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**Part 1: Reflective Paper**

The Algebra classroom being observed at John Adams Middle School is reminiscent of what some may consider an old fashioned classroom. Students sit in rows of five, with a total of five rows. Students face forward, where an older teacher sits behind an overhead projector. When compared to the technology available nowadays, this particular classroom setup is as rare as a typewriter in a computer age. Nevertheless, upon closer examination, one discovers a few learning theories at play.

On a typical instructional day, the teacher would lecture for the greater part of the period. Students would take notes while the teacher spoke, a rather uninteresting process when one considers the technology available to enhance said lesson. It is during this seemingly dull process that students learn. At the top of the lectures, the teacher begins by asking students to identify words or symbols. Then, a new topic is introduced using previous knowledge. Further into the lesson, after a few examples are worked out, the teacher produces a similar, yet more difficult problem. The task for students now becomes to solve new (though familiar-looking) problems. Eventually, through trial-and-error, students seem to reach the desired result. According to Ormrod (2011), this classroom scenario is rooted in Constructivist Theory.

Constructivism, although the principal learning theory evident, is not the only one present. During instruction, the teacher creates an environment that fosters learning by limiting conversations among students during lectures. This is one aspect from Behavioral Learning Theory. Also, whenever new tasks are assigned, the teacher provides adequate scaffolding to facilitate learning. Gradually, students are able to solve problems on their own. In some respect, this provides evidence of Sociocultural Theory. Unfortunately, at least through limited observations, there is a lack of group/partner activities.

For this portion of the project, a class set of homework was used. The section was 2.4 Solving Multi-Step Equations. The assigned problems, which will later be analyzed, were questions 1-6 and 11-22. The main standard in focus,

//Standard 4.0 – Students simplify expressions before solving linear equations and inequalities in one variable//

concerns simple operations. There is a second standard (Standard 5.0) which involves word problems with justifications, but it is not reflected in the assigned homework.

**Part 2: Reflective-Narrative**

Mathematics is objective. Problems have a single solution, but there may be multiple ways of finding said solutions. As a result, arriving at the solution becomes secondary to the process used in solving the problem. Teaching math should be rooted in showing students the methods they may use to solve problems and so, assessment becomes a much more complicated process. Grading assignments by checking for the correct answer is simply not enough. Granted, it becomes difficult to grade 100+ students and being thorough may be an impossible task. Therefore, a combination of assessment strategies would be used. First, student work would be checked for completion. Secondly, using a rubric, a few problems would be checked for accuracy. This would ensure students are completing assignments and doing it correctly. Finally, a rubric will be used to combine both completion and accuracy to determine a final score.

The rubric was designed in a way that mathematical processes would account for 40% of the assignment grade. The math notebook, which encompasses the collection of all assignments graded using the rubric, becomes the most important part of the assessment process. The purpose of the math journals is to allow students the opportunity to explain their understanding of mathematical processes. The journals, then, would contain no computational work, only entries written in students’ own words. Since the journals are housed within the notebook, it becomes clear why such importance is placed on the notebooks in the first place

** Part 3: Assessment 1 – Analytic Rubric **

Using the Arter and McTighe Six-step Process, we obtain the rubric below:

Using this rubric, students would earn a maximum of four points each for Completeness and Accuracy and a maximum of two points for Format. An assignment would have a maximum value of ten points. Generally speaking, it would be possible (down the line) to integrate Completeness with Format and adjust the scales accordingly to closer mimic the Analytic Scoring Rubric sent via email.
 * || COMPLETENESS || ACCURACY || FORMAT ||
 * <span style="color: #000000; display: block; font-family: 'Times New Roman','serif'; font-size: 20px; text-align: center;">SCALE || <span style="color: #000000; display: block; font-family: 'Times New Roman','serif'; text-align: center;">40% || <span style="color: #000000; display: block; font-family: 'Times New Roman','serif'; text-align: center;">40% || <span style="color: #000000; display: block; font-family: 'Times New Roman','serif'; text-align: center;">20% ||
 * <span style="color: #000000; display: block; font-family: 'Times New Roman','serif'; font-size: 20px; text-align: center;">4/2 || <span style="color: #000000; display: block; font-family: 'Times New Roman','serif'; text-align: center;">Most problems attempted/completed. Steps are complete and answers clearly identifiable. || <span style="color: #000000; display: block; font-family: 'Times New Roman','serif'; text-align: center;">Shows strong evidence of process comprehension. There is very little evidence of erasures, indicating a firm grasp of methods, calculations, and understanding. || <span style="color: #000000; display: block; font-family: 'Times New Roman','serif'; text-align: center;">Assignment is properly formatted with title, name, date, period, and assignment information. Work is clearly organized with problems arranged in proper sequential order. ||
 * <span style="color: #000000; display: block; font-family: 'Times New Roman','serif'; font-size: 20px; text-align: center;">2/1 || <span style="color: #000000; display: block; font-family: 'Times New Roman','serif'; text-align: center;">Missing a few problems or left unanswered. Some steps missing. || <span style="color: #000000; display: block; font-family: 'Times New Roman','serif'; text-align: center;">Minor computational mistakes. Some problems may have been copied incorrectly, but process remains accurate. || <span style="color: #000000; display: block; font-family: 'Times New Roman','serif'; text-align: center;">Missing one or more elements (heading, assignment information). ||
 * <span style="color: #000000; display: block; font-family: 'Times New Roman','serif'; font-size: 20px; text-align: center;">0 || <span style="color: #000000; display: block; font-family: 'Times New Roman','serif'; text-align: center;">Failed to complete the vast part of the assignment. Too many problems missing/not attempted. || <span style="color: #000000; display: block; font-family: 'Times New Roman','serif'; text-align: center;">Problems are generally complete, but incorrect. Student attempts problems, but failed to understand process. || <span style="color: #000000; display: block; font-family: 'Times New Roman','serif'; text-align: center;">Missing pertinent information. Problems out of order or individual problems not clearly divided (overlap). ||

