1. The Component-Attribute Approach Birgit Mayer 5 th April 2005

2. 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)

3.  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

4. 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

5. 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

6. 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,..)

7.  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

8.  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 Example

9. {a}{a} {a, b} {a, c} {a, b, c} {b}{b}{c}{c} {b, c} Set inclusion Example

10.  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}} Component structure Example

11. ( c ) ( a, b ) ( a, b, c ) ( c ) ( a, b ) ( a, ) 625 + 347 37 + 42 5 + 8 ( c ) ( a, b ) ( a, b, c ) { c } { b, c } { a, } 625 + 347 37 + 42 5 + 8 Set inclusion Example

12. 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

13. 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} {a, b} {c}{c} a b d c Component structure

14. 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

15.  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

16.  Components A and B: linear orders  A: Set of numbers used in the calculation  a 1 : Real numbers  a 2 : Integers  a 3 : Natural numbers  B: Applied operations  b 1 : Multiplication  b 2 : Addition Example a 1 a 3 a 2 b 2 b 1 x Component structure

17.  Possible combinations  A x B: 6 problem types  {(a 1, b 1 ), (a 1, b 2 ), (a 2, b 1 ), (a 2, b 2 ), (a 3, b 1 ), (a 3, b 2 )}  Establishing a surmise relation  Component-wise  Problems have to be compared in pairs regarding the attribute orders Example

18. Component-wise order Problem types have to be compared regarding the relations defined on the attributes Example

19. Exercise 2A  Form the product of E x F of the following sets  E = {e 1, e 2, e 3 }  F = {f 1, f 2 }  6 Problem types  {(e 1, f 1 ), (e 1, f 2 ), (e 2, f 1 ), (e 2, f 2 ), (e 3, f 1 ), (e 3, f 2 )}

20. 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 {e 1, f 1 } {e 3, f 1 }{e 2, f 1 } {e 1 f 2 } {e 2, f 2 }{e 3, f 2 } e1e1 e3e3 e2e2 x f2f2 f1f1 Component structure

21.  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: vs.  A more important than B (a 1, b 2 ) vs.  B more important than A (b 2, a 1 )  Previous example  A: Set of numbersB: 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

22. 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 Example

23. 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  {(b 1, a 1 ), (b 1, a 2 ), (b 1, a 3 ), (b 2, a 1 ), (b 2, a 2 ), (b 2, a 3 )}

24.  Generate the surmise relation for B x A  Lexicographic ordering rule  B > A Exercise 3B

25. 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

26. Skills S1S1 Number understanding S2S2 Understanding for the meaning of mathematical operation signs + and – S3S3 Mathematical-symbolic representation S4S4 Text comprehension S5S5 Situational comprehension S6S6 Mathematization/abstraction S7S7 Mental representation of number sequence: forward S8S8 Computation skill: addition S9S9 Mental representation of number sequence: backward S 10 Computation skill: subtraction S 11 Understanding of concrete countable sets S 12 Understanding of relations between sets Example

27.  3 Components  A: Presentation mode  a 1 : Word problem  a 2 : Numerical problem  B: Number concept  b 1 : Relationaleach problem type is characterised by  b 2 : Cardinalone property of A, B, C (a n, b n, c n )  C: Mathematical operation  c 1 : Subtraction  c 2 : Addition  Examples of problem types  a 1 b 1 c 2 : Anna has got 5 marbles.Tom has got 2 marbles more than Anna. How many marbles has Tom got?  a 2 b 2 c 2 : 5 + 4 = ?

28.  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

29. a 1 {S 1, S 2, S 3, S 4, S 5, S 6 } a 2 {S 1, S 2, S 3 } b 1 {S 11, S 12 } b 2 {S 11 } c 2 {S 7, S 8 } c 1 {S 7, S 8, S 9, S 10 } xx a1a1 {S 1, S 2, S 3, S 4, S 5, S 6 } a2a2 {S 1, S 2, S 3 } b1b1 {S 11, S 12 } b2b2 {S 11 } c1c1 {S 7, S 8, S 9, S 10 } c2c2 {S 7, S 8 } Example

31. Thank you for your attention!!

32. 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.