1. LQ: what were the results and examples dealing with functions f:IR É D® IR^m dealt with in the last lecture?  LT: objectives ...  
2. LQ: do you want examples to refresh results of the last lecture?  SA: yes  LT: let us write a function f:IR® IR^m as usual as [p\vec]:IR® IR^m, e.g. [p\vec](t)=[p\vec]_o+t([p\vec]₁-[p\vec]_o). 
3. LQ: what kind of object is {[p\vec](t): t Î IR} ?  SA: line through [p\vec]_o and [p\vec]₁ 
4. LQ: what is [p\vec]₁-[p\vec]_o ?  SA: direction vector  LT: same not only for lines in IR³ ...  
5. LQ: more examples?  SA: no  LT: let us deal with n=2, m=1 
6. LQ: how do you call z=f(x,y) ?  TA: scalar field  LT: recall scalar fields like f(x,y)=x+y or f(x,y)=x y. Let us consider another example, say z=f(x,y)=sinÖ{x²+y²} 
7. LQ: how is the graph of some f:IR² É D® IR defined? and what kind of object?  SA: graph(f)={(x,y,f(x,y)): (x,y) Î D} is a surface in IR³ 
8. LQ: how does one visualize the graph of z=f(x,y)=sinÖ{x²+y²} ?  TA: sections, perspective visualizations  LT: to get some idea, note that Ö{x²+y²} is inserted into the sine function. 
9. LQ: how do the level curves to some level z_o look like?  TA: circles 
10. LQ: which (x,y) do solve z_o=f(x,y)=sinÖ{x²+y²} ?  SA: arcsinz_o=Ö{x²+y²}  LT: E.g., let z_o=[ 1/2] 
11. LQ: what is arcsin[ 1/2] ?  LT: think of memory hook sinj_i=[(Öi)/2] for j_i Î {0,[(p)/6],[(p)/4],[(p)/3], [(p)/2]}   SA: arcsin[ 1/2] = [(p)/6] 
12. LQ: which (x,y) do solve ([(p)/6])²=x²+y² ?  LT: think of Pythagoras or rewrite |(x,y)|=|(x,y)-[0\vec]|=[(p)/6] 
13. LQ: wierd question?  SA: yes  LT: Recall vector length and distance. Consider the intersections with coordinate planes. [(p)/6], do call it r ...   SA: circles 
14. LQ: how do partial functions z=f(x,0) and z=f(0,y) look like?  SA: sine functions 
15. LQ: what kind of intersections do graphs of partial functions represent?  LT: ...   SA: intersections with coordinate planes 
16. LQ: how does z=f(x,x) look like? What kind of intersection?  TA: graph with plane, i.e. graph(f)Ç{x=y} 
17. LQ: what function of one variable is z=f(x,x) ?  SA: again a sine function 
18. LQ: what kind of intersection?  LT: ...  
19. LQ: what is the equation of the intersecting plane?  SA: x=y 
20. LQ: why is this a plane?  TA: x-y+0·z=0  LT: ...  
21. LQ: where is this plane in space?  SA: The plane contains the z-axis  LT: ...  
22. LQ: how does the intersection of x=y with the xy-plane look like?  SA: the diagonal 
23. LQ: visualize graph(f)  LT: ... think of a snapshot of a concentric water wave ...  
24. LQ: how can we model friction?  LT: ... the amplitude should be indirect proportional to the distance to the origin. Recall ordinary differential equations with characteristic polynomial with two pairwise conjugate complex zeroes ...  
25. LQ: recall: how do damped oscillations look like?  LT: ...  
26. LQ: in e^{-¼? ¼}sin(wt) fill in the exponent  SA: e^-t/tsin(wt) 
27. LQ: how can we tranfer this to our situation where damping is proportional to the distance to the origin?  LT: ...   SA: e^-Ö{x2+y2}sinÖ{x²+y²} 
28. LQ: how can we introduce time into our function?  LT: ...   LT: consider the case n=3, m=1, for example z=f(x,y,t)=A_ocos2p(nt-Ö{x²+y²}/l) ...  
29. LQ: visualize f  LT: ...   LQ: Visualize f for a fixed t  LT: ...  
30. LQ: what is the level curve to a level of [(A_o)/2] and fixed t ?  SA: {(x,y,t_o):A_ocos2p(nt-Ö{x²+y²}/l)=[(A_o)/2]} 
31. LQ: solve for x and y  SA: 2p(nt-Ö{x²+y²}/l)=[(p)/3]  LT: you can do better ...   SA: nt-Ö{x²+y²}/l = [ 1/6]  LT: you can do better ...   SA: nt-[ 1/6] = Ö{x²+y²}/l  LT: you can do better ...   SA: x²+y²=l²(nt-[ 1/6])² 
32. LQ: what are these level curves geometrically?  SA: again circles  LT: ...   LT: consider the case n=2, m=2, i.e. a function [f\vec]:IR² É D® W Ì IR² 
33. LQ: how is [f\vec] called?  SA: vector field  LT: here [f\vec](x,y)=(u(x,y),v(x,y)) 
34. LQ: what are u(x,y) and v(x,y) ?  SA: scalar fields  LT: vector fields are vectors of scalar fields, e.g. matrix transforms like
    

