The Importance of Using and Connecting Mathematical Representations

 Mathematics is often perceived as a collection of abstract concepts, but at its core, it’s a rich network interconnected ideas and representations. The use and connection of various mathematical representations—contextual, physical, visual, verbal, and symbolic—are essential for deepening students' understanding of mathematical concepts and procedures.

Why Use Multiple Representations?

Utilizing multiple representations allows students to approach problems from different angles. Each representation offers unique insights that can make complex ideas more accessible. For instance, a geometric concept can be explored through physical models, visual diagrams, and symbolic equations, providing a more comprehensive understanding. When students see how these representations are interconnected, they can deepen their understanding and gain a greater appreciation for the the coherence and unity of mathematics as a discipline.

Enhancing Understanding and Flexibility

When students engage with different representations, they not only enhance their understanding but also develop flexibility in their mathematical thinking. This flexibility is crucial for problem-solving, as it enables students to choose the most effective strategy for a given situation. For example, a student might visualize a problem with a diagram, express it through an equation, and then interpret the solution in the context of a real-world scenario. This ability to switch between representations helps solidify their understanding and builds confidence in their mathematical skills.

Fostering a Deeper Appreciation for Mathematics

Connecting mathematical representations also fosters a deeper appreciation for the discipline itself. When students recognize the relationships among different representations, they see mathematics as a coherent and unified field rather than a series of disconnected topics. This appreciation can inspire a lifelong interest in mathematics, encouraging students to explore and engage with the subject beyond the classroom.

October 2024 Focus at the Kentucky Mathematics Teacher Leader Professional Learning

During October Kentucky Mathematics Teacher Leader professional learning session, we discussed the importance of using and connecting mathematical representations in our teaching practice. We linked this approach to "attending to precision" in student practice, emphasizing how precision in language and representation supports deeper understanding.

To illustrate this concept, we engaged in a Translation Task (Van de Walle, Karp, Lovin, & Bay-Williams, 2019), where we collaboratively created four different representations for the equation $2x + 3 = 21$. This exercise not only highlighted the various ways to interpret and represent a single mathematical idea but also demonstrated how connecting these representations can enhance students' comprehension and ability to articulate their reasoning.

After creating four different representations of the symbolic equation $2x + 3 = 21$—including visual, verbal, contextual, and symbolic forms—we explored how these representations were interconnected and how one could be seen within another. Initially, much of our "explain your thinking" responses focused on reciting the procedure for solving the equation. We revisited and refined our explanations to relate the solution process back to the context of the problem. This reflection helped us become more precise in our descriptions and gave meaningful interpretations to the constants and coefficients in the equations.

Observing Student Engagement

As educators, it’s vital to observe how students interact with various representations. Look for instances where they make connections, articulate their reasoning, and demonstrate flexibility in their thinking. By promoting an environment where students can explore and connect representations, we empower them to become successful, confident mathematicians.

In conclusion, using and connecting mathematical representations is not just a teaching strategy—it’s a pathway to deeper understanding, flexible thinking, and a lifelong appreciation for mathematics. By prioritizing this approach in our classrooms, we can equip students with the skills and mindset necessary to thrive in an increasingly complex mathematical world.