Rigor

Kindergarten Mathematics

Rigor

All students deserve access to rigorous mathematics. But what is rigor? Linda Gojak's, ideas to the right highlight what rigor is and what it is not in mathematics. Rigor is a balance between procedure, concept, and application of mathematics. Her NCTM's President Corner article Links to an external site. sheds more light on rigor in mathematics.

Rigor is directly related to the tasks we use with our students. High quality tasks:

Align with relevant mathematics content standards.
Connect previous knowledge with new learning.
Encourage the use of representations.
Provide opportunities for students to develop and demonstrate the mathematical practices.
Promote reasoning and problem solving.
Allow multiple entry points (All students can begin the task. Task can be extended).
Allows for multiple solution approaches and strategies.
Engages students in explaining the meaning of the result.
Includes relevant and interesting context.

Increasing Rigor through Good Questions

We can also enhance rigor through the good questions we ask. Each standard on this site has a collection of good questions to increase rigor. Other resources for good questions listed below.

Good Questions for Math Teaching

Why Ask Them and What to Ask

by Peter Sullivan

Good Questions

Great Ways to Differentiate Mathematics Instruction

by Marian Small

Developmentally Appropriate Practice

What is Developmentally Appropriate Math?

"Perhaps the most common criticism of the Common Core State Standards-Mathematics (CCSS-M) for young children is that they are not “developmentally appropriate” (e.g., Meisels, 2011). Unfortunately, the phrase “developmentally appropriate” too often functions as a Rorschach test for whatever a person wants to see or argue against.

Often, negative evaluations are based on an implicit acceptance of the view that all “fives” can and especially cannot do certain things. However, much of the mathematical thinking that some people say “cannot be done” until age 7 (or whatever) can be learned by children—most children—in high-quality environments. Further, children learn such thinking with understanding and joy—that’s developmentally appropriate.

Let’s consider some concrete examples. One concern is that 5-6-year-olds are not “ready” to learn place value. Perhaps the phrase itself—“place value”—raises the issue. Close inspection, however, reveals little reason for worry. First, note that research has identified at least seven developmental levels of learning place value, from very early concepts of grouping to understand the exponential nature of number systems in multiple bases (Clements & Sarama, 2014; Fuson, Smith, & Lo Cicero, 1997; Fuson, Wearne, et al., 1997; Rogers, 2012). Examination of the CCSS-M shows that kindergarten children only need to “Work with numbers 11–19 to gain foundations for place value” (p. 12, emphasis added) and first graders “Understand that the two digits of a two-digit number represent amounts of tens and ones” such as knowing that “The numbers 10, 20, 30, 40, 50, 60, 70, 80, 90 refer to one, two, three, four, five, six, seven, eight, or nine tens (and 0 ones).” Those are challenging but (for vast majority of children) achievable understandings (did you notice how many times the CCSS-M’s goals involve “understanding”)?

Personally, I have many concrete experiences with preschoolers who, given high-quality learning experiences, successfully tackle these ideas and more (Clements & Sarama, 2007, 2008). And love doing it. In Boston, a mother said she wasn’t sure her preschooler could understand mathematical ideas until he told her, “Eleven. That’s just ten and one, isn’t it?”

Talking about the “levels” of place value brings up a two important points. First, when educators use such levels—organized in a learning trajectory—to engage all children in meaningful mathematics at the right level for each—developmental appropriateness is ensured. Second, the Common Core was developed by first writing learning trajectories—at least the developmental progressions of levels of thinking. (Criticisms that the CCSS-M were “top-down,” starting with high school, e.g., Meisels, 2011, are simply incorrect.) Thus, learning trajectories are at the core of the Common Core."

- Adapted from "What is Developmentally Appropriate Math" by Douglas Clements

Retrieved from preschoolmatters.org Links to an external site.

"Helping Others Understand Academic Rigor in Teachers' Developmentally Appropriate Practices Links to an external site.," The National Association for the Education of Young Children