Nix the Tricks Book Study - Chapter 5: Number Systems

 

This post is part of an ongoing book study series on Nix the Tricks by Tina Cardone. Each week, a group of math teachers meets to reflect on one chapter of the book and discuss how we can move away from shortcuts and toward true mathematical understanding. You can find our reflections from earlier chapters in previous posts and follow along as we continue exploring.

Chapter 5, “Number Systems,” pushed us to think deeply about how students build meaning around integers, rational numbers, and exponents, and how quickly shortcuts can derail that sense-making. While earlier chapters focused on specific tricks, this one made us reflect on the foundation of number itself and how our instruction shapes students’ long-term understanding.

Rethinking Our Past Practices

As we reflected on our own teaching, a few themes came up repeatedly, especially around integer multiplication and rational exponents.

Most of us admitted that at some point in our careers, we’ve leaned on oversimplified rules:

  • “Two negatives make a positive.”

  • “If the exponent is a fraction, flip the number and take the root.”

  • “Change the sign of the exponent and put it in the denominator.”

These tricks often help students get quick answers, but they rarely help them understand why the rules work. Several teachers noted that students who rely on these shortcuts struggle when they encounter more advanced ideas, like exponential functions, geometric sequences, or fractional bases, because they never understood the structure beneath the symbol manipulation.

The chapter encouraged us to think more intentionally about using number lines, patterns, and visual models before introducing rules. Students need to see multiplication as repeated addition, rational exponents as extensions of patterns, and negative exponents as a natural continuation of the properties of exponents, not as disconnected rules.

When Tricks Harm Understanding

One example that came up quickly was horizontal asymptotes. Many of us have used or heard versions of:

  • “If the top is bigger than the bottom, there’s no asymptote.”

  • “If the degrees match, just divide the coefficients.”

  • “If the bottom is bigger, the asymptote is zero.”

While these statements sometimes work, they hide the real mathematical idea: comparing growth rates and end behavior.

Students who memorize these shortcuts are often lost when they later analyze limits. They may know “the rule,” but they don’t understand function behavior, which is the actual goal.

This chapter reminded us that shortcuts can unintentionally prevent students from seeing big idea, especially in topics that depend on patterns and structure.

Overall Thoughts on the Chapter

The group agreed that this was one of the more thought-provoking chapters so far. Number systems are so foundational that any misconception here ripples upward into every high school math course.

A few shared observations:

  • Lower-level students need more conceptual grounding, not less.

  • Higher-level students often want shortcuts because they’re focused on grades and efficiency.

  • Covering less content in more depth is more valuable than racing through procedures.

  • The number system grows with students, and our instruction needs to grow with it.

This chapter challenged us to look at our instruction with fresh eyes. Even when we feel pressed for time, investing in deep understanding pays long-term dividends.

Can Tricks Still Support Special Education or Alternative Education Students?

This question sparked one of our most thoughtful conversations. Some teachers pointed out that shortcuts can occasionally serve as scaffolds for students who need support accessing grade-level curriculum. But the group agreed that:

  • Students who struggle the most need to understand the “why” even more than the “how.”

  • Memorized procedures disappear quickly, especially for students with working memory challenges.

  • When students make sense of numbers, they gain transferable skills that help them across all of mathematics.

We concluded that while strategic scaffolds can be helpful, we must avoid giving students only the shortcut. Conceptual understanding is not a luxury; it’s essential.

Big Takeaways

  • Foundations matter. Number systems underpin nearly every secondary math concept.

  • Tricks may produce correct answers, but they rarely produce understanding that lasts.

  • Students, especially those who struggle, benefit from models, patterns, and meaning-making.

  • We should aim for fewer rules and more reasoning.

  • Building number sense creates confidence, flexibility, and independence.

Next time, we’ll move into Chapter 6: Expressions and Equations, where we’ll examine some of the most persistent habits students develop, from “moving terms” to “switching signs,” and how to replace those tricks with strategies grounded in structure and logic.

Comments

Post a Comment

Popular posts from this blog

My Favorite: Doceri

Image
  My favorite classroom tool is Doceri.  It does cost $30, but I use this iPad app every day, so it was well worth the cost.  You need a computer (laptop/desktop), wireless network, and projector to use Doceri.  It basically turns your computer into a whiteboard.  You can control your computer and any files remotely from the iPad.  I use this feature for PowerPoints during class.  You can eccentrically take a screen shot of whatever is on your screen (or use a number of included backgrounds such as graph paper) and then write on that image.  I use this feature most often.  I will make a PowerPoint presentation of my notes each day and then use Doceri to annotate each slide.  You have a ton of pen style/color choices so you can color code and highlight important parts of your class notes.  Doceri can save each annotated image and you can email it to yourself in PDF form.  This can become a document to post on your cla...
Read more

Practice Structures: Error Analysis Revisited

Image
Last summer I wrote about using error analysis as one of my favorite ways to practice math skills.  Since then I've found a few more resources and created a few more that I'd like to share.  In general, error analysis activities involve purposefully making common mistakes, then having students try to find them.  Here is a link to the original post if you'd like to see it.  In that post I shared an example for the segment addition postulate and midpoint as well as a second example for multiplying polynomials . Here are some other examples that I found.  I did not create these. Grade 2: Adding and Subtracting up to 3 digits. Grade 8: Solve Linear Equations. Algebra 1: Graphing Linear Inequalities in two Variables
Read more

One Good Thing

Image
This past week was the last week of classes for this semester.  Some of my classes are full year classes while others are semester courses.  Either way, this seemed like a good time to make some changes for the new semester and to allow students to give some feedback about those changes. I gave my students a quick exit ticket on our last day of class.  I asked them to think back on the year so far and to tell me one thing that they liked about our class that they would like to keep the same for the new semester.  Then I asked for one thing that they would like to change for the new semester. Now, I must preface their responses by saying that I teach in a very non-traditional classroom.  I use hybrid learning.  You can read more about it on the hybrid page of my blog.  Basically it involves students rotating through various stations throughout the lesson.  Students have more freedom than they have ever had before and some of them fail mi...
Read more