Standards, in any setting, are intended to ensure quality, indicate goals, and promote change (NCTM, 1989). The more I read about misinterpretations of mathematics curriculum standards—be they the Common Core State Standards for Mathematics (CCSSM), the Texas Essential Knowledge and Skills (TEKS), Virginia’s Standards of Learning (SOL), or any other set of mathematics standards—the more I believe that the differences between standards and curriculum are not understood.
Standards are expectations. Expectations for all learners! Consider them as learning targets. What a district, school, or teacher does is to determine how such expectations are to be organized, sequenced—both within and across grade levels—addressed, and eventually achieved by each student. These important school- and classroom-based considerations include:
- determining the prerequisite and developmental needs of the students as they embark on daily mathematics journeys
- the relative importance of the standard, which will help define the number of lessons needed to address a particular standard or cluster of related standards
- the supporting materials (e.g., number lines, fraction tiles, area drawings) that may be needed
- how formative assessment techniques will monitor the progress of students as they engage, every day, in learning the mathematics defined by a standard or cluster of related standards
THIS is curriculum. And whether it’s related to equivalent fractions or adding multi-digit whole numbers, HOW a particular standard is addressed becomes the province AND responsibility of the classroom teacher.
As suggested above, one particular standard may require a limited number of days of instruction, while another standard or cluster of related standards may require a week or more. Some standards will require a depth of understanding different from others. Some outcomes are skill-related and others demand conceptual expertise. Regardless of the instructional intent, students should be engaged – every day – in actually doing mathematics whether that’s solving problems, reasoning, modeling with mathematics, and/or using representational tools, manipulatives, or software.
The intent here and in the example below is to recognize that whatever set of standards a state, district, or school adopts, the standards are not the curriculum. The curriculum is defined every day as particular standards and related clusters of standards are addressed in the classroom based on the needs of the mathematics learners and in the decision-making of classroom teachers as they strive to find ways to engage students in doing mathematics and developing the critical habits of mind that will move them toward becoming successful and confident mathematics learners.
Consider the following example:
Grade 3: Understand division as an unknown-factor problem. For example, find
32 ÷ 8 by finding the number that makes 32 when multiplied by 8.
- An understanding of the relationship between multiplication and division
- Prior experience with multiplication and related division facts
Thinking about division and solving division problems using unknown factors connects the operations of multiplication and division.
Engaging Students in Doing Mathematics
- Teach one or two lessons directly targeting this standard, but a “building block” lesson will be the foundation for students as they regularly use missing factors to determine quotients in multiplication and related division fact examples and problems. The regular use of unknown factors in thinking about both multiplication and division involves the structure of mathematics.
McDaniel College Professor Dr. Francis “Skip” Fennell, the inaugural recipient of the L. Stanley Bowlsbey Chair in Education and Graduate and Professional Studies, is an internationally renowned expert in mathematics education and DreamBox advisor who developed the first graduate program for elementary mathematics teachers in Maryland. You can reach him via email at firstname.lastname@example.org or on Twitter @SkipFennell.