The Component-Attribute Approach

Birgit Mayer

5th April 2005

Overview

Component-attribute approach Basics

Problem construction from components

Single valued components

Product sets: components with multiple attributes

Derivation of surmise relations on the set of problems (constructed from components)

Albert & Held Method for establishing knowledge structures Problems are represented by and can be constructed from

components Components are related to the knowledge and skills required for

solving the problems and thus, allow for characterising the problems

By systematically constructing and ordering problems from components a surmise-relation on this set of problems can be established (and hence a knowledge structure) Different ordering rules

Component-Attribute Approach

Objectives

Systematical problem construction by means of components provides

facilitated problem comparison

precise description of problems and their possible problem variations

definition of underlying cognitive structures and knowledge structures

Problem Components

Characterising problems by components and attributes Problem analysis

Knowledge demanded for solving problems e.g. operations necessary for the correct solution e.g. subgoals during the solution process

Components Dimensions describing the problems

Attributes Different values for each dimension

Problem Components

Components may correspond to Specific contents and concepts

Single valued components

Components with multiple attributes A specific content comprises several concepts Concepts may be in relation by prerequisite dependencies

e.g. addition is a prerequisite of multiplication e.g. specific vs. general concepts (is part of, subordinated,..)

Single valued components The component is either present or not present

New problems are constructed by combining these single components

Set inclusion induces a surmise relation on the set of problem types resulting from combining the components

Set of Single Valued Components

Components a, b, c: no dependencies

C = {a, b, c} a: Multiplication b: Division c: Subtraction

7 Problem types: {{a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c}} Each of these subsets denote a problem type

e.g. {a}……5 x 3 {a, c}…..5 x 3 – 2 {c}……5 – 3 {a, b}…..5/9 x 17

a

b

c

Component structure

Example

{a}

{a, b} {a, c}

{a, b, c}

{b} {c}

{b, c}

Set inclusion

179

5

Example

Components a, b, c: linear order

C = {a, b, c} a: Addition within the 100s

b: Adding tens

c: Addition between 1 and 10

3 Problem types: {{c}, {b, c}, {a, b, c}}

a

b

c

Component structure

Example

( c )

( a , b )

( a , b, c )

( c )

( a , b )

( a , b, c )

625 + 347

37 + 42

5 + 8 ( c )

( a , b )

( a , b, c )

{ c }

{ b , c }

{ a , b, c }

625 + 347

37 + 42

5 + 8

Set inclusion

Example

Exercise 1A

4 Components: a, b, c, d Find all possible problem types that result from combining

a, b, c, d by taking into account the following component structure

6 Problem types: {{a}, {c}, {a, b}, {a, c}, {a, b, c}, {a, b, c, d}}

a

b

d

c

Component structure

Exercise 1B

4 Components: a, b, c, d Generate the surmise relation (Hasse Diagram) on the

problem types {{a}, {c}, {a, b}, {a, c}, {a, b, c}, {a, b, c, d}} Based on set inclusion

{a, b, c, d }

{a, b, c}

{a, c}

{a}

{a, b}

{c}

a

b

d

c

Component structure

Components With Multiple Attributes

Problems are represented as a combination of attribute values on components New Problems are constructed by combining the

attributes of the components e.g. by forming the Cartesian Product

Derivation of surmise relations 1. Based on relations defined on the attributes

e.g. linear order

2. Global ordering by employing different decision rules e.g. component-wise order (direct product) e.g. lexicographic order ...

Components With Multiple Attributes

Components A and B: linear orders

A: Set of numbers used in the calculation a1: Real numbers

a2: Integers

a3: Natural numbers

B: Applied operations b1: Multiplication

b2: Addition

Example

a1

a3

a2

b2

b1

x

Component structure

Possible combinations A x B: 6 problem types

{(a1, b1), (a1, b2), (a2, b1), (a2, b2), (a3, b1), (a3, b2)}

Establishing a surmise relation Component-wise

Problems have to be compared in pairs regarding the attribute orders

Example

Component-wise order

Problem types have to be compared regarding the relations defined on the attributes (a3, b1)

(a1, b2)

(a2, b2)

(a3, b2)

(a2, b1)

(a1, b1)(-5) x 2

8 + 17

(5,2) x 2,3

(a3, b1)

(a1, b2)

(a2, b2)

(a3, b2)

(a2, b1)

(a1, b1)

(a3, b1)

(a1, b2)

(a2, b2)

(a3, b2)

(a2, b1)

(a1, b1)(-5) x 2

8 + 17

(5,2) x 2,3

Example

Exercise 2A

Form the product of E x F of the following sets E = {e1, e2 , e3}

F = {f1, f2 }

6 Problem types {(e1, f1), (e1, f2), (e2, f1), (e2, f2), (e3, f1), (e3, f2)}

Exercise 2B

Generate the surmise relation for the product of the attribute sets of components E and F by considering the relations defined on the attributes Component-wise ordering

{e1, f1}

{e3, f1}{e2, f1} {e1 f2}

{e2, f2} {e3, f2}

e1

e3e2

x

f2

f1

Component structure

Lexicographic order Components are classified by their effects on the difficulty of the

problem Problems have to be compared in pairs like words in a dictionary

First paying attention to the most important component Sequence of attributes:

A more important than B (a1, b2) vs. vs.

