distribution of student mistakes between three stages of solution steps in case of...

Distribution of Student Mistakes between Three Stages of Solution Steps in Case of

Action-Object-Input Solution Scheme

Dmitri LeppUniversity 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)

6 10b b

(2 4 5)a b b

2 3 2 23 2u v uuv v u vvv 3 2 2 3 2 2(20 12 4 ) : ( 4 )x y x y xy xy

2( 3) (2 3 1)u u u

Mistakes in combining like terms (operation, objects)

2 2 2

3 7 10

3 2

m m

ab a b ab

2 2 2

7 2 2 3

5 5 10

b b b b

ab ab ab

2 2 23 2 4 2

2 2 3 2

x z x z xzx x z xzx

a b a b a b b

Nature of mistake Example 7th 8th

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%

Mistakes in combining like terms(input of the result)

2 2 2 2

... 4 2 ... 2

7 12 7 2

x x x

b b b b

9 4 12

10 2 8

x x x

xy yx xy xy

2 2 4 2

2

9 4 13

10 9

x z x z x z

xyx x y xy

2 22 3

2 3 2

y y x y y y x

x x x


Error in calculating the sign of the resulting monomial

24% 9%

Arithmetical error in calculating the coefficient

27% 48%

Error in powers of variables 0% 11%

Forgets to copy some unchanged terms, copies terms with mistakes

24% 6%

2

(3 2)*4 3 8

2 *(3 2) 6 2

u u

x x x

2

16 (5 3 1) 80 48 16

( 2) ( ) 2 2

x y x y

x x y x xy x y


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)

2

(3 2)*4 3 8

2 *(3 2) 6 2

u u

x x x

(3 2 4) 3 2 4

2 ( 2 ) 2 2 4

m x y mx my m

x y x x y x y x

(2 3 ) 2 2

3 (4 3 ) 7 6

a b c ab ac

x y z xy xz

2 3 2 3 4 2 3

2 5 6

(2 3 ) 2 3

3 (4 3 ) 12 9

ab a b ab a b a b

x x y x y

3 2 2 3 2 2

2

(20 12 4 ) : ( 4 )

5 3

x y x y xy xy

x xy


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)

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-algebraT-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

Application of the rule combine like terms

Application of the rule Multiply polynomial by a monomial

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

Examples of error diagnosis

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)

Nature of mistake % of students

Unsuitable operation – rule combine like terms cannot be applied

19%

Unsuitable objects – selected objects are not like 42%

Unsuitable objects – selected monomials do not belong to the same sum, or one of the monomials is a part of product

28%


47%


23%

Mistake in calculating the mark before single monomial in the result

33%

action

objects

input

Results of the study(multiply the monomials)


Unsuitable operation – rule multiply the polynomials cannot be applied

14%

Unsuitable objects selected 23%


29%


47%


10%

action

objects

input

Results of the study(raise the monomial to a power)


Unsuitable operation – rule raise monomial to a power cannot be applied

29%

Unsuitable objects selected 19%


10%


28%


10%

action

objects

input

• 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

Distribution of Student Mistakes between Three Stages of Solution Steps in Case of

Action-Object-Input Solution Scheme

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