T-algebra: Adding Input Stage To Rule-Based Interface For Expression Manipulation By Marina Issakova, Dmitri Lepp and Rein Prank University of Tartu, Institute of Computer Science, Estonia
[email protected],
[email protected],
[email protected] Received: 4th October 2005 Revised: 12th February 2006
T-algebra is a project for creating an interactive learning environment for expression manipulation tasks of elementary algebra. Our main didactical principle has been that all the necessary decisions and calculations at each solution step should be made by the student, and the program should be able to understand the mistakes. This paper describes the design of our Action-Object-Input dialogue and different input modes as an instrument to communicate three natural attributes of the steps: choice of conversion rule, operands and result. 1
INTRODUCTION
Expression manipulation is one of the central skills needed for solving tasks in practically all fields of mathematics. But the results of learning in this area are often not satisfying. One of the reasons for poor performance is repetition of incorrect solution attempts without getting feedback. During the paper-and-pencil training process the students make many mistakes, but the teachers are not able to discover and correct them in time. Thus, the mistakes are repeated many times and can become habitual. The need to analyse information quickly implies that the training could be improved by using computerised training environments. At school, the basic types of expression manipulation tasks are usually taught together with some solution algorithms. When a student solves such tasks, he should at each solution step: 1. choose a transformation rule corresponding to a certain operation in the algorithm (or some simplification or calculation rule known earlier), 2. select the operands (certain parts of expressions or equations) for this rule, 3. replace them with the result of the operation. For proper learning of expression manipulation as well as for assessment and diagnosis of knowledge gaps, an environment should be available where all the necessary decisions and calculations at each solution step would be made by the student and the program would be able to understand the mistakes. Existing software does not address the whole complex spectrum of the problems which arise. When solving a problem using computer algebra systems, the student selects at best only the transformation rule and a part of the expression; the transformation itself is made by
the computer. In many cases it is even sufficient to apply the operation to the whole expression as the program itself selects the required operands. Rule-based interactive learning environments (Beeson, 1998 and Ravaglia, Alper, Rozenfeld and Suppes, 1998) also perform the selected transformations mainly automatically. Some programs (for instance APLUSIX, Nicaud, Bouhineau and Chaachoua, 2004) use paper-and-pencil-like dialogue design where a transformation step consists purely of entering the next line (of course, the usual editing operations are available for this). However, such input does not provide the program with information about the decisions made by the student at earlier stages. As a consequence, practically the only error that can be diagnosed is the non-equivalence of the new expression with the previous one. In 2004 we started a project for creating a new learning environment called T-algebra for four areas of school mathematics: calculation of the values of numerical expressions; operations with fractions; solving of linear equations, inequalities and systems of linear equation; operations with polynomials. Our main goal is to design a solution dialogue that allows the program to understand the decisions made by students at all three stages of the step. The design is not yet completed. We have however implemented the expression input editor for different input modes, the rules (three-stage dialogue and error checking) and most of the 51 problem types (automated solution of arbitrary tasks by means of rules from the menu; dynamic hints; comparison of a student's steps with the algorithm to be learned; checking whether the task is completed). While it is therefore possible to solve tasks using the rules we have not yet implemented error statistics (counters of different types of errors). This paper describes the design of our Action-ObjectInput dialogue and different input modes as an instrument to communicate three natural attributes of the step: choice of conversion rule, operands and result. 2
GENERAL DIALOGUE SCHEME IN T-ALGEBRA
T-algebra enables step-by-step algebra problem solving. Each solution step consists of three stages: 1 selecting a transformation rule (action), 2 marking the parts of the expression (object), 3 entering the result of the application of the selected rule (input). International Journal for Technology in Mathematics Education, Volume 13, No 1
2]
Marina Issakova, Dmitri Lepp and Rein Prank
Hereafter we will refer to this scheme as the Action-Object-Input scheme, after its three stages. The problem solution window of T-algebra is shown on Figure 1. The main part of the window contains
the previous problem solution steps and a virtual keyboard that can be used for active input. On the right side is the menu of possible actions. The lower part includes instructions for the student in this particular situation.
