My Discovery-Based Math Elective: Practice Standards (Part 4)

This is part 4, of a 15-part series of posts detailing how I developed and piloted a discovery-based high school math elective. The first, introductory, blog post for this series can be found here [Introductions]. The goal of this post is to describe how I chose and developed standards for the class.

The Common Core State Standards for math present two different ways of standardizing mathematics knowledge: Practice vs. Content

• Content Standards
• Content standards are the traditional way that we have approached the job of cataloging and talking about math knowledge. They have different grain sizes (i.e., specificity), but all tend to speak direction about the math content. For example, consider:

• The CCSS for high school math tend to focus on algebra, likely a result of the long-held tradition of high school culminating in Calculus (or Precalculus, as a close second). Some topics of Geometry and Statistics have found their way in too. They can be a decent starting place for thinking about how to construct the multi-year run-up to Calculus, especially for early career teachers.
• I don't think even the original creators of the Common Core would believe the content standards are perfect. They don't provide any real structure about math outside of the narrow band of content they address. They don't necessarily treat even those topics as thoroughly or neatly as would be ideal. They're a blunt, but sometimes helpful, tool to supplement and organize teacher content knowledge.
• Practice Standards
• To complement the articulation of content standards, the CCSS also presents the Standards for Mathematical Practice. The practice standards are much more general. For example, compare the above content standard with CCSS.MATH.PRACTICE.MP8 Look for and express regularity in repeated reasoning.

Link to full-sized image.

• Because they are more general, the practice standards can be transferred across different topics in math, beyond the traditional core content. In fact, if you look at the Next Generation Science Standards you'll notice that some of the math standards even transfer well to science and engineering. This transferability makes the practices an especially useful schema for thinking about math courses covering non-traditional topics.
• But the practice standards aren't perfect on their own either. They don't do much to articulate WHAT math students are learning across the country, which might be something you want your standards to do. Especially for early career teachers, and those new to them, the practice standards don't offer many helpful details about what to teach and how.
• A Healthy Blend (Usually)
• Unsurprisingly, the CCSS's recommendation is to have a healthy blend of the two types of standards. My personal experience in most related professional development has tended to advocate for more utilization of the practice standards. My perception here is likely biased by the fact that I'm a 3rd year teacher, and so I'm pretty new to the math PD scene. Since the formal practice standards are newer, I also imagine that people tend to give/take more PD on them, since more experienced teachers tend to be pretty familiar with the content standards. I also tend to value the practice standards more in my practice, so I probably tend to go to (and pay attention to) PD that centers "practices."
• A Radical Departure from Content Standards
• The content Modular Arithmetic, Bases, Combinatorics, Sequences, and Graph Theory all have very little coverage in the CCSS. Some standards are relevant. For example, consider CCSS.MATH.CONTENT.4.OA.A.3: Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. This 4th grade standard is technically at the center of the unit on Modular Arithmetic. But it definitely is insufficient. So I couldn't even use the content standards that much, if I wanted to.

Link to picture.

• And I didn't really want to anyways! I've already talked about other reasons I wanted to prioritize practice standards over content standards here, when I talk about the purpose of the elective. While I think the official Standards for Mathematical Practice are a good starting point, I didn't feel like they quite represented what I wanted to focus on in the course. Instead, I started with the rubric that PROMYS uses for grading their problem sets.

Link to the PSet Rubric Shown

• Explanation of the rubric
• Exploration: this work is the kind of work that happens primarily in the early stages of problem solving. This is where we notice, wonder, and try to get a feel for the situation and problem.
• Numericals: This is the kind of work we're doing where we are trying a bunch of different calculations, sketches, and diagrams to get a feel for what's happening in the problem. We're checking concrete examples.
• Logic & Reasoning: this is work is where we are reasoning about work we've done, leveraging rules, logic, definitions, and patterns to begin to generalize and prove things.
• Depth of Understanding: this is where we are connecting deep ideas to build more intuition and insight into the problem. This is also where a lot of what is ambiguously referred to as "rigor" comes in.
• If you end up teaching this class, you definitely don't have to use this rubric. You could just as easily make up your own, or pick some/all of the CCSS Math Practices. What mattered to me was that the standards be practices-oriented, instead of content-oriented.