Using Metacognition in Mathematical Modelling and Investigation Carlo MagnoProfessor of Educational Psychology De La Salle University-Manila
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.
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 ofmetacognition?• 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)
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.
MetacognitionOther 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 ofMetacognition• 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
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 inMathematical 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 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.
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 MathProblems• 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 23 + 4one digit problems.Level 2: Adding two digits with 25 + 34two digits problem (from 0 to 9)Level 3: Adding two digits with 45 + 87two digits problem (with carrying)
Incremental • Increasing operationsLevel 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 23 55 + ? - ?missing addend or minuend. 27 53Level 2: Finding two digitsmissing addends and minuend.Level 3: Finding the missing ?? ?? + 34 - 11additive or subtrahend. 48 88Level 3: Finding the missing pair 4? ?6 + ?7 -1?of the given. 58 44
Combine problems• A subset or a superset must be computed given information about two other sets.
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.
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)