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2. Conceptual Understanding Activities and Problems Solving thatPromote Conceptual Understanding Significance of Teaching through ProblemSolving in Developing ConceptualUnderstanding 3. WORKSHOP Describe a typical mathematicsclass in your school. What do you like best in thoseclasses? List at least 3. What are your wishes for thoseclasses? 4. IntroductionWhen childrenlearnedelementary mathematics, theylearnedto performmathematical procedures.The essence of mathematicsis not for a child to able tofollow a recipe to quickly andefficiently obtain a certainkind of answer to a certainkind of problem. 5. What are some of the realities that are happening in our mathematics classroom today?Many of our students tend toapply algorithms withoutsignificant conceptualunderstanding that must bedeveloped for them to besuccessful problem-solvers. 6. Why do teachers spend more time oncomputation & less time on developing concepts? Teachers believe its easierto teach computation than todevelop understanding ofconcepts. Teachers value computationover conceptualunderstanding. Teachers assume developingconcepts is astraightforward process. 7. In mathematics, interpretations of data andthe predictions made from data inherentlylack certainty. Events and experimentsgenerate statistical data that can be used tomake predictions. It is important thatstudents recognize that these predictions(interpolations and extrapolations) are basedupon patterns that have a degree ofuncertainty. 8. Conceptual Understanding What does conceptual understanding mean? How do teachers recognize its presence or absence? How do teachers encourage its development? How do teachers assess whether students have developed conceptual understanding? 9. Activity 1: 10. Content Domain: Statistics and Probability Grade Level: Grades 2 - 4 Competencies Gather and record favorable outcomes for an activity with different results. Analyze chance of an outcome using spinners, tossing coins, etc. Tell whether an event is likely to happen, equally likely to happen, or unlikely to happen based on facts Tasks Develop an activity for pupils that addresses the competencies required in grade 4. Material: A pack of NIPS candy 11. Activity1. Estimate the number of candies in a pack of NIPS.2. Open the pack and make a pictograph showing each color of candies.QuestionsSuppose you put back all the candies in the pack andyou pick a candy without looking at it.a.What color is more likely to be picked? Why?b.What color is less likely to be picked?c.Is it likely to pick a white candy? Why do youthink so? 12. Some of the Pupils Answers 13. Activity 2:Developing Connections of Algebra, Geometry andProbability 14. Problem 1:Rommels house is 5 minutes away from the nearest buststation where he takes the school bus for school. Supposethat a school arrives at the station anytime between 6:30to 7:15 in the morning. However, exactly 15 minutes afterits arrival at the station, it leaves for school already. Onemorning, while on his way to the station to take the bus,Rommel estimated that he would be arriving at the stationa minute or two after 7:15. What is the probability that hecould still ride on the school bus? 15. Successful event6:45 7:00 7:157:30 16. Problem 2:It has been raining for the past threeweeks. Suppose that the probability that itrains next Tuesday in Manila is thrice theprobability that it doesnt, what is theprobability that it rains next Tuesday inManila? 17. Let x be the probability that it rains next Tuesday. We cannow translate this word problem into a math problem interms of x. Since it either will rain or wont rain nextTuesday in Manila, the probability that it wont rain mustbe 1 - x. We are given that x = 3(1 - x). 18. Problem 3: The surface of an cube is painted blue after which the block is cut up into smaller 1 1 1 cubes. If one of the smaller cubes is selected at random, what is the probability that it has blue paint on at least one of its faces? 19. Cube with edge nunits n=1 n=2 n=3 n=4 n=5 n=6 n=7 Number of cubesfor n > 3 withNo face painted0 0 1 82764125(n - 2)31 face painted 0 0 6245496150 6(n - 2)22 faces painted0 01224364860 12(n-2)3 faces painted0 8 8 8 8 8 88No. of cubes 1 82764125 216 343n3 20. ExtensionTaskMany companies are doing a lot Write possible questionsof promotions to try to getthat you may ask aboutcustomers to buy more of theirproducts. The company that the situation.produce certain brand of milk Device a plan on how tothinks this might be a good waysolve this problem.to get families to buy more boxesof milk. They put a childrens Solve your problem.story booklet in each box of milk.That way kids will want theirparents to keep buying a box ofMilk until they have all sixdifferent story booklets. 