New School Year, New Tools!
August 2024

As the new school year starts, it’s the perfect time to consider the appropriate use of tools to support your students in learning mathematics. Tools provide opportunities for students to visualize concepts as they explore, discover, and build a conceptual understanding of mathematics. Manipulatives, calculators, and interactive websites are just a few of the tools that offer students opportunities to recognize that they can understand, rather than memorize mathematics. Unfortunately, many of these tools are not available or are not being used by students at all grade levels. As we are aware, manipulatives are often used in the elementary grades, but they should also be used in middle and high schools.

Physical manipulatives are engaging for secondary students and afford them opportunities to explore concepts as they begin to build an understanding of the material they are studying. For example, using algebra tiles or two-color counters when they are learning about adding integers can be powerful. Using these physical tools often motivates students to become engaged in finding patterns and beginning to develop an understanding. As students move among concrete, semi-concrete, and abstract stages, that understanding is deepened. With manipulative use, students aren’t passively involved in their learning but instead are actively engaged in making sense of the concepts.

Manipulatives are not a different way of doing mathematics in the secondary grades, but rather a different way of learning the content. It is vital to make the connections among the concrete, semi-concrete, and abstract stages. Students can build from their initial understandings formed from manipulatives as they make generalizations and look for structure and repeated reasoning. Some students may visualize the manipulatives, or draw a quick sketch, as they continue to work towards automaticity. As educators, we should be providing such tools to allow for increased access to content. Not all middle and high school students are able to jump straight to abstractions, and yet, too often, as educators, we don’t allow the necessary opportunities to make mathematics more concrete, thereby excluding some students. It’s fair to say that manipulatives play a critical role in making mathematics more accessible to all students, ensuring they feel included and valued in the learning process.  

Using a variety of geometric shapes and solids and other geometric manipulatives allows secondary students opportunities to actually see the solids and shapes! Too often, we ask students to visualize the three-dimensional shapes from images, but not all students are able to do so easily. As educators, we must consistently look for ways to allow all students access to concepts and geometric manipulatives that allow them opportunities to fully engage in geometric reasoning and sense-making.

Understandably, middle and high school teachers may be reluctant to use manipulatives, fearing that students may not want to use them due to their association with elementary school. However, I’ve found that if I start the year with different manipulatives from those that students frequently use in elementary school, they are willing to use manipulatives, and they become engaged in learning the mathematics. Some may make the transition to more abstract thinking quicker than they did in elementary grades, which is why providing opportunities to engage in concrete activities is so valuable. This reassurance about the effectiveness of manipulatives in the secondary grades should instill confidence and open-mindedness in educators.

In order to help all students make sense of the mathematics and develop a solid understanding of the concepts, educators must use a variety of instructional strategies. It's important to remember that every student learns differently, so we must adapt our approach to meet their individual needs. Manipulatives, both at the elementary and at the middle and high school levels, provide opportunities for each and every student to reason and make sense of the mathematics. So, feel free to manipulate the new school year for your secondary school students—in a good way of course!  

Kevin Dykema
NCTM President
@kdykema