Difficulty Analysis for Learners in Problem Solving Process based on the Knowledge Map

1. Difficulty Analysis for Learners in Problem Solving Process based on the Knowledge Map Rita Kuo, Wei-Peng Lien, Maiga Chang, Jia-Sheng Heh Speaker: Rita Kuo Multimedia Communication System Laboratory Dept. of Information and Computer Engineering Chung-Yuan Christian Univ., Chung-Li, 320, Taiwan

2. Outlines • Basic Definition • Basic Problem Model • Problem Construction Steps • Knowledge and Problem Structure • Problem Difficulty Analysis • Difficulty Features • Difficulty Dimensions • Demonstration of Item Generating System • Brief Conclusion To Develop an Item Generating System with Analyzing the Problem Model in the Inter-Correlated Knowledge

3. Basic Problem Model • Inter-correlated Knowledge • Manipulation of inter-correlation among knowledge • Ex. Physics, Chemistry, and Mathematics • Knowledge Object • Way to analyze Inter-correlated Knowledge • Basic concept of a specific domain • Basic Problem Definition • One Core Knowledge Object • Example in Physics: Physics Phenomenon • Basic Problem Model To Develop an Item Generating System with Analyzing the Problem Model in the Inter-Correlated Knowledge

4. Knowledge Base Knowledge Map Concept Selection Knowledge Map Unknown Designation Problem Matrix Proposition Construction Problem Construction To Develop an Item Generating System with Analyzing the Problem Model in the Inter-Correlated Knowledge

5. Knowledge Map • Concept Hierarchy • Hierarchical structure of concepts  Concept Map • Concept Schema • Attributes  Schema • Definitions, Ability (Can-Fly), Property (Age) etc. To Develop an Item Generating System with Analyzing the Problem Model in the Inter-Correlated Knowledge

6. Example of Knowledge Map To Develop an Item Generating System with Analyzing the Problem Model in the Inter-Correlated Knowledge

7. Problem Matrix • Manipulating concepts • Example in Physics: Physics Quantity • {“Displacement”, “Time”, “Velocity”, “Acceleration”} • Manipulating relations • Example in Physics: Physics Law • {“Displacement = 0.5 * Acceleration * Time ^ 2”,“Velocity = Acceleration * Time”} To Develop an Item Generating System with Analyzing the Problem Model in the Inter-Correlated Knowledge

8. Example of Problem Construction • Physics Phenomenon • Motion with constant velocity • Problem Matrix • tf: final time; si: initial position; ti: initial time; vi: initial velocity; sf: final position; d: distance To Develop an Item Generating System with Analyzing the Problem Model in the Inter-Correlated Knowledge

9. Unknown Designer • Select Physics Law • Set Attributes To Develop an Item Generating System with Analyzing the Problem Model in the Inter-Correlated Knowledge

10. Problem Constructor (cont.) To Develop an Item Generating System with Analyzing the Problem Model in the Inter-Correlated Knowledge

11. Difficulty Analysis • Difficulty Features According to the problem construction process • Selected Concepts • Knowledge Map • Attributes Setting • Problem Matrix • Difficulty Dimensions According to the problem solving steps • Problem Identification • Problem Elaboration • Problem Planning • Problem Execution To Develop an Item Generating System with Analyzing the Problem Model in the Inter-Correlated Knowledge

12. Difficulty Features To Develop an Item Generating System with Analyzing the Problem Model in the Inter-Correlated Knowledge

13. Basic Definition root: the root of Knowledge Map qi: ith concept Knowledge Map height(qi): the height of sub-tree from ith concept CS(qi): Concept Schema of ith concept size(CS(qi)): number of attributes in concept from full Concept Schema attrLS(CS(qi)): learning sequence attribute stored in Concept Schema Problem Matrix #given_attr: number of given attributes in the problem #unknown: number of unknown attributes in the problem attrmanip_cpt(qi): attributes extracted to manipulating concepts in Problem Matrix size(attrmanip_cpt(qi)): number of manipulating concepts in Problem Matrix attrmax_unknown(qi): maximum number of unknown in ith concept Attributes Definition To Develop an Item Generating System with Analyzing the Problem Model in the Inter-Correlated Knowledge

14. Example • In sky, raindrop proceeds the motion of constant acceleration. The initial velocity (vi) is 0 m/s. Acceleration (a) is 5 m/s^2. The final time (tf) is 5 s. Ask for the value of final velocity (vf). • Selected Concept • The motion of constant acceleration • Attributes Setting To Develop an Item Generating System with Analyzing the Problem Model in the Inter-Correlated Knowledge

