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Distribution of Student Mistakes between Three Stages of Solution Steps in Case of Action-Object-Input Solution Scheme

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## Distribution of Student Mistakes between Three Stages of Solution Steps in Case of Action-Object-Input Solution Scheme

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**Distribution of Student Mistakes between Three Stages of**Solution Steps in Case of Action-Object-Input Solution Scheme Dmitri Lepp University of Tartu Estonia**Outline**• Expression manipulation problems • Study of student mistakes on paper • Introduction to T-algebra • Decisions made for T-algebra • Study of student errors in T-algebra • Conclusions**Expression manipulation problems**• Simplifying polynomial expressions • Very important in school mathematics • Traditional study – drilling method • technical exercises are solved • errors made by the student and the teacher’s reactions to them are very important**Real cause of error**• Find the expression that is not equivalent to the previous • Student decides himself • solution algorithm • step to perform next • parts of expression • result of operation**Study of errors on paper**• Simplification problems • Two different groups of students (7th and 8th grades) • Tests solved on paper • Errors grouped by rules that students applied • Rules easily extractable • Compare mistakes of 7th and 8th grade students**Examples of problems**• 7th grade test • combining like terms (7 problems with at most one variable in a monomial) • multiplication of monomial (usually single number or variable) by a polynomial (8 problems) • 8th grade test • combine like terms (4 problems, many variables) • multiply or divide polynomial by a monomial (7 problems) • multiply polynomials (10 problems) • problems requiring application of all the mentioned operations (4 problems)**Mistakes in multiplying or dividing of a polynomial by a**monomial (operation, objects)**Mistakes in multiplying or dividing of a polynomial by a**monomial (input of the result)**Student mistakes**• Errors made by the students are of different kind • Transformation rule and objects • Calculation of the result • In some rules up to 30% students make mistakes in choosing the transformation and objects for it • While calculating and writing the result of transformation in some cases up to 50% students make errors • Depending on the rule and grade of the student up to 25% students forget to copy unchanged parts of expression to the next line in the solution • Almost disappear in higher grades**Errors in rule-based environments**• MathPert, EPGY • Opportunity to make mistakes is limited • The student selects at best only • The transformation rule (sometimes only from the list of suitable rules) • A part of expression • The transformation is performed by the computer • Mistakes only in selection of the solution step and/or parts of expression • No possibility to make errors in the calculation of the result of the operation • Suitable for learning algorithms, learning of application of the operations is passive**Errors in input-based environments**• Aplusix • The same mistakes as on paper • Solving consists simply of entering the next line • Similar to working on paper • The program can diagnose only the non-equivalence of the new entered expression with the previous one • The program cannot offer any specific error messages**Decisions for T-algebra**• Designing T-algebra we had to leave possibility for making all groups of errors and make it easy for T-algebra to detect these errors • We need information on student’s intentions for better error diagnosis of exact mistake • It is almost impossible to tell anything about exact error from two consecutive expressions • This resulted in action-object-input dialogue scheme • selecting a transformation rule (action) • marking the parts of expression (object) • entering the result of the application of the selected rule (input) • Expressions are modified using transformation rules, which are supported by the input of the resulting expression**Introduction to T-algebra**T-algebra environment, which enables step-by-step problem solving in four fields of mathematics • calculation of the values of numerical expressions • operations with fractions • solution of linear equations, inequalities and linear equation systems • simplification and factorisation of polynomials**Essential properties of T-algebra**• Student makes the same steps as in the solution algorithms taught at school • Problem solving with the program is very similar to working with pen and paper • Student makes all the decisions and calculations • The program plays the role of a teacher • monitors, whether the student works according to the algorithm • supports him with the respective dialogue • diagnoses transformation errors • offers advice on the selection of the next transformation • performs the next step by itself**Error diagnosis in T-algebra**• Error diagnosis principle in T-algebra environment: • error message is displayed as soon as error is found • student is unable to proceed to the next stage before the errors are corrected • program tries to diagnose the exact error and error position and displays it to the student**Study of student errors in T-algebra**• 21 students from 8th grade • Same problems as in paper tests • 1 hour of work in T-algebra • 15 minutes of introduction • 45 minutes of solving problems • Collected students solutions, error logs and studied them**Results of the study(combine like terms)**action objects input**Results of the study(multiply the monomials)**action objects input**Results of the study(raise the monomial to a power)**action objects input**Conclusion**• Most important typical errors that students make on paper can also be made when solving in the T-algebra • Action – Object – Input scheme and error diagnosis after each step was found useful • T-algebra is able to diagnose error before inputting the result of transformation though preventing the student from unnecessary computation and input • Approximately the same number of students made certain types of mistakes in T-algebra and on paper • Students corrected their mistakes during solving problems • Have solved at least the same number of problems as on paper**Distribution of Student Mistakes between Three Stages of**Solution Steps in Case ofAction-Object-Input Solution Scheme Questions, Comments...