Finding Elegance in Equivalence

This post was featured in an episode of 

Over the past week, we have been piloting diagnostic questions with students to get an idea of their understanding of equivalence. We gave students a series of questions, all having to do with the understanding of what the equal sign represents.

One question, though, really challenged my idea of what "higher order" strategies we were hoping to see in our students.

Here's the question:

If 4x + 8 = 52, what is 2x + 4?

Take a moment to figure it out.
What is your answer?
How did you come to that answer?

If you're like my colleagues and myself (and most of the students of whom we asked the question), your instinct might have been to solve for x using the first expression,


...and then substitute that value for x in the second expression:


So in this case, 2x + 4 = 26.

As students advance to higher order strategies when learning math, we often nudge them toward more efficient methods: generalized procedural strategies (as opposed to guess and check, or building concrete models) that can work with any case.

Solving for x in the above question is a very valid strategy, employing algebraic thinking in a method that would work for any similar question (even one that was more complex). But is it the most efficient strategy? I count four separate calculations: 52 subtract 8, 44 divided by 4, 2 times 11, and 22 added to 4.

Is there another way?

Look at the question again - can you determine 2x + 4 in only one calculation?

If 4x + 8 = 52, what is 2x + 4?

You can, if you recognize that 2x + 4 is half of 4x + 8. Because of the equivalence in the expression, the solution is therefore half of 52. Half of 52 is 26... and there's your answer. One step.

(Okay, perhaps you could argue that to make sure 2x + 4 was half of 4x + 8, you had to determine 4 divided by 2, and 8 divided by 2. But one could also argue that those are likely easy or fast calculations. There is also much less to keep track of while determining the solution, than when solving for x and substituting back. So elegant!)

Being able to look at a problem and recognize (mimic?) steps that could be taken to determine a solution is one thing. Being able to look at a problem in different ways and perhaps find a more elegant solution based on the context or the numbers given demonstrates flexibility, fluency, and comfort playing with the numbers.

When we are giving students problems to solve, what strategies are we hoping they'll employ? And are we giving them the chance to discover new strategies based on the numbers we choose?

Comments

  1. UnknownJune 21, 2018 at 11:14 AM

    In the Ontario curriculum, the second overall expectation in Patterning and Algebra provides those strategies for flexibility with numbers and expressions. This is why it is so important to cluster Patterning and Algebra with Number Sense and Numeration. Thanks for reminding me of the continuum of learning from number to algebra. You have certainly shown how the expectations around equality make their way to the intermediate levels.

    ReplyDelete
    Replies
    1. Heather TheijsmeijerJune 21, 2018 at 4:57 PM

      That's still a big area of learning for me... how the basic ideas of equivalence transcend into (and can be recognized in) intermediate and senior mathematics. Looking forward to lots of great learning through next year!

      Delete

Post a Comment

Popular posts from this blog

The Art of Questioning on Math Assessments

Image
Last year, I had a great conversation with a colleague around math assessment. We were talking primarily about how to best triangulate products, observations, and conversations to form an overall balanced assessment of student learning, but a side conversation has really stuck with me. She told me that on her math tests, for any given learning goal, she gives the students four levels of questions to choose from. The students can then decide how to best demonstrate their understanding of that expectation. I love the idea of student voice and student choice in how learning is demonstrated. But it's taken me about a year to really wrap my head around this idea, and to start working with teachers in my board to try it out. Traditionally... Traditionally, on a test, I might ask students three or four questions all on the same expectation. It might look something like this: Like most teachers, I started with an easy problem, and each subsequent problem gets a little more invo
Read more

One Word 2018

Image
It's that time of year again - a time for reflection and contemplation on the past year, and a time for setting new goals for a new year. Calm, and reflective. The past few Decembers, I've participated in the #onewordONT community, choosing a single word that summarizes my outlook and goals for the next year as I move forward in my professional practice. For 2015 , my word was JUMP - jumping into new adventures and taking risks to try new things. For 2016 , my word was REFLECTION - working more reflection into my practice as a teacher, but also helping my students reflect on their learning. For 2017 , my word was PATIENCE - practicing patience with both myself in learning a new role, and with the process of implementing change. It won't happen overnight. PATIENCE was a good focus for me this past year as I work my way toward deeper understanding. Coming out of the classroom, I've been involved in many learning opportunities in my new-ish role as a math co
Read more

Three Words I'm Eliminating from my Math Vocabulary

Image
Before we begin, take a moment and read the number sentence below out loud . Don't think about it too much, just read it as you normally would to a class or colleague. On Language This past year, I have been doing more work in looking at how aspects of literacy creep into (and affect) how we teach and learn math. One theme that reoccured in different contexts was language, and paying close attention to what we say when we are teaching math. Over time, many of us adopt "shortcuts" when it comes to talking about the symbolic representation of mathematics. We may know what we mean to say, but for students who struggle, the meanings of these shortcuts are not always apparent, and might even cause confusion. The following are three very small, deliberate changes in language I am trying to work into my practice moving forward. Though each is a simple change, I'm having a hard time undoing decades of bad habits! "Equals" Based on some of our work on
Read more