An algorithm for automatic checking of exercises in a dynamic geometry system-iGeom

An algorithm for automatic checking of exercises

in a dynamic geometry system:iGeom

Seiji Isotani a,*,Leo ?nidas de Oliveira Branda ?o b

a

The Institute of Scienti?c and Industrial Research,Department of Knowledge Systems,Osaka University,

8-1Mihogaoka,Ibaraki,Osaka 567-0047,Japan

b The Institute of Mathematics and Statistics,University of Sa ?o Paulo,Rua do Mata ?o,1010,Cidade Universita

′ria,Sa ?o Paulo,SP,05508-090,Brazil

Received 25October 2007;received in revised form 6December 2007;accepted 23December 2007Abstract

One of the key issues in e-learning environments is the possibility of creating and evaluating exercises.However,the lack of tools supporting the authoring and automatic checking of exercises for speci?cs topics (e.g.,geometry)drastically reduces advantages in the use of e-learning environments on a larger scale,as usually happens in Brazil.This paper describes an algo-rithm,and a tool based on it,designed for the authoring and automatic checking of geometry exercises.The algorithm dynam-ically compares the distances between the geometric objects of the student’s solution and the template’s solution,provided by the author of the exercise.Each solution is a geometric construction which is considered a function receiving geometric objects (input)and returning other geometric objects (output).Thus,for a given problem,if we know one function (construction)that solves the problem,we can compare it to any other function to check whether they are equivalent or not.Two functions are equivalent if,and only if,they have the same output when the same input is applied.If the student’s solution is equivalent to the template’s solution,then we consider the student’s solution as a correct solution.Our software utility provides both authoring and checking tools to work directly on the Internet,together with learning management systems.These tools are implemented using the dynamic geometry software,iGeom,which has been used in a geometry course since 2004and has a successful track record in the classroom.Empowered with these new features,iGeom simpli?es teachers’tasks,solves non-trivial problems in student solutions and helps to increase student motivation by providing feedback in real time.ó2008Elsevier Ltd.All rights reserved.

Keywords:Dynamic geometry;Automatically checking exercises;Distance education;Geometry;iGeom

1.Introduction

The computer has been used in education since its earliest appearance in 1945.1However,it was only in the 1980s,with the emergence of ‘‘personal computers ”(PCs),that the use of this machine and its resources (e.g.,0360-1315/$-see front matter ó2008Elsevier Ltd.All rights reserved.doi:10.1016/https://www.doczj.com/doc/c516031790.html,pedu.2007.12.004*

Corresponding author.Tel.:+81668798416;fax:+81668792123.

E-mail addresses:isotani@https://www.doczj.com/doc/c516031790.html, (S.Isotani),leo@https://www.doczj.com/doc/c516031790.html,p.br (L.O.Branda ?o).

1To learn more about the history of the computer,visit the timeline of computing history at https://www.doczj.com/doc/c516031790.html,/ComputingHistory

.

Available online at https://www.doczj.com/doc/c516031790.html,

Computers &Education 51(2008)

1283–1303

1284S.Isotani,L.O.Branda?o/Computers&Education51(2008)1283–1303

software)signi?cantly impacted teaching and learning processes(Oldknow,1997).Today,in Brazil and other developing countries,there has been a large expansion in the use of the Internet for teaching.Consequently, the demand for research in this area has considerably increased(Litto,2006).The large-scale distribution of cheap computers for teaching in Brazil has introduced a new stimulus for developing local software with the capability of functioning in computing equipment with low-processing capacities.

The development of environments for distance education and e-learning through the Internet have not only allowed for an easier and faster di?usion of information and knowledge,but has also helped with the intro-duction of courses using the?exibility of schedules and places(Hentea,Shea,&Pennington,2003;Litto, 2006).In this context,these online environments that support teaching and learning processes have become virtual classrooms,where students and teachers can communicate and interact using tools,including chat rooms,forums,emails,wikis,virtual blackboards,etc.

The use of the Internet and computers can bring great bene?ts to the teaching of mathematics,but in order to achieve such goals,it is necessary to choose and create appropriate programs and methodologies that take advantage of the computer’s positive characteristics.Good examples of this include dynamic geometry(DG) programs,which can bene?t teaching and learning processes(Ruthven,Hennessy,&Deaney,in press;Santos &Sola,2001).

Dynamic geometry can be understood as an alternative to traditional geometry,which makes use of a ruler and compass and produces static constructions.When using traditional methods to teach geometry,if a stu-dent,after accomplishing a construction,wants to analyze the same construction using some of the objects in another disposition,he or she needs to repeat the entire construction.However,with DG,used today to spec-ify geometric equations implemented on the computer,objects can be freely manipulated across the screen, maintaining all of the constraints and properties initially established during construction.

DG programs have proven to be an excellent resource for teachers and students(Botana&Valcarce,2002; Ruthven et al.,in press;Sinclair,2005).In spite of the great bene?ts for teaching and learning,as well as the existence of useful DG programs,including GSP(Jackiw,1995),Cabri(2007),Cinderella(Kortenkamp, 1999),C.a.R.(Grothman,1999),and Tabulae(Moraes,Santoro,&Borges,2005),DG is not commonly used in adaptive distance education environments.The main reason for this is the lack of tools for authoring and automatically checking exercises that allow communication between servers and the adaptation of their inter-face(e.g.,show/hide tools to draw objects),according to the learning context.

The importance of the evaluation or checking students’exercises is a reality in both classroom and e-learning environments.According to Hara and Kling(1999)and Hentea et al.(2003),one main frustration of students engaged in e-learning courses(including blended learning2courses)is the lack of feedback or the limitation of an immediate evaluation of completed exercises.In this context,using the usual DG pro-grams generates some di?culties for students and teachers,including the fact that the student is unaware of whether his or her solution is correct and must save the solution and send it by email(or another med-ium)to the teacher in order to receive an evaluation.Moreover,in order to gain access to the student’s solution,a teacher needs to receive the email(or other form of response),open it using a DG program and check each construction.Throughout this cumbersome process,the elapsed time eliminates one advantage of the Internet,which is to provide a faster and more interactive environment for learning.This loss of time makes a teacher’s task of helping students improve learning through the use of technology much more di?cult.

The answer to this problem is to make DG programs more e?ective when used in e-learning environments. The following can be implemented to facilitate that process:(a)generate tools that produce and automatically check student exercises,allowing for faster authoring and feedbacks;and(b)develop tools allowing for com-munication between DG programs and learning management systems(LMS)that facilitate distribution,man-agement,adaptation,and reception of exercises or any other content.

Today,two DG programs possess tools for authoring and checking exercises,including Cinderella(Kor-tenkamp,1999)and C.a.R.(Grothman,1999).The contributions of these programs inspire many teachers to include DG programs in classroom activities.However,in the current versions of these programs,there are some limitations for their use on distance education courses over the Internet.For instance,it is not pos-2Blended learning is broadly used to de?ne a course that combines usual classroom activities and online activities.

S.Isotani,L.O.Branda?o/Computers&Education51(2008)1283–13031285 sible to satisfactorily link these programs to LMS,allowing for e?ective management and/or adaptation of the created content.This means that these programs cannot personalize their content in real time nor consider teacher and student preferences that would increase success in the learning process.

