Wednesday / May 29

Words Matter—Three Ways to Avoid Harm in the Mathematical Language We Use 

I was recently watching over the shoulder of my 7-year-old “office-mate” while he attended his second grade math lesson through his computer. And yes, I realize the issue of parents listening in on online classes has drawbacks, but that’s a rabbit hole for another timeOn this day, my child’s teacher was working with the class on double-digit operations and exploring the relationship between addition and subtraction. Reliably, they reached the story problem at the end of the lesson so that they could “apply what they had learned to a real-world problem.”  

Kim has 15 buttons. Tariq gives Kim 12 buttons to fill out her collection. How many buttons does Kim now have? 

I’ll set aside for now the fact that I have never once seen 7yearolds trading buttons. The teacher then proceeded to help the kids decode the story problem and said “Now when you see the words “how many” that means you are likely going to add. Let’s underline that phrase. 

My head snapped up and my heart sankNo! I wanted to shout through the screen. Don’t say that! What about when you have the question How many more? Or how many fewer? Totally different question, totally different solution pathway. And once heard, they will never unhear that! In fact, I spent ample time that day with my son trying to make him unhear just that. Some children might also have been taught to look for key words such as “altogether” or “in all” or “in total. But notice that none of those words was in that problem? What would a student do? 

Now, I’m a big fan of this teacher, and I don’t fault her. I’m choosing to believe that this was a slip of the tongue, but it did echo something I so regularly hearMy point is that the words we use in mathematics matter SO MUCH. And when we use language incorrectly and inconsistently—not just in our own classrooms, but across a grade, or even a school building—students are harmed. Sometimes the harm is immediate, as in my son’s case, but often the damage is uncovered over the years as the mathematics students learn becomes more complex. If what they once learned suddenly stops working, they become increasingly confused and frustrated having to “unlearn” it. It’s no wonder they think math is a series of disconnected ideas and they struggle to see the bigger picture. It’s no wonder they lack confidence and self-efficacy. It’s no wonder they feel tempted to give up.  

“When we use language incorrectly and inconsistently—not just in our own classrooms, but across a grade, or even a school building—students are harmed.” 

In their book series The Math PactKaren Karp, Sarah Bush, and Barbara Dougherty offer many ideas for how teachers can work with colleagues to clarifyagree on, commit to, and consistently apply the precise and accurate mathematical languagesymbols, notations, “rules,” representations, and generalizations they use across all of their teaching. Briefly, here are three examples in the domain of language where precision can help: 

  1. Opt for precise academic languageMathematics often has terms that are similar to words we use in everyday language, but which do have a precise mathematical meaning. Think of words such as difference, expression, model, side, volume. With enough language that is already legitimately confusing, why compound that by using other confusing terms where a precise one will stave off head scratching. Here are some examples:

    Source: (Karp, Dougherty, & Bush, 2020)  
  2. Ditch coding for key words. Consider this problem: Dominique has 1markers, Monica has 12 markers. How many pencils do they have in all? Some may be tempted to point to their key words poster and explain that the phrase “in all” means “add. No harm done, right? Now consider this problem Dominique has sets of markers. There are 14 markers in each set. How many markers does Dominique have in all? Uh oh. Now we have a multiplication situation (3 groups of 14). What happens to our learner who is convinced “in all” means add? What happens to my son who also heard “how many” probably means addition? We can minimize confusion by ditching using key words as rigid, coded indicators of what a problem is asking. Instead, encourage students working through a problem by identifying what information the problem offers and what is missing; reflecting the situation using concrete models, pictures, or even acting it out; and helping them become curious about what the problem is actually asking them to think about.  
  3. Avoid unhelpful mnemonicsThe books reference a study by Delise Andrews and Beth Kobett (Andrews, 2017) in which they found within one elementary building a variety of problemsolving posters featuring popular mnemonics to help students decode and solve word problems like CUBES, SWEEP, and SUPER. In some cases ‘s’ stood for solve, or symbols, or slowly read the problem. E stood for equation, or explain with a number sentenceWhat happens when students learn CUBES in grade 2, SUPER in grade 3, and their best friend argues that no, it’s SWEEP? Why are we doing this to them? Similarly, mnemonics that describe a process but leave out the underlying understanding are problematic, such as FOIL for first, outer, inner, last. Why not just teach how the distributive property of multiplication works? How much does the mnemonic help and what does it actually mask? 

These are elementary examples, but in their books, the authors provide similar examples for middle and high school. All of this work builds from their popular series of articles published by the National Council of Teachers of Mathematics about the “Rules that Expire, which describes these are commonly used mathematical “rules” that children commonly learn but that stop working immediately or later in new or different situations. For example, the notion that multiplication makes a number bigger and division makes a number smaller is sometimes taught in elementary grades when tied to whole numbers, but falls apart almost immediately (think 1 x 0) and definitely when students get to fractions and decimals. 

Read more on how to make strong consistent agreements and align with your colleagues in your grade-level team, your building, and your district around avoiding the language and rules that expireas well as the symbols, notations, representations, and generalizations you employ, to keep kids on the same page, clear the cobwebs, and end the confusion.  


Andrews, D. & Kobett, B.M. (2017, July 18)Connection to discourse: Word Problems [Paper presentation}. National Council of Teachers of Mathematics Discourse Institute. Baltimore, MD. 

Karp, K. S., Dougherty, B. J., & Bush, S. B. (2020). The math pact, elementary: Achieving instructional coherence within and across grades. Thousand Oaks: Corwin. 

Written by

Erin Null is the Publisher and Editor for Corwin Mathematics, and is actually a huge fan of both her kids’ elementary teachers, and a firm believer in Maya Angelou’s stance of “Do the best you can until you know better, and once you know better, do better.” 

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