"Overgeneralizing commonly accepted strategies, using imprecise vocabulary, and relying on tips
and tricks that do not promote conceptual mathematical understanding can lead to misunderstanding later
in students’ math careers." - Karen S. Karp, Sarah B. Bush and Barbara J. Dougherty in their article for the National Council of Teachers of Mathematics.
In our efforts to help kids "get it" sometimes elementary math teachers and good-intentioned parents, give our children tips or shortcuts that can make their math homework easier or help them get it done more quickly. Some of these "rules" should really be called helpful hints or "guidelines with an expiration date." True math rules hold even in advanced math. Guidelines with an expiration dates may help children in kindergarten to 3rd grade get through a concept and move on, but their expiration dates come pretty quickly. Here are 6 common “guidelines with an expiration date” that sometimes get masqueraded as rules to young students.
1. “You can’t subtract a bigger number from a smaller number.”
2. “You are supposed to write the answer after the equal sign.”
3. “If you multiply two numbers together the answer will be bigger.”
4. “If you multiply any number by 10, just add a zero at the end.”
5. “In a word problem, the word ‘altogether’ means you should add.”
6. “Multiplying is just a efficient way of adding.”
Are Helpful Hints or Guidelines with Expiration Dates Useless?
Math is a precise language. The more precise we are in our way of explaining a new concept, the easier it will be for children down the road.
You might be thinking, “why not use these guidelines with an expiration date to start and then give further explanations when necessary?” Guidelines become a problem when a child internalizes them as rules. They can be hard to let go of even when they are no longer useful. This creates confusion and distrust in a child's entire foundation of mathematical understanding.
You don’t need to eliminate the guidelines with expiration dates altogether. Just remember to use precise math vocabulary and tell kids the guidelines or hints come with an expiration date. That way the kids don’t internalize them as rules.
How to Approach Guidelines with Expiration Dates
Guideline with an expiration date 1 - “You can’t subtract a bigger number from a smaller number.”
Expiration date: When students start to understand the concept of a negative quantities, usually about 3rd grade.
Common or Every Day Examples: Accountants frequently work with negative numbers $6-$100= -94. Another example of using negative numbers is if you remove 10 cubic feet of dirt from a 6 cubic foot hill you now have a 4 cubic foot hole.
What to say instead: “Subtracting a big number from a small number can get confusing because it goes into an area of math that might be hard for you to understand right now. You'll learn about this area in a few years. For now, all your math problems will be subtracting a smaller number from a bigger number.”
Guideline with an expiration date 2 - “You are supposed to write the answer after the equal sign.”
Expiration date: When kids start to learn that equal means that the values on both sides of the equal sign have the same quantity, expressed differently. The concept of equal quantity expressed differently is surprisingly difficult and is usually introduced in first grade.
Common Example: 4 + __ = 32 The answer goes before the equal sign in this problem.
What to say instead: “The equal sign means both sides must show the same amount. Let’s figure out to show 4+4 is the same as another number.” Using manipulatives, objects the child can move around, is key to understanding equal.
Guideline with an expiration date 3 - “If you multiply two numbers together the answer will be bigger.”
Expiration date: When kids start to multiply fractions, often in 4th grade.
Common Example: Fractions don’t get bigger when you multiply them. ½ x ½ = ¼. Even multiplying a whole number by a fraction results in a smaller number such as 8 x ½ = 4
What to say instead: “If you multiply two integers, or whole numbers, together the product will be bigger than the factors. Integers don’t have fractions. This year you will multiply integers. Next year we’ll talk about what happens when you multiply fractions.”
Guideline with an expiration date 4 - “If you multiply any number by 10, you just add a zero at the end.”
Expiration date: When kids start to multiply decimals, often in 4th grade.
Common Example: The zero on the far right after a decimal does not have any meaning. 4.05 x 10 = 40.5 not 4.050.
What to say instead: Don’t teach this guideline until the child has a firm understanding of place value and multiplication. When they are ready ask if they notice a pattern when multiplying whole numbers by 10. If they tell you to just add a zero acknowledge that pattern and explain that it doesn't work on numbers that are not whole. See our article on shortcuts.
Guideline with an expiration date 5 - “The word ‘altogether’ in a word problem means you should add.”
Expiration date: When word problems get more complex and more than one operation is needed to solve it, typically 2nd grade.
Common Example: Certain words in story problems often indicate a certain operation should be used to solve the problem. The word “left” often means the child should subtract and “altogether” usually means addition. But if the child doesn’t take the words in context it can be very difficult. Consider this word problem:
“Josie has 12 pennies in her left pocket. She has 3 pennies in her right pocket and some in her hand. She has 20 pennies altogether. How many pennies are in her hand?” To get the answer the child must add and subtract and the word “left” refers to the direction, not the operation.
What to say instead: “Certain words help indicate whether you should add, subtract, multiply or divide. It is important to read the word problem several times. Some words that point to adding are “total” and ‘altogether.’”
Read our article on solving word problems for more tips.
Guideline with an expiration date 6 - “Multiplying is just an efficient way of adding.”
Expiration date: When students start to learn the concept of proportional relationships, usually about 4th grade.
Common Example: 3 sq. inches x 3 = 9 square inches. A triangle that is three times the size of another triangle is different than 9 triangles. Multiplying is a way to describe a proportional relationship and an efficient way of adding.
What to say instead: “One way to solve a multiplication problem is by using repeated addition.” This may seem like splitting hairs but this guideline describes a process for solving multiplication rather than defining multiplication too narrowly.
Need more help with math? Check out some of our other articles and be sure to stop by our center, too.
This article was written by and owned by Cuttlefish Copywriting, www.cuttlefishcopywriting.com . It is copyright protected. Mathnasium of Parker has permission to use it. Other Mathnasium locations should contact Heather at [email protected] before using it.
