Strategies vs Models

Strategy: How students solve the problem. (Examples: Decomposition, Compensation, Traditional Algorithm)

Model: How students notate their thinking. (Examples: Number line, Ten Frames, Base Ten Blocks)

Important to Know:

One strategy can be explained using multiple models. One model can be used to explain multiple strategies.

Fluency with strategies evolves over time.

The Development of Mathematical Reasoning

Pamela Harris' Development of Mathematical Reasoning: Counting Strategies ⇒ Additive Thinking Strategies ⇒ Multiplicative Reasoning Strategies ⇒ Proportional Reasoning Strategies ⇒ Functional Reasoning Strategies

Students develop mathematical reasoning as they develop more sophisticated thinking. A student who solves 4 x 5 by using counting strategies or additive thinking strategies is not engaging in multiplicative reasoning. Although, this student will correctly determine the answer to 4 x 5 using other strategies, if they do not develop multiplicative reasoning, exploring more complex questions will be extremely challenging. For example, a student may use counting or additive thinking strategies as part of their process for solving 1.2 x 2.3 or (x + 2)(x - 1), but will be unable to solve it without using multiplicative thinking.

When a student develops fluency in multiple strategies and more sophisticated strategies, the student can confidently select the most appropriate strategy based on the numbers.

Naming Strategies:

Mathematical strategies should be named using the math utilized in the strategy. It might be cute to name the strategy after the student who first demonstrates it in class but when a student moves to a different class or a different school, non-conventional naming is not helpful. A grade level team and, preferably, a school, should agree upon names for the strategies that will be explored in math class. There aren't that many of them. Decomposing, compensating, partitioning by place value, give and take, and the traditional algorithm will cover almost all possible strategies.