    æ ç è x y ö ÷ ø ® æ ç è cosd -sind sind cosd ö ÷ ø æ ç è x y ö ÷ ø

    ...  
35. LQ: evaluate the product  SA: (xcosd-ysind,xsind+ycosd) 
36. LQ: what is the relation between the argument vector and the function value, the product?  LT: identify, remember C , recall Euler, remember addition theorems of sine and cosine ...  
37. LQ: what is the polar representation of (x,y) ?  SA: r(cosj,sinj) 
38. LQ: how are r and j determined?  SA: r=Ö{x²+y²} and j = arctan[ y/x] 
39. LQ: what do you get inserting the polar representation of the argument into the product?  SA: r(cosjcosd-sinjsind, cosjsind+sinjcosd) 
40. LQ: what do you get applying the addition theorems?  SA: r(cos(j+d),sin(j+d)) 
41. LQ: what happened to the argument vector geometrically?  SA: rotated by d  LT: There are important matrix transformations of the plane, e.g. scaling, shearing and even - in homogeneous coordinates - translation ... Consider in the case n=2, m=3 the vector fields [f\vec](x,y)=(u(x,y),v(x,y),w(x,y)):IR² É D® W Ì IR³ ...  
42. LQ: what is the dimension of D ?  SA: two-dimensional 
43. LQ: what is - in general - the dimension of W ?  TA: two-dimensional 
44. LQ: what do we get in W for some fixed y_o ? What is {[f\vec](x,y_o): (x,y_o) Î D} ?  TA: a curve in IR³, co-domain of some partial function  LT: ...  
45. LQ: what is the dimension of {[f\vec](x,y_o): (x,y_o) Î D} ?  SA: one-dimensional 
46. LQ: what is the dimension of W ?  SA: two-dimensional  LT: The co-domain of [f\vec](x,y):IR² É D® W Ì IR³ are surfaces in space in parameter representation ... For example, consider three vectors [p\vec]_o, [r\vec]₁, [r\vec]₂ corresponding to some plane in space 
47. LQ: what role do these three vectors play?  SA: plane through [p\vec]_o with direction vectors [r\vec]₁ and [r\vec]₂ 
48. LQ: how do we get each point of the plane?  TA: [p\vec]_o plus a linear combination of [r\vec]₁ and [r\vec]₂  LT: recall that planes in space as well as lines in the plane are hyperplanes ...  
49. LQ: what are the points of the line in IR² or IR³ through [p\vec]_o with direction vector [r\vec]₁ ?  SA: [r\vec](t)=[p\vec]_o+t[r\vec]₁ 
50. LQ: what are the points of the plane in IR³ through [p\vec]_o with direction vectors [r\vec]₁ and [r\vec]₂ ?  SA: [r\vec](s,t)=[p\vec]_o+s[r\vec]₁+t[r\vec]₂ 
51. LQ: what are the points of the plane in IR³ through [p\vec]_o, [p\vec]₁ and [p\vec]₂ ?  SA: [r\vec](s,t)=[p\vec]_o+s([p\vec]₁-[p\vec]_o)+t([p\vec]₂-[p\vec]_o) 
52. LQ: how to model the intersecting plane x=y from above?  ... SA: [r\vec](s,t)=[0\vec]+s([p\vec]₁-[0\vec])+t([p\vec]₂-[0\vec]) with [p\vec]₁=(1,1,0) and [p\vec]₂=(1,1,1)  LT: not unique, not collinear, ...  
53. LQ: why do [r\vec](s,t)=[0\vec]+s[p\vec]₁+t[p\vec]₂ and e.g. [r\vec]¢(s¢,t¢)=[0\vec]+s¢[p\vec]₁+t¢[e\vec]_z represent the same plane?  LT: ...   LQ: how to verify my claim? 
54. LQ: difficult?  SA: yes  LQ: at home or right now here  SA: here right now  LT: ...   LT: let us summarize what we have learnt today ... in the next lecture we will consider ...  

