These exams would also tap teaching performance and other capabilities unlikely to be adequately assessed using conventional paper along with pencil instruments." (Shulman, 1986, pp. 4 -- 14)
These different elements are important, because they are providing a foundation for helping the schools to become more competitive in mathematics. As, they are working together to create a basic standard for: improving learning comprehension and provide the ability to solve more complex issues. Over the course of time, this will help to increase the student's ability to understand a wide variety of concepts. This is the point that they will be more prepared to deal with the various challenges that they are facing in the 21 century. Once this occurs, it will help them to establish a foundation for adapting to the changes that they will have to deal with from: shifts in technology and through these transformations because of globalization.
To corroborate these finding with one another Ball (2008) had similar kinds of observations. As, the author believes that Shulman is providing a basic foundation for improving the student's understanding of different mathematical concepts. They found, that there were a number of similar approaches and trends that must be embraced to help dramatically improve learning comprehension. Evidence of this can be seen with Ball saying, "These lines of research indicate at least two empirically discernable subdomains within pedagogical content knowledge (knowledge of content and students and knowledge of content and teaching) and an important subdomain of 'pure' content knowledge unique to the work of teaching, specialized content knowledge, which is distinct from the common content knowledge needed by teachers and non-teachers alike. The next steps needed are to develop a useful theory of content knowledge for teaching." (Ball, 2008, pp. 389 -- 407) This is important, because it is indicating that there is support for the idea that PCK can help to improve middle school students learning comprehension when it comes to mathematics.
Yet, when you look beyond these kinds of generalizations, it is clear that there has been a lack of research in seeing how to effectively implement this kind of model on the middle school level. Part of the reason for this, is because no one has assessed the relationship between knowledge and PCK. Our study will seek to address these different gaps between the various theories and how to effectively implement this kind of protocol in the real world. At the same time, we will examine various techniques and tactics that can be used to help improve the overall amount of understanding of the different mathematical concepts. Once this occurs, it will provide specific insights about how to effectively address these challenges and create a workable strategy that will deal with the underlying problem.
To achieve this objective we will be taking an all-encompassing approach. Where, we will look at the total impact of this new philosophy through looking at: the DOK and PCK models. This will be accomplished by examining the relationship of the middle school teacher to be able to: understand math content knowledge, the ability to problem solve / reason, memorize, comprehend and establish pedagogical knowledge. This will help to improve the educator's ability to know and apply various concepts that are directly related to some of the numerous ideas in mathematics (i.e. their familiarity with numbers and algebra).
These different elements are important, because they are showing that the rationale that we will be using to understand these concepts will be based upon: looking at how the educator can have a direct impact on student performance. Once this occurs, we can begin to theorize about how we can be able to increase the overall amounts of learning comprehension in mathematics on the middle school level. This will allow us to address the various gaps and it will identify how we can create a workable solution to improve the students understanding of various facts.
Evidence of this can be seen with observations from a report released by the Bill and Melinda Gates Foundation. They found that changing and improving upon the approach that is being used by the teachers. will help to produce a number of significant long-term benefits for the student. With the report saying, "Teachers have more of impact on student learning than any other factor controlled by school systems, including class / school size and the quality of after-school programs. But currently, there is no agreement among education stakeholders about how to identify and measure effective teaching." ("Content Knowledge for Teaching," 2010) This is significant because, it is illustrating how educators must improve the tactics they are using to reach out to students. At the same time, there needs to be a workable solution to increase the ability to measure effective teaching. Once this takes place, it means that the student will begin to more effectively focus and comprehend the various ideas that are being presented.
This is the point that the educator can begin to have an actual strategy in place that will increase the total amounts of learning comprehension in relation to: their poor academic performance on this level. As, a wide variety of reforms have attempted to address these issues, yet they continue to fall short of providing some kind evidence that they are effectively reversing these trends. This is problematic, because the use of algebra is a major determining factor as to how successful someone will be later on life.
Part of the reason for this, is because understanding different mathematical concepts will help to improve the student's ability to be successful later on in the life. The reason why, is because globalization and shifts in technology mean that the ability to think logically is becoming a major area of focus. This is the point, that the ability of the student to understand technical ideas will become a major area of contention.
As a result, having a basic understanding of algebra and other mathematical concepts will help to create foundation for learning more advanced ideas. These are necessary for being successful in job markets that are increasingly competitive. As these skills can be utilized by students to create a foundation to think logically and improve their flexibility. Over the course of time, this will help to increase the productivity and innovation, with these ideas being used to introduce new concepts down that road that will help to spur innovation. Therefore, one could effectively argue that algebra is an indicator of future success and the impact that it will have on the ability of the person to adapt to a host of challenges they are facing.
One possible approach that could serve as basic foundation is the DTMAS model. This was introduced by the University of Louisville to provide, a greater understanding of the various concepts in math to improve student performance. The way that this was accomplished is by offering math teachers with an effective protocol for understanding the various challenges that they are facing. To include: memorized knowledge, conceptual understanding, problem solving / reasoning and pedagogical content knowledge. This is important, because it is showing how our basic philosophy is to improve the overall understanding of these concepts and their impact on the ability of the educator to effectively reach out to the student. "Diagnostic Mathematics Assessment," 2011)
Since the current approach is failing to address these issues, means that we must be able to effectively integrate a workable solution that will deal with these challenges in the real world. To do this, we will integrate different techniques that were implemented by: University of Louisville (DTMAS) and through the University of Michigan (a.k.a. LMT). This will provide a workable foundation that we can utilize to understand how to effectively implement a PCK approach and its long-term impact.
As a result, our focus will be on creating a workable solution that can effectively be implemented in the real world. To achieve this objective we will be answering a number of different research questions to include:
1. What is the relationship between understanding how utilizing algebra can help to effectively prepare student for the challenge they will face in the real world?
2. How the PCK methodology can be utilized to effectively deal with the various issues that are affecting students?
3. What is relationship between the ability to effectively problem solve and the role that the educator plays in helping to improve the overall amounts of learning comprehension?
Once this occurs, it will provide specific insights about how we can increase the quality of mathematics education that is being provided to middle school students. This is the point that we can be able to have a positive impact on reversing the trends and addressing the gaps in research on this subject.