- Introducing students to various tools and representations is incredibly valuable for many reasons, including but not limited to making students' abstract thinking more concrete, helping students think through problems and communicate their understanding, and making complex concepts make more sense. Although tools and representations are one of the high-impact instructional practices, we find that the usage of tools and representations are not consistent across schools or even classrooms and in some cases are not used at all, particularly in older grades. Tools and representations can consist of manipulatives like base 10 blocks, linking cubes, relational rods, pattern blocks, color tiles, or even fingers. We now also have a wide range of virtual manipulatives that can be used with students in person or online. Other tools that can be used in the mathematics classroom could range from online applications to mathematical tools like rulers and calculators. In primary and junior grades, we see teachers frequently using concrete objects or manipulatives. It is important that when students use concrete objects or manipulatives that they use the same manipulatives over an extended period of time. Children need time to engage with the manipulative and make connections to the concepts being taught. It is also necessary for students to use manipulatives that do not have distracting details. For example, when focusing on subitizing, using a single-color counter rather than counters of multiple shapes, colors, or sizes would work better. Lastly, students need guidance when beginning to use manipulatives. We cannot assume that there is an instant relationship being created between the manipulative and the students' thinking. An example of a manipulative that is quite useful but often overlooked due to misunderstandings on how they can be used are relational rods. Relational rods help students visualize mathematical concepts like fractions and proportional reasoning. They also can be used to represent length, area, and volume, which could be incredibly helpful if working on measurement concepts. Mathematical representations, on the other hand, could range from diagrams, charts, symbols, pictures, arrays, and number lines. In the classroom setting, we want to encourage our students to represent their thinking in different ways for different purposes while instilling in them the understanding and importance of making connections between these different representations and their abstract thinking and ideas. One thing we need to remember is that visual representations are not all equally as helpful. For example, asking your students to draw a picture to illustrate their understanding of a problem might actually be counterproductive if the students take a great deal of time drawing something unrelated to mathematics. As with using mathematical tools, we need to support our students in understanding how to use visual representations to help them solve a problem. An example of a visual representation or tool that has multiple uses are number lines. Although number lines are used on occasion, they are rarely used as effectively as they can be. Number lines help children develop a greater flexibility in mental arithmetic, number sense, and number relationships. In addition, number lines support students' thinking strategies. Number lines can be used with whole numbers, fractions, decimals, or integers, and supports the understanding of parts of a whole. When beginning to use number lines, it is a good idea to start with concrete models like linking cubes or number trains in order for students to become accustomed to using number lines. When students feel more comfortable, you can move on to using closed number lines with number ticks and, as understanding develops further, open number lines can be introduced. Incorporating mathematical tools and representations on a regular basis will ensure that your students gain the necessary practice they need to use these tools and representations to their full potential. Demonstrating how they might be used is a great way to encourage students to use these tools and representations themselves. Therefore, including the various tools and representations in your number routines or problem-solving lessons will show your students the value of using these mathematical tools and representations.