Metacognition and Math Education in Innovation-Driven Societies:

What’s New?

Zemira R. Mevarech

Bracha Kramarski

Bar-Ilan University, Israel

OECD, Paris, 2012

Three Warm-up Questions 1. Why teach mathematics in innovation-driven societies? The answers are self-evident: - To develop quantitatively literate citizens - To enhance students’ ability to solve problems - To encourage logical thinking 2. Does the standard school mathematics curriculum advance these goals? The answer is – Yes, to a partial extent: - basic skills are necessary, though not sufficient - It does not prepare students to solve complex, unfamiliar, non- routine problems - It is irrelevant for advancing math creativity, critical thinking, and communications - In no way does it train students to regulate problem-solving processes

3. What types of problem solving are useful for innovation-driven societies?

Problem Solving for Innovation-Driven Societies:

What Types of Problems are Useful?

• Standard, routine, textbook problems vs.

• Complex, Unfamiliar, Non-routine (CUN) problems

• Authentic problems

Large variability in CUN problems:

What is complex to one student, is simple to another, etc.

What Skills are useful in Innovation-Driven Societies?

• Mathematical problem solving • Mathematical reasoning • Mathematical creativity and critical thinking • Mathematical communications • Meta-cognitive skills for regulating the solution of

CUN problems

Two Examples: Which is the Cheapest Supermarket?

1. Before Christmas, several supermarkets advertised that they were the cheapest supermarket in town. • Your task is to decide which claim was correct. • Please give your reasoning and findings. • Please prepare a sixty-minute TV show to present your findings.

2. Before Christmas, two supermarkets advertised that they have sales. The prices in the two supermarkets were: Supermarket A – 1kg of meat for $10 and 1kg of turkey for $8. Supermarket B – 1kg of meat for $12 and 1kg of turkey for $6. The Vincent family decided to buy 3kg of meat and 2kg of turkey. • Which supermarket is cheaper?

Is the Supermarket Problem (#1) a CUN math problem?

• Is it authentic? • Is it a mathematical task even though there are no

numbers in the task? • Is it a complex task? • Is it an unfamiliar task? • Is it a non-routine task or is it based on ready made

algorithms? • Can it advance mathematical reasoning, creativity,

critical thinking, or communications? • How (if at all) can it create quantitative literate

citizens? The progress from traditional to CUN problems requires the

application of meta-cognitive processes that regulate cognitive processes

Meta-cognitive Processes for Regulating Cognition

• What is meta-cognition all about?

• The nature of “meta”

• The “meta-cognitive engine”

• Does MC develop naturally and without intervention?

• Teachers rarely emphasize the activation of MC: Why is that?

• Is meta-cognition teachable? How?

Meta-cognitive Instruction: When, How, and for Whom?

Research shows: •Like cognitive strategies, MC needs to be explicitly taught and intensively practice. •MC instruction must be embedded in subject content. •Learners must be informed of the usefulness of MC activities. •MC must be part of interactive learning environments, like: cooperative learning or ICT. •MC instruction is necessary at all grade levels: K-12, HE, adults. Veenman (2006); Mevarech and Kramarski (2012)

IMPROVE: MC Instructional Method Theoretical Basis

Feedback-correctives

Metacognitive Guidance

Cooperative Learning

IMP

RO

VE

IMPROVE Introduce new concepts to whole class

Meta-cognitive questioning practice in small groups

Practice using MCQ

Review use of MCQ

Obtain mastery over routine & CUN ps

Verification

Enrichment and remedial activities

IMPROVE:

MC Instructional Method

IMPROVE:

MC Self-Directed Questioning

Comprehension: What is the problem about?

Connection: How is the problem similar to, or different from

problems I have already solved? Please explain your reasoning.

Strategies: What kinds of strategies are appropriate for solving

the problem and why? Please explain your reasoning.

Reflection: Does the solution make sense? Can the problem

be solved in a different way? Am I stuck? Why?

Findings

IMPROVE Effects Over One Year Math Achievement & Reasoning

40

45

50

55

60

65

70

75

80

Pre-test Algebra Math Reasoning

IMPROVE

Control

IMPROVE & Long Lasting Effects: Immediate & Delayed Post-tests

Which PIZZA is the best offer? Why?

