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The teacher in this video used a variety of instructional shifts, formative assessment, facilitating discussion, and most importantly focused on understanding the answer instead of simply getting a solution.  He also used a variety of Standards of Math Practices in his lesson.  At the start of the lesson, he used Practice 1, making sense of the problem by showing students two 5 lb. chocolate bars and then he asked if any of them have ever been in a situation where they need to divide up a number of pizzas among a certain amount of friends.  Most of the students were able to relate, and even if they couldn't, they could envision having to split it up.  Right after, the students were given word problems.  Students took different approaches to solving the problem.  Some students drew a model first; others did the multiplication/ division first.  Then, they used Practice 5 which is understanding the solution process rather than on getting an answer.  ...

Everyone has their own way of understanding a math concept, whether it is concrete or abstract.  Students may be able to recognize a formula or explain solving a problem in their own words.  They may also only have partial understanding (Davis, 2006).  Understanding comes with time and it varies based on the type of mathematics being taught.  To enhance students' understanding of a concept, it is important to ask them to include symbols or models to show and interpret their work.  If they are able to explain their symbol, it is likely that they understand the concept (Davis, 2006).  Students who understand are also able to recognize how to apply a procedure in a new context.  In grade school, I struggled with knowing when to apply formulas to problems.  I could explain the process used in simple problems, but the problem came when there were multi-step problems which included new formulas.  I had to learn that some formulas only work whe...

Demands of cognitive thinking for completing math problems begins with lower level demands and moves its way up to higher level demands.  By the time a student reaches the point of higher level demands and solving complex math problems, the lower level demands are almost mindless.  At a young age, students begin to work on memorizing their addition facts and rules for adding and subtracting.  They focus on getting the correct answer without using a complex procedure (Smith and Stein, 2012).  As they become more confident and knowledgable about the facts and procedures with no connections, their problems become more demanding and require a deeper understanding.  Students in higher grade levels also begin learning new concepts by memorizing, because they are introduced to it and memorizing new formulas or patterns they see to solve problems.  Students at the level of higher demands who have memorized and made connections among problems are able to look at a p...

I had practice 4, Modeling with Mathematics.  Before I began reading the articles, I figured it was going to be a basic reading of how to incorporate manipulatives into everyday math lessons.  I learned that there are many more ways to show modeling in math than just using manipulatives, whether the teacher models how to solve a problem or students use different strategies with written visuals to solve the problems.  I also learned the importance of facilitating dialogue in the classroom while I was reading, presenting, and listening to others present in class.  Students may learn another strategy to solve from one another or they could pick up on their mistakes.  Having a verbal explanation from another perspective can also help students grasp a concept. Two things I learned from my peers' presentation are the benefit to breaking down numbers to add or multiply in order to simplify, and different strategies of showing one students' work ...

This article discusses CGI, or Cognitively Guided Approach, for students at the preschool age. It suggests that manipulatives give students the ability to think and group data when asked a question. The CGI approach includes direct-modeling, counting strategies, and number fact strategies (Shumway, J.F, & Pace, L., 2017). Direct-modeling is when children solve a problem using manipulatives or fingers to count. Counting strategies is when students realize they can count up or down from a number to get the start of the result of the question. Number fact strategies is when children apply their knowledge of number facts to problems (S humway, J.F, & Pace, L., 2017 ). The CGI method makes students think past the one-to-one correspondence to focus on a representation of the whole answer of a problem. There are two different types of problems. Joint-result-unknown compared with joint-start-unknown. With the joint-result-unknown, they know the two factors going into the re...

Practice 4 discussed models in mathematics.  Teachers play an important role in their students' development of knowing numbers and being able to represent them in multiple formats.  The PLC Journal for grades K-2 discussed the opportunities to explore real-world problems and situations (Lasron, et. al., 2012b).  Exploring these operations will allow students to solve and interpret these problems when represented mathematically.  This connects to the NCTM Problem Standard of Problem Solving as well as Connections, because students will be able to make connections to real-world situations and solve problems given about the situation (NCTM, 2000).  The article also pointed out that focusing students' attention on making sense out of the problem in order to ensure answers are reasonable is a must.  If their answers are reasonable, it indicates students used key words from the problems correctly (Larson, et. al., 2012b).  Students can use various forms to l...