Concrete Representational Abstract- Anderson Morgan

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Concrete-Representational-Abstract (CRA) Instructional Approach Assignment (28 points) To complement the Chapter Two Teaching Approaches, you will be exploring Concrete-Representational-Abstract (CRA) Learning. To begin, view these resources on Concrete Representational Abstract (CRA) instruction: Readings:  Concrete-Representational-Abstract: Instructional Sequence for Mathematics: https://www.pattan.net/getmedia/9059e5f0-7edc-4391-8c8e-ebaf8c3c95d6/CRA_Methods0117  CONCRETE REPRESENTATIONAL ABSTRACT SEQUENCE: https://www.maneuveringthemiddle.com/difficult-math- concepts/  AACPS: https://www.aacps.org/site/handlers/filedownload.ashx? moduleinstanceid=4623&dataid=13171&FileName=CRA%20-%20One%20Sheet.pdf  NCTM: https://www.nctm.org/Handlers/AttachmentHandler.ashx?attachmentID=cI1Ll4bzKXI%3D (Pg 11-12) Videos:  Planning and Implementing a Conceptual Lesson: https://youtu.be/6tzZxBX1_U8  Using The CRA Approach To Teach Fractions: https://www.youtube.com/watch?v=GLZ92IuDWSM  Concrete Representational Abstract-Multiplication: https://www.youtube.com/watch?v=AmA5HG_hkW4 Part One: (15 points) Consider the following problem: Holly wants to make necklaces for her friends. She has 4 friends. Each necklace needs 10 beads. Holly has 51 beads. Will Holly have enough beads to make necklaces for all her friends? Complete the prompts below as you solve the above problem. 1. How could a student solve this problem at the concrete level? What would the teacher use to make it concrete? Is this a hands-on activity? (3 points) To solve this on a concrete level, blocks or beads can be utilized. Each student would need 51 beads/blocks and have to put the beads they have into 4 groups of 10 to determine if they have enough beads for all four friends. If there are leftover beads not being used or each of the four piles has 10 beads each, then Holly will have enough beads to make each of the four friends a necklace. This is a hands-on activity because it is concrete which means they are working with objects to represent the problem. 2. Solve this problem at the Representation level (i.e., pictures, organized lists, chart, graph, table, etc.). You may need to insert an image to show your representation. Explain how you arrived at the solution. What did you use that made it representational? (3 points) Friend 1 (10 beads) Friend 2 (10 beads) Friend 3(10 beads) Friend 4 (10 beads) Leftovers (11 beads) MTHED 322 E1 Teaching Elementary/Middle School Mathematics University of Wisconsin-Superior Teacher Education Department

To make this solution representational I used a table to distribute the beads among the 4 friends and included a column for the leftover beads after making 4 bracelets with 10 beads each. The beads are represented by the individual circles in the table. I arrived at this solution because I drew out all 51 beads represented with 51 circles as indicated in the chart above. With each bracelet having 10 beads, there are 10 circles underneath each friend with the remaining beads in the last column showing there are 11 beads leftover after making the 4 friends each a bracelet. This is representational because I described and modeled how to perform the skill by drawing or using pictures that represent the concrete objects (in this case the beads). 3. Explain how a graph and/or table could be used for organization in the representational level? Can you make a connection to your concrete example? (2 points) A graph or table can be used in organizing at the representational level by showing through a picture how many beads are in each group (4 friends) and a column for the remaining beads after making those 4 bracelets for her friend. With a graph, we can create 5 sections or parts and mark the 4 friends with 10 beads each and then use the last section to graph the remaining or leftover beads. I utilized a chart in the question above to represent the concrete idea of the beads with the circles in my table to show exactly how many beads (10) Holly will need to make ach bracelet and use the last column of the chart to put the remaining number of beads. We can clearly see 10 circles in the representational model for each friend and exactly how many are leftover. To check the problem, the students can count all of the circles total and ensure all 51 beads are accounted for and that 10 beads are in the columns for each of the 4 friends. The circles are equal to the concrete model for the beads where students can physically count each one to ensure all beads that Holly has to make the bracelets are represented in the model. 4. Solve this problem at the abstract level (i.e., number, formula, equation, etc.). Explain how you arrived at the solution. Can you make a case for the efficiency of using an abstract method such as numbers or equations? (3 points) The equation that can be used to represent this problem on an abstract level is 10x + 11= 51, where x is the number of necklaces Holly wants to make. By solving for x, Holly can see that she can make 4 necklaces with 11 beads leftover after making the 4 bracelets. The efficiency of making a model at the abstract level is that it would be able to be solved quicker because you are solving for the one variable rather than having to account for all of the beads by counting them in the representational model and drawing a circle for each one or the concrete model where we would need to count out 51 beads for every student to model this particular equation. Solving this equation to ensure she has enough beads can be done by subtracting 11 from 51 to get 40 and dividing that by 4 to see she has 10 as the value for x. It can then be checked by plugging x back into the solution to ensure it is equal on both sides. 5. Describe why it is important to guide students from the concrete to the abstract. Use information from your opening internet search to support your point of view. Be sure to cite sources as appropriate. (4 points) In order to assist students in developing a deeper comprehension of mathematical topics, it is crucial to lead them from the concrete to the abstract form of representation. Students who encounter mathematical ideas in many contexts, such as concrete, representational, and abstract, have a better grasp and retain the knowledge longer, according to studies and based on our readings from this assignment. Students can use this foundation later to MTHED 322 E1 Teaching Elementary/Middle School Mathematics University of Wisconsin-Superior Teacher Education Department

