Although it is the third mathematical operation that students are exposed to, multiplication tends to be more challenging to students than their previous encounters with addition and subtraction. Yes, a student may be able to memorize their multiplication facts with ease. But what happens when they are asked to go beyond the scope of the multiplication table? Whether it be finding the product of two or more values or simplifying more complicated expressions, there is plenty of room for error. Here are four misconceptions students have when performing multiplication:
1) Assuming multiplication always results in a larger value
Because many students initially learn that multiplication is repeated addition, it makes sense that students generalize that the product of two values will always be greater than both of the multipliers. This assumption only holds when dealing with whole numbers. Introducing a decimal or fractional multiplier throws a wrench into this generalization since multiplying a positive number by a fraction less than one actually produces a smaller value. Adding negative values into the mix contradicts this assumption as well. Of course, the product of two numbers will always be smaller when you multiply a positive number by a negative number.
2) Multiplying numbers in the order they are listed
The commutative and associative properties of addition are presented in some math curriculums as early as second grade. Once multiplication is introduced, students learn that these properties also extend to this new operation. In many cases, these properties are only stated as rules rather than being taught in context to assist in mental math calculations. Consider the product 5 x 13 x 2. Students who are not familiar with multiplying by two-digit numbers may have difficulty finding the product of 5 times 13 in their head. Using the commutative property to rearrange the numbers, it is much easier to compute this mentally by first multiplying 5 times 2 to get 10 and then multiplying it by 13 to get 130. Which leads us nicely to the next misconception…
3) Adding zeros when multiplying by a power of 10
Students first learning about multiplication are usually told that when you multiply by a power of 10, you just need to add that many zeros onto the number being multiplied. Those who accept this shortcut without a true understanding of the underlying mathematics often apply this rule incorrectly. While merely adding zeros works when multiplying whole numbers by powers of 10—for example, 345 x 10 = 3450, this method is not appropriate when multiplying a decimal value by a power of 10 (4.5 x 10 isn’t 4.50). Adding a zero here results in a value that is exactly the same as what you started with. Students with this misconception fail to see that multiplying by a power of 10 actually indicates a shift in the decimal point.
4) Improperly applying order of operations
Ask any student how to evaluate an expression using order of operations and you’ll probably get in response “Please Excuse My Dear Aunt Sally,” or PEMDAS, meaning first simplify the parentheses, then apply exponents, followed by multiplication, division, addition, and subtraction. Since M comes before D, many students incorrectly assume that you must perform multiplication before any division. If an expression involves both multiplication and division, the correct method actually has students performing these two operations from left to right in the order in which they appear. Similarly, when simplifying an expression such as 3(4-1)2, some students may first distribute and multiply the 3 to both the 4 and the 1 because they recognize the 3 being connected to the parentheses—which, according to the mnemonic device, should be simplified first. In reality, this coefficient of 3 indicates multiplication, which should only be performed after simplifying what’s inside the parentheses and squaring this value.
Of course, another slew of misconceptions arises when students begin multiplying two-digit numbers by hand. What other common errors do students make when multiplying?