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In my host teacher’s instructional program, a myriad of learning theories are reflected that aid in enhancing student learning. For instance, while reviewing the homework from the previous night, my teacher modeled the behavior of placing a box around each final answer. Thus, she aimed to demonstrate a desired behavior to her students so that they would imitate it on their assignments in the future, which ties into social cognitive learning theory. Later in the block period, while the students were completing an in-class assignment that would later be turned in, I observed numerous students placing boxes around their final answers, which reflects their learning through live modeling, according to social cognitive theory (Ormrod, 2011). In addition, behaviorism was reflected in my host teacher’s classroom. While the students took notes on absolute values, she verbally praised the students that highlighted each note heading, saying, “You guys are the best! You guys make me happy!” By offering this form of verbal praise, my teacher was positively reinforcing this desired behavior in hopes that the students would duplicate this behavior in the future. Hence, her praise served to increase the students’ motivation to continue to highlight note headings in the future, which directly aligns with behaviorist learning theory (Ormrod, 2011). Even more, cognitivism and constructivism were both reflected throughout my host teacher’s lesson on absolute values. First, she presented a mnemonic rhyme to the students in order to help them remember when solutions to absolute value equations would involve one “and” solution or two separate “or” solutions. She offered, “If the original equation is ‘less than,’ then form an ‘and.’ If it is ‘greater,’ then form an ‘or.’” According to cognitivism, mnemonics aid students in moving information from their short-term memory into their long-term memory and ultimately allow for more effective retrieval of information to be applied in the future. After offering this mnemonic rhyme, she attempted to activate the students’ prior knowledge by asking them, “Do you remember using number lines in Algebra I?” Thus, she was attempting to use the students’ previous knowledge to build new knowledge when graphing number line solutions to inequalities, which directly aligns with constructivist learning theory (Ormrod, 2011). Various instances in my host teacher’s lesson reflected sociocultural learning theory, as she provided the students with numerous scaffolds in order to help enhance their learning. Multiple times, she asked the students questions to help them progress through a problem or to encourage them to explore whey certain solutions would or would not satisfy an original equation. Additionally, she provided them with tips on how to most effectively solve these equations and offered hints when the students could not reach certain solutions. For example, after posing the task of finding an absolute value equation that has only one solution, she noticed the students discussing and forming possible equations that they thought would satisfy this task, but failing to reach the correct solution. Therefore, she hinted, “What number does not have an opposite?” Immediately after presenting this question, many of the students came to the realization that zero was the number that she was referring to. Then, as a result of her scaffold, they were able to successfully form equations with only one solution, as she had initially desired of them. After presenting the class with this initial task, my host teacher realized that, in order to help push them into their zone of proximal development, she needed to provide them with a scaffold. Thus, she offered them a hint, which ultimately helped them complete the task that they would not have been able to complete on their own. This ties in to sociocultural learning theory, as she presented the students with a difficult task and then provided a scaffold to help them complete the task and engage them in learning (Ormrod, 2011). A final strategy my host teacher employed in her classroom was encouraging the students to utilize self-talk while solving these equations and checking their answers. She preached, “Write yourself a note. Why do you have to check your answers?” Then, she added, “Tell yourself why,” as she aimed for the students not only to talk themselves through the process of checking their answers but also to understand the reasoning for checking their answers. Thus, she promoted the students to self-regulate their behavior in order for them to help guide themselves through difficult problems and ultimately arrive at the correct solutions, which is supported by sociocultural learning theory (Ormrod, 2011).
 * __Part 1__:**