The purpose of this paper is to present the development of algorithms and tools for authoring and auto-matically checking exercises in a DG program,called iGeom,which can be used for both web pages and as a stand-alone version.Conventional algorithms,based on automatic theorem proving techniques,to check geometry constructions,are very time-and machine-consuming and,sometimes,cannot check complex con-structions in real time(Chou,Gao,&Zhang,2000;Gallier,2003;Gilmore,1970).The developed algorithm of this paper is not as formal as a proof,but does enable the instantaneous checking of a geometric exercise by comparing it with a solution’s template.The main bene?ts of this approach include a program that is(a) fast enough to be used over the Internet to support the use of DG programs in distance education courses;

(b)able to check any kind of geometric construction and return whether the construction has the necessary properties to solve an exercise;and(c)able to run on computers with low-processing capabilities,which is very important in Brazil,where many computers used in public schools are out-of-date.Furthermore,we will present results using this algorithm,implemented on iGeom,together with an under development learn-ing management system,referred to as SAW(Moura,Branda?o,&Branda?o,2007).This paper is structured as follows:In Section1,we will brie?y describe the DG program,iGeom,compare its functionalities with other DG programs and outline its potential use when together with LMS.In Section2,we show iGeom functionality for authoring and automatically evaluating exercises.In Section3,we present the automatic evaluation algorithm.In Section4,we show how geometric constructions can be exported to the Internet. In Section5,we present the results of the application in a compulsory discipline o?ered for an undergrad-uate mathematics course at the University of Sao Paulo.Finally,in Section6,we present the conclusions of our study.

1.1.iGeom:DG on the internet

The iGeom program is a complete,multi-platform,DG system(DGS)developed at the Institute of Math-ematics and Statistics at the University of Sa?o Paulo(IME–USP).It has been under development since2000, under the direction of Professor Leo?nidas O.Branda?o(Branda?o&Isotani,2003).The iGeom program is implemented in Java and can be used as a stand-alone program or as an applet.Like other DG programs, the current version of iGeom allows users to perform all of the basic operations of DG,including the creation of geometric objects,such as points,lines,circumferences,and dynamic measures(e.g.,angles);the modi?ca-tion of characteristics of the objects(e.g.,color,size);options for saving or recovering constructions in di?er-ent formats;and other advanced resources.Besides these,iGeom has special resources for creating scripts (functions used to create automatic geometric constructions)and fractals.Fig.1depicts the interface of the iGeom,including fractals created using a recursive script.

The iGeom has been developed considering the computational restrictions of computers in Brazil.This phi-losophy makes iGeom run e?ciently in both residential and personal computers that have low-processing capabilities(e.g.,computers using a microprocessor similar to Intel i386).Furthermore,it can be used without restrictions with any other learning management system(Branda?o,Isotani,&Moura,2005).

In Table1,3we show some of the resources available with the iGeom program,in comparison with other DG programs.The column Portability refers to the DG programs that can be executed in any platform.The column ADDs refers to DG programs that allow for the opening of multiple drawing areas.The column Script contains those DG programs that possess resources for script creation.In the column Rec,those DG pro-grams that allow for the creation of recursive scripts(allows for the construction of fractals)are listed.In the column labeled Web are DG programs that allow for the unrestricted Internet use of these resources. In the column labeled Com are DG programs that possess communication resources.In the column titled AA are DG programs that possess resources for authoring and the automatic evaluation of exercises.Finally, the column License refers to the license type each program possesses.

3The analysis of resources was done in2004.

In 2004,a project was initiated to develop a learning management system that could easily incorporate applets into educational modules.This system was referred to as SAW (Branda ?o,Isotani,&Moura,2004;Moura et al.,2007).The iGeom program has been used together with SAW in di?erent courses.One of these includes a course directed towards public school teachers and people interested in teaching mathematics using computers.Another use includes an obligatory discipline o?ered for a degree course in mathematics,titled ‘‘Notions of teaching of mathematics using computers ”.Each year,more than a hundred students and teachers are enrolled in these courses.The use of iGeom +SAW,including some of the obtained results using the auto-matic checking tool on the Internet,will be shown in detail in Sections 4and 5.

2.Authoring and automatically evaluating exercises

The action of evaluating or checking exercises is a complicated process (Chou et al.,2000;Gelernter,1963;Kortenkamp,1999).Because of this,automation is not a trivial matter.It requires the

application

Fig.1.The interface of the iGeom showing fractals created using recursive script.

Table 1

Comparison between the resources of some DGS

Program

Portability ADDs Script Rec Web Com AA License iGeom

X X X X X X X Free Cabri

X X Commercial C.a.R

X X X X GNU Cinderella

X X X X Commercial GSP

X X X Commercial Tabulae X X X X Commercial 1286S.Isotani,L.O.Branda ?o /Computers &Education 51(2008)1283–1303

of heuristics or simpli?cations.One of the di?culties in developing an algorithm that enables automatic checking in GDS is the multiplicity of di?erent (and correct)solutions for the same problem.To illustrate this point,consider the following problem:Given two points,A and B,build a middle point between them .Fig.2shows two di?erent constructions used to solve this problem.Many other constructions are possible,whether minimal 4or not.

We have identi?ed three techniques used by DG programs in the automatic evaluation of exercises:the automated (geometry)theorem proof (Gallier,2003),the technique based on the theory ACT (Atomic Com-ponent of Thought)(Anderson,1993),and the numerical evaluation (Isotani &Branda ?o,2004).

According to Gallier (2003),an automatic theorem proof (ATP)can be summarized as a computer program that shows whether or not a sentence (a conjecture)is a logical consequence of a group of sentences (axioms and hypotheses).The language used by these programs should be formal in order to eliminate ambiguity.The veri?cation of a sentence produced by an ATP program is known as a proof.This proof describes a sequence of steps (logical consequences)that validate a conjecture.The steps,followed by an ATP program (also known as a proof tree),can be understood by other ATP programs or even a person.Some pioneering works in the development of automatic evaluators of geometry theorems have been reported by Gelernter (1963),Gelern-ter,Hanson,and Loveland (1963)and Gilmore (1970).

The DG programs based on ATP are capable of checking the validity of di?erent constructions,which in turn,generate di?erent types of proofs that depend on the methods used to prove a scenario (Botana &Valc-arce,2002;Chou et al.,2000).At present,several methods exist for accomplishing an automatic proof in geometry.Some of them are (Gao &Zhu,1999):the method of Wu,the area method,the base of Groebner,the vector method and the angles method.Among the DG programs that use ATP methods are Geolog (Hol-land,1993),GeometryExpert (Gao &Zhu,1999)and Discover (Botana &Valcarce,2002).

The ACT-R technique is a current version of the ACT theory,a general theory on the human cognition attempting to reproduce how an individual acquires knowledge.It was developed by a group of researchers under direction of John Anderson at Carnegie Mellon University (Anderson,1993)and has been applied to intelligent tutoring programs focused on the teaching of Algebra and Geometry (Aleven,Koedinger,Sin-clair,&Snyder,1998).

Although these two techniques produce excellent results,they cannot be e?ectively applied to web-based courses that check geometry exercises.This dilemma is the result of these techniques being time-

and Fig.2.Two di?erent constructions of the middle point.

4

Minimal constructions/solutions are those from which we cannot remove any object without compromising the properties that satisfy a given problem.S.Isotani,L.O.Branda ?o /Computers &Education 51(2008)1283–13031287

1288S.Isotani,L.O.Branda?o/Computers&Education51(2008)1283–1303

machine-consuming.Furthermore,if a teacher needs to create an exercise,depending on the complexity of the solution,these techniques have di?culties to determine,rapidly,whether the solution is correct or not(Chou et al.,2000;Gallier,2003).For online courses,the time available for response is extremely limited.Feedback should be given almost instantaneously(real time).Additionally,in Brazilian public schools,most computers have low-processing capabilities,which demand programs and algorithms that do not consume a great deal of computer processing time.