2.3  Types of Questions

Model, Construct

here e.g. model friction/damping, model free form 3D objects
modelling of course is the essential part in solving any problem (of problem based learning) - especially when application oriented problems posed in the real life, non-mathematical manner are to be attacked. And, geometric models are constructed as well as algorithms, bases, estimators, measures, etc.

Classify

here e.g. level curves
but also e.g. classification of differential equations to determine the method by which to solve the differential equation

Evaluate, Solve, Apply

here e.g. arcsin[ 1/2], addition theorems, Pythagoras
but also e.g. solutions of systems of linear equations, minima/maxima, integrals, zeroes of characteristic polynomials, Laplace transforms etc. - in the end, any mathematical procedure or algorithm

Identify, Recognize

here e.g. plane or space curves in parameter representation, scaling transformation
but also e.g. the other matrix transformations, time and space requirements of algorithms etc.

Typify, Find examples

here both lessons were dedicated to illustrate scalar and vector functions of one or several variables by pertinent examples
but also e.g. to get a first idea of how an algorithm works, to illustrate theorems, to find counter-examples etc.

Generalize

here e.g. visualisation of vector fields by visualisation of each of its scalar fields
but also e.g. criteria for continuity or the like hold for real valued as well as complex valued functions etc.

Recall

here e.g. polar representation,
but also e.g. recalling vector rotation in the plane is essential to understand and implement CORDIC algorithms - precognition of the basics is a prerequisite to learn any mathematical procedure or algorithm.

Verify, Falsify

here e.g. check whether or not the points of the parameter representation of the sphere [r\vec](j,y) = r(cosycosj, cosysinj, siny) lie on the sphere
but also e.g. verification whether solutions do solve the given problem or any consideration of plausibility

Visualize

here e.g. any scalar field or a coordinate of any vector field, 3D model
but also e.g. visualization of any solution, algorithm etc.

3  Mapping Questions to Tests

3.1  Inherent Limitations

The set of Basic Item Types 'without any clues' consists of STR1, STR2, STR3, NUM1, and NUM2. All other Basic Item Types present possible solutions and hence clues. Here, the learner is able to identify the correct answer by elimination.

Other Basic Item Types may be extended to types 'without clues', e.g.

• LID8, i.e. Multiple Choice (slider-based options) –- slider-based rendering
  may be extended to real valued sliders: for example, to approximately set Ö2 on a slider with ticks 0, 1, 2 and 3 or to set the most frequent color in a false color image on a slider with the range of colors.

• STR1, i.e. Standard Single Fill-in-Blank – FIB-based rendering
  may be extended so that formulae can be entered by using the input notations of e.g. T_EX, Mathematica, Maple, mathML, OpenMath [1] etc. Then questions like Give a definition of p. can be answered by 4 arctan1 or 2arcsin1 or the like.
  Even without systems powerful as computer algebra systems JavaScript functions embedded into pdf documents allow not only to implement all sorts of numerical algorithms (www.weblearn.hs-bremen.de/risse/MAI/docs/numerics.pdf) but also to check the correctness of symbolic manipulations of algebraic expressions (www.weblearn.hs-bremen.de/risse/MAI/docs/vorkurs.pdf)

3.2  'Limited' Mapping

Model, Construct

Questions 24, 27, 28, 52, and in the second lecture: 3D modeling
It is obvious that the modeling process cannot be mapped to IMS Basic Item Types. The modeling result may be mapped to extended STR1 if it can be represented as e.g. set of differential equations. If the model consisted e.g. of a labeled graph the input of became even more clumsy so that LID3 represents a last resort with clues. Otherwise a graphical input had to be processed, an input graph recognized, and the isomorphy between correct graph and input graph checked.

