- Episode Highlights
- Transcript
In this episode, we take a closer look at the kinds of shortcuts, tips, and language we use with the best of intentions (especially in early grades) that can cause confusion down the line. What starts as helpful guidance can quietly set up misconceptions that are tough to unlearn.
We’ll talk about what it means to “attend to precision” in how we talk about math and how to help students build understanding that sticks.
Links Mentioned
- “13 Rules That Expire” by Karen S. Karp, Sarah B. Bush, and Barbara J. Dougherty
Hello, Meaning Makers! I’m excited to be with you for this episode of the Meaningful Math Podcast, and we’ve got another interesting topic for us. Today we’re examining what happens when rules that students once learned in math class are no longer true, or in other words, they expire.
Now, I know what you might be thinking, “If it’s a rule, how could it ever not be true? That seems like a contradiction.” And that’s a fair point. Let’s take a look at some examples and talk about it!
Rethinking Early Math “Rules” That Don’t Hold Up
I can’t talk about this topic without thinking back to one of my first years as a first-grade teacher, in the weeds teaching a lesson about shape attributes. I started the lesson with one of my favorite routines, “Same But Different”.
The students examined two images side by side and discussed what was the same and what was different about them. One image was a green and orange striped rectangle with its two longer sides positioned vertically. The other image showed a rectangle larger than the first one. It was solid green, and its two longer sides were positioned horizontally.
The students noticed so many things that were the same and different between the images. One student eagerly shared, “They are both rectangles!” I asked how the student knew these shapes were rectangles and he said, “Because they have two long sides and two short sides.” If you’re an early educator, you’ve probably heard this several times.
Encouraging Students to Rethink and Reexamine
I drew a picture on the board of a four-sided shape with two short sides and two long sides, but this time it looked like a kite.
I asked, “So is this a rectangle then? It has two short slides and two long sides.”
“No!” the students shrieked.
I smiled and asked, “What do you notice about where the long sides and short sides are in our original two images?”
The student responded, “They’re not next to each other like in the one you drew… they’re across from each other.” I pointed to the opposite sides of each rectangle.
Another student chimed in, “Yeah, it’s like the two long sides are looking at each other and the two short sides are looking at each other!”
As the class chuckled, I smiled and said, “That’s so interesting! The sides across from each other are the same length. Is that true for all rectangles?” I challenged students to look around the room to find other examples of rectangles to see if this was always true, and the conversation continued.
Laying the Groundwork for Deeper Understanding
A Common Misconception: What Makes a Rectangle a Rectangle?
From my experience, characterizing a rectangle as having two long sides and two short sides seems to be the number one response students give when asked to describe a rectangle. And why would we expect anything else? Since preschool, students have seen posters and images with rectangles that match this description.
But is that really what makes a rectangle a rectangle? Technically, no. By definition, a rectangle is a four-sided polygon with four right angles and opposite sides that are congruent and parallel.
So… by that definition, a square is technically a rectangle. Yes, you heard that right: a square is also a rectangle. A square has four sides and four right angles and the opposite sides are congruent (meaning they are the same length, and in this case, all four sides happen to be the same length) and parallel.
Now, don’t panic – students aren’t expected to understand this concept until much later when they learn about the hierarchy of shapes. But it’s important for us as teachers of students in the prior grades to be aware of this expectation because we don’t want students to develop this notion that a rectangle HAS to have two long sides and two short sides.
When we solidify this as the definition of a rectangle, it creates a “rule” that this is what rectangles must have. While it doesn’t matter in the context of first grade, if this definition becomes solidified year after year, it becomes harder for students to undo and think critically about it.
How to Respond When Students Use Oversimplified Math Rules
So, how can we push students when they describe a rectangle this way?
- First, ask students to notice and wonder and describe what they see.
- When students share an idea, ask them how they know this and why they think this.
- Challenge your students’ thinking. I did this by drawing a shape with two long sides and two short sides that wasn’t a rectangle.
- Don’t establish a firm definition for students to learn. Instead, let students drive the conversation with their ideas and you can guide the conversation accordingly.
In my story above, the student eventually shared that the sides across from each other were the same length in the rectangle. This leads to the idea that the opposite sides are congruent.
Students don’t need to know this now. But this conversation, while short in the scheme of things, began to lay the foundation for this concept to be developed over time.
The Problem with Imprecise Math Language
In August 2014, the National Council of Teachers of Mathematics (NCTM) published an article titled, “13 Rules That Expire” by Karen S. Karp, Sarah B. Bush, and Barbara J. Dougherty.
The tagline for the article reads, “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.”
While NCTM does not include the definition of a rectangle as one of their 13 rules that expire, I think it fits under the idea of “using imprecise vocabulary”.
Language That Expires
In addition to rules that expire, the article points out the idea of language that expires. It’s important for us to pay attention to the language we use and the language we allow our students to use because it carries connotations that can lead to misconceptions and misuse by students over time.
For example, if we say or allow our students to say that a rectangle must have two long sides and two short sides, it will create a misconception about what a rectangle is.
The use of precise and accurate vocabulary plays a big role in Math Practice 6: Attend to Precision. This Practice calls on students to try to communicate precisely with others while using clear and accurate vocabulary and definitions. This is pivotal in developing students’ understanding of math concepts as they increase in complexity over time.
Alternatives to Math Language That Expires
Here are a few other examples from NCTM of expired language and some ideas about what you can say instead. Even though they don’t all apply to the early grades, I think they’re still interesting to think about.
- Using the terms “borrowing” when subtracting or “carrying” when adding. Instead, use “trading” or “regrouping” to support the actual action of exchanging one place value unit for another.
- Using the phrase “__ out of __” to name a fraction, such as saying “1 out of 4” to name the fraction “one fourth”. Instead, name the fraction and what it’s describing. For example, instead of saying “1 out of 4 pieces of pizza”, say “one-fourth of the pizza.” The use of the phrase “out of” leads students to think a piece is being subtracted from the whole amount.
- Referring to the equal sign as “makes”, such as saying “3 plus 2 makes 5”. Instead, use the term “is the same as”. The use of the term “makes” develops the misconception that the equal sign must refer to an action when it actually refers to a relationship.
- Using the terms “top number” and “bottom number” when discussing the numerator and denominator of a fraction. Instead, just use the terms “numerator” and “denominator”. Referring to each one as its own “number” establishes the misconception that a fraction is made up of two numbers, when in fact, it is one number.
These differences are SO subtle. But wow, when you really stop to think about it, they do make a difference when it comes to being precise and accurate when talking about math.
Now that we’ve examined the idea of language that expires, next time, we’ll finish up this discussion by diving into some of NCTM’s rules that expire.
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Until next time, Meaning Makers, have a great one!
