I am excited to complete my series unpacking the Standards for Mathematical Practice (SMP). Using a great text about Common Core Math & PLC’s edited by Timothy Kanold (see Resources) and Heinemann’s SMP series as inspiration, I have been sharing & synthesizing the articles for my staff. See below to read about Standards 5-8, and click here to checkout Part 1 that covers Standards 1-4. Enjoy, and feel free to share some of your own bite-sized best practices in the comments section below!
SMP #5: Using Appropriate Tools Strategically – In his article about SMP #4, Meyer reminds us that we should avoid doing “the most interesting parts” for our students…they must be the ones who predict, question & explain! This holds true when working on Standard #5 as well! As SanGiovanni unpacks SMP #5, he states that while the selection of tools (which can be manipulative, general or technology based) depends on the “mathematical maturity of our students” – the thinking that accompanies this process is essential for all students! In other words, our students must be the ones thinking strategically as we model the “metacognitive process of tool selection” with us providing guidance & choices (not just selecting tools for them).
Below, you will find some fantastic questions to guide us through this process and help get us started. Remember that tools are there to help us make sense of mathematical processes. They do not produce “understanding, problem solving & solutions,” but our coaching & questioning strategies will!
SMP #6: Attend to Precision – Standard #6 is designed to ensure that our students “talk the talk” as mathematicians, with our focus being on “clarity & simplicity in both speech and writing.” As Mark et. al. state in their article, the outcome for this process is to have our students refine their understanding as they use & apply content area vocabulary. In order to achieve this we must use math vocabulary naturally & correctly; therefore, making these terms a part of how we always talk. Click here to visit my Curriculum Corner article on Vocabulary Development for some bite-sized best practices that will help you to embed authentic vocabulary routines into your instruction. Using math as a lens, Kanold, et al., breaks this Standard down into three areas:
- Model Accurate & Authentic use of Mathematic Vocabulary & Symbols: One basic example of this that Kanold et al. provides is using the term “regrouping” vs. “borrow.” They go on to explain that the term “borrow” implies that the “digits are not a part of the same number” & is therefore an inaccurate explanation of the concept.
- Provide Opportunities for Student to Share Their Thinking: This oral language and/or written component is interwoven throughout the SMP’s & the Common Core in general. Click here to view my Curriculum Corner series on Oral Language Development.
- Prepare Students for Further Study: This concept is at direct function of vertical articulation. As we work with our students we must ensure that our instruction is consistent with future math concepts. One example, shared by Kanold is how subtraction is typically taught, with teachers stating “The big number is always on top.” This creates confusion when students begin to learn about negative quantities, for example. Through ongoing collaboration & curriculum mapping we can prevent such inconsistencies from happening.
SMP #7: Look for & Make Use of Structure – Mathematics has a far more consistent structure than our language, but too often it is taught in ways that don’t make that structure easily apparent. This is how Goldenberg, et al., begin their article on Standard #7 & what they are referring to is numeric patterns. As we work with our students, we must provide opportunities for them to construct and deconstruct these patterns through multiple means of representation (i.e. numbers, manipulatives, visually, etc). Therefore, our instruction must not only include opportunities for our students to practice “random-order fact drills [that] rely on memory” but should also include opportunities for “patterned practice [to] develop a sense for structure as well.” Kanold et al., break down the teacher’s role into 3 areas:
- Draw Student’s Attention to Patterns: This is where the multiple means of representation come into play as we present examples that are “conductive to exploring structure.”
- Engage Students in Exploring Patterns: Here we provide students with opportunities to “create their own examples of structures to share & discuss with one another.”
- Facilitate the Application of Patterns: Once students are confident in recognizing structure, we must expose them to authentic problems, ensuring that they develop an understanding that “no matter the context, problems of a certain structure are worked exactly the same way.”
See below for some examples of numeric patterns in math facts (click through to view a larger version of each graphic) & feel free so share some of your own bite-sized best practices for representing numeric patterns in the comments section below!
SMP #8: Look for & Express Regularity in Repeated Reasoning – In his article on SMP #8, Max Ray introduces us to George (who quickly skips any math problem that becomes a challenge, despite his strong math skills) & Shana (who may not complete tasks due to the lengthy strategies she employs, despite her perseverance). He stresses that being a mathematician means balancing the qualities of both these students and that we must promote perseverance without “getting stuck doing the same inefficient methods every time.”
Ray calls this being a “lazy mathematician” but I prefer to consider one of the guiding principles of Judo, which is known as seiryoku zen’yō. When translated into English, this simply means: MAXIMUM EFFICIENCY WITH MINIMUM EFFORT. Listed below are some ways we can help students move beyond just solving problems to generalizing & determining shortcuts (Kanold, et al.).
- Scaffolding Examples to Highlight Repeated Reasoning: We must provide examples and questioning that helps “students notice if calculations repeat.” This will allow students to begin making sense of the process & to determine “general methods for calculations.” This should include multiple examples & varying contexts.
- Establishing Routines for Sharing & Working With Various Methods: This should include opportunities for students to “share multiple methods for solving the same problem” along with requests to use someone else’s method to solve a similar problem. Most importantly we must “playfully & explicitly add an element of time” as students work to ensure efficiency.
In the video below, Max Ray himself, walks us through this process:
In short, the Standards for Mathematical Practice are designed to ensure students move beyond a “factory model” of mathematical learning (simply turning out solutions via one method), to justifying, collaborating & generalizing as we construct/deconstruct numbers and mathematical processes. Click here for a nice set of tools that will support team planning & collaboration as you synthesize the Standards for Mathematical Practice to develop bite-sized best practices for your students!
Click on the graphic below to visit Heinemann’s site & view each of the SMP articles that influenced this piece.