As parents, you may have noticed less emphasis on algorithms and memorization in mathematics and more emphasis on “sense-making” since the time when we were kids in elementary school. A big piece of this process is using manipulatives– toys that are used as teaching tools to engage students in the hands-on learning of math– in the classroom.  

The teaching philosophy behind this is to start with the concrete (manipulatives), move to representational (pictures and diagrams), and finally move to the abstract (algorithms and formulas). Let’s run through what this might look like in a classroom when learning how to find the area of a rectangle. We would start out with rectangles of different sizes (I might use straws with twist-ties connecting them or better yet, Anglegs) and one inch square tiles. The kids would then have the chance to explore how to find area (knowing already that area is the space the shape covers) using these tools. We would discuss how they used these tools to find area, as well as what problems they might have encountered. Then we could move to pictures—in this case, drawing rectangles on grid paper and finding the area, afterward discussing what patterns they might have noticed (ex: instead of counting each individual square in their rectangle, they might have seen 3 rows of 6, which would be an area of 18). We would then use what the students had noticed to develop a formula that always works, length x width = area, testing it out with more rectangles to be sure that it works. This way, they are not just memorizing a formula, but have the understanding to back it up.  

This first concrete step is so important in helping kids understand why things work in mathematics. Manipulatives like base ten blocks, fraction tiles, or tangrams (anything concrete that can be viewed and handled by kids) allow students to construct their own understanding of the mathematics behind the algorithms they will eventually learn. Manipulatives give them the opportunity to observe and model more abstract math concepts. Jean Piaget, a well-known educational researcher of the 20th century, said that children only begin to understand symbols and abstract concepts after discovering and working with the ideas in a concrete way. And studies have found that the long-term use of manipulatives can increase achievement in mathematics.  

Manipulatives also help engage students, increasing their interest as they work with the manipulatives. Working with them feels more like playing, especially since it often involves exploration, discovery, and experimentation. Manipulatives can make solving problems both easier and more interesting. They also engage the senses—using sight and touch to work through a problem, and encourage discussion and communication skills more so than completing a worksheet.  

While children can remember information taught using books and lectures in the short term, a thorough understanding and the ability to apply that understanding requires conceptual comprehension and sense-making from work with concrete objects. 

You can find some of the manipulatives my colleagues and I have found effective in using with kids on this site; but also keep in mind that manipulatives don’t necessarily have to be math-specific. They can be anything concrete that kids can work with to better understand math concepts, whether it’s using buttons for sorting and classifying or using baking brownies as an opportunity to think about fractions.  

Have you come up with any everyday items that you have used as manipulatives for teaching math?