Following The American Psychological Association’s Guidelines
Paolo A. Volpi
Having a universal teaching model for students can be ideal, although not always practical (Stigler, 1999). Students across North America learn various subjects throughout the course of their academic lives. Some courses offered in some schools do not even exist in other schools. The teaching methods for each grade vary at the academic level and can be concentrated and streamlined, while other programs are more broad and generic. For the purposes of this topic, educational policies of diverse teaching methods have been researched and collaborated on a comparative level between similar lessons to give the students a vision of problem solving from varied angles, instead of from one specific path. In this report, the measure of the outcomes of theoretical and practical learning have been exercised with a list of questions to assist in the students’ ways of thinking multi-dimensionally in order to face challenges of real world applications by beginning their lessons in the classroom. The end results achieved were comparisons gained in these progressive lessons within the same subjects (mathematics) and used to define the arguments enrolled in the criteria of the quality teaching model. The impressive part was how the students retained the knowledge gained in each lesson to utilize for subsequent lessons making them capable of applying themselves further into solving a problem of higher complexity. The references shown at the end of the document have been meticulously selected as many sources were identical for the teaching model; the differences only being each author’s perspective of improved teaching models.
Keywords: mathematics, deep knowledge, quality teaching,
Introduction
Providing a consistent education of divers students can be viewed as next to impossible because of the academic rate which students learn. The question then remains, why do students not achieve high intellectual quality? The reasons do not fall within the innovative learning capability to the written or spoken word by the teachers. The factors include the levels of learning requirements that may not be present in every school because some schools do not require students to perform work of high intellectual quality (Oakes, Gamoran & Page, 1992). Simply put, a simple, yet an advanced teaching model, needs to be developed, rather than merely illustrating lessons on a blackboard or reading from a textbook. An acceptable and renowned practice entailing deep knowledge and deep understanding to improve diverse student learning experiences have demonstrated that student academic performances increased as opposed to utilizing conventional teaching practices. Evidently the roots of the Quality Teaching Model with many sources have been researched to monitor the successful practice of educators to promote higher order thinking, substantive conversation, and connected learning environments to the real world (Newmann, 1988).
One proven and tested teaching method to magnify the intellectual, social and moral development of students is lesson sequencing, or consecutive lessoning, with the purpose of optimizing learning goals as a whole or individual fragmented learning. Three (3) consecutive lessons here using Stage 6 General Mathematics Syllabus for class exercises consisting of mid- primary levels students (Grades 3 and 4) to demonstrate the effectiveness of teaching strategies. Each lesson, with the first two where the students were divided into groups, shall be explained and illustrated separately to distinguish the variety of teaching strategies in order to assist teachers in improved future planning of each curriculum.
Consecutive Lesson 1
The teacher proposed the first lesson in which the students were provided with an addition and subtraction exercise by gathering diverse numbers to reach the same end result, and how the given evidence contributes to the solution of the problem. As a visual aide, models and concrete materials for counting numbers achieved the desired results. This strategy provides students choices of using different constituent numbers illustrating the everyday use of quantitative applications. Both sentences and symbols paralleled the explanation of each question in each learning stage. The simplest forms illustrated were physical objects (wooden square blocks) and the mathematical number line, or jumping strategy. Students were divided into two equal groups to play out each exercise and then alternate. The purpose was to observe the transformation of information of each arithmetic tool to the end result, both sequencing the same lesson, but with different approaches to the final answer.
The students in each group categorized the objects in relation to a single quantity each forming a larger quantity. The combination of quantities represented the whole or end result, hence the meaning of addition and subtraction. Each group rotated to perform the second exercise of a similar nature to study the behavior of each number. Each group was treated as an ensemble group to provide students with equal opportunity learning. Although each group first managed to learn the basics of arithmetic operations with their respective exercises, the transition to the second exercise initially proved challenging. The deduction was tools with similar functions affected thinking behavior on the same topic despite of each exercise falling under the same learning tactic. The usage of each similar exercise also played on students’ psychologically of adding and subtracting. The students had to reason why the end result was the same in each exercise since both tools used provided a practical and physical representation of the numbering system.
