Certain mathematical “rules” expire
and other imprecise mathematical language to avoid
A coherent progression of concepts both within and across grade levels is one of the major shifts
represented in the design of the CCCS. Learning is connected in careful trajectories so that
students can build new understandings onto existing foundations.
It’s critical that when we’re developing lessons, we’re setting students up for future success by
making sure nothing we teach them now will prove untrue later.
Sometimes, in the effort to clarify or simplify, we teach students rules, tricks, or definitions or use
language that inadvertently undermines a student’s later ability to tackle new concepts and that
breed misconceptions that must be undone.
Some examples of “rules” that expire include:
1) “When you multiply a number by 10 just add a zero to the end of the number.”
2) “Use keywords to solve word problems.”
3) “You can’t take a bigger number from a smaller number.”
4) “Addition and multiplication make numbers bigger.”
5) “Subtraction and division make numbers smaller.”
6) “You always divide the larger number by the smaller number.”
7) “Two negatives make a positive.”
8) “Multiply everything inside the parentheses by the number outside the parentheses.”
9) “Improper fractions should always be written as a mixed number.”
10) “The number you say first in counting is always less than the number that comes next.”
11) “The longer the number, the larger the number.”
12) “Please Excuse My Dear Aunt Sally.”
13) “The equal sign means ‘find the answer’ or ‘write the answer.’”
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Other imprecise language to avoid:
What teachers say or do …
The problem…
What should be said…
“Multiplication is repeated addition.”
When multiplying rational numbers
such as 12 x 13 , it is not helpful to
think of multiplication as repeated
addition. Believing multiplication is
repeated addition will hinder some
students from developing
multiplicative reasoning.
“One ​
strategy​
we can use to solve
some​
multiplication problems is
repeated addition.”
Use the words “borrowing” or
“carrying” when subtracting or
adding respectively.
Using the words “borrowing” or
“carrying” promotes a digit approach
to subtracting or adding rather than
place value understanding.
Use “trading” or “regrouping” to
indicate the actual action of trading
or exchanging one place value unit
for another unit.
Use the words “reducing fractions”
The language “reducing” gives
students the incorrect impression
that the fraction is getting smaller or
being reduced in size.
Use “simplifying fractions.”
Ask how shapes are
similar when students are
comparing a set of shapes.
By using the word “similar” in these
situations, there can be eventual
confusion with the mathematical
meaning of “similar” that will be
introduced in middle school relating
to geometric figures.
Ask, “How are these shapes the
same? How are the shapes
different?”
Read the = as “makes” (For
example, “2 + 2 makes 4” for 2
+ 2 = 4.)
The language “makes” encourages
the misconception that the equal
sign is an ​
action​
or an operation
rather than representing a
relationship.
Read the equation 2 + 2 = 4 as “2 +
2 equals or is the same as 4.”
Indicate that a number
“divides evenly” into another
number.
The language “divides evenly” gives
students the incorrect impression
that the quotient will be an even
number.
Say that “a number divides another
number a whole
number of times” or “it divides
without a remainder.”
“Plugging” a number into an
expression or equation.
“Plugging in” encourages mindless
answer getting and may hinder
students from understanding the
difference between an expression
and an equation.
Use “substitute values” for an
unknown, or an amount that varies.
Using top number and bottom
number to describe the
numerator and denominator
of a fraction, respectively
A fraction should be seen as one
number, not two separate numbers.
Use the words numerator and
denominator when discussing the
different parts of a fraction.
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What teachers say or do …
Use the phrase “___ out of
__” to describe a fraction (For
example, “Two out of three to
describe 23 ”)
“Another name for average is mean.”
The problem…
What should be said…
Such language may lead to (a) an
overreliance on a pie model for
fractions, (b) a continued focus on
whole numbers which can lead the
application of whole number
characteristics, as 25 > 12 because 2 >1
and 5 >2, and (c) inadequate
development of the concept that a
fractions a number rather than two
numbers above and below a line.
Actually, there are many “averages” in
statistics, the three most common
averages being mean, median, and
mode.
Use the fraction and the attribute.
(For example, “ 23 of 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).
Use mean to indicate the measure of
central tendency we use to describe
a data sets balance point, or a data
sets equal share value (the result of
an equal reallocation of the total of all
the data)
“When you multiply a number by a
fraction between 0 and 1, the product
will be smaller than the other factor.”
“When you multiply a number by a
fraction, the answer is always
smaller.”
Not when you multiply by an improper
fraction or a fraction equal to one!
“You use division to find how many
groups of equal size you can make.”
That works for quotative or
measurement division, but not to
determine the size of an equal share
(partitive division)
“One way we can use division is to
find how many groups of equal size
we can make from a starting
amount.”
“You always divide the first number
you are given by the second number
you are given.”
This works for 72 ÷ 9 but not for all.
Use the words “dividend” and
“divisor” when discussing the order
of ​
t​
numbers in a division expression.
“To compare two numbers, first
compare their leading digits; if they
are the same, move to the second
digit.”
This works only when the numbers have
the same number of digits and the
decimal point is in the same position.
Thinking about only the leading digits
would have a student think 9,999 is
larger than 10,0003.
Use place value language when
comparing numbers.
“Move the decimal point before
dividing.”
Students may misinterpret this rule
and think that 5.5 ÷ 2.5 = 5.5 ÷ 25
rather than the correct statement 5.5 ÷
2.5 = 55 ÷ 25
Use place value language instead
(e.g., “To divide 5.5 ÷ 2.5 multiply
both the dividend and the divisor by
10 to create a whole number
expression that will have the same
solution; 10 times as much as 5.5 is
55 and 10 times as much as 2.5 is
25.”)
Adapted from:
Karp, K., Bush, S. & Dougherty, B. (2014) Avoiding rules that expire. ​
Teaching Children Mathematics.​
21(1) 18-25.
Cardone, T. (2014) ​
Nix the Tricks, ​
A guide to avoiding shortcuts that cut out math concept development.
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