Manipulatives

Liz Caffrey is a master middle school math teacher in the Boston area. She is the author of last year’s Lab Gear, the Great Connector, a guest post on this blog. Today, she shares her big-picture thoughts about manipulatives.

I inserted some links to pages on my website — you can find many more links on my Manipulatives home page.

— Henri

Manipulatives 

by Liz Caffrey

Recently I had the chance to lead a workshop on hands-on math for middle school teachers at the Atrium Summer Math Institute. I featured a wide array of manipulatives in the workshop, and it was clear that some were better received than others. 

What is a manipulative? I would define it to be anything you can move, sculpt, or build that helps you visualize or model mathematics. Is a card sort a manipulative? Not by this definition – you’re moving the cards around, but their placement doesn’t usually help you visualize or model the math. Is Play-doh a manipulative? Depends how it’s used. How about a ruler? Not when used alone. How about Desmos? There are lots of virtual manipulatives integrated into Desmos activities (like Polypad), and even the sliders on the graphing calculators fit my definition. 

I say all this to indicate that a) something kinesthetic does not necessarily mean it’s a manipulative and b) you do not need a huge classroom budget to utilize manipulatives in your classroom. Many household items make great manipulatives when used in a certain way. I use Play-doh to model fraction division, my own kids’ Magna tiles and Legos for some surface area and volume explorations, and I do a lot with paper as well. That said, there are several manipulatives for math that are worth investing in: base-10 blocks for elementary teachers; Lab Gear for middle and high school teachers; geoboards, pattern blocks, and interlocking cubes for all ages.

Used well, manipulatives can provide a tangible, visible, concrete entrypoint for students. They allow students to actually construct their own understanding. Manipulatives can also provide added challenge – there are many ways that students can deepen or extend their knowledge. For this reason, I always give time at both the beginning and end of a lesson for kids to “just play” with the manipulatives. At the start of class, this allows students to enjoy the new tool without being told what to do with it right away. Some kids just enjoy being playful, but most are also noticing math – how the pattern block pieces fit together or how the side lengths of each Lab Gear block are unique. I love watching students create glorious tiling patterns with pattern blocks, intricate designs with rubber bands on the geoboards, or silly skyscrapers with Lab Gear. At the end of a lesson, once students understand the purpose and power of the tools in their hands, I love to see what problems they can come up with. Sometimes the very thing they built at the start of class gets built again, with a new layer of mathematical meaning layered onto it. Every year, I threaten to make students find the surface area of silly Lab Gear skyscrapers they build during the lesson. And every year, I have at least one student take me up on this! 

In my workshop this past summer, many of the teachers were using the Desmos curriculum, which they praised for its inquiry-based lessons, emphasis on collaboration, and strong ability to make math visible. But their kids were on screens a lot. Beyond just a desire for screen-free time, there was something to physical manipulatives that was much more engaging for the teachers in my workshop. I think this is due to them being both kinesthetic and more open-ended in terms of how they can be used. 

When I offered a Desmos version of Pythagorean Theorem proofs without words, teachers enjoyed the activity. But when I showed them how it could also be done with puzzle pieces printed on card stock, they vastly preferred that option. There was much more freedom in being able to move the pieces around on the table versus being restricted to certain types of rotations and translations within the Desmos activity. 

By far, the most popular manipulative tool teachers learned about was the Lab Gear. When teachers themselves were able to see the powerful connections between the geometry of the blocks and algebraic expressions, they were blown away. Whether it was combining like terms by adding the blocks together, solving equations by actually removing blocks from both sides of an equation, or using the area model to understand completing the square, participants marveled at how deep the math was and how much more tangible it made variable operations for them and their students.

During the summer institute, in addition to my workshop for middle school math teachers, we also had workshops for PreK-2nd grade math teachers and upper elementary math teachers. We had a daily Math Circle for all, which many of the elementary math teachers found intimidating at first. On our last day, we did Angles Around a Point, the first lesson in Henri’s Geometry Labs book, and Henri’s pattern block dodecagon activity, both of which were met with great appreciation from all levels of educators. These activities were able to provide an entry point for even the most timid teachers – they could get building right away, using a manipulative they were familiar with. They didn’t even realize that the activity was for middle or high school students! There was room for both creativity and added rigor in both activities, which kept everyone engaged. Recently one of the more timid early education teachers wrote to me and shared that she has recently incorporated weekly Math Circles into her 2nd grade class! 

In short, there’s no better bang for your buck in an inquiry-based math class than manipulatives. They bring the math to life in front of students’ eyes, allow myriad ways to differentiate, and they are highly, highly engaging. 

If you are a PreK through middle school math teacher who’s interested in working with me next summer, email institute@atrium.org to get on our mailing list! We have different workshops every summer, and we open signups in April.