Blog Post #6: Video Reflection Comparing Linear Functions

Planning

                In the lesson planning section of the video, the three educators began by discussing the idea of a re-engagement lesson. According to their definition, a re-engagement lesson enables the teacher to determine what students are confused on or have misconceptions about by analyzing student work and using a series of questioning techniques. The objective of this re-engagement lesson was for the students to feel comfortable with producing multiple representations of cost analysis problems and gain an understanding that all representations are equal mathematically. The educators also wanted the students to practice representing a series of data verbally and for students to be able to read a table and a graph.

Lesson  

                The lesson began with the teacher reiterating the content that was learned in the previous lesson. He verbally reminded students that they had discussed the economic status of the world as well as how to make responsible decisions for handling money. Next, the teacher informed the students that in the current lesson they would be using the DVD plans packet and to follow along as the teacher went through page by page. On the DVD plans packet, students were asked to explain how they began their table from the previous lesson. Students were to write the explanation first, then share that explanation with their shoulder partner, and end with a whole class discussion.

                The rest of the lesson was highly repetitive and mostly involved the DVD plans packet and the white board. Students remained in their seats for the entire period and were given brief opportunities to discuss ideas with a partner when the teacher told them to do so.  The students were also given some opportunities to discuss their ideas with the entire class. Therefore, the lesson was almost entirely teacher centered. The teacher directed the lesson from start to finish and never relinquished his authority role for students to take responsibility for their own learning.

Faculty Debriefing

                In the faculty debriefing video, the four educators discussed both the positives and negatives of the Comparing Linear Functions lesson. The educator that actually taught the lesson was impressed by the students’ growth and their progress in debunking misconceptions about tables. He also discussed how students began making connections between the ideas that things can make mathematical sense even though they did not follow the DVD plan. On the other hand, the teacher noticed that students were not making the connections between the original writing prompt and the table they were working with. In order to aide future students in making this connection, the teachers decided to include all three plans and the original prompt on the DVD plan packet to remind students to refer back to the plans when analyzing the material. The teachers also discussed the student misconceptions about the differences between T charts and tables as well as the role and value of zero.

Overall Thoughts

                Though I believe the Comparing Linear Functions lesson was adequately designed to meet the intended objectives, the lesson lacked student engagement and was largely teacher centered. Students were provided with opportunities to share their ideas with their classmates but these sharing times were extremely short. Similarly, students were given specific prompts about what to discuss with their partners rather than allowing them to speak freely with one another. The teacher was in control of the entire lesson and maintained a strict schedule for every element of the lesson. Additionally, the teacher provided the students with all of the materials for the lesson. Students were given a packet that contained completed tables, directions, writing areas, etc. The students were not expected to create anything new but rather were asked to follow this packet page by page.  This lesson could have been improved by asking students to work through the problems in small groups. These small groups could have been instructed on determining whether each table was mathematically reasonable and followed the DVD plan. Then, the entire class could have come together to share their ideas with the students being in lead of the discussion and the teacher serving to clarify or expand their ideas.

Blog Post #5: Journal Article Summaries 1

Beyond Cookies: Understanding Various Division Models (Teaching Children Mathematics)

                The instruction of division in today’s classrooms generally deemphasizes the relationships that the operation of division has between multiplication, fractions, and algebraic concepts. Teachers often provide students with one type of division problem that asks students to determine the number of items in a particular group when provided with the whole and the number of groups. This type of problem is known as partitive division. The authors of the article encourage educators to also include measurement division problems in which the number of groups is unknown but the whole and the equal number of items is provided in the problem. This type of division problem requires students to be more flexible in their thinking and encourages them to make connections to multiplication. Additionally, the article discusses the importance of creating clearly written story problems. Teachers need to keep in mind these three criteria: 1) questions should be clear, 2) all groups should be equal and the equality should be emphasized in the problem, and 3) problems should include a variety of contexts. Teachers should avoid only creating problems that involve the division of food items and instead should include several real-world situations that students can relate to. It is also suggested for teachers to ask students to create their own story problems to help expand their understanding of how division works. When applicable, teachers are encouraged to provide students with manipulatives to work through division problems as well.

