Here are two word problems with different levels of difficulty and how I would use metacognition to teach them: Level 1 (Compare, compared quantity unknown): - Mary has some apples. Joseph has 8 more apples than Mary. How many apples does Joseph have? - I would have students identify the known (Mary has some apples) and unknown (how many apples Joseph has) quantities (Declarative Knowledge). Then have them derive the equation to represent the problem (Procedural Knowledge) and solve for the unknown. Finally, have them check their work (Monitoring). Level 2 (Change, starting set and results set unknown): - Sam originally had some oranges. He gave 4 oranges to Joseph. Now

1. Using Metacognition
in Mathematical
Modelling and
Investigation
Carlo Magno
Professor of Educational Psychology
De La Salle University-Manila

2. Case analysis
Jane is a college student taking up her algebra class. Every
time her teacher presents word problems that need to be
solved she stumbles, stops, panics, and doesn‟t know what to
do.
For example the teacher writes on the board the problem:
The period T (time in seconds for one complete cycle) of a
simple pendulum is related to the length L (in feet) of the
pendulum by the formulas 8T2= 2L. If a child is on a swing with
a 10 – foot chain, then how long does it take to compete one
cycle of the swing?
It takes around 30 to 40 minutes for her to stare at the word
problem and everytime she attempts to write something she
suddenly stops and is uncertain in what she is doing.

3. Case Analysis
RJ whenever faced with mathematical word
problems make himself relaxed. He thinks
of the steps on how to solve the problem.
He determines what is asked or required,
extracts the given, translates the problem
into an equation. He represents the
unknown into „X‟ or „?‟. He proceeds to
solve the problem. Checks his answer. He
reviews his answer by rereading the
problem and checking his computations.

4. Objectives
• Uncover the definition of metacognition
• Indentify specific metacognitive processes
• Use metacognition strategies to teach
mathematical investigation

5. Metacognition
• “Thinking about thinking” or “awareness of one‟s
learning”
• Metacognition is an executive system that enables
top down control of information processing
(Shimamura, 2000).
• According to Winn and Snyder (1998),
metacognition as a mental process consists of two
simultaneous processes: (1) monitoring the
progress in learning and (2) making changes and
adapting one‟s strategies if one perceives he is not
doing well.
• Schraw and Dennison (1994): knowledge of
cognition and regulation of cognition

6. What is the benefit of
metacognition?
• Majority of studies in metacognition are
related with outcome performance such as
students‟ achievement in different domains
(i. e. Magno, 2005; Al Hilawani, 2003;
Rock, 2005)
• Metacognition is related with different sets
of attitudinal variables such as self-efficacy
(Narciss, 2004; Chu, 2001; Cintura, Okol, &
Ong, 2001; Jinks & Morgan, 1999;
Schunk, 1991)

7. Model2: Effect of Metacognition (8 factors) on Critical
Thinking
DELTA2 DELTA3 DELTA4 DELTA5 DELTA6 DELTA7
100.43* 71.46 57.11 34.94 71.92 88.10
Conditional *
Procedural *
Planning *
Monitori *
Information *
Debugging
Knowledge Knowledge Management Strategy
ng
6.88*
7.07* 9.25* 7.91*
7.24*
Declarative 9.03* Evaluation
Knowledge
6.27*
25.12*
82.57 Metacognition 78.39*
*
DELTA1 DELTA8
2.10*
5.19*
Critical ZETA1
Thinking
1.00 0.67* 0.86* 0.74* 0.40*
Inference Recognition of Deduction Interpretation Evaluation of
Assumption Arguments
7.27* 2.06* 6.15* 5.03* 3.57*
EPSILON EPSILON EPSILON EPSILON EPSILON
1 2 3 4 5
2=1382, df=78, P<.05, RMSEA=.05 PGI=.95

8. Metacognition as an outcome
E
1
1.0
Self-efficacy
.17* .51* E
E
2 4
1.0 1.0
-.13* .30*
School Ability Deep Approach Metacognition
E
.14* 3 .28*
1.0
Surface Approach
• Magno, C. (2010). Investigating the Effect of School Ability on Self-
efficacy, Learning Approaches, and Metacognition. The Asia-Pacific
Education Researcher, 18(2), 233-244.

9. Metacognition
Other Models:
• Ridley, Schutz, Glanz, and Weinstein (1992)
recognize that metacognition is composed of
multiple skills.
• Ertmer and Newby (1996) specified that the
multiple components of metacognition are
characteristics of an expert learner.
• Hacker (1997) made three general categories of
metacognition: cognitive monitoring, cognitive
regulation, and combination of monitoring and
regulation.

10. Two components of
Metacognition
• Knowledge of cognition is the reflective aspect
of metacognition. It is the individuals‟ awareness
of their own knowledge, learning
preferences, styles, strengths, and limitations, as
well as their awareness of how to use this
knowledge that can determine how well they can
perform different tasks (de
Carvalho, Magno, Lajom, Bunagan, &
Regodon, 2005).
• Regulation of cognition on the other hand is
the control aspect of learning. It is the procedural
aspect of knowledge that allows effective linking
of actions needed to complete a given task

11. Components of Metacogniton
Knowledge of Cognition
• (1) Declarative knowledge – knowledge
about one‟s skills, intellectual resources, and
abilities as a learner.
• (2) Procedural knowledge – knowledge
about how to implement learning
procedures (strategies)
• (3) Conditional knowledge – knowledge
about when and why to use learning
procedures.

