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Why and how: Do students truly understand the concepts behind the procedures?

By EMILY NUTTALL posted 01-03-2018 11:43


Getting correct answers does not support conceptual understanding. We need to shift away from the practices of rewarding students whose learning style and personality are compatible with algorithms, following steps, and getting-answers. Getting answers does not support conceptual understanding - these students do not achieve true understanding or depth of knowledge on which they can build and connect ideas.  We need to build depth of knowledge that goes beyond answer-getting, as highlighted by Phil Daro (

A student’s ability to memorize an algorithm and answer a question correctly does not indicate conceptual understanding.  In fact, I contend that students who can only solve mathematical problems by using algorithms have not achieved a fundamental understanding of the mathematical concept.  A prime example that comes to mind is teaching students how to add and subtract when regrouping (2.NBT.B.7).  When I first began teaching this topic I relied on one fixed and antiquated algorithm that I remembered learning as a child, which really didn’t have anything to do with mathematics.  I learned that when subtracting, “if you don’t have enough, go next door and borrow one more.”  This rhyme did nothing to support conceptual understanding, nor did it develop my own understanding of why we regroup and how regrouping works mathematically. Somewhere along my learning journey, I came to understand that I was actually borrowing from - and making sense of - place value, but I don’t remember explicitly learning this.  Further, I don’t recall this ever being represented to me in different ways. When I reflect on my own learning experience as a student, and now on my practice as a teacher, I know that understanding place value and decomposing numbers are fundamental conceptual understandings.

During my first year of teaching 3rd graders I saw students presenting misconceptions and misunderstandings about regrouping, I was completely taken aback. I’m embarrassed to admit that I began to form some negative thoughts about my young scholars’ learning abilities. I wondered how or why they didn’t know what I deemed “simple” addition and subtraction?  Thankfully, the experiences I am writing about pushed me to think further and to see my students as capable learners. I realized that it must have taken me years to make sense of regrouping:  you don’t just borrow “one more” or “look next door.

In fact,  it wasn’t until teaching students how to regroup that I realized just how complex the concept of regrouping is.  As I watched so many of my 3rd graders struggle with what I deemed “simple math,” I realized that I needed to find a way to help all students make sense of and understand how and why we regroup.  Why is borrowing or carrying one, and not two or five, significant when regrouping?  What are we even doing when we regroup? What is the purpose of this mathematical concept?  When thinking on these questions, it became clear how important it is for students to truly see how and why (purpose) the math—or algorithms work. In this example, I found that giving students opportunities to represent the math problems (using, but not limited to, base 10 blocks, number lines, and number sets), they were more accurately able to articulate what they were doing and why.  Consequently, as they gained conceptual understanding through tangible representations of the problems, they answered questions correctly - and were able to make sense of their thinking along the way.  From this example, as an educator, I realized that helping students learn and become strong mathematicians has nothing to do with how many problems students can “get right.”  Instead, helping young scholars become exceptional mathematicians is more about giving them opportunities to explore the math and represent it in ways that make sense to them. Ultimately students obtain solid, conceptual understanding upon which they can reason, think critically, and build more mathematical knowledge. As noted in Principles to Actions, “Conceptual understanding establishes the foundation, and is necessary, for developing procedural fluency.” Of course, a great byproduct of this process is that young mathematicians will get correct answers!  However, they will understand and be able to explain how and why the math works.





11-21-2019 14:57

I wholeheartedly agree! I would appreciate some advice on working with some of my highest-performing 5th students with the long division algorithm. Here's the back story: 

These students learned the long division algorithm in 4th grade. They've familiarized well with the procedure, but they aren't connecting the algorithm to place value or the meaning of division. When I asked them to model their process with an area model, a few were dismissive, saying "We already know how to divide. A model is a waste of time."

They do know how to divide, but they're missing why that method works. Any suggestions on filling the gaps in a way that won't trigger resistance?

01-08-2018 15:21

It is always engaging to hear from those in the field, doing the work with our students, and reflective on their own practices. Great read, Emily!

01-05-2018 09:55

I really appreciate the insights of this blog. I plan to use this blog in my math methods course to highlight how Emily reflected on her practices. I also appreciate how she umpacked the complexity of teaching regrouping.