                The discussion of creating clearly written story problems was the section of the article that I gained the most from. As a result of being extremely busy during the school year, it is easy for a teacher to find a pre-made worksheet online and use it for the next day’s lesson without thoroughly reviewing the provided problems for the three criteria listed above. I think it is important for teachers to take the time to analyze their worksheets and story problems or ask other to review them in order to ensure that the wording of the problem is not confusing for students. If students are not provided with adequate information or are confused what the problem is asking them to do, they are inevitably going to struggle and do poorly on assignments. It is also important for teachers to create multiple versions of division problems to hold student interest and to avoid simple memorization of the process of solving the problem.

 

Technology Helps Students Transcend Part-Whole Concepts (Mathematics Teaching in the Middle School)

                Most American students learn fractions through a part-whole concept. Students are provided with a section of an object and asked to identify how many parts of the whole object is that section. This is the most common method for teaching fractions in the United States, but it comes with certain limitations especially concerning more complex fractions such as improper fractions. The author s of the above article stress the importance of including partitioning and iterating in the instruction of fractions. Partitioning is similar to the part-whole concept such as how to divide a food item equally among a specified number of friends. Iterating, on the other hand, is using multiplication to understand fractions. For example, students can understand that 3/5 is the same as 1/5 x 1/5 x 1/5. Using this strategy, students are more likely to be able to perform more advanced fraction problems in the classroom.

                The authors of the aforementioned article suggest an iPad application to help students with both partitioning and iterating fractions called the Candy Factory App. In this game, students serve as employees in a candy factory helping to serve customers with their candy orders. The customers ask for a specific size of candy bar and students are expected to use their knowledge of fractions in order to accurately cut the appropriate size the customer asked for. There are three levels to the game that increasingly get more difficult and the student continues playing. This application enables students to be engaged and interested in learning more about fractions and provides them with essential practice.

                Teachers have to be extremely careful when selecting games to use in the classroom. On the surface a game may seemed to provide the needed practice for a particular kind of skill, but upon further notice, the teacher may find that the game is more flashy and colorful than educational. The Candy Factory app, since it was created by The Learning Transformation Group in Virginia, is more than likely an appropriate and effective game for students to use in learning fractions. Games and applications not created by credible institutions should be explored thoroughly before being used in the classroom. Student engagement and interest is not the same as student learning and understanding of mathematical concepts.   

Blog Post #4: NCTM Standards and NCSS Standards

Problem-Solving

1)      CCSS.Math.Practice.MP1 Make sense of problems and persevere in solving them

      The NCTM Problem-Solving standard discusses the necessity for students to use their prior knowledge in order to find solutions to given problems. Additionally, the standard describes the idea that students, as they age, should be becoming proficient in selecting appropriate strategies to solve problems. This is similar to the Standard for Mathematical Practice of making sense of problems and persevering in solving them. This standard emphasizes the idea of providing context to students when asking them to solve problems. Students need to analyze the givens of the problem as well as the final goal before beginning to solving the problem. Students then use their prior knowledge to select the appropriate mathematical strategies to solve the problem accurately and efficiently. While attempting to solve the problem, students are monitoring their progress and checking their answers for reasonableness.

2)      CCSS.Math.Practice.MP2 Reason abstractly and quantitatively.

      Problem Solving can also be related to the standard of reasoning abstractly and quantitatively because the Standard for Mathematical practice asks students to represent problems in multiple ways, to understand the meaning behind the steps to solve the problem, and to be able to explain why the steps were taken to solve the problem. Both standards require students to analyze problems and to utilize their prior knowledge of mathematical strategies in order to solve the problem. Problem solving requires complex and flexible reasoning along with an understanding of the mathematical principles being addressed in the problem.

Reasoning and Proof

1)      CCSS.Math.Practice.MP3 Construct viable arguments and critique the reasoning of others.

            In order to construct a mathematical argument, students need to be able to determine the mathematical concepts and content necessary to accurately back up their ideas.  Any counterarguments against the claim of a peer are only valid if the student is able to provide mathematical proof of why they disagree with the idea of a classmate. Constructing viable arguments requires making predictions, justifying those predictions with logical reasoning, and then sharing those ideas with others. Students need to understand how to construct an effective argument using reasoning and mathematical proof, so that their classmates can fully understand their claims and learn from them.

