The Common Core State Standards for Mathematics have delivered on the promise to provide math teachers in the United States standards that are clear, specific, and focused. As I support the implementation of the Math Standards within my district, I look for three major changes that are happening within our planning of lessons and units, daily teaching practices, and assessment of students – all of which are guided by our materials.
Shift 1. Focus strongly where the Standards focus.
It is important that the materials we are using ensure that students learn all of the content for their given grade level. We must focus deeply on the major work of that grade so that students can build strong mathematical foundational skills. Within a given grade not all content is emphasized equally in the Standards. For example, in grade 2, materials should:
- provide extensive work using addition and subtraction to represent and solve problems,
- build understanding of place value,
- include instruction on how to use properties of operations to measure and estimate lengths in standard units and solve problems using lengths, and
- help allocate class time, and reflect the fact that 65-85% — but closer to 85% — should be devoted to the Major Work of the Grade to allow students time to develop their understanding and master the skills necessary for the challenges of later grades.
The emphasis in grades K–5 is on numbers and operations. The emphasis in grades 6–8 is on pre-algebra, algebra, and functions. Two important documents that complement the Standards are the Focus by Grade Level document for K-8 math teachers and the Widely Applicable Prerequisites for high school math.
Shift 2. Coherence: Think across grades and link to major topics within grades
It is important that our materials allow teachers to design lessons and units that carefully connect new content and skills to those learned earlier in the year or in previous grades. This way, each new standard is an extension of previous learning. For example, in grade 3, fractions are introduced as a new unit; instead of counting by ones, students can count by sixths. In grade 4, students learn to add those sixths. In grade 5, students add and subtract fractions with unlike denominators by replacing them with equivalent fractions to produce an equivalent sum or difference. A strong, well-designed textbook will relate grade-level concepts intentionally to previous knowledge from earlier lessons during the year and from earlier grades. In grade 8, students are expected to understand and apply the Pythagorean Theorem. A well-aligned high school geometry textbook will connect the equation of a circle with the distance formula and the Pythagorean Theorem.
Reading the Progressions Documents for the Common Core Math Standards might be helpful in understanding why the Math Standards are ordered in this way.
Shift 3. Rigor: In major topics, pursue with equal intensity: conceptual understanding, procedural skill and fluency, and application.
Now that we have a focused set of standards, our students are given ample time and opportunity to develop conceptual understanding. Learning mathematics will no longer be about relying on meaningless procedures or mnemonics. The Standards also call for speed and accuracy in calculation. As students have time and practice problems within the Major Work of their grade, they will build their procedural skill and fluency. Students also need to see the mathematics they are learning in problem-solving contexts. It is important that we provide opportunities for students to build a strong conceptual knowledge and procedural fluency background while continually applying the mathematics with each grade. For example, the 8th grade problem below illustrates how teaching procedures alone is not enough for students to be able to apply math concepts to both conceptual and application problems.
Using the equation P=2t +2, find values for P when t = 0,1,2,3, and 4.
Using the equation P=2∙2t , find values for P when t = 0,1,2,3 and 4.
This carefully vetted task and other resources for teachers can be found on https://www.illustrativemathematics.org/content-standards/8/EE/A/1/tasks/395.