The Common Core Math Standards adopted by many US states are an attempt to establish consistent, age-appropriate, research-backed goals for math learning. This allows more easeful transitions for children between grades and between schools, both when moving within or between states and when graduating to the next school. It establishes a common expectation of knowledge that each level teacher begins to work from each new school year (though teachers absolutely do understand that not every student mastered every standard, so they absolutely know that they will have to do some review and relearning).
To this end, the math standards have two main parts. Most people are aware of the grade-level standards about the specific math skills students will be learning each year. What you might not be aware of as parents, is the other portion of the math standards: the eight Mathematical Practices.
In the Common Core Math Standards, the Mathematical Practices are the same across all grades and mathematical skill levels. These Practices are included to build general thinking and problem solving skills and to develop students as lifelong leaners and problem solvers. While they are phrased in terms of learning mathematics, they are life skills that transcend math class. However, math class is a great place to practice each of them.
Each of these eight Practices begin with the phrase “Mathematically proficient students…” I would venture to go a step further and propose that we as parents instead think of them as “Well-rounded thinking citizens strive to…”
These Mathematical Practices are never achievable in an end-goal sense. They are not a destination to reach. Instead, they are practices in multiple senses of the word. They are activities that can be practiced and improved. They are not things that either we nor our children will ever do perfectly or consistently. Instead, they are ideas that can ease our path through the world and help us find our own confidence and success, however we define success. They are ideas that can be practiced and learned and polished at any age and will provide continual learning throughout our lives. Our skill levels in these Practices influence our abilities for independence in problem solving and thoughtful observation throughout our lives.
Each of the eight Mathematical Practices below is discussed more deeply in a post dedicated to itself. However, it’s also useful to see the bigger picture and look at them all together. I have summarized or occasionally rephrased each standard in language aimed for home use and the viewpoint of general thoughtful development (not just math). Feel free to use the link at the top of this page or your own internet search to see the original language if desired.
This post was first published at mathteacherbarbie.com. If you are viewing it somewhere else, you are viewing a stolen copy.
MP1: Making Sense and Persevering
Sense making might include figuring out what operations are needed and how information, data, and numbers fit together. Perseverance refers to endurance through the sometimes-long haul of finding reliable solutions to problems as well as editing and revising strategies and explanations.
Math Practice at Home: Sense-Making and Perseverence
Many of us think of math as a series of rules and procedures we have to learn to perform computations. However, with powerful computers and calculators constantly available, these computation routines become ever less important. Instead, a deep understanding of numbers, data, and patterns becomes absolutely critical. We must learn not only how to tell these machines what we want them to do, but also how to interpret and use the results they report. We need to be able to evaluate the output, know when mistakes were made, and figure out how to course-correct. In this Information Age, careers and procedures of one decade are completely transformed by the next decade. With each shift, however, the skills of making sense out of information and persevering in learning and correcting emerging ideas are constant needs.
MP2: Abstract vs Quantitative; Forest vs Trees Thinking
A person who has mastered [this] can move back and forth smoothly between the structure and relationships … and the sense-making of the quantities themselves, including units of measure. [They] can use both the abstract structure and relationships as well as the quantitative context as a sort of checks-and-balances throughout the problem solving process to a) use the efficiency of abstraction and b) Make sense of the problem and answer.
Math Practice at Home: Abstract and Quantitative Reasoning
We already begin to see here that the Math Practices overlap. Even in how I described this second Practice, we see the idea of “sense-making” reappear. This is one strategy that contributes to making sense of a problem, how to begin solving it, and the solution that results. Yes, sense-making is critical at all three of these stages! So often, practicing mathematics and skill-building in math class feels like being “lost in the weeds,” simply going through motions, performing algorithms and routines that feel unconnected to the rest of life.
This Practice reminds everyone involved in teaching students that both the context and general patterns and routines are important to learn. It serves as a reminder to practice connecting the routines and computations and algorithms back to their real-world uses. It emphasizes that being able to make those connections between abstract routines and realistic contexts is a skill in and of itself, and not necessarily automatic.
[Making] connections between abstract routines and realistic contexts is a skill in and of itself, and not necessarily automatic.
MP3: Constructing Arguments and Critiquing Reasoning
The third CCSS Mathematical Practice, Constructing and Critiquing Arguments, refers to the practice of justifying or explaining your solutions and recognizing what’s good and what’s flawed about other people’s explanations.
Math Practice At Home: Constructing and Critiquing Arguments
Media literacy, evaluating information, identifying facts from opinions, deciding for oneself: all of these fall under this practice. All of these are exceedingly critical in this time of divisive politics and politicized social ideals. Thus, this practice extends far beyond the mathematics classroom. However, math is a great place to practice these skills. Mathematics provides both large and small, both complex and simpler arguments to practice making and practice evaluating. It also provides structures in which to do this evaluating (such as the mathematical/philosophical field of formal logic) that then can abstract into real-world dilemmas.
MP4: Modeling: Rephrasing and Reformatting
[Though some] problems can be solved without a formal … model, it’s important to learn how to model … in order to solve all kinds of problems, as well as being able to solve some of these “easier” problems in a more general way (saving yourself time and energy down the road).
