If you’ve been tasked with teaching the Common Core math standards, you have probably been struggling with its increased conceptual focus. What does it mean to teach math conceptually? How are we supposed to teach for conceptual understanding while continuing to demand procedural fluency? How can we shift our own mindsets, and then those of our students, to see how procedures and concepts work together in math?

We don’t have the answer, but two important pieces of research might help.

First up: a 1999 study called “Conceptual and Procedural Knowledge of Mathematics: Does One Lead to the Other?” by Bethany Rittle-Johnson and Martha Wagner Alibali. In this study, researchers tested 4th and 5th graders’ ability to understand, solve problems, and transfer their understanding of the concept of equivalency and its related procedures. They gave some students instruction in the procedures, and others a heavier dose of the concept.

The students were then asked to solve some basic problems, such as “3 + 5 + 7 = 3 + __ “. After the students solved the problems, the researchers interviewed them about their conceptual understanding (What does it mean for two things to be equal? What does the equals sign mean?) and asked them to solve problems in an unfamiliar format to test transfer.

The findings? Teaching students about the concept of equivalency led to increased understanding and the ability to generate and transfer a procedure for determining equivalency. Teaching students the procedures also helped develop their conceptual understanding and ability to adopt the correct procedure. However, students who only received procedural instruction had difficulty transferring their knowledge to unfamiliar situations. For instance, they may not try to use the same procedure for the problem 3 + ___ = 3 + 5 + 7 as they would if the missing number was on the other side of the equal sign.

The key takeaway was this: Increases in procedural knowledge lead to increases in conceptual knowledge, and vice versa. However, transfer of knowledge depends much more on conceptual knowledge than it does on procedural.

Second: a study called “Re-conceptualizing Procedural Knowledge: The Emergence of “Intelligent” Performances Among Equation Solvers” by Jon Star of Michigan State University. In this study, Star gave two sets of students identical problems to solve, but then used class discussion to focus either on procedural accuracy and fluency, or conceptual understanding. The discussions were the same length, but in the procedural discussion the focus of conversation was finding the right numeric answer, following the right steps, and how to check to see if the numeric answer was correct. In the conceptual conversations, students were asked to compare different approaches to the problems and come up with alternative methods for solving the problem. The result? Students who participated in more conceptual discussions and had focused on finding alternative ways of solving the same problems were more flexible and innovative in their solution methods after the experiment.

Although neither of these studies solves the problem of balancing conceptual understanding and procedural knowledge in our math classrooms, they do point out the important influence that conceptual understanding can have on procedural transfer, flexibility, and innovation.

Loved this. So true! Lynn Erickson

Some senior kids and I were discussing chemistry the other day (they’d had a particularly difficult exam but they’d taken it a year early so…) anyhow, we arrived at the discussion of problem-solving in chemistry, physics and maths. One of the kids who had studied in Syria before the troubles bemoaned that it was easier in his old school – they did tougher problems, more practice and he could pass it easily whereas at our school we weren’t up to the level of his past school, teachers didn’t give enough practice drills and he was not doing so well. I’ve heard similar stories from kids and worried teachers. We have to keep up with the syllabus content – and that’s tough but we’re getting results – look at the test scores – the good kids are getting good grades, the average are doing ok and those who aren’t passing just aren’t maths sort of people – maybe they are good a art or something. The worst case I experienced recently was taking on a class of grade 11 who had come straight from the national system which is quite teacher-led and content -driven and traditional. We started off with a discussion and some big questions to winkle out their ‘prior-knowledge’. They didn’t seem to be able to answer questions that were too big apart from relating what they knew from watching TV. We focused a bit and an equation came up – and they all recognised it with excitement. Then they were disappointed again because we realised they recognised it but didn’t know it, couldn’t use it (apart from substituting in numbers) and didn’t know the consequences of it. Again there are other cases that have led me to realise that of course conceptual understanding gives better results but classes on conceptual understanding might not be the way to go – just as with isolating study skills and giving classes on it, isolating the concept bits and teaching them explicitly might not be very transferable. The best way to go it seems is to blend and differentiate – to approach teaching as though the conceptual and procedural are part of what we do, how we think and how the subject and others are approached. This means creating an atmosphere where resources (text-books, programmes, teachers, and learners themselves) are infused with the two approaches. The transformation of education necessitates that we educators fix our sights on developing learners within the contexts of our subjects and good citizenship.