As a mathematics education researcher, I study how math instruction impacts students' learning, from following standard math procedures to understanding mathematical concepts. Focusing on the latter, conceptual understanding often involves understanding the “why” of a mathematical concept; it’s the reasoning behind the math rather than the how or the steps it takes to get to an answer.
So often, in mathematics classrooms, students are shown steps and procedures for solving math problems and then required to demonstrate their rote memorization of these steps independently.
As a result, students' agency, knowledge and ability to transfer the concepts of mathematics suffer. Specifically, students experience diminished confidence in tackling mathematical problems and a decreased ability to apply mathematical reasoning in real-world situations. In addition, students may struggle with more advanced mathematical concepts and problem-solving tasks as they progress in their education.
While procedural fluency is important, conceptual understanding provides a framework for students to build mental relationships between math concepts. It allows students to connect new ideas to what they already know, creating increasing connections toward more advanced mathematics.
If we want mathematics achievement to improve, we need instruction to begin focusing on concepts instead of procedures.
Why Concept Matters More Than Procedure
Conceptual understanding builds on existing understanding to advance knowledge and focuses on the student’s ability to justify and explain. Procedural fluency, on the other hand, is about following steps to arrive at an answer and accuracy.
When considering how students will learn more advanced mathematics concepts, it is important to consider how they will engage with the problems presented to them in class and how those problems will contribute either to their greater conceptual understanding or greater procedural fluency. For example, consider these two math questions and ask yourself: What knowledge is needed to solve each problem?
The first problem requires more reasoning and broader thinking about fractions than the second problem, which requires more knowledge of procedure, fact recall and recognition.
When planning to develop conceptual understanding, imagine four groups of students typically in mathematics classrooms: one group who gets the correct answer with the correct reasoning, another group who gets the right answer but with wrong reasoning, a third group who gets incorrect answers but with correct reasoning and a final group who tends to have both incorrect answers with incorrect reasoning.
We tend to pay attention to the groups of students who get the correct answers, regardless of their reasoning. However, the groups of students who have the correct reasoning tend to be closer to conceptual understanding—despite the fact that they may have arrived at the incorrect answer. Said differently, only paying attention to correct and incorrect steps and answers is appropriate for determining if a student completed a procedure, but not so telling for students’ understanding of a concept.
In addition to paying attention to the types of reasoning your students have to develop conceptual understanding, it is also important to pay attention to the type of instruction you are providing for students to advance their conceptual understanding. Direct instruction, the gradual release model, or “I do, we do, you do,” is a teacher-led type of instruction in which teachers model procedures and processes for students to then memorize and follow — the opposite of developing concepts.
To guide students toward more robust conceptual understanding, teachers need to give students opportunities to explore, engage in productive struggle, explain their thinking and connect their existing knowledge to the new content.
Inquiry-based instruction, which tends to be more student-centered, helps students develop conceptual understanding instead of simply copying the teacher’s modeled steps. For example, the perception-action cycle allows students to develop mathematical reasoning, understanding, justifications, and their own solutions and strategies, which can result in positive math identities beyond the classroom.
Moving the Needle Toward Conceptual Understanding
Teaching for conceptual understanding is easier said than done, as it requires deep content knowledge and the ability to make connections between student responses and concepts in real time. As a teacher, I know it can be difficult to know how to respond in the moment of teaching to make sure the students' learning continues to improve toward the goal. Additionally, it demands a high level of flexibility and creativity from educators as students can contribute many unanticipated responses as they work through solving math problems.
Fortunately, mathematics education research and instructional practice have developed some instructional strategies to promote conceptual understanding for students:
Use Open-Ended Tasks
One way to help students develop conceptual understanding is to provide learning opportunities that involve working with and solving open-ended tasks. These types of tasks have more than one right answer, solution or outcome, along with multiple entry and exit points, making them great for classrooms with students of varying abilities and existing knowledge.
Students can focus on the solution strategies that best fit them, and you can facilitate solution strategies that are more complex and efficient. Open-ended questions also allow students to compare different ways of thinking, providing an opportunity to see more efficient lines of reasoning.
Attend to Students’ Operation on Units
Another element to focus on with students is the units they work on within a problem and how they operate on those units. Three questions you and your students can ask and answer are:
- What are the units you are working with in this problem?
- How are the units related?
- How can you picture the units to help you think about the relationship between them?
For example, think about the answer to those questions in relation to this problem:
You are flying from Boston to New York City. Your departure is 6:47 p.m. and the flight time is an hour and 25 minutes. What time will you land?
The units in this problem are minutes and hours, and we would want to pay attention to whether students are operating on ones (individual minutes) or groups of ones (e.g., counting 60 minutes at a time and understanding it as one hour). The units are related to each other in that there are 60 minutes in every hour, and one way a student might visualize them is as an actual analog clock passing time, though others may have different ways of picturing the problem.
Encourage Mathematical Discourse
A third way to help promote conceptual understanding is to encourage mathematical discourse between you and your students and between students. When implementing mathematical discourse in instruction, it is key to think about the math students already know as their entry points into the conversation. It is quite difficult to have a conversation about something that you know nothing about. However, if you are asked something you know, there is a lot more you can contribute.
I have found it helpful to ask, “What math do my students know that will help them engage in the conversation?” If the problem is above their level, what entry-point questions can I use to help them connect what they know to what they are trying to learn? How will students advance the math they already know as they work together (with facilitation) to solve the task?
There are nuggets of conceptual understanding in almost every math learning resource. As educators, our job is to help students find a lens that allows them to examine and explore those nuggets.
While procedures are often part of math lessons, let them evolve from conceptual understandings over time. In doing so, we can help students improve their mathematics achievement and their agency in becoming mathematical thinkers.
