Synthetic Division, Nix the Tricks, and Conversations that Matter

 

I used to declare that I hated synthetic division. It always felt like a jumble of steps with no real meaning behind them, and my students never seemed to remember the process anyway. For years, I told my Algebra 2 classes that long division made more sense because it connected to what they already knew about dividing numbers. I also believed that synthetic division only worked when the divisor was a linear expression.

Recently I came across a blog post explaining that synthetic division can, in fact, be used in all cases. I was skeptical at first, but after reading and experimenting, I realized I had been wrong. I immediately shared the post with my department, confessing my mistake and sparking an unexpectedly lively discussion about math instruction, “tricks,” and conceptual understanding.

That conversation prompted us to revisit Nix the Tricks by Tina Cardone, a book that challenges educators to think critically about the shortcuts we teach and why. One simple email turned into a multi-month book study that reminded me why I love collaborating with fellow math teachers: the shared passion for improving how we help students make sense of mathematics.

We decided to read one chapter each week and meet every Monday after school to discuss guiding questions and main takeaways. I borrowed the discussion prompts from another educator’s blog (I’ll update this post with credit once I find it) and used them to keep our conversations focused on teaching decisions rather than just personal opinions.

Our group included about ten teachers from various grade levels, each bringing their own experiences and perspectives. The outcome was an open, honest discussion about what it truly means to move away from “tricks” and toward genuine mathematical understanding.

Here are some of our collective reflections and lingering questions from Chapter 1: Introduction.

Chapter 1: Initial Reactions & Discussion Highlights

• What are your initial reactions to the start of this book?

• Tricks vs. shortcuts: not all shortcuts are bad.
  

• Definitions are okay (for example, “SOH CAH TOA”).
  

• Do elementary teachers rely on tricks because they are uncomfortable with math?
  

• Time is a major factor. Shortcuts appear too early because we’re pressed to cover everything.
  

• Can we promote more consistent notation from the start (e.g., not using “x” for multiplication in early years)?
  

• Is a formula a trick? (Quadratic formula, point-slope form?)
  
  

• How can you relate your own experiences to teaching “tricks”?

• Many of us are moving away from tricks and focusing on conceptual understanding.
  

• Introducing shortcuts too soon prevents students from developing deeper reasoning.
  

• Our own children are learning math from multiple angles. Will that lead to a stronger understanding in the long run?
  

• When will high school teachers start seeing the effects of students who’ve grown up learning under the newer Core standards?
  
  

• What’s your favorite math trick (or least favorite)?

• The elimination method for systems, often taught before students grasp what’s happening.

• Memorizing unit circle values without understanding how to derive them.

• Multiplying conjugates, recognizing why the middle terms cancel.

• Using common denominators for rational expressions.

• “Guess and check” (reverse FOIL) for factoring trinomials.

• Given two sides of a triangle, find the possible range for the third side using addition/subtraction.
  
  

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Leading this book study reminded me how powerful it is to learn with other educators, not just alongside them. Our conversations were thoughtful, sometimes challenging, and always focused on helping students truly understand the “why” behind the math. I’ll share our reflections from the remaining chapters in upcoming posts, along with the discussion prompts we used, so you can easily run your own Nix the Tricks book study. In the meantime, you can download the free PDF version of the book and start exploring how to replace tricks with understanding in your own classroom.