ICME 11 – TSG 22 Theme 3
Abstract. T-algebra is an interactive learning environment for exercises in four areas of school mathematics: calculation of the values of numerical expressions; operations with fractions; solving of linear equations, inequalities and linear equation systems; operations with monomials and polynomials. T-algebra implements more than 50 task types. The student solves the task step by step. At each solution step the student selects the operation, indicates the subexpression(s) to be changed and enters the result of conversion. The program checks all three stages of the step, gives feedback and hints.
The Introduction of the paper provides brief motivation for the design of the three-stage solution interface. The subsequent sections describe the user interface in more detail, exemplify the features of error diagnostics, and describe the information recorded about the solution process and mistakes.
1 Introduction
Expression manipulation is one of the most important mathematical skills. Learning of expression manipulation begins already in basic school. The students learn necessary properties of different kinds of expressions and solve hundreds of exercises with fractions, monomials, polynomials, equations, inequalities and systems of equations. Written solutions contain a large number of small details and there exist numerous possibilities to make mistakes. When using traditional paper-and-pencil technology the mistakes can often remain unrecognised and/or not corrected. The teachers are not able to check the solutions of all students promptly and the common method of verification of the solution is comparison of
‘my answer’ with ‘correct answer’. In many types of expression manipulation tasks, the final answer is not a uniquely defined sequence of symbols. This means that verification of correctness requires from the student quick application of some of the same skills that cause the answers to be wrong. In other types of tasks, the presence of mistake(s) can be understood quickly but it is quite difficult to find the solution step that was incorrect. The students could work more purposefully if the feedback they get during the solution process would be more systematic. What sort of computer environment could help in this case?
When the students solve an expression manipulation task, they should at each solution step
1) choose a certain operation in the algorithm for the active task type (or some simplification or calculation rule known earlier),
2) select the operands (certain parts of expressions or equations) for this operation,
3) replace them with the result of the operation.
Some task types (such as factorisation) are solved without having a ‘textbook algorithm’ but the solutions steps require from the student to make the same decisions. For economy of writing, the written solutions on paper usually do not record the first two stages of the solution step (and the result is that the teacher is often unable to decipher them quickly).
When we compare available computerized expression manipulation environments with the three-stage model of solution step, we discover that there exist two types of environments: a) rule-based environments, b) input-based environments.
A typical rule-based environment is MathXpert [1]. Using this program, the student begins each conversion step by marking some subexpression he/she intends to change. The program
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then displays a menu with a list of rules, which are applicable to the marked subexpression.
Next, the student selects a rule and the program applies the rule to the marked subexpression
(or to suitable parts of the marked subexpression). A similar conversion scheme is used in the mathematics software developed in the framework of the Stanford Educational Program for
Gifted Youth [2], but this program asks in addition the student to fill sometimes some input boxes with coefficients, exponents, etc. Some smaller programs for more restricted topics
(Cognitive Algebra Tutor by Carnegie Learning [3], AlgeBrain [4]) are likewise built around applying the rules. Solving tasks in a rule-based system requires the student to specify the steps of solution. However, each conversion step is performed by the program and learning of low-level manipulation skills is passive. Most of the common errors are simply impossible. It is clear that in order to learn to execute new operations we would need an environment where the result of operation is entered by the student.
Input-based systems use paper-and-pencil-like dialogue design where a conversion step consists mainly of entering the next line. For example, Aplusix [5] copies the content of previous line (expression, equation or system of equations) to the next line and the student should use the expression editor to convert it into the result of the step. The program displays an indicator of equivalence between the two lines, giving the student feedback about correctness of the step. MATH-TEACHER [6] is another example of input-based solution environment but the expressions in this program are entered in an old-fashioned linear form.
An early version of Cognitive Algebra Tutor [7] also used merely entering of the result and the program tried to figure out what step was performed. In input-based environments, the students have the possibility to make almost arbitrary mistakes, but as a result of this, the program does not know now what operation was applied to what part(s) of previous expression. The cases where the student has applied several operations in one step are particularly difficult. It would be very complicated to achieve a more precise error diagnosis beyond the statement that “expressions are not equivalent”.
T-algebra is an attempt to integrate the two approaches. We ask from the student more information and use this information to create more detailed feedback. In section 2 of this paper, we give brief description of a solution step in T-algebra. A more exhaustive description can be found in [8, 9]. Section 3 describes the task types. Section 4 provides an overview of error diagnostics for three stages of the step (more details can be found in [10]). Section 5 presents some outlines of hints in T-algebra and Section 6 adds a few facts about experiments with T-algebra. A more detailed description of T-algebra is published in [11].