Instead of attempting to prove a student’s construction/solution,a numerical evaluation compares the stu-dent’s answer-objects with the corresponding teacher’s answers-objects.This comparison is made using a dis-tance criterion between the objects.For instance,if the problem is to determine the middle point between points A and B,the distance is veri?ed between the point of the student’s construction and the point of the teacher’s construction according to an established criterion.Programs that use a similar method include Cin-derella(Kortenkamp,1999),C.a.R.(Grothman,1999),and iGeom(Isotani&Branda?o,2004).The numerical evaluation method is not a formal proof,like ATP or ACT-R,and demands a solution template(a previously provided correct construction).However,this type of evaluation possesses advantages by using less computa-tional processes and by not restricting the application domain.This allows a proposed exercise to be solved using any technique without harming the evaluation process.

2.1.An overview of the authoring and checking processes of iGeom

The functionalities for authoring and automatically checking exercises in iGeom are incorporated so that an exercise may be completed using the application(which facilitates the execution of tests for the de?nition of the exercise)or web pages.

The authoring of exercises in the iGeom program has?ve steps:(a)the construction of a solution tem-plate;(b)the selection of initial objects(including the selection of a statement);(c)the selection of answer-objects;(d)the use of the unable/enable buttons;and(e)saving the exercise or exporting it into HTML format.

The construction of the solution template is accomplished the same as any other construction.For exam-ple,to construct the median line of two points,one possible construction requires the authors(a)to create points A and B;(b)to create a segment s0that connects A and B;(c)to create a middle point M between A and B;and(d)to create the line r(median)perpendicular to the segment s0,which passes through M, shown in Fig.3.

When construction is complete,the author can initiate an authoring window(to the right of Fig.3)and select the initial objects(those that will appear at the beginning of the exercise)using the‘‘marcador”button (in Fig.3,the icon with a red square around it).The selection of the answer-object is completed in a similar fashion.

In the authoring window,it is possible to see which objects have been selected,as well as remove or add other objects.To disable/enable buttons in the student main interface,one can click on the buttons shown at the bottom of the authoring window.Thus,teachers can select tools that will be available to learners during the resolution of the exercise.For example,the buttons‘‘middle point”,‘‘perpendicular”and‘‘parallel”can be disabled and,thus,students will be unable to use these tools to solve the given problem.Finally,authors can save the exercise in a stand-alone version of iGeom,or save it as an HTML?le.

Along with the authoring tool,a resource for automatically checking a student’s solution has been devel-oped.When a student opens an exercise in iGeom,similar to the example shown in Fig.3,he or she will only see points A and B and the message,‘‘Given two points A and B,build the median line”.Furthermore,the but-tons disabled by the teacher during the authoring phase will not appear in the menu.Thus,the student will be unable to use them.

After a student completes a given exercise,he or she needs to select the answer-object that he or she believes to be the correct answer(e.g.,select the median line for his or her construction).Then,the evaluation algo-rithm begins.If the student’s solution is evaluated as incorrect,the iGeom?nds a counter-example or a con-?guration(instance)of the construction where the selected answer-objects do not satisfy the desired properties (more details on how to?nd the counter-example is shown in Section3).The use of counter-examples aids both teachers verifying a student’s construction and students who need to visualize their mistakes.Another

advantage of the numerical method implemented in iGeom is that a student can use any geometrically valid solution to solve an exercise,even those not imagined by a teacher.

The tools of authoring and automatically checking,together with exporting resources for the Internet,facil-itate the creation of interactive exercises that can be used for free access.By using this export feature,a teacher can produce a set of web pages using several exercises and o?er their students an immediate evaluation of their constructions,without the need of personally verifying each construction.

A practical example of the use of these resources can be found with iMa

′tica,a project making digital mate-rials for mathematics available in Portuguese (http://www.matematica.br/igeom/docs/exemplo1).This web-page possesses several activities (divided into classes,topics and exercises)that are helpful in teaching geometry (Fig.4).All of them can be completed online and the evaluation (whether it is correct or not)of each solution is supplied by iGeom.

3.Proposing an algorithm for automatically checking exercises using a numerical evaluation

Proposing a fast and accurate algorithm for checking an exercise that can verify whether a construction is correct or not is intrinsically dependent on how the concept ‘‘solution ”is de?ned.For example,in the C.a.R.program,a solution is de?ned as any static object.With such a de?nition,the algorithm used to check

exer-Fig.3.Construction of a solution template for a median line exercise.In the left-hand pane of the display,the solution template is presented as the construction and the statement ”Given Figure/table legends two points,A and B ,build the median line ”(in Portuguese).In the right is shown the authoring window.

S.Isotani,L.O.Branda ?o /Computers &Education 51(2008)1283–13031289

1290S.Isotani,L.O.Branda?o/Computers&Education51(2008)1283–1303

Fig.4.Activity example for use through Web,with the iGeom program,developed by students at IME-USP in2004.

cises in C.a.R.basically tests if the objects selected as a solution for an exercise have similar objects in the solu-tion template.This algorithm somewhat works in practice,however,it constrains the dynamicity of the sys-tem.Because a solution is de?ned as a static object,C.a.R.does not allow for the movement of objects while an exercise is being solved.In order to simplify the automatic checking of exercises,such algorithms ignore the term‘‘dynamic”of a DG system.An easy test can exemplify one weakness of this algorithm.For example,if an exercise asks an individual to?nd a middle point between the given points A and B,then one way of cheat-ing is to create a point with the same coordinates as the middle point.Although this solution is not desirable, the automatic checking in C.a.R.will determine that such a construction is correct,even though no valid con-struction has been completed.

In Cinderella,the concept‘‘solution”is de?ned as a theorem that needs to be checked.Thus,the algorithm for checking exercises is known as‘‘automatic theorem checking”,based on the Schwartz–Zippel Theorem (Schwartz,1980).This theorem states that if it is possible to estimate the probability of choosing a coun-ter-example(if there is one),it is then possible to bound the probability of failing if a theorem is true,based on a few random examples.By utilizing such a theorem when checking an exercise,the Cinderella algorithm tries to prove if objects in a construction coincide by chance or because a theorem drives them to coincide.

Finally,in iGeom,a solution is de?ned as a function applied to any object(initial objects)and returning objects(answer-objects).Such a de?nition allows for high?exibility when comparing two di?erent construc-tions.Thus,when given initial objects,it is possible to verify if di?erent constructions provide equivalent results without the necessity of comparing each object of every construction or the construction steps.To establish the de?nition of solution and equivalence between solutions,and thus,creating an algorithm based on numerical evaluation,it is necessary to measure the distance among objects of a construction.In the fol-lowing sections,the basic formalization for these de?nitions and an algorithm for checking exercises are proposed.

3.1.Numerical evaluation

The result of the evaluation is a measurement of the distance between the student’s solution and the tea-cher’s solution(a correct solution given for a problem).To complete this evaluation,it is necessary to de?ne the distance criterion among the geometric objects.We have de?ned the distance criterion only for pairs of objects of the same family or type.To simplify,we only name examples from the more common families in a construction,including the family of points(F p),the family of circumferences(F c)and the family of seg-ments(F s).The sets of all families of objects will be represented by F og.

By doing so,the distance criterion is the function dist that receives a pair of geometric objects(og1, og2)2F og X F og and returns a value of Rt:

dist:eog1;og2T!Rte1TThe computational description of the objects is de?ned for each family of F og as a list of numerical val-ues.For instance,a point can be represented by(x,y),where x and y are the coordinates of a point;a circumference can be represented by(x,y,r),where(x,y)are the coordinates of its center and r is its radius; and a segment s=[(x1,y1),(x2,y2)]can be represented by(x1,y1,x2,y2).Thus,if we consider just the fam-

ilies of points,circumferences and segments,given two objects of a same family,og1e og2,if‘1

1;‘1

2

;...;‘1

i

àá

and‘2

1;‘2

2

;...;‘2

i

àá

,represent og1e og2,respectively,then we can de?ne the function dist according to the Eq.(2).

disteog1;og2T?

j‘1

1

à‘2

1

jtj‘1

2

à‘2

2

j;eog1;og2T2F p XF p

j‘1

1

à‘2

1

jtj‘1

2

à‘2

2

jtj‘1

3

à‘2

3

j;eog1;og2T2F c XF c

min

P4

i?1

j‘1ià‘2i j;

P4

i?1

j‘1ià‘2

eit1T%4t1

j

8

><

>:

9

>=

>;;eog1;og2T2F s XF s

8

>>>

>>>

<

>>>

>>>

:

e2T

The symbol%is used to obtain the rest of the division(function module).