The work sample from my fieldwork classroom reflects the standard: Students will be able to solve equations and inequalities with absolute value. In the teaching and learning of this mathematical topic, I value that the students will be able to understand not only the nature and definition of absolute value, but moreover how to solve equations and inequalities involving absolute value and how its definition applies to these steps and solutions. More specifically, I do not merely value that the students will be able to recall that the absolute of a number is its literal distance away from zero. Rather, I place a higher value that they will be able to understand that any combination of numbers and variables inside of an absolute value must equal a positive number. In addition, I value that the students will be able to incorporate this knowledge to equations that involve two solutions, one solution, no solutions, or “all” solutions. I also value that the students will know how to set up two separate equations when solving these problems and when these two equations will produce two separate solutions. Even more, I value that the students understand how to graph these solutions. While I place high value on the mechanics and understanding of solving these equations, I also want my students to be able to visualize what their solutions represent and how they apply to real-world problems, which graphs will help facilitate. Finally, I value that my students will be able to check their answers by plugging their solutions back into the original equation in order to ensure that all of their solutions satisfy this original equation. Checking solutions is vital in the entire realm of mathematics and applies to various disciplines outside of mathematics, which is why I place such a high value on this area. The assessment tools that I have selected demonstrate all of the values that I have listed in numerous ways. The analytic rubric that I have designed clearly defines the four traits that I highly value in relation to this topic. While I aim for my students to be able to solve equations and inequalities involving absolute value (mechanics), I also want them to understand all of the ideas and processes involved in this topic (demonstrated knowledge), and be able to graph and check that their solutions satisfy the original equations (graph and check). The portfolio that I have selected will contain the students’ notes on the steps involved in solving these types of equations (mechanics). In addition, it will contain numerous examples of each different type of solution and written explanations of why certain equations contain these solutions. Even more, their portfolios will contain numerous graphs and checks to their solutions, with written explanations of how to differentiate the different types of graphs and why solutions must be checked. This relates to my value that they understand not just the mechanics of solving these equations, but also the reasoning and visualizations for these solutions in relation to absolute value. The group project will encompass a wide range of my values for this topic, too. The students will have to apply their knowledge and understanding of this topic to a real-world problem and demonstrate it through the use of graphs, visuals, and written explanations. Thus, this type of assessment will assist them in their comprehension of how absolute value applies to real-life problems, which will increase their motivation and learning, according to constructivism (Ormrod, 2011). I highly value that students understand the mechanics in solving, graphing, and checking these equations, in addition to truly comprehending this concept’s real-world applications, all of which will be demonstrated through the assessment tools that I have selected.
 * __Part 2__:**


 * __Part 3__:**

|| || **__Foshay Learning Center__** // **__Analytic Rubric__** // ||
 * // **__Solving Equations and Inequalities with Absolute Value__** //
 * // **__Solving Equations and Inequalities with Absolute Value__** //
 * || **__Name:__** || **__Teacher: Mr. DePutron__** ||  ||
 * || **__Date Submitted:__** || **__Title of Work:__ _** ||  ||
 * ||  ||   || **Scale** || **Points** ||   ||
 * ** Weights ** || ** Traits ** || ** 4 ** || ** 3 ** || ** 2 ** || ** 1 ** ||  ||   ||
 * **30%** || **Mechanics** || **No mathematical errors.** || **No major mathematical errors or serious flaws in reasoning.** || **May be some serious math errors or flaws in reasoning.** || **Major math errors or serious flaws in reasoning.** ||  ||
 * **30%** || **Demonstrated Knowledge** || **Shows complete understanding of the questions, mathematical ideas, and processes.** || **Shows substantial understanding of the problem, ideas, and processes.** || **Response shows some understanding of the problem.** || **Response shows a complete lack of understanding for the problem.** ||  ||
 * **20%** || **Graph** || **Clearly labeled graph with correct shading and bubbling of solution area.** || **Clearly labeled graph with correct shading of solution area.** || **Graph contains errors in labeling or shading around solution.** || **Graph contains errors in labeling and shading or graph is not present.** ||  ||
 * **20%** || **Check** || **Checks that all solutions satisfy the original equation and eliminates those that do not.** || **Checks all solutions but fails to eliminate those that do not satisfy the original equation.** || **Only checks one of the solutions; does not eliminate solutions that do not satisfy the original equation.** || **Does not check any solutions.** ||  ||
 * ||  ||   ||   ||   || **Total>** ||   ||   ||   ||
 * ** Teacher Comments :** ||
 * ** Teacher Comments :** ||



__**Part 4****:**__ __An alternative assessment for solving equations and inequalities using absolute value would be a math portfolio, consisting of a variety of notes and assignments that would encompass the students’ understanding and progress through the entire domain of this mathematical topic. In their portfolio, I would require that the students include their notes on the steps involved in solving these types of equations and a clear example of each different type of type of solution with a corresponding graph and check. I would also require written explanations of when each type of solution is formed and why their solutions must always be checked in order for the students to demonstrate their understanding of the reasoning behind each type of solution. Also, I would require that the students include each homework assignment, quiz, and test from this unit, with all errors and incorrect answers corrected on a separate sheet of paper. Even more, I would require that the students include written explanations of their errors in mechanics or reasoning in order to help them eliminate these mistakes in the future and help illustrate their progress and understanding of this topic. Finally, I would require that they would include a self-assessment at the end of their portfolio in which the students would discuss their growth throughout the unit, which would further help me assess their true comprehension of the topic. Prior to the start of the unit, I would provide the students with a rubric detailing all components and expectations of the portfolio to promote the standard of openness (Brahier, 2008).__

__**Part 5****:**__ __A third alternate assessment would be a group project where I required the students to investigate a real-world problem involving equations and inequalities with absolute value and present their findings in a presentation to the class. I would provide each group with a different problem that encouraged them not only to find a solution, but also to create visualizations of the processes involved in arriving at their solutions through equations, graphs, checks, and written explanations. This assessment would encourage the students to work together to arrive at an answer to a challenging problem, which is supported by sociocultural learning theory (Ormrod, 2011). Even more, since the problems will directly apply to the real world, this form of assessment will also help increase their motivation to engage in learning, according to constructivism (Ormrod, 2011), and ultimately help open their eyes to the vast applications of mathematics outside of the classroom.__


