12 Math Rules That Expire in the Middle Grades

  • 12 Math Rules That Expire in the Middle Grades

    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.

    2015-11-02 Karp 

    Karen S. Karp, kkarp1@jhu.edu, 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. 


    2015-11-02 Bush  

    Sarah B. Bush, sbush@bellarmine.edu, 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.



    2015-11-02 Dougherty

    Barbara J. Dougherty, barbdougherty32@icloud.com, 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.




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    Rachel Archie-Haliburton - 5/16/2017 10:58:21 AM

    The "plus sign" and the "minus sign" being applied as addition or subtraction no matter what is one of the first difficulties to overcome for many students in the middle grades.

    Scott Brown - 2/10/2016 2:13:06 PM
    You state correctly the acronym FOIL expires in high school, when binomials are multiplied by trinomials. I would agree with Usiskin's emphasis on applying the "extended distributive property." Reason? I just had two university students state 2x+14+y=2(x+7)+y with the accompanying explanation, "foil out the 2."

    Erin-Rose Schneider - 1/13/2016 10:43:30 PM
    "The Alligator eats the bigger number" when working with inequalities sends me over the edge. I have to break my honors geometry kids of this.

    Alicia Luna - 11/28/2015 7:27:01 AM
    I agree with Ryan Stones regarding the usage for "point". I discuss with my students both the "proper" way to read a decimal number and the "common" way to read it by using the word "point". Speaking of decimals, another rule that needs to expire is "when multiplying by a power of ten, add zeros to the end of the number". While this rule works for whole numbers, it does not work for decimal numbers. For example, 5.7 x 1,000 does not equal 5.7000. I understand that, when students first start multiplying by powers of ten, they have not yet been exposed to decimal numbers. I suggest that elementary school teachers explain that, when multiplying by a power of ten, the digits move, so that we need to add zeros as placeholders. The result of multiply 4 by 100 is that 4 "ones" become 4 "hundreds". Moreover, when students start to work with decimal numbers, the rule stays the same: the digits move when multiplying by powers of ten. The decimal point doesn't really move (this is another "rule" that should not be established); it only looks as if the decimal point has moved.

    Anna York - 11/11/2015 3:48:42 PM
    When I taught grades 6, 7, 8 and Algebra I (private school), we had a "retirement party" in 6th grade for the x symbol as a multiplication sign. It was a great way to help students make the jump to using x as a variable and using different representations for multiplication. We only "retired" x because we would still use it in scientific notation. A rule I have run across that I would like to expire is KCC or "keep, change, change" when subtracting with integers. I stress to my students that you "add the opposite" which is the mathematical process . I think it is also problematic to tell students that you can determine which way the ray for graphing an inequality should go by matching the direction of the symbol with the shape of the arrow on the number line. It only works if the variable is on the left side which is not always the case. I also agree with Lori Burch about "cross multiply." If I had a quarter for every time my 8th graders and Algebra 1 students (boo-hoo) tell me that you multiply fractions by cross multiplying, I could retire tomorrow!

    Ryan Stones - 11/11/2015 3:28:23 PM
    First, in the “Expired language” section, I would disagree with the comment on the usage for “point”. In the elementary school phase, it makes sense to say 3.4 as “three and four-tenths” because of the goal of preparing them for fractions. But at higher levels--certainly by high school--“three-point-four” is no more mysterious, incomprehensible, or potentially fraught with pitfalls than the numerical expression itself “3.4”. At that stage of the game, it is ubiquitous that “three-point-four” means 3.4. In fact, I would say that the recommendation to say “three and four-tenths” itself expires by high school: there is no way I am going to read 0.1453948 as a fraction to my students in class. I’m saying “point-blah-blah-blah” or perhaps even “zero-point-blah-blah-blah.” Second, also in the “Expired language” section, I had to laugh out loud when I saw “Reducing fractions” replaced with “simplifying fractions” or possibly “fraction in simplest form”. I sat through a discussion led by educators and test-makers backing Common Core where those leading the discussion were adamant in excising the word “simple” and its derivatives (“simplest”, “simplify”, “simplifying”) from the curriculum on the basis that the word was too vague, was overused, was too context-dependent, and was imprecise. They actually followed a test-item drafting rule that required them to never use a derivative of “simple” in the prompt of a test question. (E.g.: “simplify the polynomial expression” needed to be changed to (the more wordy) “write the polynomial expression in standard form” and—which is why I laughed—“simplify the fraction” needed to be changed to “write the reduced fraction” or “write the fraction in lowest terms”.) I laugh because it seems that not all math reformers are on exactly the same page, apparently. Lastly, in a comment to the article by Sue Mascioli –11/3/2015 11:49 AM, Sue suggests that remainders disappear after elementary grades and that division be taught with the requirement to express remainders as fractions. As a Secondary Math 3/Math 1050/Pre-Calculus instructor, I would add that remainders reappear later in polynomial division: in long division, in the remainder theorem (the functional value is equal to the remainder of the polynomial divided by the linear factor that accompanies the root), and in the synthetic division algorithm that is useful for locating roots of higher degree polynomials. It is very useful to demonstrate to students that polynomial division works EXACTLY THE SAME as the long division they did back in elementary school, even up to the choice of how to write a remainder—as a remainder itself, or as a fraction with the divisor as the denominator. So in my experience, I would say that both methods of writing remainders should continue to be taught from the beginning—the remainder concept doesn’t expire like Sue thinks it does; it just takes a while before it resurfaces later in polynomial division.

