Question 4 a) Learners have access to high-quality, engaging mathematics instruction. Knowledgeable teachers have adequate resources to support their work and are continually growing as professionals. Learners confidently engage in complex mathematical task chosen carefully by teachers. They draw on knowledge from a wide variety mathematical topics, sometimes approaching the same problem from different mathematical perspectives. They value mathematics and engage actively in learning.
Teachers help learners make, refine, and explore conjecture on the basis of evidence and They use a variety of reasoning and proof techniques to confirm or dis 1 introduction using classroom assessment to improve learner learning 2 prove this conjectures. Learners are flexible and resourceful problem solvers. Alone or in groups and with access to technology, they work productively and reflectively with the skilled guidance of their teachers. Orally and in writing, learners communicate their ideas and results effectively. b) Conceptual understanding: refers to an integrated and functional grasp of mathematical ideas. Learners with conceptual understanding know more than isolated facts and method. They understand why a mathematical idea is important and kinds of contexts in which is it useful. Example: suppose learners are adding fractional quantities of different sizes, say 1/3+2/5. They might draw a picture or use concrete materials of various kinds to show the addition. They might also represent the number sentence 1/3+2/5 =? As a story. They might turn to the number line, representing each fraction by a segment and adding the fractions by joining the segments. Procedural fluency: refers to knowledge of procedures, knowledge of when and how to use them appropriately and skill in performing them flexibly, accurately, and efficiently. Learners need to be efficient and accurate in performing basic computations with whole numbers (6+7, 17-9, 8×4, and soon) without always having to refer to tables or other aids. Example: learners with limited understanding of additional would ordinary need paper and pencil to add 589 and 647. Learners with more understanding would recognize that 589 is only 2 less than 600, so they might add 600 to 647 and the subtract 2 from that sum. Strategic competence: refers to the ability to formulate mathematical problems, represent them and solve them. Although in school, learners are often presented with clearly specified problems to solve, outside of school they encounter situations in which part of the difficulty is to figure out exactly what the problem is.
Example: an initial procedure for 86-59 might be to use bundles of sticks. A compact procedure involves applying a written numerical algorithm that carries out the same steps without the bundles of sticks Adaptive reasoning: refers to the capacity to think logically about the relationships among concepts and situations. Learners draw on their strategic competence to formulate and represent a problem, using heuristic approaches that may provide a solution strategy, but adaptive reasoning must takeover when they are determining the legitimacy of a proposed strategy. Example: it is not sufficient for learners to do only practice problems on adding fractions after the procedure has been developed. Productive disposition: refers to the tendency to see sense in mathematics, to perceive it as both useful and worthwhile, to believe that steady effort in learning mathematics pays off, and see oneself as an effective learner and doer of mathematics. If learners are to develop conceptual understanding, procedural fluency, strategic competence and adaptive reasoning abilities, they must believe that mathematics is understandable, not arbitrary, that, with diligent effort, it can be learned and used and that they are capable of figuring it out. Example: as students build strategic competence in solving nonroutine problems their attitudes and beliefs about themselves as mathematics learners become more positive.