2)      CCSS.Math.Practice.MP1 Make sense of problems and persevere in solving them

            The standard of making sense of problems and persevering in solving them relates to reasoning and proof because making sense of a problem requires reasoning and perseverance. Making sense of problems requires students to monitor their own progress and to check their answers to ensure that they are reasonable. Peers should be able to see the steps that the student took to solve the problem and understand the reasoning behind each step that was taken. Similarly, making sense of problems requires collaboration and reasoning through problems can be more easily understood with the assistance of others.

Communication

1)      CCSS.Math.Practice.MP2 Reason abstractly and quantitatively.

      Reasoning abstractly and quantitatively requires students to explain their reasoning behind solving problems to their peers. The ability to effectively and clearly communicate one’s reasoning is an essential skill in the mathematics classroom. Students need to be able to communicate orally and in writing the steps to how they solved a problem using logic and mathematical concepts. If peers are confused of the student’s meaning, the student also needs to be able to explain in a different way to ensure that the communication was effective.

2)      CCSS.Math.Practice.MP6 Attend to precision.

      The standard of attending to precision asks students to communicate to others using clear and detailed language. Students who are able to attend to precision can effectively communicate their reasoning and arguments by specifying units, providing clear definitions, and making accurate calculations. The more precise a student can be in his or her explanations, the more likely that his or her classmates will understand and learn from them. Inaccurate or unclear explanations will only seek to confuse other students or to discount the credibility of the student’s reasoning.

Connections

1)      CCSS.Math.Practice.MP1 Make sense of problems and persevere in solving them

            Making sense of problems provides students with the context to a problem. The context of a problem allows students to see where they are going and where they just came from in terms of mathematical concepts and how they relate to one another. Students are making connections between their prior learning and what they are currently studying. Connections allow students to see the subject of mathematics as a constant continuation and buildup of ideas and concepts. Students will be more likely to understand mathematical concepts if they can mentally visualize how the concepts are related and how they fit into the mathematic umbrella.

2)      CCSS.Math.Practice.MP4 Model with mathematics.

            Modeling with mathematics asks students to use their prior knowledge to solve problems and apply their knowledge to real world situations and problems. Essentially, students are required to make connections between what they know and the real world problem they are being faced with solving. Students are applying their mathematical knowledge to the real world problem in order to solve it. Additionally, students are asked to use tools, diagrams, graphs, etc. to aide them in solving problems. Students must make connections between how they were taught to use these tools and the situations in which each tool is the most effective.

Representation

1)      CCSS.Math.Practice.MP5 Use appropriate tools strategically.

            The standard of representation involves how students think about and write mathematics. In today’s classrooms, pieces of paper and the chalk board are no longer the only mediums in representing mathematics visually. Technology provides both students and teachers with a myriad of ways to represent mathematical concepts and ideas. However, students need to be instructed on the appropriate use of these tools including their capabilities and their limitations. Mathematical tools can be very useful in aiding student understanding as long as students can manipulate them effectively.

2)      CCSS.Math.Practice.MP7 Look for and make use of structure.

            Looking and making sure of structure asks students to identify different ways of solving problems, to look at problems from various perspectives, and to break down complicated problems into more manageable steps. Representation plays a key role in how students are able to accomplish this standard. Essentially, students are analyzing how a problem is represented and asked to create new ways to represent the same problem. Students are encouraged to locate patterns or commonalities in representations and analyze those patterns to solve new problems.

Blog # 3: NCTM Process Standards

Problem- Solving

                Problem-solving is an essential skill for all students to have in not only a mathematics classroom but in all subject areas. Problem solving consists of utilizing one’s prior knowledge and experiences to determine the appropriate solution to a problem. Teachers should take great care in selecting problems for students to work through in the classroom. Appropriate problems are those that contain real-world applications and challenge students to expand and connect their prior knowledge to new ideas. Additionally, problem solving instruction should not be separated from the rest of the mathematics curriculum but should be fully integrated into all aspects of the math content. Students should be able to identify specific strategies to use when problem solving and determine when it is appropriate to use such strategies.