The need of the minimum(min),when objects are from the family of segments,is based on the desire to compare not only the distance between segments AB and CD,but also their permutations AB and DC, BD and CD,BA and DC,without di?erentiation between them.In other words,for segments,the distance criterion does not distinguish segment AB from segment BA.

Once the distance among geometric objects is de?ned,we can de?ne the distance between pairs of geometric constructions,OG p and OG a.If OG p and OG a are two constructions and their objects are represented by

og p

1;og p

2

;...;og p i

eTand og a

1;og a

2

;...;og a

i

àá

,respectively,then the distance between OG p e OG a,can be expressed as shown in Table2.

A solution(geometric construction)can be represented as a function that receives a list of geometric objects (input)and returns another list of geometric objects(output):

S:OG i!OG fe6TAn instance of construction S is the application of S over a given con?guration of objects.Once the dis-tance among pairs of constructions and the representation of a solution is determined,we can decide when the two constructions(solutions)are equivalent,as shown in Table3.

It is worth noting that if two constructions are equivalent,the distance between them is invariant in relation to the initial con?gurations.In other words,the distance computed for any instance is always zero.Therefore, if we do not consider numerical errors,a good evaluation criterion states that construction S a is correct when-ever it is equivalent to a correct given construction,S p(for example,a student’s solution,S a,can be considered correct if it is equivalent to solution S p,given by the teacher).However,there is a practical problem:how to implement a version that is su?ciently fast and takes into account numerical errors?

As di?erent constructions applied to the same initial objects generate the same answer-objects(output)with small numerical di?erences,we have opted to relax the equivalence criterion that allows,for instance,two points very close in a number of di?erent instances of the initial objects to be considered almost equivalent. In principle,this solution allows for false positives(a wrong exercise evaluated as correct)or false negatives S.Isotani,L.O.Branda?o/Computers&Education51(2008)1283–13031291

(a correct exercise evaluated as incorrect),but as we show in session 4,in practice,it works well.It is also pos-sible to increase the number of instances tested or implement other modi?cations of parameters that would reduce or eliminate wrong evaluations.

3.2.The evaluation algorithm

From now on,when it is implicit or indi?erent as to which list of geometric objects should be used,the simpli?ed notation S will be used.

Based on the numerical evaluation presented in the previous section,we have created an algorithm com-posed of four main steps.These steps include numerical transformation ,evaluation ,instantiation,and valida-tion .This algorithm has been implemented into iGeom as a way of verifying its performance and accuracy.As mentioned in Section 2,in order to check an exercise,it is necessary to create a solution template by pro-viding a correct construction for the exercise and selecting which objects will be checked (answer-objects).At the end of a resolution,a student selects which are their answer-objects.The lists of the student’s answer-objects and the solution template’s answer-objects will be the input data for the evaluator algorithm.The result will be a natural number between 1and 3,with (1)being the correct solution,(2)being a partially incor-rect solution and (3)being an incorrect solution.

When applied on a geometric object,the ?rst step of our algorithm,the numerical transformation ,returns a list of scalars that represent the object.In most cases,this transformation is simple.However,besides the numerical transformation,some objects,such as the polygon,need the ordination of points representing it.With this list of scalars,we can make a comparison between objects.Then,in order to compare the solution template,S p ,and the student’s solution,S a ,we transform objects marked as answers in lists of scalars and then begin the evaluation,as schematized in Fig.5.

Table 2

De?nition of distance among pairs of constructions

If the cardinality of the lists OG p and OG a (#OG p e #OG a )are di?erent,then

dist eOG p ;OG a T?t1

e3T

If #OG p =#OG a =n ,

then I p =(p 1,...,p n )and I a =(a 1,...,a n )two permutations on the ?rst natural n ,dist eI p ;I a T?P n i ?1dist og p p i ;og a a i ;

if h tipo og p p i ?tipo og a a i àá;i 2f 1;...;n gi

t1;c :c :8><>:e4T

tipo(og)being a function that returns the type of geometric object og .

Then,

if P n is the set of all of the permutations of ?rst natural n :

dist OG p ;OG a àá?min dist I p ;I a àá;8I p ;I a àá2P n XP n èée5T

In other words,among all of the permutations of objects of OG p e OG a ,dist OG p ;OG a àáis the sum of the distances among each pair of

objects of same type that results in the smallest value.

Table 3

De?nition of equivalence of solutions/constructions

De?nition of equivalence :

Be S p and S a two constructions applicable over the same list of geometric objects OG .Then S p and S a are equivalent if,and only if,for any con?guration OG 0of the list OG ,dist S p OG 0eT;S a OG 0eTàá?0.

1292S.Isotani,L.O.Branda ?o /Computers &Education 51(2008)1283–1303

The evaluation is made in two stages.The ?rst stage consists of mapping between the lists,while the second is the comparison of the distance criterion.The ?rst stage allows students to have the freedom of making a selection of answer-objects in any order.For instance,if the solution template contains points A and B and the student’s answer-objects are points C and D ,the mapping stage will identify if points C or D correspond to A or B .

The mapping of objects S a e S p is accomplished by comparing each og element of S a ,with all elements of S p that belong to the same type (family),while minimizing the distance between them.Then,an object of S a will be mapped with an object of S p if both belong to the same family of geometric objects,have not been mapped before and the distance among them is the smallest possible in relation to other objects of S a .This mapping is made only for the ?rst instance (the initial con?guration of an exercise).The second stage of the evaluation consists of a comparison between pairs of mapped geometric objects eog a i ;og p i T.In comparison,we use a relaxed version of the de?nition of equivalence,presented in Table 3,that takes into consideration the numer-ical imprecision of using di?erent solutions.Then,we propose a de?nition of Quasi Equivalence,as shown in Table 4.

The pseudo-algorithm implemented in iGeom that corresponds to these two steps (mapping and evalua-tion),is shown in Table 5.Lines 1through 12represent the mapping of objects,while lines 13through 19rep-resent the exercise evaluation.

Notice that the part of the algorithm shown in Table 5represents just one instance of the solution,which considers a ?xed position for each input object.If this algorithm is used to check only the initial con?gu-ration,it can frequently generate mistakes,such as false positives.For instance,in the problem using the middle point (Section 2),a student could try to put a free point on segment AB and move it so that

it Fig.5.Numerical transformation and evaluation.

Table 4

De?nition of quasi-equivalence of solutions/constructions

De?nition of quasi-equivalence :Fixed a list of input objects OG ,be S p a construction on OG and S a another construction on the same OG .Then S p and S a are quasi equivalent if,and only if,for any con?guration OG 0of the list OG ,we have

dist S p OG 0eT;S a OG 0eTàá

S.Isotani,L.O.Branda ?o /Computers &Education 51(2008)1283–13031293

is close enough to the position of the medium point to generate a false positive,without making a valid geometric construction.In spite of the di?culty involved in positioning the point so that the algorithm eval-uates the solution as correct,it is possible to identify such problems in the algorithm implemented in C.a.R.This false positive problem is illustrated with the following example:Given two points,A and B,build an equilateral triangle D ABC .

