Diagnostic Models of an Intelligent Tutor System for Teaching Skills to Solve Algebraic Equations

Transcription

1 Diagnotic Model of an Intelligent Tutor Sytem for Teaching Skill to Solve lgebraic Equation Kulik, Chukhray, M Chukhray National eropace Univerity Kharkov, Ukraine btract In thi paper one olution for teaching kill to olve n-power algebraic equation by Lobachevky-Greffe- Dandelen method i decribed Student mitake are dicovered and claified Baed on ignal-parametric approach to fault diagnoi in dynamic ytem mathematical diagnotic model which allow detecting mitake clae by comparing tudent calculated reult and ytem calculated reult are created Feature of propoed diagnotic model application are preented Intelligent tutor ytem i developed and ued on utomatic Control Theory practical training by third year tudent of National eropace Univerity Index Term Intelligent tutor ytem, tudent mitake, diagnotic model I INTRODUCTION key intellectual part of the computer tutor ytem, which incorporate the pedagogic matery and experience of the bet teacher, i it diagnotic ervice It allow not only detecting a tudent' mitake, but alo identifying the caue of the mitake in order to chooe an educational equence adapted for that tudent quantity of cientific work i devoted to intelligent tutor ytem diagnotic ervice development Part of them i baed on the perturbated and reference model comparing approach [,,3,4,5] Other part of work i baed on Baye approach [6,7] Different implementation of intelligent tutor ytem are decribed Such implementation a LEKS, Carnegie Learning' Cognitive Tutor, M Lindquit, ctivemath [8] include diagnotic function and teach to mathematical tak olving Diagnotic ervice in thee ytem i decompoed on two level: one for pecific tudent tep analyzing during concrete mathematical tak olving and other for common tudent kill analyzing There are alo opponent of diagnotic ervice in tutor ytem For intance, in work [9] author hold the opinion that diagnotic apect in t neceary at all becaue educational proce i divided on mall tage and preventing from mitake accumulation intantaneou feedback i ued We don t hold uch opinion by following reaon Firt of all, intermediate reult input lead to additional time conumption Secondly, in ome cae there are everal correct calculation order [0] and uniquely defined educational proce decompoition in t poible Thirdly even on mall tage there i a poibility to make eriou mitake, for intance, to approximate 3445 to 35 or to 4 Thi reearch i motivated by following reaon In publication devoted to intelligent tutor diagnotic function decription of tudent mitake refer uually to primitive mathematical action Decription of higher education intitute tudent profeional mitake, which appeared during practical tak olving are almot abent Mitake diagnoi mathematical tool are alo weakly preented In thi paper one olution for teaching kill to olve n- power algebraic equation by Lobachevky-Greffe- Dandelen method i decribed It i ued on utomatic Control Theory practical training by third year tudent of National eropace Univerity Lobachevky-Greffe-Dandelen method [] a a bae to olve n-power algebraic equation wa choen becaue of it relatively imple calculation chema and importance for olving tability tak due to poibility to determine both real and complex root II PROBLEM DEFINITION To analyze n-power algebraic equation olution written by tudent and on the bai of reult dicover and claify tudent mitake On the bai of the ignal-parametric approach to fault diagnoi in dynamic ytem [3] to create mathematical diagnotic model which allow detecting mitake clae by comparing tudent calculated reult and ytem calculated reult 3 To develop intelligent tutor ytem for Lobachevky-Greffe-Dandelen method learning which include following main function: a) upport tudent by guide which covered theoretical quetion appeared during Lobachevky-Greffe-Dandelen method learning; b) tak olution equence control; с) tudent knowledge and kill tep-by-tep diagnoi; d) tudent knowledge and kill remediation by feedback introducing III COMPUTER MODEL DESCRIPTION OF LOBCHEVSKY-GREFFE-DNDELEN METHOD Let n, g be integer and poitive number, h be integer nonnegative number, b { 0,,,, {0,,, n}, l {,, n}, k {,,, g} v { g, g} Let u conider the two implet cae, when equation n n a 0 x + ax + + an = 0, which coefficient are real number, ha all real root or only one pair of complex root Let' introduce an auxiliary function r _ f ( x, to preent by the rounding rule any real number x with a floating decimal point to within h digit after the point: ijet International Journal of Emerging Technologie in Learning - wwwi-etorg