Figure 1 The problem-solution window of the T-algebra program The example shows, how the student would solve problems using pencil and paper according to the solution algorithm taught at school and how the student would complete the same stages of dialogue in the program. Consider the following initial problem: solve the equation 3u − 1 u + 7 1 − 8u − = − 1 . According to the algorithm the 5 3 15 student would multiply both sides of the equation by the common denominator 15 and then move terms to other side (variable terms to the left side, constant terms to the right side). After these steps the student would obtain the equation 9u − 5u + 8u = 1 − 15 + 3 + 35 . When solving this problem further on paper, the student would at first examine the expression. Suppose that he decides to combine like terms. Then he would underline the like terms he wants to combine and write the resulting equation on the next line. Figure 1 shows the same problem being solved in T-algebra. The first two solution steps have been completed (multiply both sides, move terms to other side). When applying the Combine like terms rule the program follows principally the pencil and paper scheme of actions. The corresponding solution step consists of the following
2006 Research Information Ltd. All rights reserved.
three stages (the first two are already completed in Figure 1): 1 Selecting a transformation rule – the student selects from the rule list the rule of combining like terms – the program allows selecting any rule without checking whether it is possible to apply such a transformation at that stage or not. 2 Marking the parts of the expression – the student marks the monomials similar to u , using the mouse – the program checks whether the selected parts of the expression are actually like terms, and it also checks whether these terms can be combined (i.e., whether they belong to the same sum). Students do not have to select all suitable terms at one time – the minimum selection needed is two similar monomials. 3 Entering the result of the application of the selected rule – the program copies unchanged parts of the expression onto the next line and asks the student to enter the resulting monomial or its parts depending on the solution mode. The third stage has the greatest potential for mistakes, because the student must apply the rule for the marked parts and enter the result. Three different input modes were designed for each rule to achieve better diagnosis of errors (Issakova, Lepp and Prank,
T-algebra: Adding Input Stage To Rule-Based Interface For Expression Manipulation 2005). Different input modes are described in more detail in the next section. This example should provide an idea of the connection between the actions of the student and the program, what is checked by the program and when. If an error message is displayed at any checking stage during solution of a problem, the student must first correct the error himself or let the program correct the error in order to proceed to the next stage. For example the student cannot proceed further if he has entered the wrong coefficient of monomial (Figure 1) – the program will diagnose this standard mistake and display an appropriate error message. For each action of the student, the program gives specific instructions (‘Choose the rule to apply next’, ‘Select terms to combine’, etc.). The student can cancel the step at any time. It is also possible at any stage of the step to ask the program for help and let the program complete certain stages automatically. During the input of the result the student can press the special button with the computer image and the program will put the right answers into the boxes (Figure 1). The same help button is also available when marking the operands – the program will select appropriate operands itself. Before every step the student can ask the program which rule should be applied at this moment according to the algorithm by pushing the button Hint (Figure 1). The same button will indicate if the problem is solved. 3
ENTERING THE RESULT APPLICATION OF THE RULE
OF
THE
The Action-Object-Input scheme was first developed in the Master’s Thesis of D. Lepp in 2003 (Lepp, 2003), which serves as a prototype of T-algebra. The author designed the input forms separately for each conversion rule, trying thereby to minimise the input and requiring entering only critical information for any particular operation. The form and number of parts that could be entered became too varied for different rules and the user interface of the program became too confusing. In T-algebra we try to design three fairly uniform and standard input modes for all rules. The three input modes are named free input, structured input and partial input. Free input mode is easily comprehensible (it is similar to working on paper) and it can be designed for each rule. Structured and partial input modes are more specific. The program helps the user in a certain way, whether by indicating the structure of the result or even filling out a part of the result. It turned out that there are some rules for which it is impossible to apply both of these modes. At the third stage (Input) of each step, the student should enter some parts of the expression that result from the previously selected operation. The program generates the expression in the next line based on the selected rule and marked parts, and leaves blank certain important parts of the resulting expression. When working with paper and pencil, the students themselves have to write the whole resulting expression. Consequently, they try to reduce the amount of routine rewriting by making several transformations at once. The program makes the work
[3
easier for the students by copying the parts of the expression that remain unchanged so that the students would have to enter only the parts that were modified. Only one transformation can be made in each step. This makes it easier for the program to check the solutions and gives the teacher a better overview of the student’s solution. The results can be entered on the keyboard or on the virtual keyboard (see Figure 2).