This lesson resulted in the students magnifying their critical thinking because they were forced to break their routine in each exercise. Their envelope of normal routine thinking was pushed to go beyond the simple reproduction of knowledge (Fields, 2005). Deep knowledge was practiced because the students innovated the experience of the first exercise into the second exercise of a similar problematic nature. Because this lesson started up with basics, a lot of teacher interaction was involved. Regardless of how a school is structured, not all students have the prerequisite knowledge to transition into different teaching models (Gore, Griffiths, & Ladwig, 2001). It can fall within their domestic teaching influences as well. Since learning the number line method was a conventional tool for numeric understanding, the teachers agreed to continue with the same tool for students to remember the basics of the numbering system rather than a reverse engineering method, where the fragments of the exercise would not be comprehended easily. The reverse method would only be acquired if the basic learning tool were implemented primarily, and intermediate results of complex applications could act as a springboard for better learning. Analogous to this learning curve is having infants learning how to crawl, and then walk before learning how to run and race. The need for organizing the numbers itself was not necessary since the students already learned counting and the numerical system in earlier grades. Structuring a higher step to apply the numbering system enhanced their viewing angle of numbers from a different perspective; an example where when one were to stand on the floor or upon a table in the same room, the view would differ although the dimensions, height, size, etc. remained constant. Hence the number line lesson drew the students’ view from a different standpoint of the numbering system.
Consecutive Lesson #2
Students have been presented with a 2D modeling exercise representing geometric shapes. The two groups of students were divided with one physically studying and handling the shapes. The second group was presented with a larger scale model of an architectural building where geometric shapes were present and made up the constructed the building, and then swapped for additional testing. The teacher categorized each shape by the number of lines, vertices, and surfaces that constitute each geometric shape and placed them on an overhead projector for the students to compare to when they identified the shapes in each of their exercises. The teacher’s purpose was to test the initiative of student responsiveness in discerning the fundamental pieces of geometry and how the addition or subtraction of each piece can alter the very properties of each geometric shape; something reminiscent of the first consecutive lesson. Exercises for each group were pitted against each other to discern the representation of geometric shapes in theory and in practice. The teacher would follow up and have a question and answer discussion of each project and record how each group’s perception of geometric shapes related to the increase or decrease in numeric value by merely changing on quantitative measure of each shape. In the first exercise, the students were given a piece of paper to write down the constituent parts of each shape they examined. For simplicity, the range began from the simplest to the most complicated shape. The teacher instructed them to add an extra line, or side, with the shape staying closed and reconnected, and having them identify the newly formed shape. Although this lesson was program lengthy, as the practically elapsed to be, the students attentively expressed sustained focus on the topic at hand, as both exercises involved the significance of geometric shapes. As the identity of these shapes prevailed the exercise, the complicated solving problems were grasped in a systematic way in which the theory and practice concept gap closed, and persuaded the students in associating what they previously learned with something new.
The results revealed mirrored improvement in deep knowledge to both exercises reflecting from each group. Little interaction from the teacher gave the students the incentive to exercise complete understanding of numeric manipulation. At the end, the teacher revealed the motive of identifying various geometric shapes is to discern them as objects all around them, as in the classroom, for example. Following the lesson the teacher spotted various objects in the classroom matching the shapes providing a practical learning environment in a similar 3D mode. Teachers found the exercise more difficult to observe, as the students were challenged to higher levels of understanding than originally anticipated by the teachers. As a further test on their knowledge, teachers followed up with timed performances to measure the learning curve. This lesson had to performed more than once to record the timing accuracy of deep thinking between each exercise.
