As a mathematics education researcher, I study how mathematics instruction affects student learning, from following standard mathematical procedures to understanding mathematical concepts. Focusing on the latter, conceptual understanding involves: The “why” of mathematical concepts;It’s not the reasoning behind the mathematics how Or the steps it takes to arrive at the answer.
Often in math classes, students are shown the steps or procedures for solving a math problem and then expected to demonstrate to themselves that they have memorized these procedures.
As a result, students’ agency, knowledge, and ability to transfer mathematics concepts are impaired. Specifically, students are less confident in tackling mathematics problems and less able to apply mathematical reasoning to real-world situations. Furthermore, students may struggle with more advanced mathematics concepts and problem-solving challenges as they progress in their education.
While procedural fluency is important, conceptual understanding provides a framework for students to build mental connections between math concepts. This allows students to connect new ideas to what they already know, strengthening connections to more advanced math.
If we want to improve math achievement, we need to start focusing instruction on concepts, not procedures.
Why concepts are more important than steps
Conceptual understanding focuses on the ability to build on existing understanding, justify and explain, whereas procedural fluency focuses on Steps to arrive at the answer And accuracy.
When considering how students learn more advanced math concepts, it is important to consider how students approach the problems presented in class and how those problems contribute to students’ higher-level conceptual understanding and higher-level procedural fluency. For example, consider the following two math problems and ask yourself: What knowledge is needed to solve each problem?
The first problem requires more reasoning and extended thinking about fractions than the second problem, which requires more knowledge of procedures, factual recall, and recognition.
As you plan to develop conceptual understanding, imagine the four student groups that are typical in a math class: one that gives correct answers and correct inferences, another that gives correct answers but incorrect inferences, a third that gives incorrect answers but correct inferences, and a third that tends to give both incorrect answers and incorrect inferences.
We tend to focus on the group of students who get the answer right, regardless of their reasoning. However, the group of students who make correct reasoning tends to be closer to understanding the concept, even if they may have arrived at the wrong answer. In other words, focusing only on the correct and incorrect steps and answers is adequate to determine whether a student completed the steps, but it does not provide much insight into the student’s understanding of the concept.
It is important to pay attention not only to the inferences students make to develop conceptual understanding, but also to the instruction you are using to deepen students’ conceptual understanding. Direct instruction is: Phased Release Modelor “I do, we do, you do,” is teacher-directed instruction in which the teacher models procedures and processes and students memorize and follow them, which is the opposite of developing concepts.
To guide students to stronger conceptual understanding, teachers need to provide students with opportunities to explore, engage in productive engagement, explain their thinking, and connect existing knowledge to new content.
Inquiry-based teaching tends to be student-centered and helps students develop conceptual understanding rather than simply replicating a teacher’s model. For example: The cycle of awareness and action It empowers students to develop mathematical reasoning, understanding, justification, and original solutions and strategies, leading to a positive mathematical identity that extends beyond the classroom.
Moving Towards Conceptual Understanding
Teaching for conceptual understanding is easier said than done because it requires deep content knowledge and the ability to connect student responses to concepts in real time. As teachers, we know it can be difficult to know how to respond to instructional moments so that student learning continues to improve toward goals. Additionally, students can have many unexpected responses when solving math problems, which requires educators to have a high level of flexibility and creativity.
Fortunately, mathematics education research and instructional practice have developed several instructional strategies to promote students’ conceptual understanding.
Use open-ended tasks
One way to help students develop conceptual understanding is to provide hands-on, problem-solving learning opportunities. Open-ended tasksThese types of tasks have multiple correct answers, solutions, or outcomes – and therefore multiple entry points and exit points – making them ideal for classrooms with students with different abilities and existing knowledge.
Students can focus on the solving strategies that work best for them and teachers can encourage more complex and efficient solving strategies. Open-ended questions allow students to compare different ways of thinking and provide the opportunity to identify more efficient trains of reasoning.
Accompany students on operations at the unit
Another element to focus on for students is the unit and its manipulatives that are addressed in the problem. In these unitsHere are three questions you and your students can ask and answer:
- What units are used in this question?
- How are the units related?
- How can we imagine the units to help us think about the relationships between them?
For example, consider the answers to these questions in relation to this issue:
I’m flying from Boston to New York City. I depart at 6:47pm and my flight time is 1 hour 25 minutes. What time do I land?
The units in this problem are minutes and hours. It is important to pay attention to whether students are working in units of minutes (minutes at a time) or in groups of ones (for example, counting 60 minutes at a time and understanding that as an hour). The units relate to each other in that there are 60 minutes in an hour, and students can think of this as one way to visualize an actual analog clock passing time, although other students may imagine the problem in a different way.
Encourage mathematical discussion
A third way to promote conceptual understanding is to encourage mathematical dialogue between teachers and students, and among students themselves. When incorporating mathematical dialogue into your instruction, it’s important to think about the math students already know as a way to start the conversation. It’s very difficult to have a conversation about something you know nothing about. But when you’re asked about something you know, you have a lot more to contribute.
I’ve found it helpful to ask, “What math can help my students join the conversation?” If the problem is beyond their level, what are the entry questions that can help them connect what they know with what they are trying to learn? How will students build on the math they already know as they solve the problem together (with your help)?
Almost every math learning resource has cores of conceptual understanding, and our job as educators is to help students examine those cores and find lenses through which they can explore.
Procedures are often part of a math lesson, but building on conceptual understanding over time can improve students’ math achievement and increase their ownership in becoming mathematical thinkers.
