Although George Brown and Robert Quinn provided research evidence showing that the introduction of rational numbers should occur during a first-year algebra course, the authors of the Common Core State Standards for Mathematics (CCSS-M) apparently did not agree. The introduction of rational numbers occurs in grade 6, which is at least two years before any first-year algebra course, assuming that a first-year algebra course does not occur until high school.
Grade 6: Ordering and Extending
After completing a careful treatment of fractions, rational numbers are introduced at grade 6 as an extension of the set of fractions. The CCSS-M states that students should “extend their previous understandings of number and the ordering of numbers to the full system of rational numbers, which includes negative rational numbers, and in particular negative integers. [Students should] reason about the order and absolute value of rational numbers and about the location of points in all four quadrants of the coordinate plane. (pg. 39)”
Fractions, during this discourse, are defined as the set of positive rational numbers. Grade 6 students explore whether the properties of fractions, which are similar to those of whole numbers and integers, extend to all rational numbers.
Grade 7: Operations
In grade 7 students perform operations with the complete set of rational numbers. First, all of the number sets (whole numbers, integers, fractions) are unified under the set of rational numbers. Then, students extend previously-developed properties of operations to the set of rational numbers. Students then begin to manipulate algebraic expressions and linear equations. The CCSS-M authors define the work to be completed at grade 7 this way.
Students develop a unified understanding of number, recognizing fractions, decimals (that have a finite or a repeating decimal representation), and percents as different representations of rational numbers. Students extend addition, subtraction, multiplication, and division to all rational numbers, maintaining the properties of operations and the relationships between addition and subtraction, and multiplication and division. (CCSS-M, pg. 46)
Grade 8: Moving Beyond Rational Numbers
Students, again prior to high school and presumably prior to a first-year algebra course, move beyond rational numbers to come to an understanding of irrational numbers. Students define irrational numbers and discuss some important examples, then learn how to approximate an irrational number using a rational number.
Question: Have the authors of the CCSS-M challenged the validity of the research claims put forth by Brown and Quinn? If so, can students be ready to define and work with rational numbers at sixth grade? Why or why not?
References
Brown, G., & Quinn, R. J. (2007). Fraction Proficiency and Success in Algebra: What Does Research Say? Australian Mathematics Teacher, 63(3), 23-30.
National Governors Association Center for Best Practices, Council of Chief State School Officers, (2010). Common core state standards for mathematics. Retrieved from website: https://www.thecorestandards.org/Math/
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