Problem Solving in Mathematics
GCSE or the General Certificate of Secondary Education is basically a system that is present in England, Northern Ireland and in Wales. In this system, a student is awarded an academic qualification based on the grades that they attain. The qualification that a person attains is equivalent to either a level 2 or Level 1 key skills qualification. Normally, a student can uptake as many subjects as he or she wants. However, different systems set a requirement for how many subjects or GCSEs a student must take. There is present an international system of IGCSE as well and these subjects can be up taken anywhere in the world. This was just a precise history of what exactly the GCSE system is all about. Interestingly enough, the GCSE system was not the first one of its kind. Prior to this, GCE and the English Baccalaureate System were also present.
The GCSE system as complicated as it is would defiantly require some sort of revision plan in all subjects. There is a revision plan designed which is specifically targeted to developing mathematical problem solving skills. Problem solving is an essential component of any subject and all the students must be able to have a good grasp over this. In Mathematics, problem solving isn’t only required in word problems but the approach becomes necessary in solving equations as well. There were the choices of choosing between Number development in study skills or problem solving. When one thinks of math, then problem solving seems like the more appropriate skill required.
Problem solving in math
Problem Solving is made up of interpreting, reasoning, analyzing, predicting, evaluating and reflecting. In many countries, this concept is either a crucial component of school mathematics curriculum or an extended objective of the curriculum in many countries. Turning a student into skilled problem solvers requires a lot of disposition and skill. (Stacy, 2005) The students not only need mathematical knowledge but also require a lot of reasoning ability. That is to stay that the students should be able to come up with better heuristic ways of solving other problems as well. (Anderson, 2009) Along with having good skills in math and good reasoning abilities, the students should be able to work in good environment as well. The students should be confident and persisted. They should be able to organize their knowledge and then put it to good use.
One thing that should be highlighted is that problem solving itself cannot really be taught. Problem solving is taught and developed in the students by presented them with different problems. (Lester, Masingila, Mau, Lambdin, dos Santon and Raymond, 1994) In this approach, teachers mainly focus on different topics of math by going through them in a problem solving manner. It is the aim of the teacher to explain and develop a deep understanding of the topic or the problem. The students will only go on to solve the problem and work on the situation if they are well aware of the concept.
As mentioned earlier, there should be interaction between the students and interactions between the teacher and the student as well. (Van Zoest et al., “1994) This is one of the concepts that I am assuring is present in the course designed. Regardless of what the course or the curriculum, effective communication is what makes the entire course successful.
Apart from having a good relationship with the students, problem solving as a course become effective if the students are given a purpose. Since the 1970s, problem solving has been less subjective. That is to say that they are more focused and revolve more around the basic ideas. Students are explained the importance of mathematical reasoning and are given proof as well. This goes onto answer a very common query about math: Are we ever going to use this?
Problem solving skills are thus developed through better interactions and a sense of purpose in all the students learning it. Schoenfeld also went on to suggest that the students must meet these three criteria’s of mathematical thinking. The students should go on to know the process of mathematization and abstraction to such an extent that they are not afraid to apply this knowledge. The tools that are developed in them such as tools of service and trade should be used in getting a good idea of the mathematical world. Lastly, it was recommended that problem solving should be developed within students so that they are motivated to reorganize the different themes taught to them through activity. With ideas such as these, the students only developed their own confidence but they are able to think mathematically. (Schifter and Fosnot, 1993) Problem solving is a guide for students to construct, analyze and alter their own ways and theories in math and in other subjects.
Curriculum Models
It should be noted that the content for any course is picked out on a vocational basis. That in short means that it is linked with what the student needs to know. In this scenario, it is mandatory that the students get a good review of all that they have learned. Not only does this scheme hopes to revise what the students have learned but also goes on to focus on areas where they still need some clarification. As mentioned earlier, problem solving is a complex skill and one needs clarification of concepts before he or she goes on to develop it further.
