When I travel around the country and visit teachers in classrooms, one of the things I notice is that the word strategy and model are sometimes used interchangeably.
What is the difference between a strategy and a model? Why do we care? How does the distinction help us teach more effectively?
A strategy is how you mess with the numbers, how you use relationships and connections between numbers to solve a problem. There are a handful of important strategies for each operation. Often a strategy is categorized, described, or named by the first thing you do with the numbers. Sometimes a strategy is best described as the overall plan on how you will use the relationships.
A model is a representation of your strategy, the way the strategy looks visibly. Modeling your strategy makes your thinking more clear to others because they can see the thinking and the relationships that went into your process. The model might also be the tool you used to actually do the computation. Models and modeling have many different meanings and interpretations, but for our purposes here we’ll refer to models as representations of strategies.
This distinction between strategies and models is very important and has important implications in mathematics teaching. Interestingly, we have found that if the teacher is clear on the difference, the students tend to focus correctly on strategies and use models appropriately to represent those strategies. However, if the teacher is not clear in their own understanding, the students languish in unsophisticated strategies, and use multiple models rather than multiple strategies.
At its heart, students need to be most concerned with strategies, with how they are using relationships and connections to solve problems. Teachers need to use models to represent student thinking so that the student thinking can be made visible and accessible to everyone; so the strategy can be considered, commented on, dissected, and compared for efficiency. As student thinking is made visible, there is the potential for students to clarify, make generalizations, and compare, and for other students to take up that thinking (the strategy). We use models to help students see and access relationships between the numbers they might not have otherwise considered. This allows them to consider more efficient ways to solve problems.
Here is a common addition strategy, shown on 3 different models:
Here are 2 different addition strategies, represented on the same model.
Try these CARD SORTS at Desmos.
Addition Card Sort: Click Here for Addition Card Sort
Subtraction Card Sort: Click Here for Subtraction Card Sort
Multiplication Card Sort: Click Here for Multiplication Card Sort
Division Card Sort: Click Here for Division Card Sort
You can DOWNLOAD pdf versions of the CARD SORTS here.
How does it impact your instruction?
Stay tuned for more posts to answer these and more important questions.