Respond to discussion 3

Ryan

Conceptual knowledge is "knowing more than isolated facts and procedures. They can take what they have learned and extend it to other topics to which they are applied as scholars acquire a deep understanding of the notion. Using their basic knowledge of the concepts and abilities of mathematics, students will use this base layer and add or relate their conceptual knowledge to abstract problems and learn how to solve them. (CRA) means concrete, figurative, and abstract. It (CRA) also presents an ideal basic strategy for future problem-solving in their fields. In other words, CRA will encourage interaction, but for scholars, it will still be enjoyable. We all notice that when they are more involved, scholars appear to learn more, and the teacher has developed a learning atmosphere conducive to learning. CRA would be incorporated into the teaching of students, and they would have a greater understanding of how to apply it. There are many advantages from the principles of (CRA). It offers a standardized way for scholars to learn math concepts. When moving through the layers of understanding from concrete to abstract, scholars can create a more vital link. The instructor makes learning more available to all students, including those with mathematical learning difficulties. It can be used in small groups or the whole class when addressing (CRA). Teach learners how to think, and they're not going to go stray. Students will have a better way of learning, especially for teachers seeking new teaching methods. How can I apply elements to my teaching of mathematics? I would begin by discussing mathematics to help the students understand and think about the relationships of the numbers. I would definitely introduce the use of manipulative materials because I feel. However, this gives the students the best way to represents their solutions to the problem, and I can try to identify the depth of understanding while they perform the task. The use of manipulative materials also helps the students learn to count. I would incorporate open-ended problems and extended problem-solving projects to drive deep thinking and problem-solving. I would create multiple representations of the same issues or situations to introduce multiple ways of solving problems or situations." Conceptual knowledge is "knowing more than isolated facts and procedures. They can take what they have learned and extend it to other topics.

Vantashia

The CRA model of instruction is made up of three different parts. Those parts are concrete representation, representational figures or diagrams, and abstract formulas or equations. The article Manipulatives Enhance the Learning of Mathematics gives great examples of how the CRA model of instruction supports student’s development of deep conceptual understanding. This article breaks down the effects of using manipulatives or models used to represent math problems during a lesson. The article states that one thing manipulatives do is help students develop conceptual understanding of mathematical ideas by representing ideas in multiple ways. This eliminates confusion in students minds. 

The CRA model of instruction emphasizes the need for hands on materials being implemented when teaching mathematical concepts.  This allows teachers to use concrete materials when teaching abstract standards.  Students may struggle to visualize concepts when they do not have the prior knowledge gained from previous instruction or learning experiences.  This is most often seen in elementary classes where students use manipulative such as colored counters, fraction strips, and beans.  The hands on experience reinforces the abstract ideas and thought processes needed in order to conceptualize various math standards.

Even though these manipulative are most often found in elementary classrooms, the same concept can and should be applied to middle and high school classrooms as well.  These older students may have prior knowledge on some concepts, but not all.  Prior knowledge does not have to be the only baseline or prompt used.  Using real-world examples and experiences to reinforce new concepts is important for students to gain deeper knowledge and understanding.  Without the real world examples and situations being used, they may not develop as much fluency and application skills as needed. 

The use of manipulatives also encourages engagement and deeper development of math skills. According to the article written by Dr. Shaw, students need opportunities to test their skills and knowledge in order to demonstrate understanding and efficiency.  Teachers may be able to easily gauge students understanding of concepts by watching students put the skills to use in a hands on way.  This process allows the students to demonstrate their abilities to take abstract ideas and make them more concrete in their usage.

Samoria

 CRA model of instruction support students' development of deep conceptual understanding by giving students different ways of learning. The CRA model is consist of three instructional parts; concrete, representational, and abstract. In the concrete stage, students use manipulative to learn new concepts. In the representational stage students are drawing instead of using manipulative. In the abstract stage, students are taught how to solve the problem. In this stage the teacher will model the problem doing the abstract stage. The purpose of the CRA model is to make sure students develop a well understanding of math concepts or skills learned. 

      In this week reading, Manipulative Enhance The Learning of Mathematics mention that manipulative help students  develop conceptual understanding of mathematical ideas by representing the ideas in multiple ways. I agree, because manipulative can students see the concept for themselves then just looking at the problem. Every student learn different, so teachers cannot except all students to know the concept that they are introducing. Students may not have had that concrete experience that they should have been taught in previous grades. Most of the time teachers find that majority of their students need additional concrete experience to support current concepts. For example, an 8th grade math teacher wants to teach a lesson on linear equation in one variable. The teacher first need to make sure students know the concept of solving equation from 6th and 7th grade first. Manipulative could be use show students how to solve one-step equation before moving on the multi-step equations.  

  Using the concrete model, students will be able understand the mathematical concept and foundation to be able to use it later in the abstract stage. Most students learn better through visual  and kinesthetic experience. Then they would later develop into the representational and abstract stage of learning.