In Fig.6,two solutions for this problem are shown.The ?rst construction is correct and is shown in Fig.6a.The second,shown in Fig.6b,is an incorrect solution,but contains the same properties of an equilateral tri-angle.In this case,the second solution depicts the construction of an isosceles triangle,with the free point C on the straight line s .Coincidentally,in this construction con?guration (instance),point C is in such a position that AB ?AC ?BC .Thus,the construction of the isosceles triangle can be erroneously used to produce an equilateral triangle.To avoid such problem,it is possible to move all free points of the construction,in par-ticular,point C of the isosceles triangle,so that it becomes clear that triangle D ABC in Fig.6b is not the cor-rect construction for an equilateral triangle.

Therefore,a simple way to detect such mistakes is to create a mechanism that analyzes the exercise in sev-eral instances (instantiation )and only after a considerable number of evaluations does the system return the result of the evaluation (validation ).In Fig.7,we show the procedure that is employed by the evaluation algo-rithm.For each application of the automatic checking algorithm,the positions of the geometric objects are Table 5

An pseudo-algorithm to solve the problem of checking exercises in DG

1.Receive two lists S p e S a of geometric objects

2.Creates a new list of geometric objects S t /

3.For each element og p

i of the list S p

4.SmallestFoundDistance 1

5.MappedObject /

6.For each element og a j of the list S a

7.If og p i e og a j belong the same family of geometric objects then

8.If dist og p i ;og a j

9.MappedObject og a j 10.SmallestFoundDistance dist og p i ;og a j 11.If MappedObject =/

12.Return False //mapping between S p and S a failed

13.If not

14.S t S t MappedObject //concatenation

15.Remove MappedObject de S a //object is mapped

16.For each element og p

i of the list S p 17.Be og t j the ?rst element S t 18.If dist og p i ;og t j

19.Remove og t j of S t

20.If not

21.Return False //solutions are not equivalent

22.Return True //solutions are

equivalent

Fig.6.Two constructions of an equilateral triangle.The construction (a)is the correct construction for an equilateral triangle and the construction (b)is the construction of an isosceles triangle in a instance where all the sides of the triangle have the same measure.

1294S.Isotani,L.O.Branda ?o /Computers &Education 51(2008)1283–1303

changed.This alteration is made through the random movement of all free points of the construction.The choice for the new position of a point is made by modifying the coordinates (x ,y )for (x 0,y 0)and x àk

At the end of the automatic checking process,the algorithm returns an integer value:(1)means the solution is correct;(2)means the solution is incorrect,however,it found instances considered correct and incorrect;and

(3)means the solution is incorrect.

3.3.Identi?cation of ambiguity

The concept of functions has been explored in order to deal with the problem of automatically checking geometric exercises.These functions are constructions that receive a set of input objects and return a set of answer-objects.Such functions are considered ambiguous when they are not bijective functions,or in other words,they possess more than one set of answers for the same input data.In DG,this problem can appear whenever the proposed exercise has an implicit or explicit free point needed to solve the problem.An example is presented in the following problem:Given two points,A and B,build an isosceles triangle D ABC.

This problem is intrinsically ambiguous,possessing in?nite solution instances,because point C can be located on any position of the median line A and B .Independent of A and B ,point C can be moved,changing the instance of triangle D ABC,yet maintaining the desired properties,as can be observed in Fig.8.

To ?nd this ambiguity problem during the authoring of an exercise,we have implemented an algorithm that checks the construction of the solution template.It informs the author which geometric object makes the construction ambiguous.This warning is very useful when correcting teachers’mistakes during the cre-ation of an exercise.Other programs,such as C.a.R.and Cinderella,lack similar algorithms or functionalities.

During the construction,when selecting the initial objects and answer-objects,the algorithm of ambiguity veri?es if some answer-object is dependent of some free point,P .If such point P does not belong to the selected initial objects and cannot be determined by them,then,we verify whether moving point P will change the position of some answer-objects.In the example of Fig.8,during the creation of the solution template selection,only points A and B are initial objects,while segments AC and BC are answer-objects.The

algorithm

Fig.7.Sequence of steps of the proposed algorithm for automatic checking of geometry exercises.

S.Isotani,L.O.Branda ?o /Computers &Education 51(2008)1283–13031295

identi?es the ambiguity and returns a message to the teacher informing him or her that point C needs to be selected as an initial object to remove the ambiguity of the solution template.

It is worth noting that an answer-object,dependent on a free point that does not belong to the initial objects of the solution template,is not always modi?ed when the free point is moved.An example can be observed by creating the solution template for the following problem:Given the straight line r and point A,build a straight line s by passing through A and forming an angle of 60°to r.

The solution for this problem can be seen in Fig.9.The objects selected as initial objects are straight line r and point A ,while the object selected as the answer will be straight line s .Note that in this construction,straight line s is dependent on point C .Yet,this point is not selected as an initial object.However,when mov-ing point C ,the position of straight line s does not modify and,therefore,the solution is not ambiguous.The developed algorithm detects this case using part of the mechanism of the automatic evaluation,making the comparison of the construction in di?erent

con?gurations.

Fig.8.For a same position of the points A and B ,potentially we have in?nite con?gurations for the problem of construction isosceles

triangles.

Fig.9.Construction of an angle of 60°.In this construction the movement of the point C (free point)does not interfere in the result.

1296S.Isotani,L.O.Branda ?o /Computers &Education 51(2008)1283–1303

S.Isotani,L.O.Branda?o/Computers&Education51(2008)1283–13031297 https://www.doczj.com/doc/c516031790.html,munication between the iGeom program and a web server

The tools for automatically checking and authoring exercises have been built directly into the iGeom appli-cation.To ful?ll our goal of e?ectively using DG programs in distance education courses,it was necessary to build a tool that could communicate with learning management systems(LMS).Among the several advanta-ges of the communication between iGeom and a LMS we emphasize the following:

(a)Teacher can produce the exercises with his or her machine and send them to the server or create them

directly on the Internet through an iGeom applet,working together with a learning management system (LMS);

(b)iGeom can exchange data related to the resolution of an exercise with a LMS;

(c)By receiving data from a server,iGeom is able to personalize the content according to learner’s behavior.

Today,there are several complex systems used to manage Web-based courses.Some of these are commer-cial,while others are freeware,including SAW(Moura et al.,2007),AulaNet(Lucena,Fuks,Raposo,Gerosa, &Pimentel,2006)and Moodle(Moodle,2007).Usually,these environments lack specialized resources that would boost learning speci?c contents(e.g.,geometry).Thus,the use of iGeom allows any LMS that supports Java applets to incorporate speci?c functionalities that will help the teaching–learning of geometry in a very interactive way.

The possibility of integrating iGeom into LMS o?ers many advantages for students and teachers.For stu-dents,besides the possibility of interactively accomplishing exercises online,solutions can be checked almost instantaneously and a counter-example can be given when a solution is incorrect.For teachers,we o?er resources that facilitate authoring interactive content,evaluating exercises and collecting data related to the student’s interaction with the content(exercises,examples,etc.)for a subsequent analysis.

The exchange of messages between iGeom and the server(LMS)is made using the method POST(Raggett, Hors,&Jacobs,1999).This method allows iGeom to send variables in the form of chains of characters.In this way,di?erent information is sent for the server using the same message.In this case,the program located in the server should be responsible for the treatment of the received data,in order to recover the values of the sent variables.

An example of sending messages happens after the evaluation of an exercise is accomplished on the Inter-net.When the exercise is solved,iGeom requests a connection to an address on the server.Data are stored using di?erent variables.Each variable,de?ned by a group of characters,is packed using the method POST. After the reception of the data,the server stores the messages received in local variables so that they can later be manipulated by the program manager.Yet,through the same connection,the server can request changes in the active Web page interface or open a new Web page.Some of the variables manipulated by iGeom include the following:

$envWebValor:This variable indicates the result of the evaluation of the exercise.If its value is equal to 0(zero),then the exercise is evaluated as incorrect;if it is equal to1(one),the exercise is evaluated as correct.