Price for

suppleme

nts (NIS)

Diameter Price

per

PIZZA

(NIS)

Type

of PIZZA

PIZZA BOOM

4.00 15 3.50 Personal

PIZZA

7.75 23 6.50 Small

11.00 30 9.50 Medium

14.45 38 12.50 Large

SUPPER PIZZA

9.95 30 8.65 Small

10.95 35 9.65 Medium

12.95 40 11.65 Large

MC PIZZA

1.00 25 6.95 Small

1.25 35 9.95 Large

IMPROVE for solving Authentic Tasks &

Transferring to Routine Tasks

0

10

20

30

40

50

60

70

0

10

20

30

40

50

60

70

80

Authentic total Routine tasks

Coop+IMP Coop

Online Mathematical Literacy Discourse Motivation and Attitudes

towards Problem Solving, Reasoning, Communication

0

10

20

30

40

50

60

70

80

90

100

• Motivation:

“Online learning aroused my interest

in mathematical problem solving”

• Reasoning:

“Online problem solving encouraged me

to explain my reasoning”

• Communication:

“ I look forward to my friends’ reactions

to my online solutions”

0 1 2 3 4 5 6

Online +IMP Online

Development of Scientific Literacy

by Group and Time

ALN 6.98 10.55

ALN+meta 6.97 11.47

before after

ALN+meta 5.47 10.36

F2F+meta 5.6 9.38

ALN 5.76 8.94

F2F 5.47 8.02

0.770206 0.580645161

מדידות מדידות תרשימיםתרשימים הבנת מידע הסקת מסקנותהסקת מסקנותהבנת מידע תכנון ניסוי

לפני אחרי לפני אחרי לפני אחרי לפני אחרי לפני

1 6.83 6.94 6.94 6.52 9.9 11.24 9.83 11.22 9.94

2 6.58 6.88 6.04 6.94 9.12 13.5 9.06 13.44 9.7

3 6.3 6.74 6.06 6.74 9.15 15.21 9.15 15.11 9.75

4 6 6.17 6.17 6.94 10.05 18.35 10.11 18.64 10.350

2

4

6

8

10

12

ALN+meta F2F+meta ALN F2F

mean

scores

before

after

ALN+Meta > F2F+Meta = ALN > F2F Range 0-15

Impact of IMPROVE at the College

56

58

60

62

64

66

68

70

72

74

76

Math

Knowledge

Math

Reasoning

IMPROVE

Control

3.5

3.6

3.7

3.8

3.9

4

4.1

Decla K Proc K Cond K

IMPROVE

Control

3.5

3.6

3.7

3.8

3.9

4

4.1

4.2

Plan Info Monitor Debug Eval

IMPROVE

Control

Research shows: • IMPROVE advances students’ CUN problem

solving without harming students’ abilities to solve “standard” problems.

• Positive effects were found for K-12, HE, and Professional Development, with or without ICT.

• Teaching strategies alone is not enough • IMPROVE is suitable for all students: both lower

and higher achievers. • IMPROVE showed similar positive effects in

science education. • IMPROVE helps to increase motivation, self-

confidence, judgment of learning.

Challenges: What next? Evidence-Based Policy Making

• International cooperation – don’t reinvent the wheel

• To be effective teach MC directly and practice it intensively – We know how to do this

• MC will be effective in the national curriculum

• Include CUN problems in textbooks, teacher guides, and professional development

• Pre- and in-service professional development followed by workshops and in-class mentoring

Challenges: What next?

Evidence-Based Policy Making (cont.)

• ICT: Students find it particularly difficult to apply MC in ICT environments. It is therefore essential to reconstruct these environments by embedding MC in them.

• Assessment and evaluation – “You teach what you assess”

• MC pedagogies in OECD countries

• Teaching for understanding can be achieved by implementing evidence-based MC pedagogies

If you think education is

expensive, try ignorance!

Please contact us:

mevarz@mail.biu.ac.il; Bracha.kramarski@biu.ac.il

Many Thanks!