Archive for December 2011
For teachers who are proficient at navigating their current state standards, the Common Core Standards may have a completely different look. In the area of mathematics, the content of the document is divided into domains, clusters, and standards. Domains are the overarching term and refer to a large group of related standards. Clusters are groups of related standards. Because mathematics is a connected subject, standards from different domains and clusters may be closely related. Finally, the standards define what students should understand and be able to do. An example from fourth grade would be: Domain – Operations and Algebraic Thinking; Clusters – Use the four operations with whole numbers to solve problems (three separate standards), Gain familiarity with factors and multiples (one standard), and Generate and analyze patterns (one standard).
Within the document, each grade level does not necessarily have the same domains or number of domains as the preceding grade. However, the standards are aligned vertically from one grade to the next. Unlike some current state standards, the Common Core State Standards do not spell out every step in the instructional process and some do not have example problems. Therefore, educators are going to have to investigate the new standards thoroughly and translate them into new instructional practices.
“Sum” thing new to think about!
For the last several months, I have been privileged to work with hundreds of teachers as they unpack the Common Core State Standards. In each session, which have been held in grade level groups, we have worked through the meanings of the 8 Standards for Mathematical Practice and each time one point becomes extremely obvious. These eight standards, or measures of student behavior, are inextricably linked. It is no surprise really, as they were developed using information from the National Council of Teachers of Mathematics (NCTM) Process Standards and the National Research Council’s (NRC) Strands of Mathematical Proficiency. The point is driven home, however, when teachers work through the process of identifying for themselves what each standard means and does not mean for their classroom.
NCTM’s five process standards include:
- Build new mathematical knowledge through problem solving
- Solve problems that arise in mathematics and in other contexts
- Apply and adapt a variety of appropriate strategies to solve problems
- Monitor and reflect on the process of mathematical problem solving
Reasoning and Proof
- Recognize reasoning and proof as fundamental aspects of mathematics
- Make and investigate mathematical conjectures
- Develop and evaluate mathematical arguments and proofs
- Select and use various types of reasoning and methods of proof
- Organize and consolidate their mathematical thinking through communication
- Communicate their mathematical thinking coherently and clearly to peers, teachers, and others
- Analyze and evaluate the mathematical thinking and strategies of others;
- Use the language of mathematics to express mathematical ideas precisely.
- Recognize and use connections among mathematical ideas
- Understand how mathematical ideas interconnect and build on one another to produce a coherent whole
- Recognize and apply mathematics in contexts outside of mathematics
- Create and use representations to organize, record, and communicate mathematical ideas
- Select, apply, and translate among mathematical representations to solve problems
- Use representations to model and interpret physical, social, and mathematical phenomena
The five stands of mathematical proficiency are:
(1) Conceptual understanding refers to the “integrated and functional grasp of mathematical ideas”, which “enables them [students] to learn new ideas by connecting those ideas to what they already know.” A few of the benefits of building conceptual understanding are that it supports retention, and prevents common errors.
(2) Procedural fluency is defined as the skill in carrying out procedures flexibly, accurately, efficiently, and appropriately.
(3) Strategic competence is the ability to formulate, represent, and solve mathematical problems.
(4) Adaptive reasoning is the capacity for logical thought, reflection, explanation, and justification.
(5) Productive disposition is the inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy. (NRC, 2001, p. 116)
When you combine the two you end up with these 8 Standards for Mathematical Practice:
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
As teachers begin to discuss each of these eight standards they see how the process standards and mathematical proficiencies are interwoven throughout. Examples of what that looks like for their classrooms and their practice can be found on videos at: http://www.insidemathematics.org/index.php/common-core-standards
“Sum” thing to think about!