$envWebArq:This variable possesses two functionalities.The?rst relates to when a teacher is creating an exercise and the other to when a student is solving an exercise.When a teacher creates an exercise, directly on the Internet,the variable stores all of the information of the exercise,including the solution template that will be used later.For students,it stores every construction accomplished during the reso-lution of an exercise.If the solution is considered incorrect,then the counter-example is stored to show that the student has made a mistake.If the solution is correct,it stores the construction in any con?guration.

$envWebGeoResp:This variable is used to indicate which geometric objects have been selected as answers to an exercise.

$envWebGeoOuvidor:This variable stores the data created from interactions between the user and iGeom. For instance,the buttons frequently used may be stored.It can also recall speci?c situations in which the user used such buttons.

1298S.Isotani,L.O.Branda?o/Computers&Education51(2008)1283–1303

5.Results

In order to analyze the e?ciency,e?cacy and accuracy of our algorithm,since2004,iGeom has inte-grated a learning manager system,referred to as SAW.iGeom together with SAW has been used in several didactic experiments to o?er geometry e-courses for undergraduate students,math teachers,and students in private high schools.The analysis of e?ciency refers to the possibility of applying our algorithm to check exercises in class or over the Internet,using personal computers with low-processing capabilities and o?ering immediate feedback to alleviate student frustration.The analysis of e?cacy means to determine if the algo-rithm can check a student’s solution,given any exercise in which the solution is not ambiguous.Finally, checking accuracy refers to checking how many times our algorithm will return false positive and false neg-ative results.

The tools of authoring and automatically checking exercises in iGeom have been tested on several occa-sions.The?rst instance occurred in2004for an obligatory discipline,titled‘‘Notions of teaching of mathemat-ics using computers”(MAC118),for undergraduate students in mathematics.The goal of this discipline is to teach future teachers how to use technologies that support students learning mathematics.This discipline com-prised of two teachers,three teaching assistants and more than150students,divided into three groups.The authoring and checking tools were used during a period of six months,until the end of the semester.Due to the success of the use of iGeom and SAW,these tools were incorporated into this discipline.Another test was done at the beginning of2005for courses for senior math teachers.This test included the participation of more than25junior high and high school teachers.These teachers acted as authors of content and problem solvers. During a two month period,they were instructed on how to use iGeom and SAW to construct e-courses for geometry,as well as how to bene?t from the functionalities of DG programs to create more interactive content for Web-based courses.In2006and2007,a qualitative and quantitative analysis of SAW,using the iGeom program,was completed by Moura et al.(2007),following the methodology proposed by Yin(2002).

In MAC118,before the implementation of the automatic checking tool,the exercises were solved using iGe-om(or a similar program),but exercises were corrected manually by teaching assistants and teachers.Correct-ing consumed a great deal of the teaching assistant’s time and feedback of these corrections was given two or three weeks later.Before2004,approximately20exercises were applied and solved per semester.With the use of automatic checking and SAW,it was possible to reduce the teachers’and assistants’work.Both were able to spend more time taking care of student’s personal di?culties raised during the attempt to solve an exercise. Furthermore,in2004,it was possible to apply and solve more than40exercises in class and over the Internet with immediate feedback.In2006,there were more than70exercises completed in one semester.Through the application of the automatic checking tool,students have eliminated misconceptions immediately after the res-olution of an exercise.As a?rmed by one student,‘‘...in the case that the construction is evaluated as correct, it was already recorded(in the SAW),and in the case it was wrong,I could start it again and would remove my doubts immediately...”.

According to the work of Moura et al.(2007),which analyzes the impact of iGeom and SAW in the learn-ing process,the increased number of exercises given in the course is‘‘...a consequence of the use of automatic assessment resources which have encouraged learners in their studies(in2006,82%of students that answered the second questionnaire think that the use of SAW+iGeom has encouraged them during the course).Also,it reduced tutor workload and encouraged him/her to introduce preparatory exercises to serve as an auxiliary for the solution of the main problem,as well as leaving related exercises as homework.”

The experiments completed in MAC118have shown that our algorithm is e?cacious in checking exercises and e?ciently runs in computers with low-processing capabilities.In2004,the computers used in class had processors similar to a K6II450MHZ with256Mb of RAM memory.For any proposed exercise,our algo-rithm responded almost instantly(milliseconds),immediately returning output.In comparison,GEOLOG (Holland,1993),another geometry software program that uses an algorithm based on theorem-proving meth-ods,required more than?ve seconds to check the exercise of a middle point,using the solutions presented at the right of Fig.2.For more complex exercises,such as the construction of a pentagon(see Table6),GEO-LOG did not answer until several minutes had passed.Such a delay(ten seconds and greater with no response) is not acceptable when exercises must be accomplished over the Internet,where instantaneous feedback is key to maintaining student motivation(Hentea et al.,2003).

S.Isotani,L.O.Branda?o/Computers&Education51(2008)1283–13031299 Table6

Sequence of steps(algorithm)performed by one student to construct a pentagon.The result is show on Fig.10.

c0:=Circle(P1,P2);

c1:=Circle(P2,P1);

A:=Intersection south(c0,c1);

B:=Intersection north(c0,c1);

s0:=Segment(A,B);

c2:=Circle(A,P1);

C:=Intersection(c1,c2);

D:=Intersection(s0,c2);

r:=line(C,D);

E:=Intersection(c0,c2);

s:=line(E,D);

F:=Intersection(c0,r);

c3:=Circle(F,P1);

G:=Intersection north(c1,s);

c4:=Circle(G,P2);

s1:=Segment(P1,F);

H:=Intersection north(c3,c4);

s2:=Segment(F,H);

s3:=Segment(H,G);

s4:=Segment(G,P2);

Finally,the accuracy of our algorithm has been proven robust enough to not provide false positive and false negative evaluations.Di?erent instructors from schools and universities have produced hundreds of exercises and thousands of solutions which have been checked by our algorithm.So far,none of them have been given a false positive or false negative result.A very good example of the accuracy of our algorithm occurred in MAC118in2005.The teacher had asked the students to create a pentagon,starting with a given segment(initial object).One of the students tried to solve the exercise using a solution(construction) that he knew and believed to be correct.The steps to create this solution are presented in Table6and the corresponding geometric construction is shown in Fig.10.This solution was considered incorrect by our algorithm.The student resent it again and again,complaining that our algorithm was not accurate enough, given the numerical imprecision.In this case,even the teacher was in doubt about the veracity of the answer given by our algorithm.However,a bibliographic research made by one of the teaching assistants veri?ed that the solution given by the student was,in fact,wrong.According to Fourrey(1924),this con-struction is an approximation of the correct solution.In this solution(Fig.10),the internal angles of points F and G have109°202800,and not108°(in a regular pentagon all internal angles should have108°).In such situations,without the automatic checking tool,the student would never know that such a construction is incorrect.

The di?culties and obstacles regarding the use of iGeom have been analyzed using answers from distrib-uted questionnaires.Through the application of automatic checking,students have the possibility of removing any doubt they may have concerning about the resolution of the exercise rapidly.Ninety-seven percent of stu-dents are very satis?ed with this functionality.However,we observe that it is necessary to provide a more pre-cise statement for each exercise.Many students have di?culties solving some exercises due to an imprecision in statements.According to the data obtained during the use of iGeom during MAC118,about69%of the stu-dents whose exercises are evaluated as incorrect have di?culty to understand the statements.Other di?culties listed by the students include the selection of correct answer-objects and di?culty in building the solutions due to the complexity of the exercises.