 * __Part 6__:**

Brahier (2008) suggests that the first NCTM assessment standard, the Mathematics Standard, “relates to the idea that assessment should focus on what we value in the classroom” (p. 297). As described in my reflective narrative (Part 2), the three assessments that I have chosen clearly measure what I value in relation to the standard of solving equations and inequalities with absolute value. Thus, my assessments will allow me to measure whether my students will understand the mechanics and reasoning behind solving, graphing, and checking these equations, and how this topic applies to the real world, all very important in mathematics. My assessments address the second NCTM standard, the Learning Standard, in a myriad of ways. By providing the students with real-life problems, the group project serves to “engage students in relevant purposeful work on worthwhile mathematical activities” (NCTM, //Assessment Standards for School Mathematics,// 1995, p. 13-14), which is an important part of this standard. Even more, through the students’ self-reflection in their portfolio, my assessment provides them with “opportunities to evaluate, reflect on, and improve their own work—that is, to become independent learners” (NCTM, //Assessment Standards for School Mathematics,// 1995, p. 13-14). The third NCTM assessment standard, the Equity Standard, is highly addressed through my assessments. Brahier (2008) offers that, “one of the major reasons why we use a variety of assessment strategies is to give every student an opportunity to demonstrate an understanding of the content and processes” (p. 298). By straying away from solely using written quizzes and tests as assessment and choosing instead to incorporate portfolios and group projects into assessment, I am promoting equity by allowing my students to demonstrate their understanding of content in multiple ways. Ultimately, through my assessments, I aim to provide them equal opportunities to exhibit their learning that will fully allow me to measure their progress and understanding of material. My assessments address the fourth NCTM assessment standard, the Openness Standard, because each one of them will involve a rubric being provided prior to assessment. The analytic rubric will be handed out to the students prior to a homework review or quiz similar to that of the student work sample that I collected. For the portfolio, a detailed rubric describing each component and its expectations will be handed out to the students at the beginning of the unit. In terms of the group project, a detailed project description with a rubric for each of its parts will be given to the students as well. All of these rubrics promote openness because they will clearly explain how the students will be graded and lay out my expectations of the students in relation to each category that I am grading them on. Brahier (2008) explains how “a variety of assessment strategies can be used together to generate a holistic sense of a students’ progress over time” (p. 299). The portfolio that I have chosen helps measure the students’ progress and understanding of this topic throughout the course of the unit, as the students must include explanations of their mistakes and reflect on their growth at the end of the unit. Thus, this assessment addresses the fifth NCTM assessment standard, the Inference Standard, because it consists of a variety of data that combine to illustrate a broad spectrum of student learning. My assessments address the last NCTM assessment standard, the Coherence Standard, as they all align with instruction and serve to evaluate students and chart progress (Brahier, 2008). Each one of the assessments ties directly in to what I will teach the students and expect them to know throughout the unit. Moreover, these assessments all allow for holistic evaluations of student understanding and progress, which align with the expectations that will be outlined for the students in the beginning of the unit. In addition to knowing what I aim for the students to learn, they will also be informed of how these expectations align with the state and national standards, which these assessments will address and measure in numerous ways.
 * __Part 7:__**


 * __Part 8__:**


 * **Key Design Questions **
 * What is the evidence of the desired results?
 * In particular; what is appropriate evidence of the desired understanding? || **Chapters of the Book **
 * Chapter 7 – Thinking like an Assessor
 * Chapter 8- Criteria and Validity


 * //Other resources// || **Design Considerations **
 * Six facets of understanding
 * Continuum of assessment types || **Filters (Design Criteria) **
 * Valid
 * Reliable
 * Sufficient || **What the final Design Accomplishes **
 * Lesson (unit) anchored in credible and useful evidence of the desired results ||
 * Assessment 1 (Rubric)
 * Students demonstrate ability to solve equations and inequalities with absolute value
 * Students understand when certain equations will produce different solutions (two solutions, one solution, no solutions, “all” solutions)
 * Students demonstrate ability to graph solutions
 * Students demonstrate ability to check solutions and explain why solutions satisfy the original equation || * Chapter 7 – Thinking like an Assessor
 * Chapter 8- Criteria and Validity || * Explanation || * Validity: addresses 4 traits of high value pertaining to topic that all will be taught in class
 * Reliability: measures 4 traits on a clearly defined scale, so students who demonstrate knowledge in each area should be able to demonstrate that knowledge on similar assessment in the future || * 4 traits of high value measured, with corresponding percentage assigned to each
 * Each trait important in full understanding of topic ||

