Part 8 of 8, Strategies for Integrating the Mathematical Practices into Instruction
We have compiled all strategies from this 8-part series into a complimentary white paper that can act as a guide for teachers.
Last but definitely not least in this series on Integrating the Mathematical Practices into Instruction is MP#6: Attend to Precision, which often impacts a solution more than any other practice. This practice is generally understood to be about accuracy of solutions and good estimations. While these ideas are certainly part of the practice, it includes so much more.
“Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions.” —Common Core State Standards (CCSS), MP#6: Attend to Precision
Part 7 of 8, Strategies for Integrating the Mathematical Practices into Instruction
This mathematical practice involves the ways students can explain to themselves the meaning of a problem and the ways they find to enter into solving it. It might feel like being a detective who is looking for clues or evidence on how to solve a problem. Students proficient with this practice believe they are mathematicians and try several methods to come to a solution.
This blog series will conclude by examining Practice #1 in this post and Practice #6 in the final post. These two practices can be thought of as overarching habits of mind that productive thinkers use as they work with mathematics.
Look for and Express Regularity in Repeated Reasoning
Part 6 of 8, Strategies for Integrating the Mathematical Practices into Instruction
By Dr. Michele Douglass
Of all the mathematical practices, I find this one hardest to implement. It is probably from my years of being taught how to just manipulate numbers versus how to use patterns to generalize.
This practice reverses the thinking of the previous practice (MP#7). For this practice, we want students to use patterns that we might give them to generalize a situation. For example, instead of teaching rules for adding integers, how could students look at patterns to generalize or come up with the rule? Think about a lesson that has students examining graphs and matching equations in order to generalize the slope-intercept equation.
Part 5 of 8, Strategies for Integrating the Mathematical Practices into Instruction
Looking for and Making Sense of Structure means using deductive reasoning. In other words, I recognize the pattern and can apply it to solve a specific problem. This is the one practice that I am most often asked about by teachers in primary grades. They want to know what happened to all the standards about patterns. My response is always the same: Math is all about patterns, so it isn’t something that should be taught as a single standard, but rather as a practice that we use when thinking mathematically.
This practice is about how we work with students so that they are always looking for and making sense of repeated structures. For example, a sequence of numbers begins with 5. The next term is found by adding 4, and the next term is found by multiplying by -1. If this pattern continues, what is the 25th term in this sequence? Do you have to write the first 24 terms in order to figure this out? Seeing and using patterns moves beyond the primary standards of the past of recognizing AB or ABBA patterns.
Part 4 of 8, Strategies for Integrating the Mathematical Practices into Instruction
By Dr. Michele Douglass
There are few times that students in math classes or on assessments are asked which tool they should use to complete a problem. Think about the test questions that ask students to measure something. If it’s a length, the ruler is aligned to the object within the test question. If it’s a temperature, a thermometer appears in the question. We even provide the manipulative that students should use to solve a given problem.
Although the mathematical practice of Using Appropriate Tools Strategically is one that should be easy for most of us to implement, our testing world has never required us to use this practice as it is intended.
Fast-forward to classrooms teaching this practice or, better yet, classrooms where students are using this practice independently. They know how to use the tools and when to use them appropriately. Tools can be anything from mental math; pencil and paper; physical tools such as rulers, protractors, compasses, etc.; to calculators and computers. Mathematical tools also include graphic organizers, charts, tables, and manipulatives. What is critical in the development of this practice is that students are given opportunities to use each tool and to learn when its use is appropriate.
Our series on the mathematical practices continues by looking at a practice that is often grouped with the one we will discuss in our next blog. Both require an understanding of the content in a way that allows you to represent it. If you are like me, when you read this blog post’s title, “Models with Mathematics,” you think manipulatives. However, the practice we will discuss here is not about manipulatives; it is about using mathematical symbols to represent a situation.
As with the practice of Reasoning Abstractly and Quantitatively, which was the focus of Part 1 of this series, the practice we will focus on in Part 2 also asks students to reason with mathematics. As I am teaching a class and working to integrate the mathematical practices, I think of these two practices together, even though there are distinct differences. In this blog, we will look specifically at how to get students to Construct Viable Arguments and Critique the Reasoning of Others. I am sure that after you look at these examples, you will see how reasoning is used in two well-defined ways.
This year’s work has been deeply concentrated on the implementation of the Common Core State Standards. As I work with school sites as they are planning and designing lessons, we end up in conversations about the mathematical practices.
As we focus on the key shifts of Focus, Coherence, and Rigor, lesson planning adjusts to how to use the mathematical practices as a way to address these shifts. It isn’t easy to do when most of us have had our own instruction delivered to us through very traditional approaches that use one procedural method.
This series of blogs will examine each practice individually with special attention to implementation. The first practice is Reasoning Abstractly and Quantitatively.
Student achievement in the areas of measurement and geometry has been lacking for years, as evidenced by TIMSS data as well as state-to-state student achievement data. Two contributing factors are that textbooks typically spend less time developing these concepts and teachers often don’t spend the instructional time that is needed for them.
Now is the time to address this deficiency in mathematical understanding by beginning to understand the Common Core State Standards (CCSS).
Multiple researchers discuss the best practices to use to maximize student achievement in mathematics. The good news is that the authors share many of the same big ideas.
One thing that stands out is a strong focus on procedures that has existed for years in the elementary grades. This emphasis is linked to the language and testing of our state standards, but lacks problem-solving development and the foundations of knowledge needed for higher-level mathematics.