2 Solution Step in T-algebra
Designing the step dialogue of T-algebra, we combined the dialogues of rule-oriented and input-oriented programs. Each step in T-algebra consists of three actions:
1) selection of the operation from the menu,
2) marking the operand(s) in expression,
3) entering the result of the operation.
The program guides each action, displaying the corresponding instruction under the solution window. Figure 1 demonstrates the solution window during execution of the first two stages of a step.
The upper lines of the window contain the first two steps of the solution. For the third step of the solution, the student has already selected the operation Combine like terms and has marked three terms for combining. The first two stages of the step are “mixed together” in the user interface of T-algebra and the student can perform them in arbitrary order.
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Figure 1. Solution window during execution of the first two stages of a step
T-algebra starts the verification process when the student confirms both stages by clicking the green check button on the virtual keyboard. If T-algebra detects a mistake, an error message is issued and the student should correct the mistake. If the first two stages of the step are acceptable T-algebra copies the unmarked parts from the previous expression onto the next line and asks the student to enter the result of application of the selected operation to the marked operands. T-algebra has three input modes for entering the results of operations: free input, s tructured input and partial input . In different input modes, T-algebra offers different number and types of boxes for entering the result. Figure 2 presents the three different modes of an input stage resulting from the situation presented in Figure 1. The coloured boxes are initially empty and should be filled by the student during this stage.
Figure 2. Three input modes
In the free mode, the result of conversion is to be entered in a single box (or in two boxes in case of some operations with fractions and equations).
In the structured mode , the program calculates the result of application of the operation before input and offers a set of boxes that corresponds to the structure of the expected result.
Different kinds of boxes enable entering specific parts of the expression: signs, numbers, variables, monomials, powers, etc.
In the partial input mode, T-algebra displays some parts of the result automatically – the user has to fill only the boxes with critical parts of the result.
Input in a prescribed structure helps the student to some extent. On the other hand, it creates the possibility to formulate more precise error messages and to indicate the position of errors.
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3 Task types
T-algebra implements more than 50 types of expression manipulation tasks. The types have been taken directly from textbooks. Some of them are used to practice a specific skill
( Combine like terms , Open parentheses ) and some integrate the content of a whole chapter in the textbook ( Solve linear equation ).
The task type of T-algebra determines allowed form(s) of initial expression, required form of answer, and set of rules available for solution steps. For some types, the mandatory attributes contain more items than only the main expression/equation/system. For example, the task type
Check the solution of inequality requires an inequality and a number.
For each task type, T-algebra has a built-in solution algorithm. The program uses it for finding the right answer, for generating the demo solution, and for giving dynamic hints in any situation.
4 Error Diagnostics in T-algebra
We now describe the main features of error diagnostics in T-algebra. The dialogue with three stages of the step gives the program the opportunity to distinguish between the mistakes made during the step planning phase and the mistakes of implementation. Additionally, separate verification of the first two stages prevents needless work by the students for calculation and input of defective expressions. Consider now what items does T-algebra check.
4.1 Operation
The menu of operations/rules depends on the task type. It contains operations from the textbook algorithm for the active type and necessary calculation and simplification rules.
T-algebra checks the rule only in combination with the marking of operands.
Correspondingly, if the selected rule is inapplicable, the error message does not tell this directly but belongs usually to types 3-6 in the list of subsection 4.2. However, if the student has selected an inapplicable rule and asks T-algebra for a hint on marking the operands the message about inapplicability appears. Current version of T-algebra does not check whether the selected rule corresponds to the algorithm for the actual task type.
4.2 Marking of Operands
For diagnostic purposes, T-algebra requires precise marking of operands. For example, a student combining like terms should not mark the entire sum of monomials but only the like terms that will be combined.
Secondly, T-algebra does not allow ‘parallel’ application of selected operation to two or more groups of operands. Only one group of like terms can be collected in one step, only one fraction can be reduced, etc. This makes the solution scripts longer but does not require additional work from the student because unchanged parts of the expression are copied onto the next line automatically.
T-algebra diagnoses the following errors in the selection of operands:
1) Marked part is not a syntactically correct expression.
2) Marked expression is not a proper subexpression.
3) Number of marked operands does not correspond to the selected operation,
4) Operand does not have the form required for the selected rule.
5) Operands do not satisfy the compatibility requirements (are not like terms, etc).
6) Operands do not satisfy the location requirements (do not belong to the same sum, etc).
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The actual error messages for errors 3-6 depend on the selected operation. Figure 3 demonstrates an error message displayed when a student trying to find the reciprocal value has selected only the fractional part of mixed number.