Math Practice At Home: Modeling with Math
Modeling is a further means of practicing sense-making and both quantitative and abstract reasoning (MPs 1 and 2). It involves cutting to the core of a complicated problem and creating a structure (or structures) that is (or are) solvable. It then involves interpreting and evaluating (MP3) the results of solving that model. Modeling can help us see both differences and similarities between and among problems we might face. It helps us improve both our skill and our efficiency in problem solving. It helps us communicate both the core of the problem itself as well as explaining and defending (MP3) the solution. Each of the computational algorithms and major ideas learned in math class can become part of a model, and part of practicing the skills is learning when and how they are appropriate to use. Modeling is where both knowledge and creativity meet in mathematics, helping us frame our own views of a problem and thus the world. There is seldom just one potential model for any problem, though occasionally there may be a “best” or “most efficient” or “easiest to understand” model. Building models is “where the rubber meets the road” in problem solving.
MP5: Use Appropriate Tools Strategically
I like to think of the fifth practice as building and learning to use a personal “problem solving toolbox.”
Math Practice At Home: Strategically Choosing and Using Tools
We’ve never before had so many tools to choose among. We still have arithmetic, sketches or drawings, and other mathematical procedures. We also have powerful electronics that can organize our data (such as spreadsheets and databases), perform complex computations in seconds (such as computers and calculators), show us step-by-step solutions (such as Wolfram Alpha and an ever-evolving range of apps), create graphs of information at the touch of a button (spreadsheets again and graphing software such as Desmos and Geogebra), and more. Beyond that, so many tools and classroom manipulatives (both physical and virtual such as at Polypad, Math Learning Center, and the NLVM) allow us to interact with our models, to touch and revise and make abstract models more concrete. And these are only mathematics tools. We also have interest and amortization calculators, comparison tables, thought organizers, and more for non-mathematical problem solving.
As useful as these are, the choices can be overwhelming. Thus, it’s important to develop skills of evaluating the needs of a problem and choosing the tools that are most appropriate. These choices typically rely on a combination of personal preference and specifics of the problem. Thus this Mathematical Practice exists to remind all involved in mathematics instruction and learning that this is a skill in and of itself, and worthy of intentional and targeted practice.
MP6: Precision, or Communicating Clearly
We tend to think of both math and the word precision numerically. While precision of computation is a part of this practice, the sixth practice is mostly about communicating ideas clearly and precisely. Learning to communicate with right levels of detail to avoid confusion and maintain clarity is a useful skill throughout life
Math Practice At Home: Attending to Precision
Precise communication is clear, detailed enough to be specific, general enough to avoid tedium, and using common language when possible while defining special terms as needed.
Math Practice At Home: Attending to Precision
This Practice is not just about details, but about the right details and the appropriate details. Giving too much detail can sometimes be just plain distracting. Giving too little detail can leave too much open to interpretation, confusion, or misunderstanding. Giving wrong, minor, or unimportant details is often a tactic used to sway audiences to a desired viewpoint that may not be supported by the actual source. This is a skill learned only through practice, through verbal and nonverbal feedback, through making and correcting mistakes. Thus, this Practice reinforces that mistakes are not bad; mistakes (and correcting them) are learning experiences, and often the most efficient learning strategies we have.
MP7: Finding and Using Structure
The seventh principle is ultimately about making life (and math) easier, more manageable, and more connected by observing common patterns and using them to simplify or better understand problems.
Math Practice At Home: Finding and Making Use of Structure
To my mind, this Practice is very closely related to the second and fourth practices as described above. The abstraction (MP2) and modeling (MP4) of a problem distill it to its essential core structure (MP7). This practice takes those ideas and extends them to looking across problems to recognize when one problem is like another and thus when we can use a similar problem-solving technique. This means that our problem solving can become more efficient since we don’t have to “recreate the wheel” every time. It also opens up a variety of arguments and reasonings (MPs 3 and 6) that we can use to communicate our ideas and solutions.
This Practice does require an understanding of the meaning and purpose of various mathematical tools (MP1) such as multiplication or graphing, not just the howto. In the last century or two of mathematical classroom instruction, the focus has been so heavily on the how that many students have lost the why and when. These practices together, and this practice most specifically, reminds us that the why and when are just as important, if not more important than ever before. Moreover, it reminds us that these whys and whens are no longer innate skills that come up naturally as often as they used to through childhood play, and thus direct instruction is essential.
MP8: Using Regularity and Repetition
Students … abstract (observe and describe general patterns) from repeated mathematical actions. … This idea of repeated reasoning has already come up as pattern recognition [MP7] and repetitive practice to experience advantages and disadvantages of tools in [MP5] and models in [MP4], among others. We see the idea of abstracting these repeated observations in [MP2]. What Standard 8 adds to this discussion is the idea of moving a subconscious learning to a conscious recognition of the patterns with the ability to describe and abstract (observe and describe in general terms) that repetition.
Math Practice At Home: Regularity and Repetition
This Practice is again about making problem solving more efficient as well as communicating effectively. It also emphasizes a very natural means of learning, especially for very young children. Mathematics has often been called the study of patterns. This idea brings repetition and regularity to the very forefront of mathematics past, present, and future. But patterns surround us, not just in math class. Patterns in the world can connect ideas back to math, but we need not make that connection explicit to use the patterns in solving problems. Observing patterns, observing repetitions, observing regularities help us to learn about structure (MP7) and both contribute to and use human knowledge.
All of these Mathematical Practices are thinking and problem-solving ideals that transcend math class. They are continually developed throughout our lives. We make forward progress, and sometimes we regress in practicing them. So do your children. They are reminders of skills that are critically important for an educated citizenry and for decision making. They are reminders that these skills also must be practiced in order to be learned; it’s not enough just to practice computations and algorithms. They also give a framework for understanding the teaching and learning of problem solving and how it goes far beyond simply teaching traditional and dreaded “word problems” in math class.
You’ve Got This!
