Developing conceptual understanding of probability
Conceptual Understanding Activities and Problems Solving that Promote Conceptual Understanding Significance of Teaching through Problem Solving in Developing Conceptual Understanding
WORKSHOP• Describe a typical mathematics class in your school.• What do you like best in those classes? List at least 3.• What are your wishes for those classes?
IntroductionWhen children learned elementarymathematics, they learned to performmathematical procedures.The essence of mathematics is not for achild to able to follow a recipe to quickly andefficiently obtain a certain kind of answer toa certain kind of problem.
What are some of the realities that are happening in our mathematics classroom today?Many of our students tendto apply algorithmswithout significantconceptual understandingthat must be developed forthem to be successfulproblem-solvers.
Why do teachers spend more time on computation & less time on developing concepts? Teachers believe it’s easier to teach computation than to develop understanding of concepts. Teachers value computation over conceptual understanding. Teachers assume developing concepts is a straightforward process.
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.
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?
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
Activity 1. Estimate the number of candies in a pack of NIPS. 2. Open the pack and make a pictograph showing each color of candies.Questions Suppose you put back all the candies in the pack and you 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 you think so?
Activity 2:Developing Connections of Algebra, Geometry and Probability
Problem 1: Rommel’s house is 5 minutes away from the nearest bust station where he takes the school bus for school. Suppose that a school arrives at the station anytime between 6:30 to 7:15 in the morning. However, exactly 15 minutes after its arrival at the station, it leaves for school already. One morning, while on his way to the station to take the bus, Rommel estimated that he would be arriving at the station a minute or two after 7:15. What is the probability that he could still ride on the school bus?
Problem 2: It has been raining for the past three weeks. Suppose that the probability that it rains next Tuesday in Manila is thrice the probability that it doesn’t, what is the probability that it rains next Tuesday in Manila?
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 won’t rain nextTuesday in Manila, the probability that it won’t rain mustbe 1 - x. We are given that x = 3(1 - x).Solving for x: x = 3 – 3x 4x = 3 3 x= 4 3The probability that it will next Tuesday in Manila is . 4
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?
Cube with edge n units n=1 n=2 n=3 n=4 n=5 n=6 n=7Number of cubes for n > 3 withNo face painted 0 0 1 8 27 64 125 (n - 2)31 face painted 0 0 6 24 54 96 150 6(n - 2)22 faces painted 0 0 12 24 36 48 60 12(n-2)3 faces painted 0 8 8 8 8 8 8 8No. of cubes 1 8 27 64 125 216 343 n3
Extension Task Many companies are doing a lot Write possible questions of promotions to try to get that you may ask about customers to buy more of their products. The company that the situation. produce certain brand of milk Device a plan on how to thinks this might be a good way solve this problem. to get families to buy more boxes of milk. They put a children’s Solve your problem. story booklet in each box of milk. That way kids will want their parents to keep buying a box of Milk until they have all six different story booklets.
Use ofCommunication Technology Connections EstimationProblem Solving Visualization Reasoning
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 flexible thinking and number sense. Problem Solving ◦ Trough problem solving students can develop a true understanding of mathematical concepts and procedures when they solve problems in meaningful contexts.
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, transform and recreate different aspects of the visual-spatial world” (Armstrong, 1993, p. 10). The use of visualization in the study of mathematics provides students with opportunities to understand mathematical concepts and make connections among them.
Questions ◦ Can procedures be learned by rote? ◦ Is it possible to have procedural knowledge about conceptual knowledge?
Is it possible to have conceptualknowledge/understanding about something without procedural knowledge?
What is Procedural Knowledge?◦ Knowledge of formal language or symbolic representations◦ Knowledge of rules, algorithms, and procedures
What is Conceptual Knowledge?◦ Knowledge rich in relationships and understanding.◦ It is a connected web of knowledge, a network in which the linking relationships are as prominent as the discrete bits of information.◦ Examples of concepts – square, square root, function, area, division, linear equation, derivative, polyhedron, chance
By definition, conceptualknowledge cannot be learnedby rote. It must be learnedby thoughtful, reflectivelearning.
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 relationships between pieces of information.”
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 today’s students. This will also allow them to make the connections to the use of mathematics in other disciplines.
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.
What we value most about great mathematiciansis their deep levels of conceptual understanding which ledto the development of new ideas and methods.We should similarly value the development of deep levels ofconceptual understanding in our students.It’s not just the first person who comes upon a great ideawho is brilliant; anyone who creates the same ideaindependently is equally talented.
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.
On the other hand, if proceduralknowledge is the limit of apersons learning, there is noway to reconstruct a forgottenprocedure.Conceptual understanding inmathematics, along withprocedural skill, is much morepowerful than procedural skillalone.
Procedures are learned too, but not without aconceptual understanding.
"It is strange that we expect students to learn, yet seldom 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.
"We should be teaching students how to think.Instead, we are teaching themwhat to think.“Clement and Lochhead, 1980, Cognitive Process Instruction.
If we have achieved these moments ofsuccess and energy in the past then weknow how to do it – we just need to do it more often.
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.