12. Examples of knowledge of cognition in
Mathematical Investigation
• Declarative Knowledge
– Knowing what is needed to be solved
– Understanding ones intellectual strengths and
weaknesses in solving math problems
• Procedural knowledge
– Awareness of what strategies to use when solving math
problems
– Have a specific purpose of each strategy to use
• Conditional knowledge
– Solve better if the case is relevant
– Use different learning strategies depending on
the type of problem

13. Components of Metacogniton
Regulation of cognition
1) Planning – planning, goal setting, and allocating
resources prior to learning.
(2) Information Management Strategies – skills and
strategy sequences used on- line to process
information more effectively (organizing, elaborating,
summarizing, selective focusing).
(3) Monitoring – Assessing one‟s learning or strategy
use.
(4) Debugging Strategies – strategies used to correct
comprehension and performance errors
(5) Evaluation of learning – analysis of performance
and strategy effectiveness after learning episodes.

14. Examples of regulation of cognition
• Planning
• Pacing oneself when solving in order to have enough time
• Thinking about what really needs to be solved before
beginning a task
• Information Management Strategies
• Focusing attention to important information
• Slowing down when important information is encountered
• Monitoring
• Considering alternatives to a problem before solving
• Pause regularly to check for comprehension
• Debugging Strategies
• Ask help form others when one doesn’t understand
• Stop and go over of it is not clear
• Evaluation of learning
• Recheck after solving
• Find easier ways to do things

15. Case Analysis
RJ whenever he is faced with mathematical
word problems makes himself relaxed. He
thinks of the steps on how to solve the
problem. He determines what is asked or
required, extracts the given, translates the
problem into an equation. He represents
the unknown into „X‟ or „?‟. He proceeds to
solve the problem. Checks his answer. He
reviews his answer by rereading the
problem and checking his computations.

16. Example
• Objective: Write verbal phrases using
algebraic symbols
• Reminder: It is very important to learn to
state problems correctly in algebra so that a
solution might be obtained (DK). Each
statement must be made in algebraic
symbols, and the meaning of each algebraic
symbol should be written out in full,
common language (CK).

17. • Follow these steps (PK):
• 1. Read the problem carefully. Look for
kewords and phrases.
• 2. Determine the unknown. If there is only
one unknown, represent it by a letter. If
there is more than one unknown, the letter
should represent the unknown quantity we
know least about. (CK)
• Determine the known facts related to the
unknown.

18. • Give students a list of keywords that they
can recognize in word problems
(information management)
• Provide exercise:
– Write an algebraic expression representing each
of the following phrases.
• Checking of answers (self-evaluation)
• Ask some students what item did they have
a mistake and what was the mistake.
(debugging)

19. Increasing Difficulty of Math
Problems
• Spiral Progression Curriculum
– Building n the schema of the learners
– Focusing in student mastery
– Assessing if students can work tasks from
simple to complex
– Test if the basic skills are met and readiness to
move on to the next level

20. Incremental
• Adding another skill in the next level
• Increasing values
Level 1: Adding two digits with 23
+ 4
one digit problems.
Level 2: Adding two digits with 25
+ 34
two digits problem (from 0 to 9)
Level 3: Adding two digits with 45
+ 87
two digits problem (with carrying)

21. Incremental
• Increasing operations
Level 1: One operation problem 21 – 20 =
Level 2: Two operations problem 21 – 20 +12 =
Level 3: Three operations problem 21 – 20 + 12 x 11 =

22. Reversibility
• Finding the unknown to complete the
equation
Level 1: Finding a one digit 23 55
+ ? - ?
missing addend or minuend. 27 53
Level 2: Finding two digits
missing addends and minuend.
Level 3: Finding the missing ?? ??
+ 34 - 11
additive or subtrahend. 48 88
Level 3: Finding the missing pair 4? ?6
+ ?7 -1?
of the given. 58 44

23. Combine problems
• A subset or a superset must be computed
given information about two other sets.

24. Combine problems

25. Change problems
• A starting set is changed by transferring
items in or out, and the number of starting
set, transfer set or the results set must be
computed given information about two of
the sets.

26. Change problems

27. Change problems

28. Compare problems
• The number of one set must be computed
by comparing the information given about
sets.

29. Compare problems

30. Compare Problems

31. Workshop
• Write 2 word problem items (Combine,
change, compare) with 2 levels of difficulty.
• Indicate in bullet points how will you use
metacognition to teach it. Label which
specific metacognitive strategies are used.

32. Example
• Compare (compared quantity unkown)
• Mary has 4 pens.
• Joseph has 8 more pens than Joe.
• How many pens does Joseph have?
• Compare (referent unknown)
• Sam has 5 books
• He has 4 books more than Brittney.
• How many books does Brittney have?

33. • Use real objects (Declarative)
• Derive the given (planning)
• Represent the unknown (Declarative)
• Derive the equation and solution
(procedural)
• Checking (Monitoring)