21. DISCUSSION 22. Use ofCommunication Technology ConnectionsEstimationProblem Solving VisualizationReasoning 23. Communication The students can communicate mathematical ideas in a variety of ways and contexts. Connections Through connections, students can view mathematics as useful and relevant. Estimation Students can do estimation which is a combination of cognitive strategies that enhance flexiblethinking and number sense. Problem Solving Trough problem solving students can develop a true understanding of mathematical conceptsand procedures when they solve problems in meaningful contexts. 24. Reasoning Mathematical reasoning can help students think logically and make sense of mathematics. This can also develop confidence in their abilities to reason and justify their mathematical thinking.Use of Technology Technology can be used effectively to contribute to and support the learning of a wide range of mathematical outcomes. Technology enables students to explore and create patterns, examine relationships, test conjectures, and solve problems. Visualization Visualization involves thinking in pictures and images, and the ability to perceive, transformand recreate different aspects of the visual-spatial world (Armstrong, 1993, p. 10). The use ofvisualization in the study of mathematics provides students with opportunities to understandmathematical concepts and make connections among them. 25. Questions Can procedures be learned by rote? Is it possible to have procedural knowledge about conceptual knowledge? 26. Is it possible to have conceptualknowledge/understanding about somethingwithout procedural knowledge? 27. What is Procedural Knowledge? Knowledge of formal languageor symbolic representations Knowledge of rules,algorithms, and procedures 28. What is ConceptualKnowledge? Knowledge rich in relationships andunderstanding. It is a connected web of knowledge, a networkin which the linking relationships are asprominent as the discrete bits of information. Examples of concepts square, squareroot, function, area, division, linearequation, derivative, polyhedron, chance 29. By definition, conceptualknowledge cannot be learnedby rote. It must be learnedby thoughtful, reflectivelearning. 30. What is conceptual knowledge of Probability? Knowledge of those facts and properties of mathematics that are recognized as being related in some way. Conceptual knowledge is distinguished primarily by relationshipsbetween pieces of information. 31. Building Conceptual Understanding We cannot simply concentrate on teaching the mathematical techniques that the students need. It is as least as important to stress conceptual understanding and the meaning of the mathematics. To accomplish this, we need to stress a combination of realistic and conceptual examples that link the mathematical ideas to concrete applications that make sense to todays students. This will also allow them to make the connections to the use of mathematics in other disciplines. 32. This emphasis on developing conceptual understanding needsto be done in classroom examples, in all homework problemassignments, and in test problems that force students to thinkand explain, not just manipulate symbols.If we fail to do this, we are not adequately preparing ourstudents for successive mathematics courses, for courses inother disciplines, and for using mathematics on the job andthroughout their lives. 33. What we value most about great mathematiciansis their deep levels of conceptual understandingwhich led to the development of new ideas andmethods.We should similarly value the development of deeplevels of conceptual understanding in our students.Its not just the first person who comes upon a greatidea who is brilliant; anyone who creates the sameidea independently is equally talented. 34. Conclusion:One of the benefits to emphasizingconceptual understanding is that aperson is less likely to forgetconcepts than procedures.If conceptual understanding isgained, then a person can reconstructa procedure that may have beenforgotten. 35. On the other hand, if proceduralknowledge is the limit of apersons learning, there is noway to reconstruct a forgottenprocedure.Conceptual understanding inmathematics, alongwithprocedural skill, is much morepowerful than procedural skillalone. 36. Procedures are learned too, but notwithout a conceptual understanding. 37. "It is strange that we expect students tolearn, yetseldom teach them anythingabout learning." Donald Norman, 1980, "Cognitiveengineering and education," in Problem Solving and Education: Issues inTeaching and Research, edited by D.T. Tuna and F. Reif, Erlbaum Publishers. 38. "We should beteachingstudents how to think.Instead, we are teaching themwhat to think.Clement and Lochhead, 1980, Cognitive Process Instruction. 39. If we have achieved these moments ofsuccess and energy in the past then weknow how to do it we just need to doit more often. 40. References:Carpenter, T. P., Corbitt, M. K., Kepner, H. S., Lindquist, M.M., & Reys, R. E. (1981). What are the chances of yourstudents knowing probability? Mathematics Teacher, 73, 342-344.Castro, C. S. (1998). Teaching probability for conceptualchange. Educational Studies in Mathematics, 35, 233-254.MacGregor, J. (1990). Collaborative learning: Shared inquiryas a process of reform. New Directions for Teaching andLearning, 42, 19-30.