# 19.7: Mathematical representations and manipulatives

$$\newcommand{\vecs}{\overset { \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$

Mathematics instruction, at all grade levels, should begin with developing a conceptual understanding of mathematical ideas. This can be accomplished through the use of concrete and representational models before moving to abstract representations. When planning instruction, teachers should consider how to sequence the learning to support moving from concrete representations to the symbolic and abstract. Visual representations of the mathematics are critical in laying a strong foundation of mathematical ideas. Students need experience using concrete manipulatives and then moving to representational models to solidify the use of imagery in problem solving before moving to abstract symbols. The connections students make throughout these stages are essential and should be an intentional design of any lesson.

The first stage is the concrete stage in which students experience math by physically manipulating various objects. The second stage engages students in using representational models to solve math. During this stage, students represent concrete objects as pictures or drawings. Using abstract symbols is the third stage.

Instructional practices should support students moving from the concrete and representational stages to using numbers and symbols to model and solve math problems. Students need opportunities to develop mathematical thinking at each stage and to make connections between the stages to develop the ability to move flexibly among the different representations.

Traditional mathematics instruction has historically focused on computation and students’ ability to apply procedures quickly and accurately. According to National Council of Teachers of Mathematics, procedural fluency includes the ability to apply, build, modify, and select procedures based upon the problem being solved. This definition of procedural fluency pushes the bounds of traditional mathematics instruction, as it requires foundational knowledge of concepts, reasoning strategies, properties of numbers and operations, and problem-solving methods.

*CRA will be addressed further in chapter 20

19.7: Mathematical representations and manipulatives is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.