At the Preconventional stage, children begin rote counting. They identify shapes and patterns. They begin to use comparative language such as more or less. Children participate in data collection.
At the Emerging stage, children begin demonstrating one to one correspondence. They construct shapes and pattern. They begin to use algorithms to compare (how much more). Children are data collectors and participate in data analysis.
At the Developing stage, children are counting, reading and writing numbers up to 100, and beginning to grasp the concept of base ten number system by breaking down numbers in the hundreds, tens, and ones. They are comfortable adding numbers, combining and subtracting sums up to 20. They can identify coins and know the values up to $1.00, as well as write cents and dollars notation. Developing mathematics students are transitioning from measuring with non-standard tools (unifix cubes, etc.) to measuring to the nearest inch or cm. They are competent in copying, extending and creating patterns.
At the Beginning stage, children are fluent reading and writing numbers up to 100, including adding and subtracting numbers without regrouping. They are using the concept that addition and subtraction are inverse operations as they work through problems. They notice patterns in the number + operation facts. They are also developing the concept of place value by breaking down numbers into hundreds, tens and ones. Their problem solving practice leads them to discover more than one way to solve mathematical problems. Beginning mathematicians are introduced to more precise measuring units and also to basic fractions. They are engaged in collecting data and making graphs and charts and beginning to interpret the data.
At the Expanding stage, students use their fluency in number and place value to expand operations to multiplying and dividing. They are able to use number lines with equal jumps, arrays, and groups to process multiplication and strategies such as successive subtraction, partitioning and sharing for processing division problems. They are working through equivalent fraction problems with manipulatives. Expanding mathematicians are introduced to the metric measurement system and its power of ten, as well as other units of measurement involving temperature and volume. They are introduced to the process of finding area and perimeter of two-dimensional shapes.
At the Bridging stage, students draw picture representations for the process in problem solving; numbers & operations. Students begin to transfer the process into algorithms. In guided situations, students transfer mathematic skills into real world and story problems. Students begin to engage in group discussions or debates comparing problem-solving strategies. Students are able to explain how they got from start to finish in a given problem.
At the Fluent stage, students can apply knowledge of multiple operations to solve problems. They are aware of how to discover mistakes in their process and start to correct them. Students are able to find solutions for problems that they may not have been exposed to the algorithm for. They connect prior knowledge to new concepts.
At the Proficient stage, students start to expand application of math concepts into other subject areas. Students can fluently shift from one process to another when problem solving. They incorporate more of the language of math into explanations of solutions. They select appropriate tools, and strategies to solve problems.
At the Connecting stage, students employ a variety of strategies to solve problems. They recognize the value of estimation when appropriate in the process. They are able to move procedurally forward and backward when solving problems. They are able to explain why the rules of different aspects of mathematics make sense.
At the Independent stage, students adapt prior strategies to reframe an equation while maintaining the integrity of the solutions. They conceptualize and work with very large numbers (billions, trillions) and very small numbers (ten billionths). (irrational?) The independent mathematician uses mathematics to describe and explain the world. They apply mathematics to the real world, develop a capacity for abstract thinking, and ask and answer questions involving numbers or equations. They demonstrate persistence in problem solving.