Using Metacognition

in Mathematical Modelling and Investigation

Carlo MagnoProfessor of Educational

PsychologyDe La Salle University-Manila

Case analysisJane 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.

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.

Objectives

• Uncover the definition of metacognition

• Indentify specific metacognitive processes

• Use metacognition strategies to teach mathematical investigation

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

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)

Model2: Effect of Metacognition (8 factors) on Critical Thinking

7.24*7.91*

6.88*9.25*7.07*

9.03*

6.27*

82.57*

34.94*

0.40*0.74*0.86*0.67*

3.57*

EPSILON5

5.03*6.15*2.06*7.27*

5.19*

57.11*

2.10*

Planning

Metacognition

CriticalThinking

Deduction Interpretation Evaluation of Arguments

Inference Recognition of Assumption

DELTA5DELTA4

ZETA1

EPSILON1

EPSILON2

EPSILON3

EPSILON4

Monitoring

Information Management

Procedural Knowledge

Conditional Knowledge

71.46*

DELTA3

100.43*

DELTA2

88.10*

DELTA7

Declarative Knowledge

Debugging Strategy

Evaluation

71.92*

DELTA6

DELTA1

78.39*

DELTA8

25.12*

2=1382, df=78, P<.05, RMSEA=.05 PGI=.95

1.00

Metacognition as an outcome

• 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.

1.0

1.0

1.0

.30*

.14*

.17* .51*

Self-efficacy

Surface Approach

Deep Approach Metacognition

.28*

1.0

E4

School Ability-.13*

E1

E2

E3

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.

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 (Carvalho & Yuzawa, 2001).

Components of MetacognitonKnowledge 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.

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

Components of MetacognitonRegulation of cognition1) 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.

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

Case AnalysisRJ 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.

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).

• 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.

• 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)

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

Incremental

• Adding another skill in the next level

• Increasing valuesLevel 1: Adding two digits with one digit problems.

23+ 4

Level 2: Adding two digits with two digits problem (from 0 to 9)

25+ 34

Level 3: Adding two digits with two digits problem (with carrying)

45+ 87

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 =

Reversibility• Finding the unknown to

complete the equationLevel 1: Finding a one digit missing addend or minuend. Level 2: Finding two digits missing addends and minuend.

23 55+ ? - ? 27 53

Level 3: Finding the missing additive or subtrahend.

?? ??+ 34 - 11 48 88

Level 3: Finding the missing pair of the given.

4? ?6+ ?7 -1? 58 44

Combine problems

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

Combine problems

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.

Change problems

Change problems

Compare problems

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

Compare problems

Compare Problems

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.

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?

• Use real objects (Declarative)• Derive the given (planning)• Represent the unknown

(Declarative)• Derive the equation and solution

(procedural)• Checking (Monitoring)