After the needs are looked at, the course should put its main objective on what the learning outcomes should be. Learning outcomes are basically what the learning class or the students would be able to do when the whole class is over. The objectives that are usually written out are done so with increasing level of specificity. (Davies, 1975) By starting from a very general objective and making way towards the specific targets of the course, the course managers can hope to cover a vast variety of goals and aims. In this way, the objectives and the targets are ‘operationalised’ and the entire course is directed in a very organized manner.
If hierarchies are applied to the targets and the objectives listed, then proper activities can be made that target the students. These activities can be in the form of lectures, assignments, quizzes and even assessments. Here the major idea is to start off with very simple objectives and targets and then move on to more complex tasks. This idea of starting from simple and then moving towards complex all falls under Bloom’s taxonomy. (Bloom, 1956) It is up to whoever is designing the course to come up with outcomes or objectives based on the student population. This reflects on what their capabilities are, what it is that they need help with and how many the students can be actually achieved. The objectives model
The major obstacle is cleared when the students are able to steer easily through any tests or assessments thrown their way. Thus a curriculum model turns out to be exemplary or a good model where the assessment targets all the objectives in a very clear way and thus goes on to test all the objectives listed. If any student is not able to rightly fulfill an objective and answer it in a proper way, then it is solely the teachers or the curriculum planner fault. Many a times, what happens in schools or colleges is that when a curriculum is made, it is tested in a proper way by a group of typical students. These students can either be very diverse such that a certain group of children are not targeted.
This course that is made is for ages fourteen to nineteen years old. If a typical group of children were chosen for this curriculum, then they might belong to different race or casts however they would be in the aforementioned age group. After the typical students go through the objectives and give the proper feedback, the curriculum is then fixed to whatever alteration might be required. The Keller plan is a plan in which the students function at their own pace and once they have become skilled in a certain objective, they move on to the rest of the objectives.
Pedagogy theory
Pedagogy is basically the art and science of education. In the simplest terms, this can be defined as how the teachers teach and what the students learn. The major objectives of pedagogy are that whenever a teacher or any instructor is wishing to teach the students a certain course, the objectives should be met and the students should learn all the things taught to them. There are many factors that come between and affect the interaction that students and teachers might have. These could mean the instructive strategies and the own beliefs that a teacher has. Where every student has a different learning style, a teacher has a different teaching style as well. The teachers thought about the entire course holds a lot of importance. Not only that, the teacher should have certain beliefs and a positive attitude. Problem solving in this course would be developed effectively, if the student’s right and wrong response is met with encouragement. Any sort of discouragement or negative feedback can render them incapable of solving further problems.
As for the students, their knowledge and background holds a lot of importance. Surely, if a student is more learned and has more knowledge, then his understanding of new concepts would be better as opposed to the different students. Learning habits, previous knowable experience and the environment also makes a lot of difference. If the student is constantly discouraged by his peers and his parents, then he might not be able to do well in the problem solving. This in turn affects how the student learns and thus fulfils the objectives told to them. Many might think that a successful pedagogy is when the student learns everything. However, a teacher who has been successful in teaching to a child with known disabilities is far more accredited than the one who taught to an already learned student.
Context of the course
This course is being targeted for learned ages fourteen through nineteen. Many studies have been done to show why the development of problem solving skills is crucial in these age groups. These are the age groups where the students are learning to be more aware of the outside world. They are learning to choose what they like and consider different prospects. It is this consideration and reasoning that is being polished when their problem solving skills are looked at. Students go on to get a better idea of reasoning and looking at different problems and ideas in their own perspective.
The students can look at different scenarios and problems and be able to come up with the best solution. They are presented with different problems and choices, and it is only with a better reasoning skill that the students will not be dependent on a thing else. Stemming from independence comes the art of being confidence. It has been shown that the more students solve problems on their own, the better their confidence gets.
