Multiplication is the third mathematical operation students learn. Many standards-aligned curricula introduce multiplication around the third grade. Mastery of multiplying multi-digit numbers, decimals, and fractions is usually expected by the end of the fifth grade. Many ideas make sense for third-grade multiplication but don’t remain true for fifth-grade (and beyond) multiplication. Being unable to forget these third-grade misconceptions can hold students back in middle school math. So it’s key that publishers and providers design their curricula to prevent these misunderstandings. This will give students a strong foundation for upper-level math.
4 Misconceptions Students Have About Multiplication
1. Assuming Multiplication Always Results in a Larger Value
Many students initially understand multiplication as repeated addition. Therefore, it makes sense that they then generalize that the product of two values will always be greater than both of the multipliers. However, this assumption only holds when dealing with whole numbers. So, how can providers combat this misconception?
- Make sure students have time to develop a conceptual understanding.
- Use concrete examples at all levels of multiplication instruction.
- Provide opportunities to discuss assumptions.
2. Multiplying Numbers in the Order They Are Listed
Students learn that the commutative and associative properties extend from addition to multiplication when they first learn about the new operation. However, without frequent practice and review, students may not remember these properties. This causes difficulties when they later problem-solve using mental math. For example, consider finding the product of 5 x 13 x 2. Students who are not familiar with multiplying two-digit numbers may need help finding the product of 5 times 13 in their heads. It is much easier to use the commutative property to compute this mentally by first multiplying 5 times 2 to get 10 and then multiplying it by 13 to get 130. Consequently, publishers should make sure to:
- Include reviews of these properties periodically
- Promote the use of mental math strategies
- Create opportunities for discussion of strategies
3. “Adding” Zeros When Multiplying By a Power of 10
Students first learning about multiplication may think that when you multiply by a power of 10, you just need to “add” that many zeros onto the number being multiplied. Merely placing zeros in the last place works when multiplying whole numbers by powers of 10 such as with 345 x 10 = 3450. But this method is not appropriate when multiplying a decimal value by a power of 10 (4.5 x 10 isn’t 4.50). Students who think of this rule as “adding zeros” will struggle with higher-level math. But providers can use some strategies to prevent this misunderstanding by:
- Connecting multiplying by powers of 10 to place value concepts, not quick tricks
- Providing opportunities for students to review place value concepts before teaching increasingly complex multiplication concepts
- Promoting visualizations of multiplying by a power of 10
4. Improperly Applying Order of Operations
Ask any student how to evaluate an expression using order of operations and you’ll probably hear something about, “Please Excuse My Dear Aunt Sally,” or PEMDAS. This means first simplifying the parentheses, then applying exponents, followed by multiplication, division, addition, and subtraction. It sounds simple, but you’ve undoubtedly seen adults fighting about how exactly to apply PEMDAS in social media feeds. There are a couple of ways students typically misunderstand PEMDAS. Since M comes before D, many students incorrectly assume that you must perform multiplication before any division. However, 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. Another misconception arises when simplifying an expression such as 3(4-1). 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. According to the mnemonic device, this should be simplified first. In reality, this coefficient of 3 indicates multiplication, which should only be performed after simplifying what’s inside the parentheses. Here are a few ways providers can fix these misconceptions:
- Help students understand that math isn’t just a series of rules to be memorized
- Break down the underlying meaning of each step of PEMDAS
- Link the order of operations to real-world problem solving
- Give ample opportunities for practice evaluating expressions in many different contexts
Publishers and providers can help students avoid misconceptions about multiplication by threading place value and deep understanding throughout their curricula. Opportunities for classroom discussions and concrete examples can also help students develop a deep understanding of multiplication that will allow them to succeed in math class.