Classify

Questions 3, 6 - 8, 33, 34
All questions of the type "What kind of function is a given one?", "What class of differential equation is a given one?" etc. are mapped to LID1 or - if one wants not to give clues - STR1. The same holds for the variant with several correct answers, e.g. "How to visualize vector fields?".

Evaluate, Solve, Apply

Questions 10, 11, 12, 19, 30, 31, 35, 39, 40, 52
As in the case of modeling, all numerical results or algebraic formulae can be mapped to extended STR1. All other results again pose the same problems, e.g. Äpply a spanning tree algorithm"

Identify, Recognize

Questions 4, 7, 15 - 18, 21, 22, 32, 36, 41 - 47, 53
If it is accepted that the recipe for success in answering these questions is training then the implementation of such question/answer pairs in e-learning applications is rather important. Fortunately, mapping to LID or STR is adequate as long as STR takes synonyms into consideration which may be preventive since e.g. a plane can be verbally described in very many ways.

Typify, Find examples

Questions 24, 29, and in the second lecture: find examples for free form surfaces
The idea to let the students come up with any free form surfaces they might think of cannot be mapped to Basic Item Types. Again, LID or STR may act as a makeshift where STR prevents automatic marking.
To separate degenerated cases from non-pathological ones input of certain model parameters may be sufficient. This can be mapped to STR or NUM Basic Item Types.

Generalize, Transfer

Questions 27, and in the second lecture: generalize e.g. Bézier-curves in the plane to Bézier-surfaces in space
Generalization and transfer save work, generate ideas and insight. However, both are based on arguments which are hard to map to Basic Item Types, e.g. real valued functions exhibit properties which complex valued functions exhibit as well, e.g.
From     ln(a b)=lna+lnb     we derive     ln1=0 and ln(a/b)=lna-lnb
can be transfered to
From     (f g)¢=f¢ g+f g¢     we derive     (f/g)¢=(f¢ g-f g¢)/g².
Again, the result can, but the process of generalization and transfer cannot be mapped to Basic Item Types.

Recall

Questions 1, 6, 7, 20, 25, 26, 33, 36 - 38, 47 - 51
Without recall of 'dormant' knowledge there is no ascent, and no consolidation and no synthesis of old and new knowledge! Especially e-learning application can be designed to stress recall wherever needed individually. Depending on the nature of the recall the remarks of this list apply.

Verify, Falsify

Questions 7, 20
Verification checks whether or not some solution solves a given problem, i.e. whether or not a solution solves e.g. a set of linear equations, a set of differential equations, whether or not an interpolating function passes through the given points etc. Falsification invalidates some hypothesis. Depending on the input, verification and falsification are mapped to STR1 or only extended STR1.
No computer algebra systems can check whether or not an input definition is correct. Disregarding the notational problem, the question to give some definition can be mapped only to STR without automatic marking.

Visualize

Questions 8, 9, 14, 16, 22, 23, 29, and in the second lecture: certain sections of the shere, certain triangles in the shere etc.
The only candidate is LID3 with clues. In general, processing any graphical input in order to automatically check correctness seems not feasible!

3.3  Dealing with the Limitations of Basic Item Types

The ability to graphically thumbsketch say function graphs, 2D and 3D objects - even paradox ones, that is any graphical representation is essential in order to communicate ideas, designs, etc. Thinking of thumbsketches of surfaces in space is seems not feasible to test such graphical inputs on 'correctness'. Hence this example circumstantiates a severe limitation of tests in e-learning application.

