Once students have mastered the concrete level of success, introduce appropriate drawing techniques where students solve problems by drawing simple representations of concrete objects they have previously used (such as counts, points, and circles). “By reproducing movements that students have already used with concrete materials, drawing simple representations of these objects supports the development of an abstract understanding of students` concept/ability” (Special Connections, 2005). The overarching goal of the CRA`s pedagogical approach is to “ensure that students gain a tangible understanding of the mathematical concepts and skills they are learning.” (Special Connections, 2005) Through their concrete understanding of mathematical concepts and skills, students are able to use this foundation later and complete/link their conceptual understanding to abstract problems and learning. As students go through these three stages, students gain a deeper understanding of mathematical concepts and ideas and provide an excellent basic strategy for solving problems in other areas in the future. (Special Connections, 2005). One of the first and most important steps in implementing the CRA`s classroom approach is to “use appropriate real-world objects to teach certain mathematical concepts and skills. Discrete materials (p. e.g., counting objects such as beans, chips, unifix cubes, popsicles, etc.) are especially useful because students can see and feel the attributes of the objects they use. (Special Connections, 2005). In the classroom, this approach is a framework that makes it easier for students to make meaningful connections between concrete, representative, and abstract levels of thought and understanding.
Students` learning begins with visual, tangible, and kinesthetic experiences to gain basic understanding, then students are able to expand their knowledge through pictorial representations (drawings, diagrams, or sketches) and finally move on to the abstract level of thought, where students exclusively use mathematical symbols to represent and model problems (Hauser). Studies have shown that “students who use concrete materials develop more accurate and complete mental representations, often show more motivation and behavior in the task, understand mathematical ideas, and better apply those ideas to life situations” (Hauser). Manipulative materials are concrete models or objects that contain mathematical concepts. The most effective tools are those that appeal to multiple senses and can be touched and moved by students (no demonstrations of materials by the teacher). Manipulative documents must relate to the students` real world (Heddens, 1997). Posted in pedagogy | Tags: abstract, concrete, representations, tools Finally, after a student has demonstrated a thorough understanding of the representative level, use appropriate strategies to help students move from that level of representation to the more abstract level. If students have difficulty moving to the abstract, “reseed the mathematical concept/ability with appropriate concrete materials, and then explicitly show the relationship between concrete materials and the abstract representation of materials.” (Special Connections, 2005) If students already have a concrete understanding of this concept or skill, “offer them the opportunity to use their language to describe their solutions and their understanding of the mathematical concept or skill they are learning” (Special Connections, 2005). Special Connections, (2005).
From the concrete to the representation through the abstract. Retrieved on April 9, 2009, from the Special Connections website: www.specialconnections.ku.edu/cgi-bin/cgiwrap/specconn/main.php?cat=instruction&subsection=math/cra. Hauser , Jane. Concrete-representational-abstract pedagogical approach. Retrieved 9. Access Centre April 2009: Improve outcomes for all K-8 students. Website: www.k8accesscenter.org/training_resources/CRA_Instructional_Approach.asp · Snap cubes are similar to Unifix cubes, but are linked to each other on all sides (in six ways). They are available in a variety of colors How can I implement the CRA approach in my classroom? Tips for teachers for using math manipulation tools in the classroom Other useful math tools that can be implemented in the classroom are: List/use of suggested math tools (manipulators) · Two-sided counters are usually circular chips of different colors on each side. · Unifix cubes are colored, nested cubes that are connected to each other in only one way.
They are available in ten spot colors, which makes them very visual for demonstrations and makes it easy to create patterns and sort. www.eaieducation.com/?gclid=COb69aTk35kCFRwwawodjSVyWA · Ten frames are a rectangle consisting of ten squares (5 by 2). They can contain points that represent values 1 through 10, or they can be empty. Concrete lessons: coe.jmu.edu/mathvidsr · For more information and details, visit the following Web site: www.netrox.net/~labush/nctm.htm#Pattern · For more information, visit the following Web site: www.christiancottage.com/articles/HundredChart.html · Broken bars are colored cubes or tiles that are proportional and represent integers, halves, thirds, fourths, fifths, sixths, eighths, tenths, and twelfths. · 100 pearls are made by aligning 100 pearls on a string. They consist of 2 colors that alternate every ten. · Beans and cups consist of a cup and beans. The teacher decides how many beans to put in a cup for certain problems.
This image shows how to use the CRA`s instruction approach with the 4+5=9 issue. 7 musts to use manipulators: content.scholastic.com/browse/article.jsp?id=4003 learn more about mathematical manipulators: www.iched.org/cms/scripts/page.php?site_id=iched&item_id=math_manipulatives · Cuisenaire stems are colored wooden or plastic stems that have values from one to ten and are colored according to the number they represent: · 10 base blocks come with units (one cube), long (consisting of 10 units), apartments (consisting of 10 long or 100 units) and dice (consisting of 10 apartments or 1000 units). · These are 1″ x1″ square colored tiles. They can usually be purchased with a mixture of four colors. · Pattern blocks are multicolored blocks one centimeter thick that come in six forms: hexagons, squares, trapezoids, triangles, parallelograms, and diamonds. Each shape has a different color. Heddens, James W., (1997). Improved mathematics education through the use of manipulators.
Retrieved April 9, 2009, from Edumath: www.fed.cuhk.edu.hk/~fllee/mathfor/edumath/9706/13hedden.html. The CRA`s pedagogical approach is “an intervention for mathematics education that research finds can improve students` mathematical achievement.” (Hauser) The approach is a “three-part teaching strategy, with each part building on previous teaching to foster student learning and retention and address conceptual knowledge.” (Hauser) The three parts are as follows: o White stem = 1 cm. Red stem = 2 cm.