Figure 2 Input of the result (in free input mode) The parts of the expression that the student has to enter are highlighted with yellow boxes. The form and the number of user-definable parts depends on the selected rule, marked parts and mode. While entering the results, the program protects other parts of the expression from modification – only the highlighted locations of the expression can be modified. This makes it easier for the program to check the solution and, in addition to checking the equivalence between the new expression and the previous one it also enables the correctness of separately entered parts to be checked, thus improving the overall responsiveness of the program to errors. The input mode is selected by the teacher during problem composition. Three Input Modes In free input mode the program generates one input box (or two boxes in the case of some rules with fractions and equations) inside the expression on the next line instead of marked parts (see Figure 2). The student should enter in the box one expression replacing the whole marked part from the previous line. Even though the name of the mode is free input, the input is still restricted to some extent. The editor gives the student freedom for entering, but after the input, the program checks not only the syntactical correctness of the expression and equivalence to the previous expression, as also occurs with APLUSIX (Nicaud, Bouhineau and Chaachoua, 2004), but also the correctness of applying the rule. For example, in Figure 2 the rule Combine like terms was selected and two like terms were marked. After the input is confirmed, the program first checks whether the entered part is syntactically correct, monomial and equivalent to the marked parts. Finally the program checks the equivalence of the complete new line with the previous line. If the student enters 2 y 2 x without the leading addition sign then the entered part is equivalent to the marked parts but the whole expression is not. In some other rules the student should type brackets around the entered sub-expression. International Journal for Technology in Mathematics Education, Volume 13, No 1
4]
Marina Issakova, Dmitri Lepp and Rein Prank
In structured input mode, the program uses the information about the actual rule and operands, and itself predicts the structure of the required input using different input boxes for signs, coefficients, variables, exponents, etc (see Figure 3).
by itself. For example, Figure 4 shows the same example as Figure 3, but using partial input. The program itself writes the variables with exponents. The user should enter only the sign and coefficient of the monomial. The program also simplifies the work of the user by converting the monomial into normal form. After the input the program checks the correctness of the expression and its equivalence to the previous one as in other modes.
Figure 3 Structured input The size and position of the boxes should immediately indicate to the user what should be entered. In this mode, input into the boxes is restricted. If the cursor is in some input box, the buttons with unavailable symbols on the virtual keyboard are inactive and corresponding keys on the regular keyboard do not work. For example in Figure 3, where the rule Combine like terms was selected and two like terms were marked, the program offers a structure of monomial with six boxes in the next line. The first box is the sign input box, the next is the coefficient input box (active in Figure 3) followed by boxes for input of variables with exponents. The program generally offers the same number of boxes for the variables of one monomial as the number of variables in the marked parts. Variables can be entered in arbitrary order inside one monomial, but the program requests the user to standardise the result to some extent (for example yxy to xy 2 or y 2 x in Figure 3). It is possible to leave some boxes empty. For example, if the power of the variable is 1, then the exponent box can be left empty. When the user has finished entering, the program checks whether the new expression is equivalent to the previous one and whether the entered parts are equivalent to the parts determined by the computer. If the expressions are not equivalent, the program checks the correctness of each entered part to produce a more specific diagnosis. Structured input mode is rule-specific (each rule requires a unique input pattern of the resulting expression) and it turned out that this mode is useless for some rules. For example, it would be pointless to offer a structure for the result if the applied rule was Remove parentheses, because only signs change. The third mode (partial input) is a simplified form of the second mode, where the program fills some boxes 2006 Research Information Ltd. All rights reserved.