connect their conceptual understanding to abstract mathematics learning activities when they are encouraged to first establish a tangible degree of comprehension for any mathematics idea or skill. Drawing simple illustrations that imitate or mimic their usage of physical materials to express students' concrete understandings (representational) provides students with a supported procedure for transferring their concrete understandings to the abstract level. Additionally, teaching students how to sketch answers to problem-solving scenarios is a great method they can utilize for problem-solving in the future. Source: The University of Kansas. (2023). Concrete-to-representational-to-abstract instruction . Department of Special Education: Special Connections. https://specialconnections.ku.edu/instruction/mathematics/teacher_tools/concrete_to_representational_to_abst ract_instruction#:~:text=The%20purpose%20of%20teaching%20through,math%20concepts%2Fskills%20they %20learn . Part Two: (13 points) Find a standard in the Common Core and create a problem that demonstrates CRA similar to the problem above. Standard and Grade Level: (1 point) CC.4.OA.3 - Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding. 4 th grade standard Problem (you create): (3 points) Margo wants to buy snacks for her friends. She has 5 friends and wants to give each of them 4 snacks. Margo has a 18 snacks. Will Margo have enough snacks for all of her friends? 6. How could a student solve this problem at the concrete level? What would the teacher use to make it concrete? Is this a hands-on activity? (3 points) The students could physically count out the snacks to determine whether she has enough to distribute to all of her friends in order to address this issue on a concrete level. The teacher could use blocks, or colored chips to represent the 18 snacks Margo has total and distribute them in groups of 4 to see if all 5 friends will get the 4 snacks. This would be a hands-on activity because students would have physical objects to manipulate to solve this problem. A small snack such as skittles could also be utilized to represent the snacks that Margo has. 7. Solve this problem at the Representation level (i.e., pictures, organized lists, chart, graph, table, etc.). You may need to insert an image to show your representation. Explain how you arrived at the solution. What did you use that made it representational? (3 points) MTHED 322 E1 Teaching Elementary/Middle School Mathematics University of Wisconsin-Superior Teacher Education Department

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To solve this problem at the representation level, students could create a chart or table to keep track of the number of snacks she has and the number she needs for each friend. A table could look like this: Friend 1 Friend 2 Friend 3 Friend 4 Friend 5 In the table above after distributing the 18 snacks and putting 4 for each friend, we can see that Margo would not have enough for all 5 friends and only enough to give 4 friends 4 snacks each. This table is a representational model because it is an image that students can create in order to model the problem accurately while being able to count out each shape that represents each snack so that they know all of the snacks she has total are accounted for in the model. Using a table is one of the ways we can use to represent a problem so students can physically see on their papers the total amount of snacks and organize them into 5 categories for each of the friends and start by adding 4 to each friend. When they get to 18, they will see that there are only enough snacks to distribute to 4 of the friends if each friend receives 4 snacks. 8. Solve this problem at the abstract level (i.e., number, formula, equation, etc.). Explain how you arrived at the solution. Can you make a case for the efficiency of using an abstract method such as numbers or equations? (3 points) To solve this problem at the abstract level, students could use an equation to represent the number of snacks she has and the number she needs. the equation could be represented by: 4x +2 =18, where x is the number of friends Margo wants to give snacks to By solving for x, Students can see that she can give snacks to 4 out of her 5 friends when giving each friend 4 snacks. The efficiency of this process is more efficient because students will be able to demonstrate their understanding of how to solve an algebraic problem and could utilize a calculator to solve for this problem making it and easier more efficient way to get a solution to the problem. If the variable x is not equal or more than 5, students will know there is not enough snacks for the 5 friends. Scoring Considerations: Ensure that each component is fully addressed with meaningful detail describing what students would be doing at the concrete, representational and abstract level. The solutions and explanations should clearly align with each level (Concrete, Representation, and Abstract). The standard selected in part 2 should align with your created problem and offer the three levels of CRA in distinct solution methods. Your strategies should further your learning of the concept and integrate your internet search results. MTHED 322 E1 Teaching Elementary/Middle School Mathematics University of Wisconsin-Superior Teacher Education Department