Switching to any new standards is hard, but the CCSS make such a transition even more challenging. For starters, they aim higher than previous standards. They have a new organizational structure. And they call for different practices than past standards. The first year or two with new standards is particularly challenging — as many students encounter unfamiliar content and methodology for the first time.
Understanding Key Practices
The Standards for Mathematical Practice are an essential part of the Common Core State Standards. They describe the attributes of mathematically proficient students. These standards don’t just describe how students should use mathematics: they also provide a vehicle through which students engage with and learn mathematics. The practices draw from the NCTM Process Standards of problem solving, reasoning and proof, communication, representation, and connections and the strands of mathematical proficiency specified in the National Research Council’s report, “Adding It Up: adaptive reasoning, strategic competence, conceptual understanding, procedural fluency, and productive disposition.”
As students move from elementary school through high school, the Standards for Mathematical Practice remain the same. What changes is the way these standards look as students engage with and master new and more advanced mathematical ideas. The mathematical practices must be taught as carefully and practiced as intentionally as the Standards for Mathematical Content. Neither should be isolated from the other; impactful mathematics instruction occurs when these two halves of the CCSS come together in a powerful whole.
Understanding Key Concepts
If students ever did get by with memorization alone, they can no longer do so. Even proficiency with numerical algorithms is built on understanding. Research shows that difficulties in working with rational numbers can often be traced to weak conceptual understanding. The Common Core State Standards stress “conceptual understanding of key ideas” and call for “continually returning to organizing principles such as place value or the properties of operations to structure those ideas.” The CCSS stress not only procedural skill but also conceptual understanding, to make sure students are learning and absorbing the critical information they need to succeed at higher levels.
Identify and Rectify Misconceptions
The power of targeting misconceptions is made clear by research. All pupils constantly ‘invent’ rules to explain the patterns they see around them. Because these rules are based on limited examples, they are often flawed. No matter how we teach, students will form some misconceptions, many of which will remain hidden unless the teacher makes specific efforts to uncover them.
Some misconceptions — such as thinking that zero doesn’t matter in a number — are easy to spot. Yet, the more ingrained some misconceptions become the more difficult it is for students to let go of them. Other misunderstandings, such as thinking that variables represent objects rather than quantities, can be more difficult to identify. For example, students are prone to translate the phrase “one week equals seven days” into the equation w = 7d. They are confusing the unit ‘week” with the quantity “number of weeks.” This can lead to absurd results, which can reinforce the misguided notion that algebra doesn’t make sense.
Knowledge of the common mathematical errors and misconceptions can provide an insight into student thinking and a focus for teaching and learning
Help Students Become Comfortable with CCSS Content and Methods
CCSS calls for teaching new content areas as well as teaching existing topics earlier. To help students meet unfamiliar content and methods, connect such material to what students already know. Research tells us that students learn better when new knowledge is connected to things that they already know and understand. For example, rather than teach percents as something entirely new, students see that percent is simply another name for hundredth, a concept very familiar to them.
In a similar vein, students see algebraic notation not as something totally foreign, but as an extension of the arithmetic they already know. Algebraic symbols are an abbreviated and efficient way to describe and relate changing quantities. Students can see how the symbols can even represent things in their everyday lives.
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