As a Grade 6 mathematics teacher, I’m big on student discovery. I believe mathematics is the science of patterns—it allows students to explore problems and create their own rule statements that help them make deep, meaningful connections. This approach bridges the gap between conceptual and procedural knowledge, ensuring learners not only grasp the “how” but also the “why” behind mathematical processes.
When students discover concepts on their own, they retain the information longer and can apply it in many different situations. Rather than simply finding the greatest common factor of two numbers, they can extend this skill to three numbers or more and ultimately to factoring polynomials in high school courses.
In my accelerated classes, I see this firsthand. Learners with a strong foundational understanding are better prepared for upper-level courses. In my school, we use vertical alignment planning, starting in Grade 6. We provide opportunities to use manipulatives and build a robust foundation that supports higher-level learning. The pupils who have this strong foundation of number sense are so much more successful in middle school and beyond.
Changing Negative Perceptions of Mathematics
One major roadblock in most mathematics classrooms across America is mathematics anxiety. Often, this stems from familial attitudes—maybe a parent had trouble with mathematics and passed down this anxiety. My aim is to dismantle these negative perceptions and show students that they can excel in mathematics by connecting conceptual knowledge to procedural knowledge.
Three months into the school year, I usually see pupils who were once apprehensive to participate suddenly eager to lead number talks and demonstrate problem-solving skills using algebra tiles or other tools. Creating a love for numbers is essential. When students understand concepts deeply, they become confident in their abilities and start to enjoy mathematics, seeing it as a series of intriguing problems rather than daunting tasks.
A Dynamic Classroom Environment
I believe a mathematics classroom should be vibrant, noisy, and collaborative. Learners should be sharing strategies, debating approaches, and explaining concepts to one another. This interactive, student-centered environment fosters deeper understanding and helps them see multiple ways to solve problems. For instance, students may use the easy option of algebra tiles to model an equation, and I will then challenge them to pick another manipulative to show the same solution. Even in general and enrichment classes, I encourage manipulative use and productive struggle. I remind my students that it’s okay to fail. In fact, failure is part of the learning process. Students should feel safe taking risks and trying different approaches. They need to determine the right tools for the job, and that often involves trial and error.
Creating a Safe Environment
It’s crucial to invest time in building relationships with students, especially in the first year of middle school. Despite having more than 120 students, I try to get to know each one personally. Incorporating their interests into lessons—like creating word problems about things they enjoy, like Taylor Swift and Travis Kelce and football, makes learning more relatable and engaging.
It’s also important to support my students outside the classroom. Attending their events, whether it’s a band concert or Thursday night football game, shows them that I care. My family often joins me in these activities, reinforcing the concept that we are all invested in their success.
However, a truly safe environment in which kids can step out of their comfort zone needs to be established on day one. In my classroom, we have some non-negotiables. No ridiculing one another. I nip this behavior in the bud, so my students understand I won’t tolerate it. I teach that every wrong answer is an opportunity for analysis. No blank spaces on test papers—even if my students don’t have the answer, they know something, and that needs to be written down. I am big on learning through error analysis. When we make mistakes, we treat them like puzzles to solve, which fosters a culture of collaboration and mutual respect.
The Role of Formative Assessments
Formative assessments are essential in identifying and addressing common errors. If many students are making the same mistake, we use it as a teaching moment and a collective learning experience. End-of-lesson quizzes help identify areas for improvement and offer students a chance to correct mistakes before they take the graded assessment. This continuous feedback loop ensures learning is an ongoing process.
Transitioning from Rote Memorization to Conceptual Understanding
In college, I excelled in mathematics through rote memorization, but I didn’t truly understand the concepts. This realization prompted me to shift my teaching approach. Today, many teachers stick to traditional methods of teaching mathematics because they are familiar and it’s seemingly effective. However, this approach lacks depth.
The great news is you don’t need to overhaul your entire teaching method—small changes can have a significant impact. Start by shifting from a teacher-led to a student-centered approach. Allow students to explore problems without front-loading information. Their unique approaches to solving a problem will often blow your mind.
Creating a Collaborative and Engaging Classroom
From the outside, my classroom may seem loud and chaotic at first, but this dynamic environment is essential for deep learning. Instead of churning out 50 problems a day, we focus on solving a few rigorous problems, which requires deep thinking and collaboration. The setup changes frequently—students never know who they’ll work with next, keeping the environment fresh and engaging. Quality over quantity is key.
I incorporate innovative lessons like digital escape rooms, human number lines, and real-world data analysis projects to make learning fun and relevant. For example, plotting real-world statistics and identifying the impacts of outliers requires students to think critically and apply their knowledge in new ways.
Balancing Conceptual Understanding with Memorization
While I emphasize conceptual understanding, I also believe some memorization is necessary. Knowing basic facts like multiplication tables frees up cognitive resources for higher-order thinking. Just as fluent reading enables comprehension, fluency in basic mathematics facts supports more complex problem solving.
By fostering a love for numbers, creating a safe and collaborative environment, and balancing conceptual understanding with necessary memorization, we can transform our students’ attitudes toward mathematics and equip them with the skills they need for future success. Let’s shift from teaching tricks to building deep, lasting mathematical knowledge.
Want to learn more about the importance of conceptual understanding in mathematics? Check out this publication.
