When students are Mathematically Proficient, they possess:

1. A Productive Disposition which means they believe in one’s own mathematical efficacy
2. Adaptive Reasoning which means they can: 
1. compare two or more plausible arguments 
2. identify correct logical thought 
3. correct flawed logical thought
3. Strategic Competence which means they can formulate, represent, and solve novel mathematical problems
4. Conceptual Understanding, which means they: 
1. can see the connections among concepts and procedures 
2. can see the relationship between representations of math concepts and abstract symbols 
3. know when different representations can be used for different purposes 
4. can represent mathematical situations in different ways
5. the ability to Mathematically Reason, which means they: 
1. can formulate a mathematical problem using knowledge of mathematical solution strategies 
2. know when to use an appropriate mathematical solution strategy 
3. can solve a mathematical problem accurately 
4. can represent the essential components of a mathematical problem using numbers, symbols, drawings and mental images
6. the ability to be Procedurally Fluent, which means they can skillfully: 
1. use algorithms 
2. follow and create written procedures and rules
3. use tools to help them make calculations such as: 
1. manipulatives (concrete, representational and virtual) 
2. mental methods 
3. calculators 
4. computers 

2nd Grade Student

based on the work of Kilpatrick, J., Swafford, J., & Findell, B. (2001). Adding+ it up: Helping children learn mathematics. National Academies Press. Shannon E. Bostiga, Michelle L. Cantin, Cristina V. Fontana, & Tutita M. Casa. (2016). Moving Math in the Write Direction: Reflect and Discuss. Teaching Children Mathematics, 22(9), 546.