By Karen S. Karp, Sarah B. Bush, and Barbara J. Dougherty, posted November 2, 2015 –

Ed. note: In the November 2015 issue of Mathematics Teaching in the Middle School, authors Karen S. Karp, Sarah B. Bush, and Barbara J. Dougherty initiated an important conversation in the middle-grades mathematics education community. We are dedicating this discussion space as a place where that conversation can continue.

In our article, “12 Math Rules That Expire in the Middle Grades,” we point out 12 rules commonly taught in middle-grades mathematics classes that do not hold true over time; in fact, these rules “expire.” For example . . .

Rule 1. KFC: Keep-Flip-Change

When learning to divide fractions, students are sometimes taught to KFC (Keep-Flip-Change) or told “Yours is not to reason why, just invert and multiply.” Although both versions align with the standard algorithm, students might overgeneralize this rule to other operations with fractions. Additionally, these mnemonics and sayings do not promote conceptual understanding, making it challenging for students to apply them in a problem-solving context. Instead, division of fractions can be linked to whole-number division by asking how many groups of the divisor make up the dividend. Although students will eventually use the algorithm, they should gain a conceptual understanding of dividing fractions through the use of physical models or other methods, such as the common denominator strategy. Expiration date: Grade 6 (6.NS.1)

See the article for the other rules that expire.

We also provide instances of expired language and notation. For example, using the term “reducing fractions” may cause students to think that the fraction value is getting smaller. Instead, we should use the term simplifying fractions, or instruct students to write the fraction in simplest form or in lowest terms.

Similarly, plugging in a value for a variable is not a mathematical term. Instead, the language used should be substitute a value.

See table 2 (p. 214) in the published article for additional examples.

Use the comment section that follows this blog post to submit additional instances of “rules that expire” or expired language that our article does not address. If you share an example, please use the format of the article:

1. State the rule that has been shared with students.

2. Discuss how students overgeneralize the rule.

3. Provide counterexamples, noting when the rule is untrue or unhelpful.

4. State the “expiration date” or the point when the rule begins to fall apart for many learners.

If you submit an example of expired language that was not in the article, include both “What is stated” and “What should be stated.”

By building a schoolwide plan for the consistent and precise presentation of rules, terminology, and notation used by all teachers, students will never find that something in their past instruction is no longer accurate. As we avoid these 12 Rules That Expire, we instead find ways to present a seamless and logical world of mathematical ideas.

Karen S. Karp, [email protected], is a visiting professor at Johns Hopkins University in Baltimore, Maryland. She is professor emeritus at the University of Louisville in Kentucky, a past member of the NCTM Board of Directors, and a former president of the Association of Mathematics Teacher Educators. Her current scholarship focuses on teaching interventions for students in the elementary and middle grades who are struggling to learn mathematics. 

Sarah B. Bush, [email protected], an associate professor of mathematics 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, [email protected], a research professor for mathematics education at the University of Missouri–Columbia, is a past member of the NCTM Board of Directors and the editor for the Putting Essential Understandings into Practice series. She is a co-author of conceptual assessments for progress monitoring in algebra and curriculum modules for middle school interventions for students who struggle.