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The use of multiple representations supports and requires tasks that involve decision-making and other problem-solving skills. [2] [3] [4] The choice of which representation to use, the task of making representations given other representations, and the understanding of how changes in one representation affect others are examples of such mathematically sophisticated activities.
Problem solving is the process of achieving a goal by overcoming obstacles, a frequent part of most activities. Problems in need of solutions range from simple personal tasks (e.g. how to turn on an appliance) to complex issues in business and technical fields.
The teachers’ role in discovery learning is therefore critical to the success of learning outcomes. Students must build foundational knowledge through examples, practice and feedback. This can provide a foundation for students to integrate additional information and build upon problem solving and critical thinking skills. [citation needed]
The teacher teaches a lesson to the students and they then work in teams and ensure that they have mastered the lesson. The students take individual quizzes on the material, at which they may not help each other.
Problem-based learning (PBL) is a teaching method in which students learn about a subject through the experience of solving an open-ended problem found in trigger material. The PBL process does not focus on problem solving with a defined solution, but it allows for the development of other desirable skills and attributes.
The general consensus of large-scale studies that compare traditional mathematics with reform mathematics is that students in both curricula learn basic skills to about the same level as measured by traditional standardized tests, but the reform mathematics students do better on tasks requiring conceptual understanding and problem solving. [3]
The philosophy of problem-posing education is the foundation of modern critical pedagogy. [4] Problem-posing education solves the student–teacher contradiction by recognizing that knowledge is not deposited from one (the teacher) to another (the student) but is instead formulated through dialogue between the two. [5]
The incorporation of classroom assessment techniques is an age-old concept which teachers have been using and practicing for years. Whether a teacher uses a technique learned in training, or simply a strategy conjured up on their own, teachers need to know if their methods are successful and many feel that the desire to understand students' comprehension is instinctive.