Another implemented resource that contributes to the success of iGeom in the classroom and over the Internet is the integrated communication capability.With communication,the teacher can create the exer-cises directly on the Internet and store them in a database using any LMS that supports our protocols. Thus,iGeom has facilitated the reuse of created exercises for di?erent teachers and courses.Besides these bene?ts,each exercise accomplished by a student and other information about its solution also can be stored.When an exercise is considered incorrect by our algorithm,the iGeom sends the student’s solution

to the server/database in a con?guration that facilitates the visualization of the mistake (a counter-example).Thus,a teacher can verify a student’s solution and provide more personalized feedback.It is also possible to build a library of di?erent solutions for the same problem,including one illustrating mistakes most fre-quently made.

6.Conclusions

The possibility of increasing the bene?ts of DG programs to be used in computers with low capabilities,over the Internet,and in distance education courses,has been a great challenge,as well as a strong motivation for developing and implementing these resources.In this context,the resources of communication,authoring,and automatically checking exercises have been intentionally developed to increase the dissemination of DG programs to teachers and students.

According to Ruthven et al.(in press),one of the biggest di?culties in encouraging a wider use of DG pro-grams is the di?culty of creating content and evaluating and guiding students during learning activities.To overcome such di?culties for teachers,this work o?ers resources that facilitate the authoring and checking of exercises which can be used on simple Web pages or with an LMS.Thus,teachers’time spent creating and evaluating exercises is reduced.Moreover,resources for storing and reusing exercises are provided.For students,the use of DG is provided directly on Web pages and,because of automatic checking,fast feed-back is given for each accomplished exercise.With this,students’doubts can be immediately eliminated,which reduces student frustration over a lack of feedback after the conclusion of an

exercise.

Fig.10.Result of an incorrect construction of a pentagon sent by one student.The internal angles in F and G have 109°202800and not 108°.In Table 6is shown the steps (algorithm)used to construct the ?gure.

1300S.Isotani,L.O.Branda ?o /Computers &Education 51(2008)1283–1303

S.Isotani,L.O.Branda?o/Computers&Education51(2008)1283–13031301 In order to develop an automatic checking tool in iGeom that could be used in computers with low-processing capabilities and over the Internet,this work proposes and implements an algorithm that checks the validity of a construction,using the idea of function.Each geometric construction is considered a func-tion that receives some geometric objects(input)and returns other geometric objects(output).Thus,for a given problem,if one function(construction)is known to solve the problem,we can compare it with another function to check whether these functions are equivalent or not.Two functions are equivalent if,and only if,for the same input they have the same output.Because of the numerical imprecision of geometric constructions,some numerical di?erences between outputs of each function are accepted (quasi-equivalence).By using such de?nitions to check an exercise,we can compare a solution given by a student with the solution given by a teacher to solve the same problem.If the student’s solution is con-sidered equivalent(or quasi-equivalent)to the teacher’s solution for a variety of di?erent inputs,then we consider the student’s solution correct.This algorithm is implemented into iGeom and is shown to be fas-ter and consume less machine resources than formal approaches(rule-based approaches),which allows for an increased use in classrooms,with computers with low-processing capabilities and via the Internet(Iso-tani,2005).The main concern of the presented algorithm is the possibility of giving false positive and false negative answers.This means that our algorithm could indicate that a given solution is correct when it is, in fact,incorrect(false positive)or indicate that a given solution is incorrect answer when it is,in fact, correct(false negative).Nevertheless,an extensive evaluation of our algorithm shows that it works well in practice.So far,we have not found any case where a solution has been given a false positive or false negative evaluation.

The communication protocol implemented in iGeom allows for the exchange of information over the https://www.doczj.com/doc/c516031790.html,ing such capabilities,the iGeom has been used together with a learning management system, called SAW,to create a very interactive environment to learn geometry.Thus,teachers can create exer-cises directly through a Web page(or upload the exercise)and students can submit their solutions that will be stored on the server.Furthermore,students receive immediate feedback that determines whether their solutions are correct or not.According to Moura et al.(2007),the use of iGeom in classrooms and in distance education courses can increase student’s motivation and help teachers identify student misconceptions.

iGeom is continuously under development.Each semester,users of the program o?er feedback and sugges-tions to improve its functionalities and remove technical problems(bugs).As mentioned at the end of Section 5,we intend to create a library with di?erent solutions for the same problems,including good solutions and frequent mistakes committed by students.One of the biggest challenges facing this program is the augmenta-tion of our automatic checking algorithm.We intend to use this library to improve feedback given by our program.

Today,our algorithm runs only if the user clicks the‘‘checking exercise”button.Otherwise,no action is taken.By using a library made up of correct solutions and common mistakes,our algorithm could run in the background and identify whether the user’s solution is similar to a possible mistake,and then o?er some sort of hint to help the student realize his or her mistake(or avoid it).If the student’s solution is similar to a correct solution,then the program can o?er praise for partial accomplishment to increase his or her https://www.doczj.com/doc/c516031790.html,ually,during the problem-solving process,a user’s solution is not completely wrong.Thus,this approach allows a student to reuse part of his or her solution that is considered correct(i.e.,partially similar to a good solution in the library),to guide the user or to give some suggestions/hints in order to help him or her‘‘?nd”the correct solution to the problem.

To achieve this goal,our library needs to be formal and organized in terms of(a)a common vocab-ulary with structured de?nition of geometric concepts;(b)semantic interoperability and a high expression of each concept;and(c)a variety of contexts by which a problem can be solved.One possibility in ful?lling these requirements is to use ontologies and ontological engineering techniques that support the development of our library(Mizoguchi&Bourdeau,2000).By using these ontologies,we may fea-sibly augment our automatic checking algorithm by providing a better way of identifying necessary and desirable concepts for solving an exercise and identifying similarities between the actual construction done by the user and the constructions recorded in the library.Thus,we believe that with such improve-ments,it will be possible to o?er a meaningful and more personalized system for users,which will con-

1302S.Isotani,L.O.Branda?o/Computers&Education51(2008)1283–1303

tribute to an increase in motivation and the avoidance of possible frustrations during interactions with the system.

The latest version of the program iGeom,with all resources discussed in this paper,is freely available by visiting http://www.matematica.br/igeom.

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家用面条机的设计与分析

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目录 一、目的及背景 (3) 1.1 目的： (3) 1.2 背景： (3) 二、方案设计 (4) 2.1 背景： (4) 2.2 黑箱法分析： (5) 三、各工步设计 (6) 3.1 放面团 (6) 3.2一级输送 (6) 3.3压面 (6) 3.4二级输送 (6) 3.5 切面机构 (6) 四、各子系统设计 (6) 4.1 动力系统选取 (6) 4.2 传动系统设计 (6) 4.3 压面机构 (6) 4.4 切面机构 (7) 4.5 机械生产效率计算 (7) 五、产品前景 (8) 产品优势 (8) 产品前景 (8) 六、参考文献 (8)

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国密算法(国家商用密码算法简介)

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全自动果蔬切丁机设计

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Abstract： Since the 21st century, the development of food industry of our country greatly the reform open early in the processed food is improved day by day, people's lives with the high demand from the food packaging Food machinery of the development of the food industry The improvement of people's material life continually in order to meet the demand of the food industry to promote the development of various types of food machinery and development An advanced development in food machinery and automation of the machine and the social development inevitable in processed foodsPromoting the development of socialist economic construction constantly, people's material living next to the new requirements The quality of food, fruit quantity change in order to meet the demands of the people's daily, and convenient method of the supply system of small food packaging market supply immediately mentioned the agenda, extensibility and, Behold, no hard crusts cut of each fruit fruit julienne slices of fruit Application of food machinery's [1] this machine Application of food machine, reduces the labor intensity of the workers, the advantage is used to save time, improve efficiency, large labor production release In order to guarantee the special effect processing to realize whether the handmade, product quality, reduces the waste of raw materialsThe basic work principle of shredded machine via the worm gear worm drive motor on the dial, dial ring week exercise sword rack