2 t r _ f ( x, = ( ) ( z0 + z 0 + z 0 + h p + + z h 0 ) 0, where z b {0,,,9}, t {, }, p i a integer number, ( z 0 > 0) ( b zb = 0) Let ( 0, = a Then coefficient of reformed equation can be calculated by formula ( k, ( k, ( k, + ) h = ( k, c) =, if ( c < 0) ( c > n) = r _ f ( + ( ), ), where 0 Condition of coefficient calculation interrupting i v ( > 0) ( r _ f (, ) = ( ( v, h = r _ f ( ( g,, )) Real root calculation i performed by formula g xl = ± r _ f (,, g, at that v ( v, > 0) ( v, > 0) Sign of real root i determined by ubtitution t complex root calculation x l = α + iβ and x l + = α iβ at firt ρ = x l = xl+, g l+ ) ρ = r _ f (, i computed, where v 0) Then ( v, < a α = r _ f ( ( + x + + xl + xl+ + + xn ), and a 0 β = r _ f ( ρ α, are computed IV EXPERIMENTL RESULTS To determine concrete problem which appear when tudent olve algebraic equation we conducted Cla of the mitake TBLE I CLSSES OF THE MISTKES Number of mitake Mitake at imaginary part complex root calculation 4 3,3 Non-compliance with the condition of root quaring end,4 Mitake connected with root quaring miundertanding 0,5 Mipelling 0 9,5 Mitake at doubled coefficient product calculation 8 7,6 Rough calculation 8 7,6 Unrecognized mitake 6 5,7 Inverted formula for root calculation 6 5,7 Lack of complex root exitence condition knowledge 5 4,8 Lo of ign at calculation 5 4,8 Redundant iteration 5 4,8 Incorrect coefficient exponentiation 3,9 Mitake at g th root calculation,9 Diplaced quotient calculation,9 Mitake at exponent calculation,9 Other 6 5,7 experiment 37 third-year tudent who tudy on a peciality «utomatic control ytem» took part in it 0 different 3-power equation were prepared Every equation ha either 3 real root or real and complex root Every tudent wa offered to olve equation Neceary accuracy of calculation wa 4 ignificant poition To reveal a maximum quantity of mitake during the analyi of tudent' work we imulated calculation of the tudent even after detection of mitake In that way 05 different mitake were detected From the point of view of mitake appearance reaon all mitake can be divided into two clae: miconception and procedural mitake [] Concerning to place of appearing all mitake can be divided into two clae: general (for example, rounding mitake, mipelling and o on) and pecific for concrete place (for example, lack of complex root exitence condition knowledge) Below dicovered clae are preented ordered by number of mitake appeared in our experiment (Table) Let uncover eence of preented clae Mitake at imaginary part complex root calculation cla decribe ituation when tudent forget that abolute value of complex root i directly calculated quared and thu ubtitute ρ 4 intead of ρ at β computing Non-compliance with the condition of root quaring end cla i connected with ignorance of fact that not all ignificant poition of poitive coefficient coincided with ignificant poition of quared coefficient received on previou tep Mipelling cla include uch typical pelling or input error a ingle trancription (4 time), ymbol % ijet International Journal of Emerging Technologie in Learning - wwwi-etorg