Even if these problems are minute mathematical word problems, these give them a sense of confidence and independence. The students take more part in their class activities and are not afraid to be creative. They have that confirmation that they solved problems and their confidence is boosted to a good extent. The students go on to develop better leadership skills and they take up a much more active role in class. These four attitudes were found in individuals who were more skilled at problem solving. These individuals felt a control from the environment to go the control to the individual, there was a development of an intricate mental precise going on, they become more well informed of the social nature and origin of more complicated mental function and they resorted to using those complex mental functions. (Wretch, 1985 p 25)
Problem solving was looked into by many different countries and results of studies went on to alter their course. Singapore was one example in which a study known as TIMSS went on to affect the curriculum by a great extent. The content was decreased by about thirty percent and the center of learning in mathematics was shifted to problem solving. Similarly, in Australia, the teachers were provided with journals and publications to further enhance their knowledge about problem solving skills. In Hong Kong, presentations were shown in schools prior to developing the course outline. Here, a lot of aspects were incorporated in the course framework that I have included in this course as well. There was focus given on learning via the alignment of pedagogy, curriculum and assessment. These principles were thus incorporated into the learning so that the students have the ability to go and solve the problems of the future and the contemporary world.
These sorts of studies and speculations further motivated me to go for a better revision of problem solving skills in students of this age group. Problem solving allows the student to be able to feel a different range of emotions in the solution process. Math experts and theorists have always studied that they feel a sense of joy when they solve a problem. Not only do the persons feel happy but they also discover a newfound love for mathematics. The more the students become adept at solving problems, the more they would wish to learn mathematics and appreciate it as well.
I went through a variety of frame works of curriculum prior to designing the course work for this course. My major focus stayed on the fact that the children should not be overburdened with knowledge and concepts and they should actually enjoy solving problems. It is a general rule of thumb that whenever anyone forces him or her into doing something, they always end up not doing well in it. Also, I looked extensively into the capabilities of the class room or the group of students who would be taking this coursework. No matter how amazing my course is designed, or how it covers all the different aspects, if it is above the capabilities of the students, it would not be successful.
Curriculum framework has a lot of affect on how the course is taken by the students and the teachers alike. I think the best way is for the framework of a curriculum to be flexible and open for any alterations. Education and studying itself is a dynamic process that can go on to be enhanced by every study or every publication developed. Therefore, my scheme of work is developed however it is flexible enough to incorporate changes in it. Another thing that I noticed is the more a course or a curriculum is rushed, the worse results it provides.
Thus, along with flexibility and ease of change, the framework for this curriculum is paced on a very steady basis. There is room to even spend time on a certain concepts of problem solving. This is crucial because every teacher should go on to a different objective, only if the first objective is properly understood. This is quite a characteristic of mathematics because in mathematics all the different concepts are built on each other. In this course, information and communication technology can be used to quite an extent.
There are numerous websites available with quizzes and assessments. Along with the amount of class work and problems presented to the children, they can have more practice as they go home. Applications with simulated math problems can be installed din their phones or their iPads and they can get all the more practice that they need. Practice is the key ingredient that makes any person skilled in a subject. The more practice a student will undertake, the better his reasoning and argument skills will be. The different varieties of problems a student faces, he will become more adept at dealing with them and managing them in a certain way. Since this is a short course, the better the students becoming at using their time outside class or on weekends, the better the progression of this course will be.
A difference has been noted in the achievement of mathematics between girls and boys. Turns out that girls participate less and achieve little in mathematics as compared to boys. (Boaler, 2002, p 137) Such was the degree of the difference that many feminists and social groups made it their responsibility to address to their problems. After successive work on this issue, girls and boys are able to attain top grades in GCSE assessments.
