A presentation during the national Seminar-Workshop at UP NISMED in October 2011

2.  Conceptual Understanding
 Activities and Problems Solving that
Promote Conceptual Understanding
 Significance of Teaching through
Problem Solving in Developing
Conceptual Understanding

3. 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?

4. 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.

5. What are some of the realities that are happening in
our mathematics classroom today?
Many of our students tend
to apply algorithms
without significant
conceptual understanding
that must be developed for
them to be successful
problem-solvers.

6. 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.

7. 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.

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. 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?

12. Some of the Pupils’ Answers

13. Activity 2:
Developing Connections of
Algebra, Geometry and
Probability

14. 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?

15. Successful event
6:45 7:00 7:15 7:30
𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑠𝑢𝑐𝑐𝑒𝑠𝑠𝑓𝑢𝑙 𝑒𝑣𝑒𝑛𝑡 15 𝑚𝑖𝑛𝑢𝑡𝑒𝑠 1
𝑃 𝑠𝑢𝑐𝑐𝑒𝑠𝑠 = = =
𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑡𝑜𝑡𝑎𝑙 𝑠𝑒𝑔𝑚𝑒𝑛𝑡 45 𝑚𝑖𝑛𝑢𝑡𝑒𝑠 3

16. 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?

17. 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).
Solving for x:
x = 3 – 3x
4x = 3
3
x=
4
3
The probability that it will next Tuesday in Manila is .
4

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 n
units n=1 n=2 n=3 n=4 n=5 n=6 n=7
Number of cubes for n > 3
with
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

20. 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.

21. DISCUSSION

22. Use of
Communication
Technology
Connections
Estimation
Problem Solving
Visualization
Reasoning

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 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.

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, 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.

25. Questions
◦ Can procedures be learned by
rote?
◦ Is it possible to have
procedural knowledge about
conceptual knowledge?

26. Is it possible to have conceptual
knowledge/understanding about something
without procedural knowledge?

27. What is Procedural Knowledge?
◦ Knowledge of formal language
or symbolic representations
◦ Knowledge of rules,
algorithms, and procedures

28. 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

29. By definition, conceptual
knowledge cannot be learned
by rote. It must be learned
by thoughtful, reflective
learning.

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
relationships between
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 today’s
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 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.

33. 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.

34. Conclusion:
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.

35. 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.

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

37. "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.

38. "We should be
teaching
students how
to think.
Instead, we are
teaching them
what to think.“
Clement and Lochhead, 1980,
Cognitive Process
Instruction.

39. 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.

40. 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.