Figure 4 Partial input
Additional Input While designing the rules for T-algebra we found that it is difficult to express some rules purely in terms of ActionObject-Input dialogue (Issakova and Lepp, 2004). In order to decide which features we need to add to the dialogue we studied students’ written work – how and which steps they make while solving problems on paper. We also reviewed school textbooks to find all the rules used in the solution steps and the algorithms used for solving the problems. In virtually every topic we found some rules where adequate expression required modification of the dialogue. We extended the input stage of the dialogue by adding three new features (Lepp, 2005). Each rule may use one or several of these features at once, depending on the mode running:
• input of the rule-specific additional information, • input of intermediate result, • adding terms to the result. In some rules the result of the application is not uniquely defined by the operands but depends on some additional decision of the student. For example, Estonian textbooks suggest writing addition of fractions with different 4 3 denominators as follows: 1 + 3 = 4 + 9 . Here the students
6
8
24
first calculate the common denominator of the fractions being added and write it in the resulting fraction after the equality sign. Then the students find so-called extenders (“extenders” are numbers by which you need to multiply both numerator and denominator of the fraction for converting the denominators to the common denominator - this term is used in Estonian schools and textbooks (Nurk, Telgmaa and Undusk, 2000)) and write them to each addend. Even if this information (common denominator) is included in the final input, it could be very difficult to guess if the input is inconsistent. This information is
T-algebra: Adding Input Stage To Rule-Based Interface For Expression Manipulation also needed for checking the extenders in intermediate input stage. As we still want to check the students’ skills and identify the cause of errors, this specific information has to be entered separately. For each such rule that needs
[5
additional information a separate input window was created. When adding fractions, the student has to input one number – the common denominator of the selected fractions (see Figure 5). Similar input was used in the MathPert system (Beeson, 1998).
Figure 5 Input of the rule-specific additional information for addition of fractions with different denominators This added window is the first new feature that can be followed by other options or the usual input of the resulting expression. When the objects of the rule have been selected, the program checks whether the rule is applicable to them and after that displays this input window to the user. After the student has made the input in this window, the program checks whether the entered information is correct. If no errors are diagnosed, the student may proceed to the next stage.
Figure 6 Input of intermediate result when adding fractions with different denominators Looking at pencil and paper solutions of the students, we found that some rules are applied using two input stages: first some intermediate result is found (for example, extenders for each fraction are found when adding fractions with different denominators) and then the final result is written. We tried to follow the same pattern while extending the dialogue that is used when working with pencil and paper: at first, the common denominator is entered in an additional window, then the extenders of the fractions are entered (Figure 6) and after that the members of the final result are entered. As we wanted to keep the initial expression unchanged with the objects selected in it, after entering the common denominator the program copies the expression to a new line and provides boxes for
entering extenders. The same constraints are used here as in structured or partial input – the boxes only allow numbers to be entered. After the intermediate result has been entered, the program checks the correctness of the entered parts. In the case of an error the student is given an appropriate message and the program lets the student correct the result before proceeding. If no error is diagnosed then the program constructs the result of applying the rule based on all the information entered and lets the student enter some parts of the result, depending on which solution mode is in operation. Most rules that are used for making transformations to algebraic expressions actually shorten the initial expression. However the rules dealing with polynomial multiplication lead to the growth of the expression and the structure of added terms differs from the structure of the terms that caused this growth. In free input mode, the student has to build the structure of the result himself. In the particular form of structured input mode described above, we would be giving the student too many hints on the structure of the result – he would see the number and the kinds of terms in the result. We have found a better solution. The members of the resulting sum have the same general structure. Instead of drawing the boxes for all terms we can draw the box for the first term and give the possibility to add more terms dynamically by adding or removing monomial structures. When checking the result the program checks whether an appropriate number of terms was added and it also checks each term separately. Figure 7 shows an example of adding terms to the structure of the result in the rule of multiplying two polynomials. The result of application of this rule is also a polynomial that the student has to construct of monomials. At International Journal for Technology in Mathematics Education, Volume 13, No 1
6]
Marina Issakova, Dmitri Lepp and Rein Prank
first one monomial structure is given (Figure 7 on the left). Then the user can extend the structure by pressing the appropriate button on the virtual keyboard and the program adds one more monomial (Figure 7 on the right
shows added monomial, input boxes are filled with the parts of the result). This mode requires exact application of the multiplication rule only; combining similar terms is not allowed.