Figure 3. Error message about unsuitable selection
4.3 Result of Operation
The entered result of operation is checked in T-algebra for equivalence with the previous expression and also for matches with the possible result of application of the selected operation to the marked operands. The following errors are diagnosed:
1) Entered term is not a syntactically correct expression.
2) Entered subexpression should be preceded by a sign.
3) Entered subexpression should be put in parentheses (order of operations).
4) Entered subexpression does not have the structure required for the current operation and operands (is not a proper fraction, is not a monomial, has wrong number of members, etc).
5) Specific parts of the input (sign, coefficient, variables, exponents, denominator, etc) do not have the correct value.
6) Entered subexpression is not equivalent with the marked part.
7) Selected operation with marked terms is not performed (nothing reduced, terms are not moved to other side, etc).
The structure of the entered expression (error 4) is checked before checking the equivalence issues 5 and 6. The general non-equivalence error message 6 is issued when the program has not succeeded in establishing a more precise diagnosis of type 5. Figure 4 demonstrates an error situation with message about a specific part of the result.
Figure 4. Entered result contains sign error
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4.4 Recording the Solutions and Error Statistics
T-algebra records the work of the student performed on a specific task file in a corresponding solution file. The solution file contains the final solution path (after correction of mistakes and/or possible undoing of steps) for each activated task. If the solution is not finished then the student can continue it. The file also contains records of error situations: previous expression together with marking, selected operation, entered expression (if the mistake was made at the third stage of the step) and error message. T-algebra groups the diagnosed errors in 20 categories (most of them are listed in subsections 4.2-4.3). The Student Program enables to view the list of errors ordered by the time of occurrence, or the table where the rows are categories of errors and the columns are solved tasks. It is possible to open the description of each error in both views.
5 Hints
The student may need help at any stage of the solution process – at any step or substep. In order to generate hints, T-algebra uses its automatic Solver that implements algorithms for all task types.
The form of help provided by T-algebra depends on the actual substep that is to be performed.
When the operation is not yet selected, the student can click the Hint option on the right-hand menu and the program recommends: ‘Apply the rule …’. After selection of the rule, a button with computer monitor image appears on the virtual keyboard (Fig. 1). If the student clicks this button the program automatically marks appropriate subexpressions (or an error message is displayed if the selected rule is inapplicable). During the input stage, the same button generates automated input of the result in input box(es). It is also possible to view the entire solution of the task. The availability of each of the four hint facilities depends on the options selected by the author of the task file. T-algebra records the instances of hint use in a similar manner to the recording of mistakes.
6 Practical experiences
We started experimental classes in 2005 when only the very first task types in T-algebra were functional. We saw that the students understood the dialogue of T-algebra very quickly. After half an hour of work, the questions about the use of the program virtually disappeared. The students were able to solveat least as much exercises with T-algebra as they solve in traditional classes. In 2006, we organized the first instruction courses for teachers with 40 participants. Many of them have already used T-algebra in their schools for some topics and give us feedback about the program. In most cases, the error messages are sufficient for correcting the mistakes.
The first more systematic experiment with 126 students of seven classes from four schools was organized in the beginning of 2007. The topic of linear equations was chosen for the experiment and the experiment began when the topic had already been explained and practiced in the schools. The experiment included pre- and post-tests on paper and two 45minute exercise sessions. The experimental groups (5 classes) practiced solving the problems with T-algebra, while the control groups (2 classes) practiced solving exactly the same problems using paper and pencil. The students of experimental groups had earlier experience with T-algebra. Before counting the results, two categories of students were excluded: those who had missed at least one lesson (11 students) and those who scored less than 30 percent of the maximum points available in pre-test (9 students). The average result of experimental group (76 students) in pre-test was 29.4 points and in post-test 31.3 points from a maximum
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of 39. The control group (30 students) scored 29.3 and 29.9 points respectively.
Consequently, the two additional exercise classes with traditional technology did not produce a statistically significant difference in the results but the computerised classes did.
When we look at the types of mistakes we can see that already two hours of systematic feedback lead to a clear decrease in some technical mistakes where the student applies minus before the brackets or before the fraction only to the first member in brackets/numerator or forgets to multiply the member that is not a fraction when multiplying an equation by common denominator. However, we are not yet able to explain the somewhat chaotic changes in the number of arithmetical mistakes. A more detailed description of this experiment is published in [12]. We have conducted a longer experiment (6 exercise hours) in 2007/2008 but the results have not yet been analysed.
7 Acknowledgments
Programming of T-algebra is financed by the Estonian School Computerization Foundation
“Tiger Leap”. The authors are financed also by Targeted Financing grant SF0182712s06 of the Estonian Ministry of Education and Research and by Estonian Doctoral School in
Information and Communication Technologies.
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