There are 4 principles in cognitive apprenticeship that teachers can be apply in the design of mathematics e-course.
The first principle is the content. It was 4 types of knowledge needed for expertise which are domain knowledge, heuristic strategies, control strategies and learning strategies. The mathematics e-course does not just include the contents, but also the steps to solve the task given. After students already mastered how to use the knowledge and formula to solve the certain problem, teachers can provide some kind of problems related to the knowledge and formula that they had just learnt to other domains of problems. From here, the e-course will guide them to learn the new concepts, facts and procedures.
The second principle is the method. It was 3 kinds of ways to enhance the progression of expertise which are modeling, coaching and scaffolding. Before let the students to solve the task given, teachers will demonstrate the steps to solve the problem correctly by using the right knowledge and formula. Students will see the steps and try to understand the purpose for each step through my explanation for every step. After students already mastered about the steps, teachers will create some platform for them to demonstrate their steps of solution regarding the task given to them. This e-course is designed to enable the students to present their work to teachers. It also provided some function like forum where students are able to see their other friend’s result of work. From here, teachers can see and respond toward student’s progress about the lesson from times to times while motivate them to reach the goal of the task. To enhance the knowledge gained by students, teachers can invite students to pose and solve their own problems related to the knowledge and formula they had just learnt to see their real understanding about the lesson.
The third principle is the sequencing. It was 3 sorts of keys to ordering learning activities which are increasing complexity, increasing diversity and global to local skills. The task that teachers can put in my e-course is in the form of 2 ways which are from easy problems to hard problems and from general problems to specific problems. The tasks in my e-course also come from different domain and field related to the content as to enhance students understanding about the knowledge and its application.
The forth principle is the sociology. The sociology is very important in order to create the conducive learning atmosphere for the cognitive apprenticeship to occur in the mathematics e-course. The task given in the e-course is based on the real life problems. This will make the lesson more practical and real for students and make them interested towards the content of subject. Through the forum that created by teachers for students to post their work among their peers, this will create the community of practice as they collaborate together in there to discuss and withdraw the correct solution.

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