Bridging the Gap: Why the ‘R’ in CRA Matters
Imagine learning a new language without pictures or diagrams. That’s the challenge many students face when grappling with abstract math concepts. The “R” in CRA, the Representational stage, acts as a critical bridge between hands-on experience and the symbolic manipulation that will eventually lead to abstract thinking. Representations are also a universal language! (Want to know more about the C in CRA? Check out the last Heidi’s Hoots post: The Foundational Building Blocks.)
As a classroom math teacher, prior to computers, I lined my room with chalkboards and eventually white boards. I couldn’t really help students with mathematical misconceptions unless I could see their thinking. I would send my students to work in pairs all around the room, while I stood in the center observing their initial stages of thinking as they created visual representations to help them in solving their problems.
When I first encountered Cube and Glow, I knew I had come across tools that were visualization powerhouses! Both Cube and Glow utilize concrete manipulatives and turn them into visual representations that foster deeper connections to help solidify understanding. Let’s dig into some examples…
The screen shot to the left shows the types of math skills we can work on with Cube. Cube can go as low as Kindergarten counting and cardinality, with counting by ones and tens to 100…all the way to 5th grade with reading, writing, and comparing decimals.
For example, as students are placing the cubes into the correct place value column, they will see the respective numbers increase in value. In the example below, students have placed 6 in the 1’s column (yellow in app), 8 in the 10’s column (green in app) and 2 in the 100’s column (blue in app), to represent the number 286. You will notice that the 100’s place value column is labeled with a blue 100 on the sticker on the Cube Tower itself, and the corresponding number representation also shows a blue numeral 2. Students can see that 2 groups of 100 = the number 2 in the hundreds place. Moving from the concrete use of the cubes to the number representation on the screen provides opportunities for students to become more proficient with their number sense.
Another representational pattern students should discover is that if they need to use 17 cubes, the digits used in creating the number will also add up to 17.
When working with money, the stickers on the top of the Cube device are changed and represent the dollar on the left, and tenths and hundreds on the right. As with base ten place value, money can be represented with the same color schema for the place value it represents on the strip that is placed below the lights on Cube. Teachers might want to consider placing a dollar bill, dimes, and pennies underneath the column they represent.
Glow’s representations are just as strong as Cube’s! When working with some third grade students who were struggling with subtraction, it was amazing to see them finally understand the relationship between addition and subtraction. By physically manipulating Glow’s dials to get the representative images on the screen, students were able to see visually how the two interact. As one student stated, “like yin and yang.”
By seeing the visual representation of the red and blue squares, students were easily able to ascertain that 100 blue + another 20 red would = 120. When reasoning through the subject, one student said…”I got it! I got it! If you look at the grid as a whole with the red + blue which = 120, and then take away the 100 blue, you get 20 red left over.” This is a key concept in understanding the inverse relationship and property of mathematics!
Glow’s representations and how students construct them can give us insight into how they think and reason. When providing this problem to 3rd grade students, I saw them wrestle with the relationship, solving the problem in a variety of ways.
I saw students counting by 10’s. They would turn the left dial to count by tens for the 1st three rows. Then they used the right dial and held down the button to continue counting by 10’s from 36 – 46- 56 – 66 – 76 – 86 – 96 – 106.
Other students would get to 36 and then count 4 more to get to 40. From there, they would count by 10’s until they reached 6’s tens = 60. Then they went back and counted out 6 more so that they had a total of 70. Adding the 36 + 70 would then give them the 106.
Any way you slice it, the visualization of key math concepts will help students move from concrete to representational…to finally abstract (but more on that later!) ALLOW students to make mistakes and learn from those mistakes. Don’t correct them…but LISTEN! Ask them questions and see if they can identify their own misconceptions and then fix them. This is where true deep learning takes place!
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CONCRETE, REPRESENTATIONAL, AND ABSTRACT THINKING IN ACTION
Heidi Williams is the Computer Science Curriculum Specialist for Marquette University in Milwaukee, WI. Her focus is on the K-8 integration of computer science and computational thinking within all core content areas. She is the author of ISTE’s No Fear Coding: Computational Thinking Across the K-5 Curriculum, as well as a facilitator of ISTE-U’s Computational Thinking for Every Educator course.
Within this Blog, Heidi will be posting on how Cube and Glow can help educators to support their students’ mathematical thinking. Over the course of the next three months, you will hear about best practices in pedagogical use, CRA – concrete, representational, abstract connections, visualization of partial products, connections to mathematical standards, and MUCH more!
Within each post you will find detailed explanations, links directly back to the components and resources of Cube and Glow, as well as external links that will provide you with more background information and support in teaching a variety of mathematical standards. As you continue to read her posts, week after week, we hope you fall in love with Cube and Glow. We also hope you enjoy the wealth of mathematical pedagogical content and strategies she shares.
Heidi has an extensive background in mathematical pedagogy, with over fifteen years of experience as a 6th – 8th grade math teacher, as well as a K-8 mathematics specialist. She has had extensive training in differentiation, coaching within a Response to Intervention (RtI) framework, as well as inquiry based instruction. Birdbrain is tapping into her understanding and passion for helping K-8 teachers with mathematical thinking in the classroom. Please join us and follow Heidi’s Hoots!