15. Attributes Definition – Knowledge Map • size (CS (“Motion with Constant Acceleration”) ) = 11 • size (CS (“Free Fall”) ) = 13 • height (“Physics”) = 4 • height (“Motion with Constant Acceleration”) = 2 • attrLS(“Motion with Constant Acceleration”) = 2 To Develop an Item Generating System with Analyzing the Problem Model in the Inter-Correlated Knowledge

16. Attributes Definition – Problem Matrix • #given_attr = 3 • #unknown = 1 • size(attrmanip_cpt(qi)) = 7 • attrmax_unknown(qi) = 2 To Develop an Item Generating System with Analyzing the Problem Model in the Inter-Correlated Knowledge

17. Difficulty Features Calculation (1) • gneed_attr = size(CS(qi)) / arg maxjKMsize(CS(qj)) = size(CS(“Motion with Constant Acceleration”)) / size(CS(“Free Fall”)) = 11 / 13 = 0.8 To Develop an Item Generating System with Analyzing the Problem Model in the Inter-Correlated Knowledge

18. Difficulty Features Calculation (2) • glearn_seq = attrLS(qi) / arg maxjsibling(qi) attrLS(qj) = attrLS(“Motion with Constant Acceleration”) / attrLS(“Motion with Constant Acceleration”) = 2 / 2 = 1 To Develop an Item Generating System with Analyzing the Problem Model in the Inter-Correlated Knowledge

19. Difficulty Features Calculation (3) • gcpt_depth = (height(root) – height (qi)) / height(root) = (height(“Physics”) – height (“Motion with Constant Acceleration”)) / height(“Physics”) = ( 4 – 2 ) / 4 = 0.5 To Develop an Item Generating System with Analyzing the Problem Model in the Inter-Correlated Knowledge

20. Difficulty Features Calculation (4) • ggiven_attr = (size(attrmanip_cpt(qi)) - #given_attr + 1) / size(attrmanip_cpt(qi)) = (size(attrmanip_cpt(“Motion with Constant Acceleration”)) - #given_attr + 1) / size(attrmanip_cpt(“Motion with Constant Acceleration”)) = (7 – 3 + 1) / 7 = 0.7 To Develop an Item Generating System with Analyzing the Problem Model in the Inter-Correlated Knowledge

21. Difficulty Features Calculation (5) • gunknown = #unknown / attrmax_unknown(qi) = #unknown / attrmax_unknown(“Motion with Constant Acceleration”) = 1 / 2 = 0.5 To Develop an Item Generating System with Analyzing the Problem Model in the Inter-Correlated Knowledge

22. Difficulty Features Calculation (6) • gelb_attr= (size(attrmanip_cpt(qi)) - #given_attr - #unknown + 1) / size(attrmanip_cpt(qi)) = (size(attrmanip_cpt(“Motion with Constant Acceleration”)) - #given_attr - #unknown + 1) / size(attrmanip_cpt(“Motion with Constant Acceleration”)) = ( 7 – 3 – 1 + 1) / 7 = 0.4 To Develop an Item Generating System with Analyzing the Problem Model in the Inter-Correlated Knowledge

23. Difficulty Dimensions • According to four steps of problem solving • Identification Difficulty (gidf) • Elaboration Difficulty (gelb) • Planning Difficulty (gpln) • Execution Difficulty (gexc) To Develop an Item Generating System with Analyzing the Problem Model in the Inter-Correlated Knowledge

24. Difficulty Dimensions (cont.) • Identification Difficulty (gidf) • Position in Knowledge Map • Elaboration Difficulty (gelb) • Size of Concept Schema in Knowledge Map • Planning Difficulty (gpln) • Number of Manipulating Concepts • Execution Difficulty (gexc) • Number of Unknowns To Develop an Item Generating System with Analyzing the Problem Model in the Inter-Correlated Knowledge

25. Difficulty Dimension Transformation • gidf = w21 * glearn_seq + w31 * gcpt_depth= 0.5 * glearn_seq + 0.5 * gcpt_depth = 0.5 * 1 + 0.5 * 0.5 = 0.75 • gelb = w12 * gneed_attr + w42 * ggiven_attr + w62 * gelb_attr=0.3 * gneed_attr + 0.3 * ggiven_attr + 0.3 * gelb_attr = 0.3 * 0.8 + 0.3 * 0.7 + 0.4 * 0.4 = 0.61 • gpln= w53 * gunknown + w63 * gelb_attr = 0.5 * gunknown + 0.5 * gelb_attr = 0.5 * 0.5 + 0.5 * 0.4 = 0.45 • gexc= w54 * gunknown = 1 * gunknown = 1 * 0.5 = 0.5 To Develop an Item Generating System with Analyzing the Problem Model in the Inter-Correlated Knowledge

