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?

Introduction

When children learned elementary

mathematics, they learned to perform

mathematical procedures.

The essence of mathematics is not for a

child to able to follow a recipe to quickly and

efficiently obtain a certain kind of answer to

a certain kind of problem.

Many of our students tend to apply algorithms without significant conceptual understanding that must be developed for them to be successful problem-solvers.

What are some of the realities that are happening in our mathematics classroom today?

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 and the predictions made from data inherently lack certainty. Events and experiments generate statistical data that can be used to make predictions. It is important that students recognize that these predictions (interpolations and extrapolations) are based upon patterns that have a degree of uncertainty.

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?

Activity 1:

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.

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?

Questions

Some of the Pupils’ Answers

Developing Connections of Algebra, Geometry and

Probability

Activity 2:

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?

6:45 7:00 7:15 7:30

Successful event

𝑃 𝑠𝑢𝑐𝑐𝑒𝑠𝑠 = 𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑠𝑢𝑐𝑐𝑒𝑠𝑠𝑓𝑢𝑙 𝑒𝑣𝑒𝑛𝑡

𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑡𝑜𝑡𝑎𝑙 𝑠𝑒𝑔𝑚𝑒𝑛𝑡 =

15 𝑚𝑖𝑛𝑢𝑡𝑒𝑠

45 𝑚𝑖𝑛𝑢𝑡𝑒𝑠=

1

3

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?

Problem 2:

Solving for x: x = 3 – 3x 4x = 3

x = 3

4

The probability that it will next Tuesday in Manila is 3

4.

Let x be the probability that it rains next Tuesday. We can now translate this word problem into a math problem in terms of x. Since it either will rain or won’t rain next Tuesday in Manila, the probability that it won’t rain must be 1 - x. We are given that x = 3(1 - x).

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?

Problem 3:

Cube with edge n

units

n = 1

n = 2

n = 3

n = 4

n = 5

n = 6

n = 7

Number of cubes

with

for n > 3

No face painted

0

0

1

8

27

64

125

(n - 2)3

1 face painted

0

0

6

24

54

96

150

6(n - 2)2

2 faces painted

0

0

12

24

36

48

60

12(n-2)

3 faces painted

0

8

8

8

8

8

8

8

No. of cubes

1

8

27

64

125

216

343

n3

Many companies are doing a lot of promotions to try to get customers to buy more of their products. The company that produce certain brand of milk thinks this might be a good way to get families to buy more boxes of milk. They put a children’s 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.

Write possible questions that you may ask about the situation.

Device a plan on how to solve this problem.

Solve your problem.

Extension Task

DISCUSSION

Communication

Connections

Problem Solving

Reasoning

Use of Technology

Estimation

Visualization

Connections ◦ Through connections, students can view mathematics as useful and relevant.

Communication ◦ The students can communicate mathematical ideas in a variety of ways and contexts.

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.

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.

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.

◦Can procedures be learned by rote?

◦Is it possible to have procedural knowledge about conceptual knowledge?

Questions

Is it possible to have conceptual knowledge/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, conceptual knowledge cannot be learned by rote. It must be learned by thoughtful, reflective learning.

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 needs to be done in classroom examples, in all homework problem assignments, and in test problems that force students to think and explain, not just manipulate symbols. If we fail to do this, we are not adequately preparing our students for successive mathematics courses, for courses in other disciplines, and for using mathematics on the job and throughout their lives.

What we value most about great mathematicians is their deep levels of conceptual understanding which led to the development of new ideas and methods. We should similarly value the development of deep levels of conceptual understanding in our students. It’s not just the first person who comes upon a great idea who is brilliant; anyone who creates the same idea independently is equally talented.

One of the benefits to emphasizing conceptual understanding is that a person is less likely to forget concepts than procedures.

If conceptual understanding is gained, then a person can reconstruct a procedure that may have been forgotten.

Conclusion:

On the other hand, if procedural knowledge is the limit of a person's learning, there is no way to reconstruct a forgotten procedure. Conceptual understanding in mathematics, along with procedural skill, is much more powerful than procedural skill alone.

Procedures are learned too, but not without a conceptual understanding.

"It is strange that we expect

students to learn, yet

seldom teach them anything

about learning."

Donald Norman, 1980, "Cognitive engineering and education," in Problem

Solving and Education: Issues in Teaching and Research, edited by D.T. Tuna and F. Reif, Erlbaum Publishers.

"We should be teaching

students how to think.

Instead, we are teaching them what to think.“

Clement and Lochhead, 1980, Cognitive Process

Instruction.

If we have achieved these moments of

success and energy in the past then we

know 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 your students knowing probability? Mathematics Teacher, 73, 342-344. Castro, C. S. (1998). Teaching probability for conceptual change. Educational Studies in Mathematics, 35, 233-254. MacGregor, J. (1990). Collaborative learning: Shared inquiry as a process of reform. New Directions for Teaching and Learning, 42, 19-30.