    Dorothea Steinke - 11/9/2015 10:27:02 AM
    A pair of "rules" that were missed: Multiplication always makes the answer bigger. Division always makes the answer smaller. This understanding only applies only for values greater than 1. When multiplying or dividing by decimals or fractions less than 1, students may fail to accept the answer because it is less than the number they started with in multiplication, or greater than the number they started with in division. One way to correct these "rules" is to return to concrete, physical examples (using objects which students can manipulate) before asking students to draw representations of the multiplication and division concepts. When the concrete and representational stages are solid, then students can accept the abstraction of the concepts.

    Lori Burch - 11/4/2015 8:51:43 PM
    #3 Absolute Value My students have the wrong idea that “absolute value makes everything positive.” So when we’re solving absolute value equations, they automatically want to change |x – 3| to x + 3. Why? "Because absolutely makes everything positive.” This is actually an order of operations misunderstanding as well. #5 PEMDAS In addition to thinking that M comes before D and A before S and not understand the P to mean grouping symbols, in general. They forget that parentheses can be used for multiplication and not just grouping. For example, I have students who want to call “P Parentheses” as the first operation in simplifying a problem like 2+3(7). Yes, there are parentheses, but they aren’t the P in PEMDAS. I have seen GEMS to help students with the hierarchy as well. With this model, multiplication implies division and subtraction implies addition. Additional "rules" that expire: REDUCING FRACTIONS: “Cross out what is the same.” This approach is a problem when students start simplifying algebraic rational expressions like. They want to cross out the terms “because they match.” Instead of factoring and allowing the division to cancel common factors. MIXED NUMBERS are PREFERABLE to IMPROPER FRACTIONS: While mixed numbers definitely help students reason about the value of an improper fraction, when it comes to performing operations on fractions, improper is definitely the more useful form, and decimal approximations are rarely used. REMAINDERS AS DECIMALS: The fraction form of a remainder is much more valuable in algebra than a decimal approximation is. CROSS MULTIPLY: Students overgeneralize the “cross multiplying” concept and want to use it all the time, especially with multiplying fractions. INEQUALITIES: The alligator mouth eats the bigger value. This lets them get by in lower grades without actually learning how to read < and > symbols.

    Brady Ward - 11/4/2015 3:02:15 PM
    "You can't take the square root of a negative number." I try to be careful to clarify, "In the real number system, the square root of a negative number is undefined," or, "the equation x^2 = -k (where k is positive) has no real number solution."

    Emily Westerling - 11/3/2015 2:26:10 PM
    That is a great point Justin, and a great resource as I am going to start a unit on order of operations and combining like terms next week. Making a point to not use that language in our middle school classrooms is the first step in establishing our new rules. Not that the old rules have "expired", just that they have questionable value when we really are trying to understand mathematics. We want them to be mathematics investigators, and putting a new hat on their heads from the start will help combat some previous tricks they may have memorized.

    Justin Butler - 11/3/2015 12:11:43 PM
    If you return to the March 2011 issue, it contains an article titled "The truth about PEMDAS". This article advocates an alternative method of teaching PEMDAS that I believe will help prevent some of the misconceptions about it, that would later cause this methodology to expire. I think this article provides great insight into the misconceptions that are common at the middle grades level, but I fear that teaching students that rules "expire" could have a negative impact on their interest and confidence in the subject. Instead I think it is our job to teach concepts rather than rules because concepts can be built on. As middle level educators we lack control of students' prior knowledge, yet I believe we should work to build relationships with teachers from lower grade levels to help ensure that statements such as “yours is not to reason why, just invert and multiply" are not used as an easy way out for teachers.

    Sue Mascioli - 11/3/2015 11:49:11 AM
    I too dislike teaching rules and language to elementary students that expire later. One example is using "number" when actually referring to a digit (guilty myself!). Another is using "times" instead of "multiply", as when a student says something like, "I times 3 by 4" instead of "I multiply 3 by 4." Should we consider reading "3 x 4" as "3 multiplied by 4" instead of "3 times 4"? Maybe too fussy there. Another question I have is about teaching division with remainders. After elementary grades the remainder just disappears. Perhaps division could be taught right from the beginning to express any remainders as fractions. For example, 16 divided by 5 would result in 3 and one-fifth. Then when decimals are introduced they can be written as decimals. Just some thoughts.

    John Golden - 11/2/2015 9:33:43 AM
    Do you know about Nix the Tricks from Tina Cardone and the #MTBoS? It's a free book with teacher generated list of ideas like this, now in 2nd edition. See http://nixthetricks.com/ or on Twitter https://twitter.com/hashtag/nixthetricks?f=tweets&vertical=default&src=hash.

    Carlos Castillo-Garsow - 11/2/2015 1:33:15 AM
    "Parentheses means multiply" expires with function notation, and is nearly impossible to fix. Rise over run. Slope of 3/4 means "over four and up three" or "over one and up 3/4" definitely expires with Calculus, but also expires with step functions, average rate of change for non-linear functions, or point-slope form done with any sort of meaning at all.

    Kathleen Traylor - 11/1/2015 7:53:37 PM
    To get students to understand that PEMDAS has "expired" in middle school, I use the analogy of Crayola crayons. In first grade, you were only allowed the 8 basic colors: red, orange, yellow and so on. As you progressed in school, you were introduced to finer gradations of color: red-orange, tangerine and orange-yellow. Just as the "P" in PEMDAS now represents many grouping symbols, including the radical and the fraction bar (which also indicates division...), and "E" includes roots as well as exponents. This seems to help them understand "expiration!"