President, Colorado Council of Teachers of Mathematics
Continuing our focus on the Standards for Mathematical Practice, this message focuses on Math Practice 7: Look for and make use of structure. A good analogy for thinking about structure in mathematics is to compare it to the structure of buildings. When engineers construct a building, they pay attention to both the parts of the structure as well as to how these parts fit together to create the structure as a whole. Thus, in mathematics, we need to support our students in paying attention to both the parts and structure and the interplay between the two.
The structure of mathematics can be viewed through the big ideas of mathematics and its patterns. Big ideas, sometimes referred to as enduring understandings, are themes that often span multiple grade levels and become increasingly more sophisticated over time. Focusing on patterns helps students to develop an understanding of mathematics properties and supports them in seeing how mathematics is predictable and makes sense.
Multiplication is an example of a big idea in mathematics that spans multiple grades in K-12 mathematics. In the Common Core Standards, students in second grade begin working with equal groups to build a foundation for multiplication that supports their work in third grade as they solve problems involving multiplication and apply the properties of operations as they multiply. Their sophistication with multiplication continues to grow as they multiply multi-digit whole numbers, fractions, decimals, and integers. Their understanding of multiplication is further extended to multiplication of polynomials and complex numbers.
When approaching multiplication through the lens of Math Practice 7, we would want to support students in seeing patterns in multiplication and using properties to ensure that their understanding of multiplication became an enduring understanding as they progress from grade to grade. For example, when students are developing fluency with their multiplication facts, we would want them to notice the pattern of commutativity. As they employ partial products as a strategy to multiply multi-digit numbers, they do so by making use of the distributive property. This property can be extended to algebra and used by students to multiply polynomials (rather than using FOIL which has limitations) and complex numbers.
What do students do to look for and make sense of structure in the mathematics classroom? Students search for and identify patterns that help them to understand the structure inherent in tasks. They connect skills and strategies previously learned to solve new problems and tasks. They might breakdown complex tasks into simpler ones that are more manageable to solve. They develop the ability to view complicated quantities from both the perspective of a single quantity and a composition of quantities.
To support students in looking for and making sense of structure in classrooms, teachers would encourage students to explore and explain patterns as a way to understand the structure of mathematics. To accomplish this, teachers would support productive discourse in their classrooms and pose open-ended questions that help students to identify structure and help students to make connections to skills and strategies already learned. Such questions might include:
- What observations do you make about...?
- What do you notice when...?
- Where have we seen this idea before?
- What pattern do you see? How do you know it is a pattern?
- How is this problem similar to other problems we’ve solved?
Our support for students in developing their expertise in Math Practice 7 will ensure they see mathematics as a discipline built on structure and patterns. This view of mathematics supports them in understanding that mathematics makes sense and that they are capable of being mathematical sense makers.