In
the August 2014 issue of Teaching Children Mathematics, authors Karen S. Karp, Sarah B. Bush,
and Barbara J. Dougherty initiated an important conversation in the elementary
mathematics education community. We are dedicating this discussion space as a
place where that conversation can continue.
In
their article, “13 Rules That Expire,” the authors point out thirteen math rules commonly taught in
the elementary grades that no longer hold true in later grades; in fact, these
rules “expire.” For example—
Rule 1: When you
multiply a number by ten, just add a zero to the end of the number.
This
rule is often taught when students are learning to multiply a whole number
times ten. However, the rule is not true when multiplying decimals (e.g., 0.25
× 10 = 2.5, not 0.250). Although the statement may reflect a regular pattern
that students identify with whole numbers, it is not generalizable to other
types of numbers. Expiration date: Grade 5 (5.NBT.2).
See
the article for the other
rules.
Using
the comment section that follows this blog post, submit additional instances of
“rules that expire” or expired language that the article does not address. If
you would like to share an example, please use the format of the article (as
listed below):
 State the rule that teachers share with
students.
 Explain the rule.
 Discuss how students inappropriately
overgeneralize it.
 Provide counterexamples, noting when the rule
is untrue.
 State the “expiration date” or the point when
the rule begins to fall apart for many learners. Give the expiration date in
terms of grade levels as well as CCSSM content standards in which the rule no
longer “always” works.
If
you submit an example of expired language that was not in the article, include
“What is stated” and “What should be stated” as shown in the table below (for
additional examples, see table 1 in
the published article.
Expired mathematical language and suggested
alternatives 
What is stated 
What should be stated 
Using the words borrowing or carrying when subtracting or adding,
respectively 
Use trading or regrouping to indicate the actual action
of trading or exchanging one placevalue unit for another unit. 
Using the phrase ___ out of
___ to describe a fraction—for example, one
out of seven to describe 1/7 
Use the fraction and the attribute. For example, say the length of the string. The out of
language often causes students to think a part is being subtracted from the
whole amount (Philipp, Cabral, and Schappelle 2005). 
Using the phrase reducing fractions 
Use simplifying fractions. The language of reducing gives students the incorrect
impression that the fraction is getting smaller or being reduced in size. 

Karen S. Karp, karen@louisville.edu, a professor of math education at the University of Louisville in Kentucky, is a past member of the NCTM Board of Directors and a former president of the Association of Mathematics Teacher Educators. Her current scholarly work focuses on teaching math to students with disabilities. 

Sarah B. Bush, sbush@bellarmine.edu, an assistant professor of math education at Bellarmine University in Louisville, Kentucky, is a former middlegrades math teacher who is interested in relevant and engaging middlegrades math activities. 

Barbara J. Dougherty is the Richard Miller Endowed Chair for Mathematics Education at the University of Missouri. She is a past member of the NCTM Board of Directors and is a coauthor of conceptual assessments for progress monitoring in algebra and an iPad^{®} applet (MOTO) for K–grade 2 students to improve counting and computation skills.