3 deletion (3 time), ymbol addition (once), adacent ymbol tranpoition (once), and multiple trancription (once) Mitake connected with root quaring miundertanding are g-th root computing intead of g -th root computing (8 time) and equation root calculation without computing of radical (3 time) Mitake at doubled coefficient product calculation cla include uch mitake a loing one of multiplier (4 time), uing current left coefficient intead of top left (once), uing initial coefficient (once), uing current top coefficient intead of top left (once) and uing current coefficient intead of top (once) Rough calculation cover rounding mitake (5 time) and ignorance of computing accuracy requirement (3 time) Mitake in Inverted formula for root calculation cla are caued by initial coefficient mirroring Lack of complex root exitence condition knowledge cla i connected with complex root calculation when exitence condition (coefficient ign interchanging in one of the column) in t true Lo of a ign at calculation i one of the mot common clae of mitake [] Redundant iteration cla decribe ituation when tudent continue calculation when condition of interrupting i true Incorrect coefficient exponentiation cla mean raiing to the fourth power intead of quaring (3 time) Mitake at g th root calculation cla decribe ituation when tudent calculate g+ -th root ( time) Diplaced quotient calculation cla appeared time at ρ g,3) calculation when tudent took g,) g,) intead of g,0) Mitake at exponent calculation cla decribe ituation when ignificand i correct but exponent in t Other cla include uch mitake a incorrect 8 0 power dividing ( = 0 (once)), β i calculating by 4 0 formula for ρ (once), incorrect ign determining of real root (once), incorrect equence of operation ( ( 0) = 00 (once)), lack of main algebra theorem knowledge (calculation only root intead of 3 (once)), taking g -th root from one coefficient but not from quotient (once) Maximum amount of mitake made by one tudent wa 6 (3 time) For intance, one tudent at firt raied to the fourth power intead of quaring then mipelled one digit, later ued current left coefficient intead of top left, then ued inverted formula, later howed hi lack of complex root exitence condition knowledge and at the end made mitake at imaginary part complex root calculation V DIGNOSTIC MODELS Diagnotic model (DM) i a mathematical model which connect mitake with it ymptom and allow olving of invere problem [3, 4] Let introduce table of ymbol: x i a value calculated by tudent, x i a reference value calculated by tutor program, h m( r _ f ( x, ) = z0 + z 0 + z z h 0 and ex ( r _ f ( x, ) = p are two auxiliary function Then DM for mitake detecting i r _ f ( x, r _ f ( x, fter mitake detecting we hould identify it caue So we ue DM for cla finding For intance, DM for finding Mitake at imaginary part complex root calculation cla i 4 β = r _ f ( ρ α, DM for finding Non-compliance with the condition of root quaring end cla i ( z ) ( ) ( {,,, } 0 = z0 p = p h z z ) DM for Mitake connected with root quaring miundertanding are defined a follow _ ( x r f g l = ±,, _ (, ) l+ ) ρ = r f g h g, decribe g-th root computing intead of g -th root computing xl = ± r _ f (,, l+ ) ρ = r _ f (, are g, defined for ituation when root or ρ are calculated without computing of radical DM for Mitake at doubled coefficient product calculation cla are: _ ( ( ) = r f ( k, + k, k, + ), = looing of ; _ ( ( ) = r f ( k, + k,, looing = of right multiplier; _ ( ( ) = r f ( k, + k, k, + ), = uing current left coefficient intead of top left one DM for Rough calculation cla are defined by two following model ex( r _ f ( x, ) h ( r _ f ( x, r _ f ( x, = 0 )^ ( z h+ 5) erve for finding mitake in rounding, where z h+ i a tored reference (h+) ignificant poition For intance, rounding mitake wa made when tudent round,4445 to,444 DM for finding mitake connected with ignorance of computing accuracy requirement i defined a follow r _ f ( x, = r _ f ( x, b), 0 b < h 7 7 Uing 6 0 intead of 63 0 can be een a an example of uch mitake To find ingle trancription we ue DM which defined a b ( z ) ( {0,,, } { } b zb h b z = z ) dacent ymbol tranpoition can be found by uing ijet International Journal of Emerging Technologie in Learning - wwwi-etorg 3

4 { 0,,, h }( z + ) ( z z + z ) ( w {0,,, h} {, + } z w = z w ) For finding all of Mipelling mitake imilar tring detecting method [6] can be ued DM for Inverted formula uing for root calculation cla i _ ( g xl = ± r f, DM for Lack of complex root exitence condition knowledge cla i defined a ( I ( x ) 0) ( ( l v v, > 0), where I ( x l ) i an imaginary part of x l root We define Redundant iteration a g, g }( ( > 0) {, r _ f (, r _ f (, ))) ( ( g, = ( g, h ex( r _ f ( ( g, l ), ) ex( r _ f ( ( g, l ), ) g m( r _ f (, ) x _ ( l = ± r f 0, m( r _ f (, ) erve for finding of incorrect power dividing mitake One of the poible cenario of obtained DM application i preented in Fig by the example of tudent coefficient ( b, calculation If tudent make mitake the program will inform that mitake have been made, analyze mitake and then allow tudent to correct it If mitake i made repeatedly the program again will inform that mitake have been made, analyze mitake and offer to repeat operation by tep nd only if tudent make mitake on i-tep the program will produce diagnotic meage about mitake with pecifying it cla Such approach i agreed with a principle that tudent mitake i not given immediately Intead of it, the opportunity to find and correct a mitake i given to the tudent t the ame time reult of the mitake analyi which i performed by the program on each tep are tored in the tudent model ll conidered diagnotic model are created with taking into account only ingle tudent mitake, though tudent ometime make alo multiple mitake The argument for uch aumption are Probability of multiple mitake occurring i much le than ingle mitake occurring probability becaue per < per per per, where p er probability of ingle mitake occurring Even for not too complicated formula when we take into conideration multiple mitake huge amount of alternate olution are appeared It complicate the diagnotic ervice ignificantly [0] 3 If the diagnoi can t be determined the program will ak tudent to repeat calculation by tep [] or execute diagnotic converationally with tudent [5] Let note that everal diagnoe may be inferred, for intance rounding from 3005 to 3 can be interpreted either a rounding mitake or a ignorance of computing accuracy requirement In thi cae diagnoi i tored in tudent model a d d and can be defined more exactly by additional quetion to tudent We have implemented in Delphi 60 firt verion of uch intelligent tutor ytem creenhot of which are preented in Fig Thi ytem include all defined above diagnotic model When tudent miconception i found ytem return him to the theory learning It i ued on utomatic Control Theory practical training by third year tudent of our univerity reult of it uing we can note following Firt of all even backward and lazy tudent work in tutor ytem with pleaure Secondly every tudent correctly olve hi tak without teacher aitance hould at firt work on mitake without aitance Thu the diagnotic meage with a hint about the caue of the Figure Fragment of diagnotic model application cenario ijet International Journal of Emerging Technologie in Learning - wwwi-etorg 4