When more attention was given to these issues, the different attitudes of girls and boys towards mathematics were figured out. It has been speculated that both girls and boys have separate ways of learning and working. Women are more known to work by knowing their concepts, intuition, experience and creativity. On the other hand boys tend to work more by rigor, logic and rational arguments. Further research also went onto show how girls tend to work better in connected groups and with increased interaction among their peers. As mentioned earlier, problem solving is best learned and figured out when students work in groups. In a school, the way of learning encouraged connected thinking and group work. These ways thus showed increased success of girls because they were better skilled at working in groups. These issues can be tackled by trying out both ways that fit to both girls and boys. Instead of using one way and having one group suffer, both connected and individual learning should be promoted.
These learning ways should not only be imposed on the students, but also should be taught to them as well. For instance, many boys might find it difficult to work in groups. This can be solved by having them paired with their friends. It can be linked to certain bonuses and rewards like presents or extra football time. On the other hand, girls can be taught to work individually and be able to face problems by themselves as well. Diversity is also a key factor when comes to teaching a group of children.
The best way to know how diverse a class is to have a one to one session with each and every student. It is true that every student is different and comes from different background. I can go on to interview the student and get a closer idea about their family life. In doing so, I would have a better idea of his or her level of interest as well. With level of interest come experience and it provides me a way to realize how skilled the student already is. The more conversations and individual time a teacher spends with the student, the better she gets to know him or her.
The students who come from a different background, they can be counseled about where they are and what the general goal of the class is. This little briefing session would remove any ambiguities he or she might have. Furthermore, it would also help the student feel easier around his or her peers. Surely, not everyone comes from a same social background. However, providing a general sense of direction and a sense of class unity moves everyone on the same page.
This scheme of work has been developed after extensive study and different scenarios. It has taught me the different tactics involved in teaching and having your message reach the students. The different concept given by experts and their advice has molded the framework of this course. The context and curriculum is designed with a lot of thought and hopefully it would be sufficient for the students in the future.
References
Anderson, J. (2009) Mathematics Curriculum Development and the Role of Problem Solving. [E-Book] The University Of Sydney. Available Through: ACSA Conference 2009 Http://Www.Acsa.Edu.Au/Pages/Images/Judy%20Anderson%20-%20Mathematics%20Curriculum%20Development.Pdf [Accessed: 11th February 2013].
Bloom, B. (1971) Handbook Of Formative And Summative Evaluation Of Student Learning. New York: Mcgraw-Hill.
Boaler, J. (2002). Experiencing School Mathematics: Traditional And Reform Approaches To Teaching And Their Impact On Student Learning. Mahwah, N.J., L. Erlbaum.
Davies, I. (1975) Writing General Objectives And Writing Specific Objectives. In: Golby, M. Et Al. Eds. (1975) In Curriculum Design . 1st Ed. Open University Books .
Lester, . Et Al. (1994) Learning How To Teach Via Problem Solving. In: Aichele, D. And Coxford, A. Eds. (1994) Professional Development For Teachers Of Mathematics . 1st Ed. Reston, Virginia: NCTM, P.152-166..
Schifter, D., & Fosnot, C.T. (1993). Reconstructing Mathematics Education: Stories Of Teachers Meeting The Challenge Of Reform. New York, Teachers College Press.
Ssdd.Bcu.Ac.Uk (1975) Curriculum Models – Objectives-Based Model. [Online] Available At: Http://Www.Ssdd.Bcu.Ac.Uk/Crumpton/Curriculum-Design/Key-Concept-Map/Obj-Based-Proc-Model.Htm [Accessed: 11 Feb 2013].
Stacey, K. (2005) The Place Of Problem Solving In Contemporary Mathematics Curriculum Documents. Journal of Mathematical Behaviour, 24 P.341-350.
Van Zoest, . (1994) ‘Beliefs About Mathematics Teaching Held By Pre-Service Teachers Involved In A First Grade Mentorship Program. Mathematics Education Research Journal., 6 (1), P.37-55.
Wertsch, J.V. (1985). Vygotsky and the Social Formation of Mind. Cambridge, Mass, Harvard University Press.
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