Math “Rules” That Expire: Questioning Conventional Wisdom

“Multiplication makes bigger and division makes smaller.” “You can’t subtract a larger number from a smaller number.” “Improper fractions should always be written as mixed numbers.” “You always divide the larger number by the smaller number.”

We’ve all heard these statements…and I’ll bet we’ve all said them at one time or another. And every time we say these, we further perpetuate misconceptions that are difficult, if not impossible, to un-teach in future years.

Far too often, teachers inadvertently teach mathematics in an overgeneralized and imprecise way. We teach tricks that promote nothing more than memorization. The result is that our students misunderstand the very ideas we are trying to illuminate. How many teachers have had to contradict their colleagues from previous grade levels because what was taught was so painfully limited in scope?

So…how can we stop this practice in its tracks and work together to teach mathematics correctly and coherently? Here are a few ideas to get you started.

  1. Read Up! First, I’d like to introduce you to two recently-published articles that drive this point home: 13 Rules That Expire and 12 Math Rules That Expire in the Middle Grades. These two articles simply and eloquently outline overgeneralized, commonly-accepted, inadequate strategies we need to abolish from our vernacular. Seriously – check them out – you’ll likely find one or two or more to work on.
  2. Talk to Your Colleagues: Professional discourse provides foundational and at-your-fingertips opportunities for growth. After perusing one or both of the articles above, talk to your colleagues. Last month, I had a group of 40 teachers and leaders read and discuss these articles, and their conversations were robust.
  3. Create School-Wide Agreements: Take some time during grade-level and staff-wide meetings to discuss these ideas. Then come up with a list of five to ten school-wide agreements, identifying the concepts, vocabulary, procedures, etc. that everyone, or no one, will use on campus. This will make for a lively conversation!

I’m in the process of working through these steps in my own school district right now. Won’t you join me on this journey to abolish rules that expire by engaging in professional discourse and by creating school-wide agreements?

Kimberly Rimbey, Ph.D., works with teachers and leaders to develop system-wide change in mathematics teaching and learning.

Preview for Next Week: When selecting visuals and manipulatives to use within and across grade levels, choose wisely. Focus on ones that allow for connections throughout the grades. A great example is KP Ten-Frame Tiles. They are so superior to base-ten blocks! More on this topic next week…for now, though, check out Peggy’s latest paper which focuses on why KP Ten-Frame Tiles are superior to base-ten blocks from the get-go.


Leave a comment below.
  • Jillian Eiseman :

    I love this article and can’t wait to share it with my teachers at my school. I was wondering if you have suggestions for an engaging and thought provoking way that I could present this to my teachers. Thank you!

    • Kimberly Rimbey, Ph.D :

      I presented this article in the context of creating school-wide Math Agreements, an idea I learned from the authors at last year’s NCTM conference. During a full-staff meeting, I initiated a conversation about things we often say to kids that aren’t quite as accurate as one would want.

      To address vocabulary, I talked about using descriptions in place of terminology. For example, the term “flip-flop facts” is often used in first-grade classrooms. We discuss the notion that if kids can say tyrannosaurus rex and stegosaurus and know exactly what they mean, then they can say “commutative property” with understanding. “Flip-flop facts” may be okay as a first-grade description of the commutative property, but not a replacement for the actual term.

      I also used the fact that we say “multiplication makes bigger and division always makes smaller” as an example for ways we inadvertently teach misconceptions.

      We talk about a few examples, and then I passed out the article. They read, annotated, and discussed the article right then and there. And, finally, they wrote personal commitments about the things they were going to change in their practice. I asked them to include their commitments in the feedback survey so that I could follow-up in the coming weeks.

      As a follow-up, I met with the teachers in grade-levels to generate lists of things they think would be helpful as school-wide agreements – non-negotiables that everyone would hold to across the entire campus. Our lists included vocabulary;definitions and descriptions; symbols and other conventions; arithmetic rules; models and recording; and problem solving.

      We revisited and revised the lists multiple times over a couple of months. All of the lists were posted in our conference room so that grade levels could view one another’s lists and use them as springboards.

      When the time felt right, we met in grade clusters to come to agreement on 10 non-negotiables to include in our School-Wide Agreement, and eventually the entire staff came together to finalize it.

      The mathematical conversations that came out of this process were amazing. I hope you give it a try!

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