B more important than A (b2, a1)

Previous example A: Set of numbers B: Applied operations

A is more important than B The numbers used in the calculation effect a problem‘s difficulty in a

stronger way than the operation that has to be applied

Components With Multiple Attributes

(a1, b1)

(a1, b2)

(a2, b1)

(a2, b2)

(a3, b1)

(a3, b2)

First elements identical, attribute relation of B determines problem order

First elements not identical, attribute relation of A determines problem order

Lexicographic order

A > B

a

a

1

a 3

2

b

b 1

2

x

Example

Exercise 3A

Construct the problem types that result from forming the product of A and B when B > A

B x A: 6 Problem types {(b1, a1), (b1, a2), (b1, a3), (b2, a1), (b2, a2), (b2, a3)}

Generate the surmise relation for B x A Lexicographic ordering rule B > A

(b1, a1)

(b1, a2)

(b1, a3)

(b2, a1)

(b2, a2)

(b2, a3)

(b1, a1)

(b1, a2)

(b1, a3)

(b2, a1)

(b2, a2)

(b2, a3)

Exercise 3B

Derivation of Attribute Orders

Held Cognitive skills are associated with the

components’ attributes Used for establishing attribute orders

Skills are identified by problem analysis [solution ways] Knowledge or operations necessary for mastering a

problem may serve as relevant elements of this problem

Skills

S1 Number understanding

S2 Understanding for the meaning of mathematical operation signs + and –

S3 Mathematical-symbolic representation

S4 Text comprehension

S5 Situational comprehension

S6 Mathematization/abstraction

S7 Mental representation of number sequence: forward

S8 Computation skill: addition

S9 Mental representation of number sequence: backward

S10 Computation skill: subtraction

S11 Understanding of concrete countable sets

S12 Understanding of relations between sets

Example

Example

3 Components

A: Presentation mode a1: Word problem a2: Numerical problem

B: Number concept b1: Relational each problem type is characterised by b2: Cardinal one property of A, B, C (an, bn, cn)

C: Mathematical operation c1: Subtraction c2: Addition

Examples of problem types a1b1c2: Anna has got 5 marbles.Tom has got 2 marbles more than

Anna. How many marbles has Tom got? a2b2c2: 5 + 4 = ?

Attribute orders Skills have to be assigned to the attributes Set inclusion induces relation on attributes

An attribute consisting of a subset of skills of another attribute is the easier one

Derivation of Attribute Orders

a1 {S1, S2, S3, S4, S5, S6}

a2 {S1, S2, S3}

b1 {S11, S12}

b2 {S11} c2 {S7, S8}

c1 {S7, S8, S9, S10}

x x

a1 {S1, S2, S3, S4, S5, S6}

a2 {S1, S2, S3}

b1 {S11, S12}

b2 {S11}

c1 {S7, S8, S9, S10}

c2 {S7, S8}

Example

Example

(a1, b2, c1)

(a1, b1, c1)

(a1, b1, c2) (a2, b1, c1)

(a2, b2, c1) (a1, b2, c2)

(a2, b2, c2)

(a2, b1, c2)

Thank you for your attention!!

References

Albert, D., & Held, T. (1994). Establishing knowledge spaces by systematical problem construction. In D. Albert (Ed.), Knowledge Structures (pp. 78–112). New York: Springer Verlag.

Albert, D., & Held, T. (1999). Component Based Knowledge Spaces in Problem Solving and Inductive Reasoning. In D. Albert & J. Lukas (Eds.), Knowledge Spaces: Theories, Empirical Research, and Applications (pp. 15–40). Mahwah, NJ: Lawrence Erlbaum Associates. 

Held, T. (1999). An Integrated Approach for Constructing, Coding, and Structuring a Body of Word Problems. In D. Albert & J. Lukas (Eds.), Knowledge Spaces: Theories, Empirical Research, and Applications (pp. 67–102). Mahwah, NJ: Lawrence Erlbaum Associates.