Reasoning and Proof

                Reasoning and proof are essential skills in not only mathematics but in all subject areas. Student should be able to make conjectures and then justify those conjectures using mathematical reasoning and evidence. These skills should be taught consistently from early elementary school  all the way through high school. Teachers need to encourage students by asking questions that challenge students to think deeper. Teachers should allow students to discover answers on their own or with their peers, rather than stating claims and providing students with practice problems. Students, especially in the younger grade levels, can explore mathematical ideas and claims by working with manipulatives and other tools. Eventually, students should be able to give both the answer to a problem along with the reasoning why the answer is the correct one. Students should feel comfortable sharing their ideas in the classroom in front of their peers.

Communication

                Communication is an aspect of the mathematics classroom that is often neglected or forgotten. If students are given opportunities to communicate with one another about mathematical concepts as well as to critique the ideas of their classmates, they will expand their mathematical knowledge and take on responsibility for their own learning. Instruction on how to mathematically communicate should begin with early elementary students and continue throughout high school. By the end of high school, students should be able to make clear and precise mathematical arguments that can be easily understood by others. High School students should be able to question and analyze the arguments of their peers by using appropriate terminology and reason. The goal is for students to eventually have communication as a natural part of their mathematical learning.

Connections

                All mathematics is connected, though it is frequently taught with strong divisions. The knowledge and skills students gain in one year of schooling should continue to build upon one another the following year.  When students learn to make mathematical connections, they learn not how concepts but also how mathematics works as a whole subject. Teachers should seek to help students discover mathematical connections throughout instruction by avoiding teaching the subject in categories. Instead, teachers should build connections in every lesson by asking students to think about their previous learning and how it relates to current instruction. Students should be able to see the connections between prior learning as well as to other subject areas and daily life.

Representation

                The representation standard describes how we write mathematics and how we think about mathematics. A mathematical representation can include diagrams, graphs, symbols, etc. The choice of how to represent a problem is a major contributor to whether the student will understand the problem and be able to find a solution. They also serve to inform the teacher if the student truly understands a concept or not. Students should be taught how to create several different kinds of representations for a single problem so that they can chose the best representation that makes the most sense to them. New forms of technology have recently come available that can offer additional representations for students to use to understand mathematical ideas and find solutions to problems.

Blog Post #2: Rich Activities and Lesson Planning

In a classroom of 25+ students it is beyond difficult to create a lesson plan that is slightly above each students cognitive demand. What students gain from a lesson is dependent upon the grade, age, prior knowledge, and life experiences of each student. Teachers need to allow for all students to access the curriculum and therefore should take great care in designing and implementing lesson plans. First, a teacher should ask herself what she knows about her students. What prior knowledge do they have on the topic? Could their previous life experiences and culture impact learning on this topic? Then, the teacher should seek to include all students into the lesson by asking questions such as: What additional supports would individuals with special needs require for this lesson? How can I further challenge students who easily grasp the material of the lesson? Thirdly, the teacher should determine how she is going to hold her students accountable for the material of the lesson. The teacher should establish specific classroom norms for students to follow such as taking turns, listening to others, and disagreeing with the idea not the person. Students should know the expectations for the classroom, so that class discussions run smoothly and all students feel comfortable sharing their ideas. Additionally, teachers should emphasize the idea that there are always multiple ways to solve a problem and one way is not necessarily better than another way.

Blog Post #1: Mathematical Practice Standards

CCSS.MATH.PRACTICE.MP2 – Reason Abstractly and Quantitatively

·         Students who reason abstractly and quantitatively are able to understand the meaning behind mathematical operations and problems, not just how to compute numbers into an answer.

·         Students are able to represent problems symbolically (i.e. algebraically, pictorially, etc.)

·         Students are able to contextualize the meaning of the symbolic representations of problems while performing calculations.

·         Students are able to use a variety of different mathematical operations in order to solve a problem.