Assessment 2 (Portfolio)* Students demonstrate ability to solve equations and inequalities with absolute value
 * Students understand and can explain when certain equations will produce different solutions (two solutions, one solution, no solutions, “all” solutions)
 * Students demonstrate ability to graph solutions
 * Students demonstrate ability to check solutions and explain why solutions satisfy the original equation
 * Students recognize their mistakes and can explain their flaws in mechanics or reasoning
 * Students explain how they have progressed throughout the unit || * Chapter 7 – Thinking like an Assessor
 * Chapter 8- Criteria and Validity || * Explanation
 * Self-Knowledge || * Validity: measures students on a holistic scale pertaining to all aspects of the topic that are of high value
 * Reliability: holistic scale allows for the display of knowledge, understanding, and progress over entire course of unit || * Measures student learning and progress over entire unit
 * Uses a variety of assessment to encompass holistic understanding of entire topic ||
 * Assessment 3 (Group Project/Presentation)
 * Students demonstrate ability to solve equations and inequalities with absolute value that relate to their real world
 * Students demonstrate ability to graph solutions
 * Students demonstrate ability to check solutions and explain why solutions satisfy the original equation
 * Students explain their reasoning for their solutions and illustrate it through the use of visuals (graphs, charts, etc.) || * Chapter 7 – Thinking like an Assessor
 * Chapter 8- Criteria and Validity || * Explanation
 * Interpretation
 * Application
 * Perspective || * Validity: measures students’ application of topic in a real-world sense
 * Reliability: students’ presentations will allow them to orally explain their understanding of the topic and visually illustrate this understanding through graphs, charts, and equations || * Measures students’ abilities to demonstrate understanding and application of topic through real-world problems
 * Student presentations allow for oral and visual explanation of problems, so that they will be able to demonstrate their knowledge and understanding in multiple ways ||

Since the sample of student work that I collected consisted of a short homework review with only three questions, I strongly believe that it was not able to fully measure the extent to which the students met or did not meet the learning objective/standard. Each of the questions on this homework review was scored out of two points, so that the maximum possible score that a student could receive would be six total points. Only three students scored perfectly on this assessment, while 12 students scored a five, three students scored a four, eight students scored a three, and one student each scored a two, one, and a zero. In terms of this assessment, the first and third questions were relatively similar and only required that the students knew how to set up two separate equations and solve for the two solutions. However, a large majority of the students did not recognize that the second question involved an answer with no solution, which I believe is a very important concept for the students to understand in order to be considered proficient in this topic. Since only the three students that received a score of six answered this question correctly, I would conclude that there was still a relatively large gap in where the teacher expected the students to be and where the students actually were in terms of meeting the learning objective/standard. Thus, for this assessment, I would only consider those three students proficient, though as I mentioned above, I still believe that it was too limited to truly measure their proficiency. Nonetheless, since almost half of the class scored a four or below, I would conclude that this brief assessment was effective in illustrating the large gap between the students’ knowledge and the knowledge expected of them. Any score of four or below, in my opinion, would not only fail to meet proficiency, but it would also reveal that these students either did not understand the basic mechanics of solving these equations or made careless mistakes in algebra when solving these problems. Hence, my analysis of this student work standard brought me to the conclusion that the majority of the students did not meet the achievement level of the learning objective/standard and would need further instruction, explanations, and practice in order to build their knowledge to a level that would be considered proficient.
 * __Part 9__:**

At Foshay Learning Center, my host teacher uses the LAUSD electronic grading book, which I will most likely use when I start my student teaching. This grading book is very organized and clearly displays all assessments that my host teacher has utilized throughout the year, though she still believes that it contains flaws. She told me that Foshay forces them to input every assessment out of ten points, regardless of the total number of points that she had originally intended for each assessment to be worth. Therefore, she has to convert each one of her grades into a ten point scale and input them with necessary weights accordingly, which she insisted is very time-consuming. An alternative electronic grading book is called Easy Grade Pro, which allows teachers to easily input the classes they are teaching and the students in each class. While this may be time consuming in comparison to the LAUSD electronic grading book, which has each class and its students saved in a database, I believe that it is still advantageous, as the teacher is able to determine the total number of points and weights for each assessment. Even more, Easy Grade Pro has applications not only for scores, but also for attendance, seating, and meeting standards, all of which will be of use as a future teacher.
 * __Part 10__:**

**References** Brahier, D. J. (2008). //Teaching secondary and middle school mathematics// (3rd ed.). Upper Saddle River, NJ: Allyn & Bacon. Ormrod, J. E. (2011). //Educational psychology: Developing learners// (7th ed.). Upper Saddle River, NJ: Pearson.