The Standards for Mathematical Practice stress the importance of developing a conceptual understanding of various mathematical principles and being able to apply mathematical knowledge and skills to solve problems.
In other words, practice standards set the bar for how well students are able to apply their mathematical knowledge and skills to solve problems and understand what they're doing and why. Think of practice standards in terms of a doctor not only being able to pass a test but also being able to use that same knowledge to accurately diagnose medical conditions and effectively treat patients. The practice standards set expectations that students at all levels master the following habits of mind as they apply to the content standards at each student's grade level:
With any mathematical concept the standards introduce, any number of math practice standards come into play along with the content standards. Keep in mind that the Standards for Mathematical Practice are used on a case-by-case basis depending on which math content standards your student is learning and the specific problem your student is trying to solve.
In other words, practice standards set the bar for how well students are able to apply their mathematical knowledge and skills to solve problems and understand what they're doing and why. Think of practice standards in terms of a doctor not only being able to pass a test but also being able to use that same knowledge to accurately diagnose medical conditions and effectively treat patients. The practice standards set expectations that students at all levels master the following habits of mind as they apply to the content standards at each student's grade level:
- Understand mathematical problems and persist in solving them. Students need to have a clear understanding of what they're required to do in math problems and why they're supposed to do it. Acquiring this skill involves students asking questions as they work through problems and continually self-monitoring to make sure they understand why they're taking certain steps.
- Think about different aspects of a problem as separate and related parts. Sometimes students need to think about what something means apart from the rest of the problem. In math, this means they understand not only the operations needed to solve specific parts of a problem but also how to break down the problem to use different operations to successfully solve the entire problem.
- Understand a concept well enough to explain and defend an answer. Solving a problem correctly is great, but having the know-how to get the correct answer and then explain and defend how you got it is even better. When students can verbalize their thought process in a way that gives support to the steps they took to solve a problem, question the thought processes of others to decide whether they make sense, and ask useful questions to better understand a problem, then you have a good indication that they truly understand the concept.
- Apply understanding of a concept to a real-life situation. We've probably all had the experience of sitting in a math class and wondering when we were ever going to use the information being presented. A major step toward mastery of any concept involves understanding the applications of that concept to real life. Successfully relating math concepts to real-life situations shows that a student can transfer an isolated skill to a broad realm of possibilities. The more a student practices this skill, the more connections he is likely to see for applying it to real-world situations.
- Choose the right resources to help solve problems. When working a math problem, students usually have a variety of resources to help them solve the problem, including calculators, graphs, rulers, and scratch paper. The standards also stress the use of concrete models, digital content, and mathematical software. Knowing when and how to pick the right resources to solve a particular math problem is an indication of how well students actually understand the problem they're solving. Efficiently solving problems with the proper tools is a better indication of mastery than trial and error.
- Pay attention to the details. The world of mathematics rewards specificity. The better students become at using math-specific vocabulary and calculating efficiently and accurately to the degree of precision required by the situation, the sooner they gain proficiency and fluidity when solving problems. Math requires students to keep track of the details in order to be successful problem solvers.
- Find and use patterns in problems. As students get more comfortable with certain math concepts, they begin to pick up on patterns and structures that enable them to break apart problems with greater ease. Many younger children do this naturally when they sort and order manipulative objects such as blocks, but this habit of mind can require some practice as students move into more complex math. This skill comes with lots of practice. Isolating a certain part of the problem to make better sense of the whole is an effective skill in math. Being able to do it repeatedly in various contexts indicates that a student can see patterns and successfully use them to his or her advantage.
- Look for and use repeated reasoning. Understanding that some steps or procedures are repetitive saves students a considerable amount of time. When students grasp this skill, they're able to circumvent repetitive processes and quickly move on to solve the problem. However, arriving at a place of familiarity that allows students to recognize processes that repeat takes practice and an eye for detail.
With any mathematical concept the standards introduce, any number of math practice standards come into play along with the content standards. Keep in mind that the Standards for Mathematical Practice are used on a case-by-case basis depending on which math content standards your student is learning and the specific problem your student is trying to solve.