**<span style="color: #000000; display: block; font-family: 'Times New Roman','serif'; text-align: center;">Part 4: Assessment 2 – Math Notebook **
<span style="color: #000000; font-family: Arial,Helvetica,sans-serif; font-size: 12px;"> A math notebook, which may just be another name for portfolio, will be used throughout the semester. In this notebook, students will complete warm-ups, take notes, and glue in assignments (homework, class work, projects, etc.). The main purpose of the notebook would be to provide a space for students to house their work over the course of a semester. Most assignments would already have been assessed via use of the rubric (Assessment 1) and so the notebook would simply be checked for completion. Keeping a notebook will help students keep track of their own progress.

<span style="color: #000000; font-family: Arial,Helvetica,sans-serif; font-size: 12px;"> The math notebook would have a simple design. The first few pages would contain a table of contents, which would include dates, assignment descriptions, and page numbers. The pages, as mentioned before, would house warm-ups, assignments, and any other work done in and out of class.

<span style="color: #000000; font-family: Arial,Helvetica,sans-serif; font-size: 12px;">In section 4 of Chapter 2, the primary focus was for students to solve multi-step equations. As part of the math notebook, students could complete an in-class assignment or show their work to a presentation. Additionally, the homework assignment discussed earlier would also have a place in the notebook.

<span style="color: #000000; display: block; font-family: 'Times New Roman','serif'; text-align: center;">Part 5: Assessment 3 – Math Journals
<span style="color: #000000; font-family: Arial,Helvetica,sans-serif; font-size: 12px;">A rubric works great for problems that are rather straight forward. However, a rubric is unable to assess a student’s logic process. A second type of assessment would be in the form of a reflection journal. These journals would be assigned periodically in the form of an exit slip, which they would be required to complete in their math notebooks (Assessment 2). The journals themselves would only be graded for completion, but their contents would be invaluable to providing insights to any misconceptions or difficulties that students may experience.

<span style="color: #000000; font-family: Arial,Helvetica,sans-serif; font-size: 12px;">As further practice for solving multi-step equations, students could re-work one of the problems they found difficult and explain their reasoning in the form of a journal entry. By reading how students are thinking, an educator should be able to clear up any misconceptions as well as address common mistakes. Along with utilizing the graphs from part 6 (below), it becomes easier to see why multiple students are making the same mistakes.

= = =<span style="color: #000000; display: block; font-family: Arial,Helvetica,sans-serif; font-size: 16px; text-align: center;">Part 6: Excel chart/graph =



**<span style="font-family: 'Times New Roman','serif';">Part 7: NCTM 6 Assessment Standards **
<span style="color: #000000; font-family: 'Times New Roman','serif';">Students will be assessed on their knowledge of mathematical processes as they pertain to core standards. As described in the Rubric, student work will be checked not only for completion, but for accuracy. Furthermore, assigned work will be aligned to standards (CA standards are used in the Algebra text). The principal focus of the Mathematics Standard is on process skills such as problem solving. <span style="color: #000000; font-family: 'Times New Roman','serif';">The math notebook (Assessment 2) is crucial in determining any misconceptions students encounter with their assignments. Journals reflect a student’s understanding of concepts and can be used to guide lessons (and reteach if necessary). <span style="color: #000000; font-family: 'Times New Roman','serif';">Students are able to demonstrate their understanding of important mathematics both directly (via assigned work) and indirectly (via reflection in math journals). Hence, an educator is better able to assess a student. Furthermore, math notebooks will include group work and exams, thus students are assessed in multiple ways both formally (exams, quizzes) and informally (journals). <span style="color: #000000; font-family: 'Times New Roman','serif';">By keeping math notebooks current, students will be able to look back at past units to understand their assessment is based on multiple criteria. Again, students will be able to use their journals to express their feelings towards a particular exam/assignment along with suggestions for future work. An educator can then adjust or modify work accordingly. <span style="color: #000000; font-family: 'Times New Roman','serif';">Periodically, a student’s progress report would be provided to be included in the math notebook. A student will be able to see their grades and, hopefully, predict what grade their skills will earn them. Additionally, the inclusion of exams/quizzes/etc. in the notebook will let them become familiar with the various assessments used by the educator. <span style="color: #000000; font-family: 'Times New Roman','serif';">Students are able to see that instruction is closely tied to assessment primarily through their math notebooks. Both assessments and notes will be available to them within their notebooks.
 * <span style="color: #000000; font-family: 'Times New Roman','serif';">1) The Mathematics Standard **
 * <span style="color: #000000; font-family: 'Times New Roman','serif';">2) The Learning Standard **
 * <span style="color: #000000; font-family: 'Times New Roman','serif';">3) The Equity Standard **
 * <span style="color: #000000; font-family: 'Times New Roman','serif';">4) The Openness Standard **
 * <span style="color: #000000; font-family: 'Times New Roman','serif';">5) The Inferences Standard **
 * <span style="color: #000000; font-family: 'Times New Roman','serif';">6) The Coherence Standard **