Figure 7. Adding terms to the result in the rule of multiplying two polynomials
3
4 EXAMPLE – RULE MULTIPLY BOTH SIDES
The first step in solving linear equations according to school algorithms (Nurk, Telgmaa and Undusk, 2000) is the following: use the multiplication property to remove fractions if present. The first stage in inputting the result (input of the rule-specific additional information) is the input of a number (common denominator) in the separate window, by which you want to multiply both sides of the equation. The program then checks whether the number is suitable as the common denominator. If a suitable number is entered, then input of extenders takes place (input of intermediate result). Extenders should be entered for every term of the equation (Figure 8, expression in the second
row). The program then checks the correctness of extenders. After the right extenders have been entered, it is possible to go to the second stage of inputting the result – input the result of multiplication. In the case of free input, only an equality sign and two boxes appear in the next line. In the case of structured input, an equality sign and a number of boxes appear in the next line. There are two kinds of boxes: small boxes are for input of signs + and –, larger boxes are for entering numbers and variables. The number of boxes corresponds to the number of terms in the result of multiplication. In the case of partial input, the same boxes are given, but less data should be entered. Variables are already displayed and the user should enter only the signs and numbers.
Figure 8 Applying the rule Multiply both sides Figure 8 shows an example of the application of this rule in which the first stage is done (common denominator and extenders have been entered). In the next line there are all three input modes, where the user is entering the result of multiplication.
5
STUDENT TRIALS
2006 Research Information Ltd. All rights reserved.
When designing the program, we have taken into account the results of a study of pupils’ mistakes made when working with pencil and paper. The study was carried out in Tartu in the winter of 2005. For this study, mathematics teachers wrote a 45-minute test for each topic (operations with fractions; solving linear equations, inequalities and systems of linear equations; simplification of polynomial expressions). The topics of the test had been covered before. The same types of problem were chosen for the test as are
T-algebra: Adding Input Stage To Rule-Based Interface For Expression Manipulation available in T-algebra. Three to four classes of pupils participated in each test (93 students of 7th grade (13 years old) participated in the linear equations test, 81 students of 8th grade in the polynomial simplification test, etc.). The results and conclusions for T-algebra are described by Issakova (2005). After this, in the spring of 2005 the same pupils participated in the trial of T-algebra. T-algebra was in the development phase at that time, and therefore the objective of this trial was to validate only the user interface of the program from the point of view of its usability. Two topics were chosen for that purpose: operations with fractions and simplification of polynomials (the same topics were covered in paper tests). In this trial, the pupils were given exactly the same problems as in previously completed tests on paper. In addition, the problem set contained some demonstration examples from other chapters. The trial was conducted in two different classes. A 6th grade class was chosen for the topic of operations with fractions and an 8th grade class for the topic of simplification of polynomials. The pupils already had sufficient experience with computers (using the keyboard, mouse, Windows), but it was the first time they had seen T-algebra. The pupils could choose whether they wanted to sit at the computer alone or in pairs. For operations with fractions we had 25 computers occupied by the pupils and for simplification of polynomials 21 computers were occupied. The sessions lasted one hour. During the first five minutes we demonstrated T-algebra and the solution processes in T-algebra and wrote our general dialogue scheme on the blackboard. In the first ten minutes, the pupils asked questions concerning the use of the computer (keyboard), the use of T-algebra tools (how to mark the objects and what to enter into the boxes), and mathematical questions about the solution steps. After that, questions concerning the use of software disappeared. Questions about mathematics (on operations with fractions and polynomials) continued after the first ten minutes. Questions relating to which rule to select in the menu continued to be asked throughout the trial. At that time the concrete problem types were not yet implemented in our program and the menu contained all the rules needed for the actual topic. In most cases, the pupils even knew how they wanted to change the expression but they were often not able