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Stephanie Burroughs, Ed.D.

Curriculum Leader, K-12 Education

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The Conversation on Teaching Mathematics

Guiding Principle 2 for Massachusetts Mathematics Programs:

An effective mathematics program is based on a carefully designed set of content standards that are clear and specific, focused, and articulated over time as a coherent sequence.


At first glance, Guiding Principle 2 appears to just be a simple statement on the need to have written curriculum aligned to standards. I think this is an easy 'go-to' interpretation of the second guiding principle, but when digging deeper into the Frameworks you can clearly see that the guidelines for teaching are intentionally connected to the Mathematical Practices that support reasoning, mathematical modeling, and deeper thinking. As educators, we have to move past the practice of coverage and focus on making connections and creating deeper learning opportunities for our students. 

 

An effective mathematics program is based on a carefully designed set of content standards that are clear and specific.

In pulling out this line specifically, I think we can all agree that every mathematics program has an obligation to clearly define what it is the students need to know in each course curriculum through a detailed course curriculum mapping. I think we have to go deeper than just having a really good curriculum map, however. Curriculum maps are for teacher reference and of no indication that the students are aware of what it is they are doing and what specifically they will need to be able to do in order to master the next piece of content. I believe this becomes even more important as we explore a variety of instructional and assessment methods. The broader our assessment methods, the greater need there is for clarity and a well defined foundation of content standards and learning expectations. 

When designing curriculum, we have to look further than grabbing a checklist of standards from our frameworks and matching it up with the textbook. We must make sure that the design of our units makes sense for the conversation we wish to have with our students and we must also ensure that the grouping of content standards allows for a clear definition of mastery of an entire concept, not just an isolated skill. Teachers must then clearly identify the objectives, essential questions, and standards associated with their lessons, units, and assessments in order to provide their students with an opportunity to master the content. These content standards must be carefully grouped together in order to allow for purposeful mathematical discourse around a given topic. 

 

An effective mathematics program is based on a carefully designed set of content standards that are focused.

The Common Core State Standards calls for a thematic approach to mathematics course design, demanding that we make conversations in Mathematics meaningful for students. We should be creating opportunities for students to make connections between concepts and have a conversation on the development and application of skills related to concepts. We want students to understand the intricacies in breaking down a particular concept, looking at it algebraically and visually as they try to understand its practical application. We need to guide our students in a focused discussion of key concepts for each course.

In creating a set of content standards that are focused we are allowing for deeper learning through meaningful discourse. The days are gone where students rehearse a particular skill in isolation and are not given the opportunity to compare that solution strategy to an alternative solution strategy. The quadratic formula, for example, is a method for determining the roots of a polynomial function that cannot be factored any further. The Common Core demands that this is taught in unison with factoring polynomials, finding the roots by completing the square, or determining the roots graphically. While it is tempting to teach these methods in isolation given their complexity, for students to truly understand what it is they're using mathematics to solve, these concepts must be grouped together thematically to encourage deeper learning. Students must understand that mathematics is not just a series of skills to solve arbitrary puzzles. Instead, students must value the logic behind multiple solution strategies and, by allowing this level of understanding,  develop an appreciation for mathematics as a means to make sense of the world. 

 

An effective mathematics program is based on a carefully designed set of content standards that are articulated over time as a coherent sequence. 

Vertical curriculum conversations are the most effective use of professional development time. In order for students to make connections in mathematics, teachers must also understand why their concept is important and where it will show up again. The conversation in class with students on making connections and making mathematics meaningful is more important than any worksheet or problem set we can assign, and this conversation is most effective when teachers understand where students have come from and where they are headed in the mathematics program. Teachers have an obligation to develop a vertical understanding of content standards and their related algebraic processes in order to support student understanding.

Teaching Mathematics is not about what you know, it is about the conversation you can have with your students that allows for discourse and deeper learning. As teachers, Guiding Principle 2 demands that we look carefully at the design of our course curriculum to ensure that conversations on concepts are thematic and meaningful and that vertically there is a continuous and cohesive conversation on student mastery of concepts and their real world applications. Guiding Principle 2 calls for a mathematics program that is designed to create connections for students and carefully build off of concepts vertically.

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