The rubric order was based in this lesson in the teachers measuring up on the students’ understanding of the numbering system. The results have also indicated teacher progress in guiding the students as well as monitoring how students progressed. The general grading system incorporated the holistic model coupled with the analytical model. The results were indeed impressive, however, some students, and the teachers for that matter, required some improvement. Evidently, improvement requirements are always present as no accepted model is perfect, and can only provide excellence (A spreadsheet of the rubric order is shown on a separate file associated with this paper).
Consecutive Lesson #3
The third lesson compiled nine graded mathematical questions of varying difficulty carefully formulated to, not to solely evaluate students’ memory of studying, but to measure the presence of deep knowledge and understanding in versatile portions of sustaining continuum learning of centralized concepts and topics. The purpose of incorporating all nine graded questions in ne lesson was to keep the students focus on deep understanding of related concepts. Each question progresses in difficulty and complexity to streamline each concept of problem solving and producing new methods of knowledge acquirement of constructing explanations and drawing conclusions (Hargreaves, 2003). The test of student knowledge brought about reasoning and examining the unit plan, and eventually into component parts distinctively. As the learning curve always operated, some students grasped the questions promptly than others. The students that took lengthy periods to solve each question initiated a fair chance to themselves to solve them on their own primarily without excessive dependence of the teachers. The following questions are basic but render the students’ ability to actually read the question and analyze each piece of the question to answer progressively.
Question #1:
Explain what happens to the value of a number when you move the number along the number line left or right? (Which way does the number get smaller or larger?)
Question #2:
In which directions along the number line do you use the + symbol or the – symbol?
Question #3:
How do you compare the square blocks to the number line when it comes to addition and subtraction?
Question #4:
Use three examples, using two numbers only in each example, with the number line where the answers are 5, 13, and 18.
Question #5:
What happens to a geometric shape when you add or remove one of its lines? Give an example of the new shape created.
Question #6:
How do you compare the change of geometric shapes to the number line when you add or subtract numbers? What happens to the geometric shape when you change its numeric value?
Question #7:
Why are identifying geometric shapes important in the real world? Give a good example of important geometric shapes you see everyday.
Question #8:
Provide and name at least 6 examples of geometric shapes you see in the real world, and include the type, number of sides to each shape, and compare them to the objects used in the real world.
Question #9:
Choose any of two polygons, join them at the lines or vertices, and identify the result by the number of lines and vertices. What does the new shape tell you about the similarities between each shape?
Conclusion
Even if many teaching models, and the schools that adopt them, have proven to be successful, not only do professionals need to maintain them as long as they are expedient, but also to improve them to correspond to the quality teaching model. Level of intellectual quality, the learning environment, and the significance of students’ work are the key factors in trying and testing new and improved teaching models. They do not negate the earliest teaching models because modern teaching models would not otherwise exist. Past students have learned just as efficiently with early models. Unfortunately the teachers always needed to maintain consistent elements of teaching with the constant rate of technological growth, and must reboot older models to ensure they continue to prove effective. With forever changing practices, the issue becomes a matter of how do we develop a responsible knowledge practice (Muller, 2001).
Another conclusion drawn was the teachers who proxied these lessons were not indigenous to the classrooms assigned to them (Wenglinsky, 2000). The reason was to allow these teachers practice of the quality teaching model with students other than their own, and be evaluated of their adoptions of accepted teaching practices. Teachers were able to locate the behavior of each student’s thinking technique as well understanding their own teaching behavior as well as their colleagues under the same circumstances. Not only did these consecutive lessons observe the problem solving patters of each student, but the benefits of participating teachers became essential. At the end of each completed curriculum, the boards decided to consensus these lessons to teachers preferred to participate further to extract better quality teaching from within themselves. Participation was, of course, arbitrary for each teacher. However, the quality teaching models exercised have been attractive enough to entice many teachers to enroll in such educational programs. It merely became a portfolio for all teachers to study their own teaching abilities and how they can assist in students’ active learning.
Work Cited
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