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How we use the CRA Approach, Part II (Fractions)
Who’s ready to talk about fractions? Great-me, too! Let’s talk about how to get kids really diving into the concept of fractional parts, and let’s do it from the perspective of the Concrete-Representational-Abstract instructional approach. Here’s a quick review of this approach before the example. (Ready?)
The Concrete-Representational-Abstract sequence of instruction provides a “graduated and conceptually supported framework for students to create a meaningful connection among concrete, representational, and abstract levels of understanding.” The US Department of Education’s Access Center tells us that when students can begin “with visual, tactile, and kinesthetic experiences to establish understanding, [they] expand their understanding through pictorial representations of concrete objects and move to the abstract level of understanding.” If you’re not with me on this, please take a few minutes and listen to the last two podcasts on the Concrete-Representational-Abstract instructional approach.
OK, let’s apply this to fractions now. The whole enchilada here is about understanding one central concept: ready? “Fractions refer to parts of a whole.” That’s it. All that hair pulling and all those tantrums, and here it is in seven words. Here it is again, say it with me: “Fractions refer to parts of a whole.”
I know, I know; you’re saying to yourself, “well, yes, but you’re seriously oversimplifying things.” Hmmm…maybe. But, as the Zimbabwean proverb goes, “If you can walk, you can dance; if you can talk, you can sing.” Remember, one of the basics of the Equity principle is to get everybody-regardless of background or aptitude, in the conversation. If you can move, come dance with us. If you can communicate, come sing with us. If you can put two pieces together and make a broken thing whole again, come talk about fractions with us.
There are an infinite number of physical materials that we can use to show the meaning of a fraction as “part of a whole.” Pizza and graham crackers and chocolate bars are the yummy versions of fractional parts. In terms of math manipulatives, our schools have bags of fraction cubes, counters, fraction bars, attribute blocks, and fraction strips that can show a fraction –many of which can be homemade.
So, let’s explore with counting bears-they look like gummy bears but don’t eat them. Trust me. Let’s say we have 2 blue counting bears (part) out of 4 total counting bears (whole, the total number of bears). Some students may be ready to talk about how they see that half of the bears are blue. Other students experiment later with a larger number of bears and explore the proportion of blue bears to make it “look right,” meaning that they seem to know that they need enough blue bears to be half of the bears, but they don’t have the words to express it. The teacher then guides the students to use bear shaped stamps on paper and color in however many blue bears they have. She pulls the words from them, like “more, less, same, different” and some may use “half” and “equal.” She rewords their statements with the words “groups,” “part” and “whole” and brings everyone into the discussion. This exploration of a concept in the concrete and migration of it into the representational requires lots and lots and lots of conversation, which builds numeracy and mathematical thinking and reasoning skills. We would never attempt to build literacy without communication; students learn by “hearing” the language demonstrated in a fluent manner as much as they do by the worksheets and practice activities. Numeracy skills are no different; students need to hear fluency in mathematics and try out their own math vocabulary and reasoning skills in a safe environment.
Once our students are talking about and showing what they mean with fractional parts and wholes, then we can jump to the formalities of writing a numeric symbol of the number of bears or parts of the whole in “correct fraction form”-a step that involves assigning the order in which digits should be read or written--which number is represented on the top? Which number is represented on the bottom?
Here’s an example from the Access Center’s website. The goal is to “develop the spatial organization, visually and kinesthetically, to read and write fractions correctly” and we’re told to collect small red and large black squares to help with sequencing and number placement. So, hum for a moment and I’ll pretend I’m cutting out my construction paper squares at the paper cutter. [Hmm, hmm, hmm.] OK, now I’ll read you the script! [Press pause if you want to go make popcorn. I’ll wait.]
Teacher: “Today we are going to write and say fractions. We’ve been talking about these for a while, and today we’ll do it together. Ready?”
Concrete: Teacher points to the squares arranged on a table. “What colors are the squares?” (Students say black and red.) “Count the total number of squares (whole).” (Students point and count to 8. Students say 8.) “How many red squares are there?” (Students point and count to 3. Students say 3.)
Represent: “When we talk about fractions, we say the ‘part of the whole.’ (Say together ‘part of the whole’). We can write a fraction showing the part of the whole, as shown above. The number for the part is written on the top and the number for the whole is written on the bottom. (Say part on top and whole on bottom.) What was the total number of squares?” (Students say 8.) “Let’s call that the whole.”
Abstract: “Write the total number of squares or the whole on the bottom where the word ‘whole’ is shown.” (Students write 8.)
Represent: “How many red squares are there?” (Students say 3.) “Let’s call the red squares ‘part’ of the whole.”
Abstract: “Write the number of red squares on the top where the word ‘part’ is written. (Students write 3.)
Summary: “From this example, what did you write for the fraction?” (Students say 3 and 8.) “We say 3 out of 8 or three-eighths.”
End of script. So, what do you think?
The teacher would continue to guide students in practicing different examples with the squares, writing and reading the fractions aloud. She would change up the objects and ask students to explain why we need to be able to do this on paper, walking back along the steps they have taken. Explain, explain, explain. Discuss, discuss, discuss. We repeat the process until the child independently can read and write the numbers for a fraction.
So that’s an example of a Concrete-Representational-Abstract sequence of instruction. It showed a “graduated and conceptually supported framework for students to create a meaningful connection among concrete, representational, and abstract levels of understanding.” We know from research that when kids use concrete materials to explore mathematical concepts, they develop more accurate mental representations of what that concept really means which means they building new numeracy skills and strengthening their existing ones. We are creating more motivated kids who stay on-task because they have the skills to participate. They can join he conversation and talk about the concepts because they now have the words and pictures to do it! These students have been provided with the skills to apply math concepts to real-life situations.
Do we think that at the end of these lessons, students would be able to talk to you about why and how “fractions refer to parts of a whole”? I think they could. Remember, “If you can walk, you can dance; if you can talk, you can sing.” Here’s the Equity principle again-if you can communicate, come sing math with us.
OK, math singers, that’s all for now-in the next podcast we’re going to continue to discuss manipulatives for the concrete level with an example about algebra tiles and the area model of multiplication.
In the meantime…for more information about the research-based use of concrete, representational, and abstract representations in math instruction, visit the U.S. Department of Education’s Access Center at www.k8accesscenter.org.
views since 10-13-08
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