In addition, it became apparent that integration of visualization, animation or simulation systems is highly desirable to depict models, algorithms, procedures in action. To some extend this is achieved by interactive documents, i.e. documents which compute [8], allowing to study diverse algorithms in action, to compare precision and performance, to identify advantages and disadvantages etc., e.g. in numerical mathematics, www.weblearn.hs-bremen.de/risse/MAI/docs/numerics.pdf or in cryptography, coding and probability, www.weblearn.hs-bremen.de/risse/MAI/docs/epuzzles.pdf

The integration of virtual experiments is near at hand in science, i.e. physics, biology, medicine or computer architecture.
How processors work and how to design (pipeline) processors is demonstrated by stepwise refinement [7]. Extending processors to process new instructions leads to a deeper understanding of the design process, of performance and of verification issues. Processor emulators, like e.g. SPIM or WinDLX, s. www.weblearn.hs-bremen.de/risse/RST, ostensively demonstrate how the ensemble of control, functional units, and register file processes instructions (even so, in the Knowledge Factory for Computing Systems [6] simulations are reserved to to complement lectures and exercises). Therefore, it is near at hand to ask for integration of a working, extendable emulator into a learning unit for computer architecture. Then the students can estimate cost to benefit ratios of modifications of the hardware/software interface of a processor.

Another reaction on the limitation was the inventive and creative employment of Basic Item Types in e-learning applications. For example, in Mathe Online [2] there are besides LID, GRP (called puzzles or association) also two other forms of interactive tests, namely reading off coordinates (twice NUM1 or something like the inverse of LID6) or a cross word puzzle - only in the german version - training to locate and to position objects in coordinate systems (similar to LID7). By the way, questions of type "where is the mistake made?" - again only in the german version - are implemented by LID2 in Mathe Online.

4  Conclusion

References

[1]

Caprotti, O.; Cohen, A.M.; Cuypers, H.; Sterk, H.: OpenMath Technology for Interactive Mathematical Documents; Eindhoven University of Technology 2002
www.win.tue.nl/~hansc/lisbon.pdf

[2]

Embacher, F.; Oberhuemer, P.: Mathe online - A Gallery of Multimedia Learning Material; www.univie.ac.at/future.media/moe/

[3]

Hockemeyer, C.: References on Intelligent Tutoring Systems; University of Graz, 2001
http://wundt.kfunigraz.ac.at/hockemeyer/its-bib.html

[4]

Holzinger, A.: Basiswissen Multimedia - Lernen: kognitive Grundlagen multimedialer Informationssysteme; Würzburg 2000 www-ang.kfunigraz.ac.at/ ~holzinge/multimedia/multimedia-lernen-theorie.html

[5]

IMS Question & Test Interoperability: ASI Best Practice & Implementation Guide - Version 1.2; www.imsglobal.org

[6]

Lucke, U.; Tavangarian, D.; Vatterrott H.R.: The Use of XML for the Development of an Adaptive Multimedia Teaching and Learning System; NAISO (Natural and Artificial Intelligence Systems Organization): Proc. of NL2002, World Congress Networked Learning in a Global Environment, May 1-4, 2002 Berlin, ISBN 3-906454-31-2
www.wwr-project.de/de/infopool/veroeffentlichungen.htm

[7]

Patterson, D.A.; Hennessy, J.L.: Computer Organisation & Design - The Hardware/Software Interface; Morgan Kaufman 1997

[8]

Risse, Th: Interactive Documents -– Yet Another Way to Activate Students in Mathematics; M. Auer, U. Auer: ICL2001 Workshop 'Interactive Computer Aided Learning - Experiences and Vision'; Kassel University Press 2001
www.weblearn.hs-bremen.de/risse/papers/ICL2001

[9]

Risse, Th: e-Learning - Ideas, Intentions, Illusions vs Status, Strategies, Stakes; 18^th Scientific Colloquium, University of Pécs, 2002
www.weblearn.hs-bremen.de/risse/papers/Kolloq18

Author:

Dr. Thomas Risse
Institute for Informatics & Automation
Hochschule Bremen, Flughafenallee 10, 28199 Bremen
risse@hs-bremen.de