5 Figure Intelligent tutor ytem creenhot VI CONCLUSIONS Creating of diagnotic ervice i one of the central problem in intelligent tutor ytem development The main contribution of preent paper are firt of all the reult of tudent mitake experimental reearch and econdly baed on thee experimental reult propoed diagnotic model for tudent mitake clae finding We are going to implement diagnotic model in interpreted language and to tore it in databae Such approach will give potentialitie to diagnotic model addition/modifying without any change in program hell Moreover program hell in mode of elf-learning [] will be able to generate new diagnotic model, tore it in databae and then interpret it For intance, part of propoed model connected with miing of operation can be obtained automatically from reference model We are alo going to create diagnotic model for common tudent kill analyzing, uch a etimating of attentivene, ability to learning, ability to work on tudent own and other REFERENCES [] Self, J: Formal approache to tudent modeling // Tech Report I-59, Lancater Univerity, 99, [] Stern, M; Beck, J; Woolf, B: daptation of problem preentation and feedback in an intelligent mathematic tutor // In Proceeding of Intelligent Tutoring Sytem, 996, pp [3] Woo, C: Intructional Planning in an Intelligent Tutoring Sytem: Combining Global Leon Plan with Local Dicoure Control // Ph D Diertation, Illinoi Intitute of Technology, 99 [4] Liong Yu Tu; Wen-Lian Hu; Shih-Hung Wu: Cognitive Student Model - n Ontological pproach // International Conference on Computer in Education, ICCE 00, December 3-6, 00, uckland, New Zealand IEEE Computer Society, 00, Volume, pp - [5] Ragnemalm, E L: Student diagnoi in practice; Bridging a gap // Uer Modeling and Uer dapted Interaction, 5():93--6, [6] Nkambou, R; Tchetagni, J: Diagnoing Student Error in E- Learning Environment Uing MPE Theory // In Proceeding of the International Conference on Web-Baed Education, 004, pp [7] Millán E; Pérez-de-la-Cruz, J: Bayeian Diagnotic lgorithm for Student Modeling and it Evaluation// Uer Model Uer- dapt Interact (-3), 00, pp [8] Shute, V; Underwood, J: Diagnotic ement in Math Problem Solving Educational teting ervice, Princeton, NJ, 005, 7р [9] Patel, ; Kinhuk K: pplied rtificial Intelligence for Teaching Numeric Topic in Engineering Dicipline Lecture Note in Computer Science, 08, 996, pp [0] Self, J: Bypaing the intractable problem of tudent modeling // In Proceeding of Intelligent Tutoring Sytem, 988, pp [] ngot, : Complément de mathématique à l'uage de ingénieur de l'électrotechnique et de télécommunication Pari: Centre National d'etude de Télécommunication, 957, 836 p [] Schechter, E: The mot common error in undergraduate mathematic, [3] Kulik : Fault Diagnoi in Dynamic Sytem via Signal- Parametric pproach //IFC/IMCS Symp Baden-Baden, Sept , Vol, pp 57-6 ijet International Journal of Emerging Technologie in Learning - wwwi-etorg 5

6 [4] Kulik, ; Rainkova, N: lgorithmic Fault-Tolerance Support for a Gyrocopic Senor Unit // Proc 3 rd IFC Symp On Intelligent utonomou Vehicle Madrid, 998, pp [5] Dimitrova, V; Self, J; Brna P: Involving the Learner in Diagnoi - Potential and Problem// Tutorial at Web Information Technologie: Reearch, Education and Commerce, Montpellier,France,000 [6] Kulik, ; Chukhray, ; Zavgorodniy, : Similar tring detecting method // In Proceeding of the Eat-Wet Fuzzy Colloquium 005, Zittau, Germany, 005, pp 83-9 UTHORS natoly Kulik, Doctor of Technical Science, Profeor, Laureate of Ukraine State Prize, head of Control Sytem department, dean of Control Sytem faculty in National eropace Univerity, Kharkov, Ukraine ( kulik@d3khaiedu) ndrey Chukhray, Candidate of Technical Science, deputy head of Control Sytem department, deputy dean of Control Sytem faculty in National eropace Univerity, Kharkov, Ukraine ( chukhray@d3khaiedu) Marina Chukhray, reearch aitant of Control Sytem department in National eropace Univerity, Kharkov, Ukraine ( mchuhray@mailru) Manucript received 6 October 006 Thi work wa upported by the Preident of Ukraine grant GP/F/04/006 ijet International Journal of Emerging Technologie in Learning - wwwi-etorg 6