One unique aspect of the Common Core State Standards is that their focus goes beyond the familiar content of numbers, geometry, algebra, and statistics. They also include a set of Standards for Mathematical Practice, or SMP, that describe how people work when they're doing math.
The SMP apply across all grade levels, with kindergarteners operating at a level of sophistication appropriate to them and high school students working at a much more sophisticated level. The SMP list is fairly long — there are eight of them — and they overlap in ways that make it challenging for the average non‐math teacher to tell them apart. Fortunately, they can be boiled down to four simple statements about what students at all grade levels should be doing in math class: ask questions, play, argue, and connect.
Ask many questions
Students should ask questions such as, "What if . . . ?", "Why?" and "How do we know that?" They should also seek to answer these questions. These may not be the questions that you picture students asking in math class, but they're essential questions for learning more math.
Play to learn
When children play, they make things up and try things out. They don't worry about getting everything perfect. They repeat the same scenario many times, changing it a little bit each time to see what happens. They challenge themselves. They laugh.
All of this can happen in the math classroom, too. Math is challenging, but so are handstands, video games, and soccer. All of these activities involve risk‐taking and exploration. Math should too. Often, the line between play and work is drawn with consequences. If an activity has high stakes, it isn't so much fun and turns into work. A Common Core classroom has many opportunities for students to play with math: to try something new, to create challenges for themselves and others, and to get things wrong and try again.
Argue to uncover truths
Arguing is a highly mathematical activity. A good argument has some agreed‐upon starting point, has some rules for moving forward, and seeks to uncover the truth. In a Common Core classroom, students have to figure some things out for themselves, which means that they need to formulate an argument to support their thinking. The sophistication of these arguments increases as students age and as they gain more practice.
For example, in second grade, a student might need to convince someone else that 14 is an even number. In high school, a student might need to write a proof that the sum of the angle measures of any quadrilateral is 360 degrees. Both of these activities count as arguing.
Connect facts and ideas
Math is often taught as a long list of disconnected facts, but it shouldn't be. Mathematical ideas are connected to each other, and they're easier to use and to remember when students see connections among them.
Learning multiplication facts — an activity that is rich with connections — sometimes boils down to a conditioned response activity. Math isn't memorizing. To be sure, being able to quickly remember multiplication and addition facts is useful, but an overemphasis on memorization can keep many students from developing the even more useful skill of thinking through things they don't know right away.