1. Distribution of Student Mistakes between Three Stages of Solution Steps in Case of Action-Object-Input Solution Scheme Dmitri Lepp University of Tartu Estonia

2. 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

3. 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

4. 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

5. 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

6. 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)

7. Mistakes in combining like terms (operation, objects) Nature of mistakeExample7 th 8 th Combines non-like terms, combines terms with different variable parts 12%26% Does not take into account some signs before monomials 24%17% Doesn’t combine all terms, doesn’t recognize like terms if they are like 18%33%

8. Mistakes in combining like terms (input of the result) Nature of mistakeExample7 th 8 th Error in calculating the sign of the resulting monomial 24%9% Arithmetical error in calculating the coefficient 27%48% Error in powers of variables0%11% Forgets to copy some unchanged terms, copies terms with mistakes 24%6%

9. Nature of mistakeExample7 th 8 th Doesn’t multiply or divide one of the terms of the polynomial by the monomial 24%4% Multiplies the polynomials or polynomials by monomials instead of adding 12%20% Mistakes in multiplying or dividing of a polynomial by a monomial (operation, objects)

10. Nature of mistakeExample7 th 8 th Doesn’t multiply or divide one of the terms of the polynomial by the monomial 24%4% Doesn’t change signs of some monomials in result 42%15% Mistake in calculating the coefficient of single monomial in the result 24%6% Mistake in calculating the power of a variable in a single monomial in the result -24% When dividing the same monomials the result is 0 -15% Mistakes in multiplying or dividing of a polynomial by a monomial (input of the result)

11. 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

12. 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

13. 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

14. 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

15. 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

16. 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

17. Application of the rule combine like terms

18. Application of the rule Multiply polynomial by a monomial

19. 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

20. Examples of error diagnosis

21. Study of student errors in T-algebra 21 students from 8 th 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

22. Results of the study (combine like terms) Nature of mistake% of students Unsuitable operation – rule combine like terms cannot be applied 19% Unsuitable objects – selected objects are not like42% Unsuitable objects – selected monomials do not belong to the same sum, or one of the monomials is a part of product 28% Mistake in calculating the coefficient of single monomial in the result 47% Mistake in calculating the power of a variable in a single monomial in the result 23% Mistake in calculating the mark before single monomial in the result 33% action objects input

23. Results of the study (multiply the monomials) Nature of mistake% of students Unsuitable operation – rule multiply the polynomials cannot be applied 14% Unsuitable objects selected23% Mistake in calculating the coefficient of single monomial in the result 29% Mistake in calculating the power of a variable in a single monomial in the result 47% Mistake in calculating the mark before single monomial in the result 10% action objects input

24. Results of the study (raise the monomial to a power) Nature of mistake% of students Unsuitable operation – rule raise monomial to a power cannot be applied 29% Unsuitable objects selected19% Mistake in calculating the coefficient of single monomial in the result 10% Mistake in calculating the power of a variable in a single monomial in the result 28% Mistake in calculating the mark before single monomial in the result 10% action objects input

25. 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 Conclusion

26. Distribution of Student Mistakes between Three Stages of Solution Steps in Case of Action-Object-Input Solution Scheme Questions, Comments...