Students may develop misconceptions because we as adults do not always use mathematical vocabulary accurately when talking about familiar concepts. For example, because numbers and arithmetic operations are familiar to adults, we may be more casual in how we talk about them. No wonder kids who are learning math concepts for the first time get confused.
However, when interacting with students who have yet to develop full understandings of key mathematical concepts and relationships, this lack of precision can create or reinforce misconceptions. Here are a few examples that illustrate this issue along with suggestions for how to revise the way things are phrased.
Example 1: Clarifying the Equal Sign
Common Pitfall: Reading an equal sign as “is what” or “gives” rather than “is the same as” or “is equivalent to.” For example, 3 x 7 = ____ might be read aloud to students as, “3 times 7 gives what?” This communicates the misconception that the equal sign is an operation (something to do) rather than a relationship (something to compare).
Do This Instead: A more precise reading would be, “3 times 7 is equivalent to ____” or “3 times 7 is the same as _____.” Each of these communicates that the equal sign represents a relationship of equivalence between the expressions on either side of it. Two additional strategies to reinforce the correct meaning of the equal sign are (a) placing the missing expression to the left rather than to the right of the equal sign (e.g., ____ = 3 x 7) and (b) posing true/false statements that students must evaluate and explain (e.g., “True or false: 3 x 7 = 7 + 7 + 7”).
Example 2: Suggesting Subtraction Makes Numbers Smaller
Common Pitfall: Telling students that “subtraction makes things smaller.” Certainly, when primary grade students first learn what subtraction is, most of the cases they encounter in and out of school support this claim. However, one does not only need to think ahead to operations with integers in upper elementary and middle school to realize the problematic nature of this statement. Just consider 3 – 0 . . . the result is not smaller.
Do This Instead: Saying subtraction is about taking away (counters are useful with this) and finding the distance between two values (a number line model is particularly helpful with this) is more precise and helpful. So, 7 – 3 can be thought of as, “What is the result if I take 3 away from 7?” and “How far is 7 away from 3?” This will better prepare students for understanding subtraction with integers.
(Adults also make the mistake of telling students that “division makes things smaller.”)
Example 3: Suggesting Multiplying by 10 Is about Adding Zeros
Common Pitfall: Teaching students that “multiplying by 10 (or any positive power of 10) means we add zeros.” This does long-term damage to students’ understanding of place value—something they must know well in order to make sense of operations with decimal numbers.
Do This Instead: Although it might appear that the product of 17 x 10 is found by “adding a zero” to get 170, what is really happening is much richer. If you can imagine a place value chart with a column for hundreds, tens, and ones, the number 17 will have the “1” in the tens column and “7” in the ones column. Multiplying 17 x 10 results in the “1” shifting from the tens column one place value to the left to become 1 hundred and the 7 moving from the ones to the tens column to become 7 tens (or 70). The 0 is placed in the ones column to represent that there are no ones. So, it’s more precise and more useful in the long term to help students recognize that multiplying by 10 (or any positive power of 10) will shift each digit one or more place value(s) to the left. With this foundation, students will more easily build understanding of multiplication and division with decimal numbers when they encounter them later.
Example 4: Clarifying Meaning of the Denominator
Common Pitfall: When thinking about the value of a fraction, some adults will tell students, “The larger the bottom number (denominator), the smaller the fraction.” Although this is true when working with unit fractions (those with a numerator of 1), this claim causes confusion when the numerator is not 1. For instance, using this “rule of thumb,” students might incorrectly reason that 3/5 is smaller than 1/3 because 5 is larger than 3.
Do This Instead: It’s better to ask students questions that help them focus on making sense of unit fractions. Some to consider include:
- What is the unit fraction I am working with?
- How many of these units do I have?
- Where does this fraction belong on a number line?
- How might I create equivalent fractions with common units? (This is important for comparing fractions with unlike denominators and building foundations for fraction operations.)
Relatedly, it’s important when working with fractions to refrain from referring to the “top number” or “bottom number” because the entire fraction is a number.
Example 5: Speaking of Decimal Points with Precision
Common Pitfall: Many adults read decimal numbers without reference to place value. For example, 3.2 is read as “three point two.” In everyday life, this is efficient and generally not problematic. However, for students learning to make sense of decimal notation—who need to recognize decimals as fractions with denominators that are powers of 10—there must be more precision in how we talk about these numbers.
Do This Instead: Hearing “three and two-tenths” communicates the meaning of the decimal value and its connection to fraction notation (making it easier to recognize 3.2 as equivalent to 3 and 2/10).
This post has been adapted from Recognizing Misconceptions as Opportunities for Learning Mathematics with Understanding. Download the whitepaper to learn more about math misconceptions and how to use them to help students build deep conceptual understanding.DOWNLOAD WHITEPAPER
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