全自动一次成型挂面机操作方法技巧

全自动一次成型挂面机操作方法技巧 我们一共将制面的整个过程分为3大步骤进行操作分析 1.和面：将面粉倒入拌面机内，然后按规定25%左右的比例进行加水，启动运力，把面和水拌均匀达颗粒状(10分钟后使用最佳)，即可倒入面条机使用。 2.调整：用手轮进行调整轧辊间隙，各组轧辊间隙的参考数据是：(330以上机组例外)在调整过程中，调紧或放松手轮的多少，有时不易一次调准，关键是要弄清道理，耐心把握，不熟练时可停机调整，正常后锁紧手轮。 3.操作：把拌好的面粉放入面斗内，按刀并固定挡刀板，就可开机运转，第三组下来面板要用手引到下一组(多组照此类推)经过面条机压辊压制，出面条操作完毕。

重庆友乐乐机械专业生产各种款式的面条机，热门搜索词：重庆全自动面条机、重庆面条机、重庆压面机、贵州面条机、四川面条机、云南面条机、重庆面条机厂、面条机多少钱一台、多功能家用面条机价格、重庆面条机厂家。欢迎大家登录我公司网址详细咨询，友乐乐竭诚为您服务！！ 产品关键词：全自动面条机、重庆老牌面条机、重庆老式面条机、重庆传统面条机

2. 3. 面通过拉丝处理 结构紧凑，造型新颖，辊具有揉面功能，生产面条时不需另外熟化，广受客户的青睐。 4. 转动部位采用高精轴承，经久耐用，运转灵活，性能稳定。 5. 护板采用不锈钢，面斗木质部分用不锈钢板装

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目录

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国密算法应用流程

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(毕业设计)蘑菇切丁机的设计

毕业论文（设计） 题目：蘑菇切丁机的设计 姓名： 学院： 专业： 班级： 学号： 指导教师： 年月日

目录 摘要....................................................................... I Abstract ........................................................................................................................................... II 1绪论. (1) 1.1研究的目的及意义 (1) 1.2国内外研究现状 (3) 2切丁机总体方案的确定 (3) 2.1切丁机的总体构成 (6) 2.2整机结构设计 (6) 3主要零部件的设计与计算 (4) 3.1传动方案的确定 (4) 3.2滚筒的设计 (5) 3.3部分零件的规格设计 (5) 3.4 电动机的选择................................................................................ 错误！未定义书签。 3.5 V带的设计计算 (6) 3.6V带轮的设计 (8) 3.7直齿圆柱齿轮的设计计算............................................................. 错误！未定义书签。 3.8轴的结构设计计算......................................................................... 错误！未定义书签。 3.9轴承的选择及校核......................................................................... 错误！未定义书签。 3.10键的选择和校核 (17) 3.11切刀的设计 (18) 3.12机体结构设计 (22) 4结论............................................................................................................ 错误！未定义书签。 4.1 本设计的优点................................................................................ 错误！未定义书签。 4.2 存在的不足和建议........................................................................ 错误！未定义书签。5总结............................................................................................................ 错误！未定义书签。参考文献....................................................................................................... 错误！未定义书签。致谢............................................................................................................... 错误！未定义书签。

面条机主要结构与工作原理

面条机主要结构与工作原理 1、构造： MT5-200NT5-250型面条机是由电动机，主传动装置，二对进料压辊，一对后片压辊，一对压实薄压辊，一对实压光辊，共五对压辊。一付切机刀及断条机构组成。 2、原理： 本机采用五对压辊连续滚压法，将和好的面粉加入面斗内，第一道压辊压出两片面片，然后进入第二道压辊，将两片后成入片压实，然后通过托面槽引入第三道压辊，再次将面片压实，再经过第四道压辊将面片压光，最后通过面刀切成面条。 以上产品有两种方式控制面条长度 (1)普通型：是利用脚踏板自动断面条。 (2)配有挑条器系统，是自动断面，面条长度出厂已设定。

重庆友乐乐机械专业生产各种款式的面条机，热门搜索词：重庆全自动面条机、重庆面条机、重庆压面机、贵州面条机、四川面条机、云南面条机、重庆面条机厂、面条机多少钱一台、多功能家用面条机价格、重庆面条机厂家。欢迎大家登录我公司网址详细咨询，友乐乐竭诚为您服务！！ 产品关键词：全自动面条机、重庆老牌面条机、重庆老式面条机、重庆传统面条机

2. 3. 面通过拉丝处理 结构紧凑，造型新颖，辊具有揉面功能，生产面条时不需另外熟化，广受客户的青睐。 4. 转动部位采用高精轴承，经久耐用，运转灵活，性能稳定。 5. 护板采用不锈钢，面斗木质部分用不锈钢板装

重庆友乐乐机械有限公司位于重庆市巴南区金竹工业园内，前身为“重庆永利食品机械厂”，因自身发展需要，2015年底由重庆的合川区搬迁到了巴南区金竹工业园区内，不但交通地理环境更加便利，而且大大提升了我公司在研发、制造领域同周边“重庆大江工业集团”和“长安铃木公司”配套厂的综合协作能力。“友乐乐”是一家专业从事食品机械设计、制造、销售、服务为一体的企业，公司不断开拓创新，迅速发展。公司主打产品为面条机、米粉机和粉条机、凉皮机等，企业以创新、科技、服务和经济效益为设计理念，以优越的性能和人性化的智能设计赢得了广大新老客户的信赖，以过硬的质理和精湛的技术，在行业内享有较高的声誉! 我们的目标：创业路上友(有）乐乐！ 我们的宗旨：销售的不仅仅是产品，更重要的是服务；最终目的是帮助选择我们的朋友获取适宜的利润！！ 产品关键词：全自动面条机、重庆老牌面条机、重庆老式面条机、重庆传统面条机

国密算法

国密算法（SM1/2/4）芯片用户手册（UART接口） 注意：用户在实际使用时只需通过UART口控制国密算法芯片即可，控制协议及使用参考示例见下面 QQ：1900109344（算法芯片交流）

目录 1.概述 (3) 2.基本特征 (3) 3.通信协议 (3) 3.1.物理层 (3) 3.2.链路层 (4) 3.2.1.通讯数据包定义 (4) 3.2.2.协议描述 (4) 3.3.数据单元格式 (5) 3.3.1.命令单元格式 (5) 3.3.2.应答单元格式 (5) 3.4.SM1算法操作指令 (6) 3.4.1.SM1 加密/解密 (6) 3.4.2.SM1算法密钥导入指令 (6) 3.5.SM2算法操作指令 (7) 3.5.1.SM2_Sign SM2签名 (7) 3.5.2.SM2_V erify SM2验证 (7) 3.5.3.SM2_Enc SM2加密 (8) 3.5.4.SM2_Dec SM2解密 (9) 3.5.5.SM2_GetPairKey 产生SM2密钥对 (9) 3.5.6.SM2算法公钥导入 (10) 3.6.SM4算法操作指令 (10) 3.6.1.SM4加密/解密 (10) 3.6.2.SM4算法密钥导入指令 (11) 3.7.校验/修改Pin指令 (11) 3.8.国密算法使用示例（Uart口命令流） (12) 3.8.1.SM1算法操作示例 (12) 3.8.2.SM2算法操作示例 (13) 3.8.3.SM4算法操作示例 (14) 3.9.参考数据 (15) 3.9.1.SM1参考数据 (15) 3.9.2.SM2参考数据 (15) 3.9.3.SM4参考数据 (17)