by Kate Fanelli
Research shows that a Concrete-Representational-Abstract (CRA) approach to teaching students who are struggling and students with disabilities is an effective way to build mathematical understanding.
“Concrete” refers to using physical materials (often called “manipulatives” in the math classroom) that students can touch and move to model and build understanding of math concepts. Examples include using base-10 blocks to teach place value, two-color counters to teach integer operations, and 1-inch tiles to show perimeter and area. I prefer using manipulatives that can be used throughout a child’s education to provide consistency and structure, and to prevent the need for children to learn tools as well as concepts. For example, Algebra Tiles can be used to teach operations on everything from whole numbers through polynomials. Multilink cubes can be used to teach patterns, number, operations, and 3D geometry.
“Representational” refers to a drawing or pictorial version of the concrete. For example, once students understand how to use fraction tiles to add fractions, they can draw the tiles on a piece of paper to accomplish the same end. Representational versions of math concepts are important as students are not tied to having the physical manipulative with them, and can sometimes be more easily adapted to larger numbers, and used on tests.
“Abstract” refers to using numbers and symbols to model the mathematics. A student may use base-10 blocks by placing one ten block and two unit blocks on a mat, followed by three ten blocks and seven unit blocks on the same mat. When the student combines the blocks, he/she will see four ten blocks and nine unit blocks, and see that the answer is 49. Once a student is comfortable with that model, he/she may not need to use the blocks, but could draw the blocks on a piece of paper, and use arrows or circles to show how the quantities are being combined. When the student writes 12 + 37 = 49 he/she is working at the abstract level – using only numbers and symbols to show what is happening.
Technological advances offer a 4th layer to the CRA model: virtual. Virtual manipulatives are available, often free, on a variety of websites and apps. Research shows that using concrete manipulatives is important, and using virtual manipulatives can also be important, but using both can have the most positive impact on student learning.
Teachers should not skip steps when presenting math using the CRA model. Virtual manipulatives are not a replacement for the concrete as they cannot be picked up, handled, or used beyond the limitations of the app or website. However, virtual manipulatives, used in conjunction with the CRA model (usually in between concrete and representational) are a good addition to a math teacher’s toolbox.
Virtual manipulatives may also address learning needs beyond the conceptual. For students with fine motor control challenges, virtual manipulatives remove the need to physically manipulate small items. For learners with limited spatial skills, virtual manipulatives may help keep the objects aligned and proportionate (especially important for objects such as fraction tiles or area models).
Glencoe offers a free, user-friendly set of online virtual manipulatives. Users may use the menus on the left of the screen to choose different types of commonly used manipulatives that are easy to place, move, and delete from the mat. Virtual tools at the bottom of the screen allow students to annotate and interact with the manipulatives on the mat. There are multiple choices for work mats (under “backgrounds”) allowing users to use the same manipulative in different ways. For users unsure where to begin, resources are also filtered by grade level.
The National Library of Virtual Manipulatives (NLVM) has been around for a long time and is a common go-to site for applets of many types of manipulatives. Depending on the tool, the NLVM offers virtual manipulatives as simply manipulatives (drag, drop, and move), as applets, or as more active workspaces for users.
Finally, one teacher has figured out how to use Google Drawings to make virtual manipulatives in a collaborative, online environment, working seamlessly with her Google-tool classroom.
A variety of options exist for those ready to explore virtual manipulatives. These are just a few. For those with access to technology, looking for ways to engage and teach students who struggle, virtual manipulatives can address a variety of learning needs.
Kate Fanelli is the math accessibility specialist for Michigan’s Integrated Mathematics Initiative (Mi)2, a state of Michigan initiative that promotes and supports high quality mathematics education for ALL students. Follow (Mi)2 on Facebook (www.facebook.com/mi2.page) or on Twitter (@MI2_Math). Contact Kate at firstname.lastname@example.org.