to find the name of the necessary operation. It is clear that we should pay attention to this issue when preparing the teachers for using rule-based software. We collected the records of this trial – files with data about errors made by pupils – for further study. The collected data included initial expression, current expression, selected rule, marked objects, entered parts (in the case of an error at the input stage) and any error message shown to the student. We also had some notes taken by the observers during the trial (two mathematics teachers and the four authors of T-algebra). When reviewing the files containing the students’ mistakes, initially we noticed that almost all the pupils had made mistakes in marking the objects for applying the rule. The reason was probably that the pupils did not understand how to use the software – how and which parts of
[7
the expressions had to be marked for applying the rules. The mistakes of this type occurred two or three times in the beginning and then disappeared. Almost all subsequent mistakes were due to a lack of mathematical knowledge (how to calculate the result of applying the rule, arithmetic errors, etc.) – pupils made the same mistakes as they made in paper tests. When reviewing the trial, we noticed that many pupils preferred to mark the objects of the rule before selecting the rule itself (despite the “Select the rule” instructions on the screen and the instruction “1. Select the rule. 2. Mark the operands. 3. Enter the result” on the blackboard). At that time our program gave no opportunity for marking more than one part in the expression before the rule was selected – it confused some of the pupils and they asked questions about that. After the trial we added the possibility to select objects for applying the rule before the rule itself is selected. Yet the hints on selection of objects become available only after selection of the rule. We are also planning a study to determine which order of actions will be used the most and what could affect the order (actual rule, location of menu on the screen, etc). In November 2005 we organised one more trial, this time in one class of so-called “difficult” children, who were studying in the 8th grade for the second year. There were 15 pupils and they had 45 minutes to try T-algebra. The problem set contained 20 problems: 5 problems on combining like terms (this topic was already covered before) and 15 problems covering a new topic (5 on addition of polynomials, 5 on subtraction of polynomials, and 5 problems combining both addition and subtraction). Their teacher usually prepares the same number of problems for the pencil and paper work in the same topic. After an introduction to T-algebra and solving the first 5 problems the teacher explained the new rule – Clear parentheses. He solved one problem on the blackboard and after that the pupils solved the remaining fifteen problems (based on the new material) in T-algebra by themselves. By the end of the trial almost everyone had solved all the problems; the pupils were solving the problems with great interest although mathematics is not one of their favourite subjects. As the set of possible rules was limited, the pupils did not have difficulties in selecting the correct rules. In this session we saw that when the students made a mistake and the program displayed an error message, many of them were closing the message window without reading the diagnosis. After that they were unable to correct the error and they even thought that their result was the correct one. Therefore we have now added a small delay for the error messages – the students cannot close the window for the first 3 seconds and probably some of them will now read the message. In the teacher’s opinion, the ability of the pupils to recognise similar terms in expressions improved after this session. He thought it was probably because they had to mark similar terms explicitly when working in T-algebra. Our development program is not yet completed, and therefore we have not organised more comprehensive
International Journal for Technology in Mathematics Education, Volume 13, No 1
8]
Marina Issakova, Dmitri Lepp and Rein Prank
experiments. Summarising the results of first trials, we can say that the time required for learning the dialogue stages is quite short. In the first hour with T-algebra, most of the pupils had solved the same number of problems that were given to them in paper sessions. But unlike in the paper tests, the pupils corrected all the mistakes they made. Error messages shown by the program were clear enough for the students to correct the mistakes. Different input modes of different rules were tested during the trial – all input modes were found useful. When solving the problems, no questions were asked on why all three stages of the dialogue are needed; the idea of the first two stages was clear to the pupils. All the pupils (even the weakest in mathematics) were using the program with great interest.
