• # 13 Rules That Expire

By Karen S. Karp, Sarah B. Bush, and Barbara J. Dougherty, Posted July 14, 2014 –

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):

1. State the rule that teachers share with students.
2. Explain the rule.
3. Discuss how students inappropriately overgeneralize it.
4. Provide counterexamples, noting when the rule is untrue.
5. 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 place-value 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 middle-grades math teacher who is interested in relevant and engaging middle-grades 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 co-author of conceptual assessments for progress monitoring in algebra and an iPad® applet (MOTO) for K–grade 2 students to improve counting and computation skills.

 10. We often use the 'house' to show division of two numbers. When that symbolism is used, it is typically read as, for example, 4 'guzinta' (goes into) 8. When the same problem is written horizontally, such as 8 ÷ 4, it is read as 8 divided by 4. Reading the 'house' representation as 'goes into' often confuses students so that when they write it in horizontal form, they will write it as they say it: 4 ÷ 8. Reading the 'house' representation is 8 divided by 4 will reinforce the actual computation.Posted by: BarbaraD_89416 at 7/29/2014 12:18 PM

 You asked for other "rules that expire" that readers could think of. Here are two I thought of: "Line up the right hand side of the numbers to add them (in a column)" and "The denominator tells how many parts are in one whole." (Parts must be equal, and there is also the understanding that we can be speaking of a "whole group" which may have multiple objects)Posted by: LindaA_97041 at 8/4/2014 6:37 PM

 “Expired mathematical language” and suggested alternativesWhat is stated: Say the word "and" to separate places when reading a whole number (ex. read 187 as one hundred and eighty seven).What should be stated: Only use the word and to signify a decimal point (ex. read 187.5 as one hundred eighty seven and five tenths).While using the word "and" to only refer to the decimal point when reading a number is considered a “cultural convention” (Beckman, 2014, p. 18), and not a "rule" in mathematics, it is confusing to students when they start to work with decimal naming. Students are taught to read the decimal point as "and" to differentiate between the whole number and the decimal number. Students should recognize there is consistency in the way numbers are read (Van de Walle, Karp & Bay-Williams, 2013). Expiration date: Grade 5 (5.NBT.A.3).References:Beckmann, S. (2014). Mathematics for elementary teachers with activity manual (4th ed.). Boston, MA: Pearson.Van de Walle, J. A., Karp, K.S., & Bay-Williams, J. M. (2013). Elementary and middle school mathematics: Teaching developmentally (8th ed.). Upper Saddle River, NJ: Pearson Publications.Posted by: RebeccaH_18324 at 8/11/2014 11:30 AM

## Leave Comment

Auston Cron - 4/19/2019 9:40:19 AM

"Cancel" for dividing out common factors in any situation. Cancel should only be used when additive inverses are added and the result is zero. Yes, cancel has be used for over a century, yet it is very confusing for students when reducing/simplifying fraction operations. I was taught this in graduate (Teaching of Algebra) school 45 years ago by Dr. Rolands at TAMU; I have practiced it ever since and it work. He got doctorate during University of Illinois math projects in the 1960s.

Ginger Riddle - 7/26/2018 8:47:25 AM

FOIL. This mnemonic device teachers use to try to help students understand the distributive property fails once the student tries to multiply trinomials.

Abbie Arlinghaus - 9/30/2015 1:31:52 PM
A lot of times in earlier grades, students are taught remainders. Then in later grades, like middle school for example, students begin to learn fractions or decimals, and remainders are usually forgotten. I think students should be given a reason why we teach remainders, and what they actually represent.