<span style="color: #000000; font-family: 'Times New Roman','serif';">Individually, the assessments would not properly address all six assessment standards. However, when using multiple forms of assessments, the standards are met to a much higher degree, as was the case in using the rubric, notebook, and journals together. Score reports, as mentioned earlier, would reflect the completeness/accuracy of a student’s abilities. Furthermore, the scores would also be available to parents since students will be required to carry the notebooks at all times. Finally, it is reasonable to assume that students who receive passing scores on the assessments would certainly qualify to move ahead to the next level.

=<span style="color: #000000; display: block; font-family: 'Times New Roman','serif'; font-size: 16px; text-align: center;">Part 8: UbD Assessment Matrix =



**<span style="color: #000000; font-family: 'Times New Roman','serif';">Part 9: __<span style="color: #000000; font-family: 'Times New Roman','serif';">Gap Analysis __**
<span style="color: #000000; display: block; font-family: Arial,Helvetica,sans-serif; font-size: 12px; text-align: justify;">The homework set used was not graded by the instructor. As a result, grading was done in two ways: 1) Grading for correct answers and 2) grading with analysis of incorrect answers. For the first part, if the answer was correct, a point was awarded (up to a maximum of eighteen). For the second part, the incorrect problems in part (1) were further analyzed for errors. The main focus of the analysis, as described in the beginning, was the process used. Often times, the process used was correct and the mistake lay in minor computational errors (such as omitting negative signs). These ‘minor mistakes’ were ignored and awarded the point. Hence, the second method for scoring yielded higher results. For our purposes, the higher scores were used since the primary focus of the standard was in solving equations and not carrying out simple operations.

<span style="color: #000000; font-family: Arial,Helvetica,sans-serif; font-size: 12px;">The types of problems assigned were designed to be solved. According to Bloom’s Taxonomy of Increasing Complexity, this assignment falls under Level 2 – Application. This lower level, along with the fact that 13 of 15 students would receive passing scores on the assignment, would indicate that perhaps the problems were not challenging enough. It would be appropriate to assign more complex problems (such as word problems) or even have students create their own problems. Assigning tougher problems would challenge students, which could result in learning.

**<span style="color: #000000; font-family: 'Times New Roman','serif';">Part 10: __<span style="color: #000000; font-family: 'Times New Roman','serif';">Grade Book Example __**
<span style="color: #000000; font-family: Arial,Helvetica,sans-serif; font-size: 12px;">Easy Grade Pro, although not free, is an excellent program. The main screen features a very rich gradebook with the ability to choose terms/periods from a drop-down menu. Grades can be entered to include categories, scales and more. A second page features a seating chart that can be color-coded and moved around. A student’s name can be edited to include personal information or simple notes. Attendance can also be taken from the seating chart page or the main page. Aside from attendance, an educator can record tardies or other excused absences. There are many more features, but another noteworthy mention is on the summaries and reports. Educators have the option of printing a class progress report (with names or ID numbers) as well as individual reports that contain scores of all assignments. Easy Grade Pro definitely touts many features that can be very useful to an everyday teacher.

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 20px;">**References**
<span style="font-family: Arial,Helvetica,sans-serif; font-size: 12px;">Brahier, D. J. (2008). //Teaching Secondary Middle School Mathematics// (3rd Ed.). Upper Saddle River, NJ: Pearson.

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 12px;">Ormrod, J. E. (2011). //Educational Psychology: Developing Learners// (7th Ed.). Upper Saddle River, NJ: Pearson.

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 12px;">Wiggins, G., McTighe, J. (2005). //Understanding by Design// (2nd Ed.). Alexandria, VA: ASCD.