Intercepts, The Zero Product Property, and Other Such Mathematical Disasters

You know when students bomb on a concept so badly that you make sure to really drill it into next year’s class so it doesn’t ever happen again? Well, my plan completely backfired and then some. While grading the latest quizzes on quadratics, I discovered two conceptual misunderstandings that were emotionally triggering. The results I saw frustrated me, angered me, and had me screaming internally.

Here they are:

Scenario #1

Can you see what’s wrong with this solution?

Scenario #2

What can I do better to help students move towards more efficient strategies?

I want to highlight two facts:

  • Multiple students made the same errors in conceptual understanding
  • The error was made by students of all ability levels, including high performers

When I noticed the same errors being made repeatedly, I knew I needed to take a closer look at my instructional practices.

Where I failed as a teacher

Click here to see the detailed breakdown of the mathematics in each scenario. For the TLDR, see below.

Scenario #1: The student correctly factors each equation (even though the approach to question 1a is still massively triggering), identifies the values of x that make each equation true, and then incorrectly states their solution as an ordered pair (x, y). If we look at question 1 carefully, we see that each equation only has a single variable – x. No y’s! I failed in getting students to recognize WHEN stating an ordered pair (x, y) response was appropriate.

Scenario #2: I’d like to point out that there is nothing wrong with the work the student has presented, however, it misses a major conceptual idea (the zero product property) – entirely. This type of error is more innocuous, because, when done correctly, use of the quadratic formula will always lead to correct results. The problem, however, is that students should be able to obtain the solution to a question like this in a matter of seconds, so why isn’t that happening?

Where did I go wrong? How can I make it better?

I take for granted the prior knowledge I expect students to come into the classroom with. The truth is, many students in high school (and this includes seniors!), do not know the multiplications tables or are still unsure about how to work with fractions. This lack of fluency can become extremely detrimental, especially when it is carried over through year after year. Students need to develop procedural fluency so they can tackle harder problems and not waste brainpower on trivial matters (like figuring out what numbers multiply to 56 without a calculator).

Solving quadratic equations, like the ones presented in scenario 1, requires procedural fluency. Problems arise when we do a procedure blindly without understanding what the question is asking and what kinds of results we are looking to find. I would love to blame students for being intellectually lazy, but I must accept some of that blame for myself. As a teacher, how often do I jump right into the solution without properly developing the question?

I do not like to see students struggle in math, but they need to. Except they need to struggle just the right amount; too little, you lose interest, too much, and maths class becomes a painful and intolerable experience. Russian psychologist Lev Vygotsky coined this the “zone of proximal development”. There’s a fine line to walk here, because every student has a different threshold for facing discomfort.

We want to give student enough little nudges in the right direction so they can be successful without depriving them of the satisfaction of having put their own minds to work. The goal is for students to walk away from their high school experience knowing that they have what it takes to succeed in maths, and to truly embody that. It means getting students to understand the math, and not just blindly copy procedures. It means getting students comfortable with struggle, and being okay with not understanding things right away. It means equipping them with the right habits, like

  • Asking themselves: Does my answer make sense? How can I check my work?
  • Looking for alternate ways to approach the same problem
  • If they get stuck, reviewing their work and noting the point at which they started to become confused

I was discussing this issue with my partner last night, who pointed out that this type of rote-memorization/blindly-following-rules outcome was likely the result of a grade-oriented mentality as well. We (myself included) are so focused on getting to the answer, that sometimes we miss the mark entirely, which is understanding the process.

My Key Takeaways

  • As as a teacher, I need to find the right balance between helping my students develop procedural fluency AND conceptual understanding. This starts by spending sufficient time developing the question.
  • Help students develop a new habit: asking themselves “why?” and “how?” questions after completing a problem. Why did I write this down? Why is this step necessary? Why does this answer make sense? How did I approach the problem? How could I have solved this another way?

Next Steps

1. Collect some preliminary data:

  • Determine how many students made the same errors in each of my classes, compare this to other teachers’ in my teaching team.
  • Find out what the common denominator is for students who did NOT make the same error

2. Take this opportunity to discuss and analyze these errors with my classes, in a safe environment, and have students create a plan for how they can develop math habits that can help them avoid making the same types of mistakes in the future.

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