RMT format consists of different topics or mathematical concepts, each requires learning that involves six core mathematical concepts – quantity, relationship, representation, abstraction/generalization, precision, and logic/proof. Each topic also involves the appropriation and use of the mathematically specific psychological tools of signs and symbols and mathematical language. Because there are cognitive functions that are naturally aligned with and needed to build the core concepts, teaching the topics will require development of these cognitive processes. Teaching a certain topic provides a supportive foundation to facilitate teaching the topic that follows. However, because each topic demands its own core conceptual development along with the naturally aligned cognitive functions, the topics can be taught relatively independently of each other.

Examples of the amount of time required for RMT classroom teaching to develop and/or improve understanding and skills with regard to specific concepts/topics at respective grade levels are given below.

• Number sense for 2nd to 7th grades, 30 hours.

• Basic math operations and properties for 2nd to 7th grades, 40 hours.

• Meaning of fractions, adding, subtracting, multiplying, dividing fractions with like and unlike denominators, simplifying fractions, converting from fractions to decimals and vice versa, and ratio and proportions for 4th to 7th grades, 45 hours.

• Factoring, writing expressions, and solving simple equations for 4th to 7th grades, 20 hours.

• Basic geometric properties of two-dimensional and three-dimensional figures, area, and volume for 4th to 7th grades, 25 hours.

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