- Episode Highlights
- Transcript
Some of the “rules” we teach in math class are surprisingly short-lived. They might work for a while until they don’t.
In this episode, we’re diving into the kinds of strategies and shortcuts that seem helpful early on, but later become roadblocks to deep understanding.
From multiplying by 10 to “you can’t take a bigger number from a smaller number,” we’ll take a look at a few examples of expired rules and the misconceptions they create.
We’ll also explore how we, as teachers, can be more intentional with the language we use, laying a stronger foundation now to support the more complex thinking that’s coming later.
Links Mentioned
- “13 Rules That Expire” by Karen S. Karp, Sarah B. Bush, and Barbara J. Dougherty
What Are Expired Rules?
Hello, Meaning Makers! Welcome back to the Meaningful Math Podcast. I’m excited to be here to continue our conversation about rules and language that expire.
Last time we talked about how the language we use and the language we allow our students to use can lead to misconceptions and misuse by students over time. The National Council of Teachers of Mathematics (NCTM) explains that this “expired language” is the language that students used in previous grades that eventually is no longer true, based on students’ current learning.
Similarly, “expired rules” are rules that students learned in previous grades that eventually are no longer true, based on students’ current learning. Today we’re going to dive into some of these expired rules. Let’s get started!
Shortcuts That Don’t Hold Up
The Problem with Overgeneralizing in Math
In preparing for this episode, I had a conversation with one of the most wonderful math specialists I know. She shared the example I’m going to read for you, that perfectly illustrates this idea of expired rules. Here is what she wrote to me:
“When I was teaching fifth grade and we were working on multiplying decimals by multiples of ten, I faced a huge roadblock that I had not anticipated: my students repeatedly wanted to just add zeroes to the end of the decimal to get the answer. For example, if we were solving 1.3 x 10, they would say the answer was 1.30. Ugg. That didn’t even make sense!
Do you know why students wanted to do this? Well, in third and fourth grade they had learned that whenever you multiply a number by a multiple of ten, you just add the number of zeroes in the multiple of ten to the end of the other factor to get the answer. So for 5 x 10, you just add a zero to 5 to get 50, and for 5 x 100, you add two zeros to get 500. In other words, they learned a shortcut.
Unfortunately, that shortcut only applies to multiplying WHOLE NUMBERS by multiples of 10, so it completely falls apart when students learn to multiply decimals. In other words, this rule expires in fifth grade.”
Common Math Rules That Eventually Expire
This is actually the first rule that NCTM lists in their article titled “13 Rules That Expire” by Karen S. Karp, Sarah B. Bush, and Barbara J. Dougherty. In this article, the authors explain that rules such as the one shared before tend to expire when students expand their learning beyond whole numbers and are forced to consider other ideas about what numbers can be (think decimals, fractions, negative numbers, etc.).
The explicit teaching of rules can lead to rigid thinking that makes it difficult for students to get past as they progress in their learning. This can cause frustration for students, as it further propels the notion that math is a bunch of mysterious rules and tricks to memorize as opposed to groups of connected concepts.
Furthermore, using rules, tricks, or tips to make math easier does not encourage students to be problem solvers who can reason and think critically about math.
Let’s take a look at some of the other rules the authors point out in this article.
Using Keywords to Solve Problems
This rule encourages students to search for certain words in a story problem and perform an associated operation. Unfortunately, many words in our language can have multiple meanings.
For example, “some” (s-o-m-e) and “sum” (s-u-m) are homophones, so if a student isn’t aware of the difference in spelling between the two words, they might not understand the meaning of the word in the story problem.
Consider this story problem:
Jack took some blocks out of the basket to play with. He also grabbed 12 blocks from the table. Jack now has 28 blocks. How many did he take out of the basket?
A student reading this problem who is scanning for keywords might come across “some” and decide to add since addition is the associated operation with the term “sum” (s-u-m), not realizing that the context of the story actually requires subtraction.
This rule also expires in third grade when students begin solving multistep story problems, as students have to determine which keywords are associated with each step of the problem.
Since multistep story problems contain multiple numbers and keywords, you need to understand the story in order to know which keywords apply to which numbers. This rule is not to say that keywords aren’t helpful – they totally are! They just need to be examined in the context of the story.
You Cannot Take a Bigger Number From a Smaller Number
Students come across this rule when they are learning to subtract. When teaching students how to set up subtraction problems, you might find yourself saying something like, “The bigger number always comes first!” While students will only see problems that look like this in early grades, this rule expires when students work with negative numbers in sixth grade.
Students might encounter a problem like this:
At bedtime, the temperature outside is 2 degrees fahrenheit. Overnight the temperature will drop 6 degrees. What temperature will it be after the temperature drops?
In order to solve this, students will actually do 2 – 6 and arrive at an answer of -4. So it IS possible to subtract a larger number from a small number!
When talking with your students, identify the parts of the subtraction problem by name: the minuend is the number we start with and the subtrahend is what is taken away from the minuend.
Addition and Multiplication Make Numbers Bigger
This rule is given to students to help them understand the meanings of the operations, but it actually falls apart pretty quickly.
Consider a problem where 0 is added to a number – that causes the number to stay the same not increase!
Later on in elementary, particularly in fifth grade, this rule falls apart again when students multiply fractions, for example ½ x ¼ = ⅛. The product, ⅛, is actually LESS than either factor.
In seventh grade, students add negative numbers, which also results in an answer that is less than either addend.
Subtraction and Division Make Numbers Smaller
Just like the last rule, students learn this to help them better understand the meanings of subtraction and division, the idea being that if I am taking away from a number or dividing a number into parts, my answer MUST be less than what I started with.
This rule actually expires in sixth grade when students subtract two negative numbers and the difference is actually greater than the number they started with, such as -4- (-7) = 3. Students also see that dividing a fraction by a fraction results in an answer that is greater than either number in the problem, such as ½ divided by ¼ = 2.
Building a Stronger Foundation for Future Math Learning
After hearing about some of these rules that expire, I know what you might be thinking, “I’ve definitely said some of these rules before. How do I fix it?” And the answer is, you don’t need to fix anything. The key is now that you know better, you can do better.
Replace Absolutes with Curiosity and Flexibility
So moving forward, be intentional and thoughtful about the language you use and the language you allow your students to use when talking about math. Avoid defining these particular math concepts in absolutes by saying things like… “this ALWAYS happens” or “this is ALWAYS true” or “this NEVER happens” or “this is NEVER true” because we know that eventually, that won’t always be the case.
Even though “eventually” is not “right now”, we know that we are establishing a foundation from which all future math understanding will build on. We want to make sure the foundation we are laying is able to withstand what is coming later for our students.
Those are just some of the rules that expire as outlined in the article from NCTM, so if you are interested in reading about the others, I definitely encourage you to check out the article. Maybe you can even think of other rules that we teach our students that eventually expire!
Don’t forget to subscribe to this podcast so you don’t miss an episode. And, if you have a moment, please consider leaving a review so we can bring this information to even more teachers.