26. Demonstration To Develop an Item Generating System with Analyzing the Problem Model in the Inter-Correlated Knowledge

27. Brief Conclusions • Problem Construction Steps • Concept Selection • Unknown Designation • Proposition Construction • Problem Difficulties • Difficulty Features • Difficulty Dimensions • Item Generating System • Physics Domain To Develop an Item Generating System with Analyzing the Problem Model in the Inter-Correlated Knowledge

28. Future Works • Complex problem construction • Mathematical complexity analysis • Answer of the learners diagnosis To Develop an Item Generating System with Analyzing the Problem Model in the Inter-Correlated Knowledge

29. Thank You

30. To Develop an Item Generating System with Analyzing the Problem Model in the Inter-Correlated Knowledge

31. To Develop an Item Generating System with Analyzing the Problem Model in the Inter-Correlated Knowledge

32. Presentations of Knowledge • Concept • Basic element of knowledge • A atomic unit of knowledge pieces • Ex. “Free Fall”, “Constant Acceleration Motion”etc. • Relation • The aggregation of concepts • Ex. Is-A, Has-Aetc. • Proposition • An integration of concepts and relation • One relation with at least two concepts • Ex. “Free Fall”Is-A“Constant Acceleration Motion” To Develop an Item Generating System with Analyzing the Problem Model in the Inter-Correlated Knowledge

33. Inter-correlated Knowledge • Inter-correlated Knowledge • Manipulation of inter-correlation among knowledge • Ex. Physics, Chemistry, and Mathematics • Knowledge Object • Way to Analysis Inter-correlated Knowledge • Basic concept of a specific domain • Relations between Knowledge Objects • Ten types in The Frame Game [Clifford, 1981] To Develop an Item Generating System with Analyzing the Problem Model in the Inter-Correlated Knowledge

34. A graph composed with proposition Relation among concepts in the problem Example Concept Set{"Object", "Free Fall", Displacement", "Time","19.6", "unknown"} PropositionsObject <proceeds> Free Fall,Object <has> Displacement,Object <has> Time, Displacement <is> 19.6, Time <is> Unknown An object proceeds Free Fall. After the phenomenon, the displacement of the object is 19.6. Ask for the procedure time of the object. Problem Structure 1: Problem Graph To Develop an Item Generating System with Analyzing the Problem Model in the Inter-Correlated Knowledge

35. Problem Construction • Basic Problem Model • Problem Construction Steps • Essential Data Structures To Develop an Item Generating System with Analyzing the Problem Model in the Inter-Correlated Knowledge

36. Problem Matrix for Item Generation • Extract manipulating concepts and attributes • Set Given and Unknown Attributes To Develop an Item Generating System with Analyzing the Problem Model in the Inter-Correlated Knowledge

37. Problem Template • Problem Format • Description Sequence • Possible Sentences To Develop an Item Generating System with Analyzing the Problem Model in the Inter-Correlated Knowledge

38. Proposition Table • Possible Sentences in problems • Corresponding syntax To Develop an Item Generating System with Analyzing the Problem Model in the Inter-Correlated Knowledge

39. Item Generating System To Develop an Item Generating System with Analyzing the Problem Model in the Inter-Correlated Knowledge

40. Problem Constructor • Finding suitable Problem Template • Finding related proposition To Develop an Item Generating System with Analyzing the Problem Model in the Inter-Correlated Knowledge

41. Item Generating System To Develop an Item Generating System with Analyzing the Problem Model in the Inter-Correlated Knowledge

42. Conception Selection To Develop an Item Generating System with Analyzing the Problem Model in the Inter-Correlated Knowledge

43. Problem Matrix Construction To Develop an Item Generating System with Analyzing the Problem Model in the Inter-Correlated Knowledge

44. Physics Law Selection To Develop an Item Generating System with Analyzing the Problem Model in the Inter-Correlated Knowledge

45. Physics Quantity Setting / Item Construction To Develop an Item Generating System with Analyzing the Problem Model in the Inter-Correlated Knowledge

46. Knowledge Categories • Grammatical Knowledge • usually has syntax, which has rules and formats in it. • Ex. Courses of Chinese, English, Music, and Art • Positioning Knowledge • lays stress on the location and direction among objects. • Ex. History and geography • Inter-Correlated Knowledge • attaches more importance to the manipulation of inter-correlation among knowledge. • Ex. Physics, Chemistry, and Mathematics To Develop an Item Generating System with Analyzing the Problem Model in the Inter-Correlated Knowledge

47. Complex Problem Model To Develop an Item Generating System with Analyzing the Problem Model in the Inter-Correlated Knowledge