ACKNOWLEDGEMENTS T-algebra is developed as a project financed by the ‘Tiger Leap’ computerisation programme for Estonian schools. During the preparations for the project, the authors were supported with grant no. 5272 of the Estonian Science Foundation. Doctoral students’ studies are supported by the Estonian Information Technology Foundation.
REFERENCES Beeson, M. (1998) Design Principles of Mathpert: Software to support education in algebra and calculus, Kajler, N. (ed.) Computer-Human Interaction in Symbolic Computation, Berlin/ Heidelberg/ New York: Springer-Verlag, 89-115. Issakova, M. (2005). Possible Mistakes During Linear Equation Solving On Paper And In T-algebra Environment. Olivero, F. and Sutherland, R. (eds), Proceedings of the 7th International Conference on Technology in Mathematics Teaching, Volume 1, Bristol: University of Bristol, 250-258. Issakova, M. and Lepp, D. (2004). Rule dialogue in problem solving environment T-algebra, Böhm, J. (ed.) Proceedings TIME-2004: Montreal International Symposium on Technology and its Integration into Mathematics Education, Linz: bk teachware. Issakova, M., Lepp, D. and Prank, R. (2005) Input Design in Interactive Learning Environment T-algebra, Goodyear, P., Sampson, D.G., Yang, D.J.-T., Kinshuk, Okamoto, T., Hartley, R. and Chen N. S. (eds.) Proceedings ICALT-2005: The 5th IEEE International Conference on Advanced Learning Technologies, Kaohsiung (Taiwan): the IEEE Computer Society Press, 489-491. Lepp, D. (2003) Program for exercises on operations with polynomial, Triandafillidis, T. and Hatzikiriakou, K. (eds), Proceedings of the 6th International Conference Technology in Mathematics Teaching, Volos: New Technologies Publications, 365-369. Lepp, D. (2005) Extended Solution Step Dialogue In Problem Solving Environment T-Algebra, Olivero, F. and Sutherland, R. (eds), Proceedings of the 7th International
2006 Research Information Ltd. All rights reserved.
Conference on Technology in Mathematics Teaching, Volume 1, Bristol: University of Bristol, 267-274. Nicaud, J., Bouhineau, D. and Chaachoua, H. (2004) Mixing microworld and CAS features in building computer systems that help students learn algebra, International Journal of Computers for Mathematical Learning, 5, 169-211. Nurk, E., Telgmaa, A. and Undusk, A. (2000) Mathematics for VII grade (in Estonian), Tallinn (Estonia): Koolibri. Ravaglia, R., Alper, T., Rozenfeld, M. and Suppes, P. (1998) Succesful pedagogical applications of symbolic computation. Kajler, N. (ed.) Computer-Human Interaction in Symbolic Computation, Berlin/ Heidelberg/ New York: SpringerVerlag, 61-88.
BIOGRAPHICAL NOTES Rein Prank is an Associate Professor at the University of Tartu (Estonia). He obtained his degree in mathematical logic from Moscow University in 1982. His research interests include the development of problem solving environments and the use of computers in teaching mathematics and logic. Marina Issakova and Dmitri Lepp are both PhD students at the University of Tartu (Estonia), Institute of Computer Science. All their major studies at the university (Bachelor and Master theses) have been connected with developing educational software for studying mathematics. Some of their programs are currently used in Estonian schools.