 

Common Core Mathematics in a PLC at work (Grades 3-5)

                The mathematical practice standard that asks students to reason abstractly and quantitatively enables students to “make mathematics useable and useful” (Larson, et al. 2012). It is one thing for students to be able to perform calculations but another for students to be able to explain how they arrived at an answer as well as their reasoning behind performing the particular steps in solving the problem. Students need to have conversations with the teacher and with their peers in which they justify their thinking in solving problems. Students should consider examples and extensions to problems in order to fully understand the meaning behind the mathematical operations and processes. Teachers should seek to design and implement lessons that allow students to practice using and applying mathematical concepts and operations in multiple ways. Lessons should enable students to develop skills such as recognizing relationships among numbers, interpreting numbers within context, realizing the magnitude of numbers, and solving real-world problems using numbers.  Students will use these skills in all mathematical areas and in every grade level.

 

Establishing Standards for Mathematical Practice

                Teachers can develop student skills in reasoning abstractly and quantitatively by designing lessons that ask students to explain their reasoning to others as well as understanding the reasoning of others (Stephan, 2014). In the journal article written by Michelle Stephan, she describes the first lesson of the school year in which she begins teaching her students what she deems the “societal norms of the classroom”. The societal norms include 1) explain the reasoning to others, 2) indicate agreement or disagreement, 3) ask clarifying questions when they do not understand, and 4) attempt to understand the reasoning of others.  In this lesson, Stephan pairs students together and asks them to work collaboratively to solve a story problem. Then, students present the steps they took to solve the problem to the rest of the class. In these presentations, the presenter is held accountable for his/her explanation of the steps to the solution as well as their justification for why they performed those particular steps to solve the problem. The other students are held accountable for asking questions if they do not understand the justification along with re-explaining the presenter’s explanation in their own words. The teaching of the societal norms of the classroom is a yearly practice.  Teachers need to encourage students to think about mathematical problems conceptually so that they can explain why they performed the steps to solve the problem and what the calculations mean in terms of the problem and real-world application.

 

Stephan, M. L. (2014). Establishing standards for mathematical practice . Mathematics teaching in middle school , 532-538.

 

 

 

CCSS.MATH.PRACTICE.MP3 – Construct Viable Arguments and Critique the Reasoning of Others

·         Students utilize their prior mathematical knowledge in order to construct logical arguments.

·         Students make predictions or state their opinions about a problem and then use logic to reject or support their initial ideas or predictions.

·         Students are able to explain and defend their conclusions as well as communicate those conclusions to others.

·         Students are able to compare and contrast two or more possible arguments for a problem.

 

Common Core Mathematics in a PLC at work (Grades 3-5)

                The goal of the third standard for mathematical practice is for students to justify or explain the reasoning behind solving a problem to the teacher and to their peers. Additionally, students need to be able to critique their own reasoning and the reasoning of their peers to ensure comprehension. There are multiple ways to solve a mathematical problem and this standard allows students to understand the various methods their peers used to solve the same problem. Teachers need to create student-centered classrooms in which students feel comfortable sharing their answers with one another. Students need to be encouraged by the teacher to ask questions if they are confused about a classmate’s solution to a problem and to critique the work of their classmates if the reasoning does not make sense.

Advice for Mathematical Argumentation

                Mathematical argumentation consists of three steps: 1) make conjectures, 2) justify the conjectures, and 3) decide whether the conjectures are true or false. A conjecture is essentially a guess based upon prior knowledge. Teachers need to emphasize to students that a conjecture needs to go beyond what is known to include what is not known. Students should be encouraged to make multiple conjectures over the same problem and to avoid getting upset if a conjecture is proved to be false. Justification asks students to explain their reasoning behind their conjectures. There are three ways to justify a conjecture: 1) numerically, 2) visually, and 3) geometrically. Finally, conclusions determine whether the conjecture is true or false. Students need to critique the work of their peers and decide whether or not they agree or disagree with their classmates. Students who disagree can use counterexamples to prove the conjecture false. Once the entire class agrees on the validity of a conjecture, it should be recorded and used for future reference. Teachers need to encourage students to use mathematical argumentation in every lesson throughout the school year.

 

Knudson, J., Lara-Meloy, T., Stallworth Stevens , H., & Wise Rutstein, D. (2014). Advice for mathematical argumentation . Mathematics teaching in the middle school , 494-500.