Examining the errors made by high school students while applying the Pythagorean Theorem

Taamneh, Muntasir Ali; Díez-Palomar, Javier; Mallart-Solaz, Albert

doi:10.1590/1980-4415v39a240229

person Muntasir Ali Taamneh

All authors contributed substantially to the conception and planning of the study · to the acquisition · analysis · and interpretation of data · to the writing and critical revision · and approved the final version to be published

school Estudiante de Doctorado de Educación Matemática, Universidad de Barcelona (UB). Profesor de secundaria en la escuela Alhosh, Abu Dhabi, Emiratos Árabes Unidos. E-mail: mtaamnta7@alumnes.ub.eduschool

0009-0006-3425-1520

person Javier Díez-Palomar

All authors contributed substantially to the conception and planning of the study · to the acquisition · analysis · and interpretation of data · to the writing and critical revision · and approved the final version to be published Doctor en Educación Matemática, Universidad de Barcelona (UB), España.

school Profesor de la Universidad de Barcelona (UB). Barcelona, España. E-mail: jdiezpalomar@ub.edu

0000-0003-4447-1595

person Albert Mallart-Solaz

All authors contributed substantially to the conception and planning of the study · to the acquisition · analysis · and interpretation of data · to the writing and critical revision · and approved the final version to be published Doctor en Educación Matemática, Universidad de Barcelona (UB), España.

schoolProfesor de la Universidad Autónoma de Barcelona (UAB), Bellaterra, España. E-mail: albert.mallart@uab.cat

0000-0003-3442-1551

Editor-chefe responsável:

Prof. Dr. Marcus Vinícius Maltempi

Editor associado responsável:

Prof. Dr. Danyal Farsani

Previous Research	Type of Errors	Correspondence with our Classification
Hanggara; Agustyaningrum; Ismayanti, (2024)	Conceptual errors Procedural errors Principal errors	Conceptual error Strategy error Concept error
Sekgoma (2023)	Calculations error Misidentification error Interpretation error	Calculations error Conceptual error Conclusion error
Arivina and Jailani(2022)	Misinterpret language Misuse data Distort the theorem definition	Data error Data error Conceptual error
Setiawan (2022)	Misconception error Miscalculation error Fact errors	Conceptual errors Calculations errors Conceptual error
Supardi (2021)	Word use error Visual mediator error Narrative error	Data error Data error Conceptual error
Fahrudin; Pramudya (2019)	Data error Conceptual error Strategy error Calculation error Conclusion error	Data error Conceptual error Strategy error Calculation error Conclusion error
Hutapea; Suryadi; Nurlaelah, (2015); Sari; Wutsqa (2019); Zulyanty; Mardia (2022)	Decoding Comprehension Transformation Process skill Encoding	Data error Conceptual error Conceptual error Calculation error Calculation error
Veloo; Krishnasamy; Wan Abdullah (2015)	Conceptual errors Careless errors Problem-solving errors Value errors	Conceptual errors Careless errors Strategy error Calculation error

Types of Errors	Indicator
Data Error	Students are writing down existing problem data wrongly. Students are translating existing problems wrongly.
Concept Error	Students are determining the formula or theorem or definition to answer the problem incorrectly. Students do not write formulas or theorems or definitions to answer problems.
Strategy Error	Students try to operate at the right level on a problem but use procedures or methods that are not appropriate. Students try to operate at the right level but choose inappropriate data information.
Calculation Error	Students are giving or writing signs of mathematical operation incorrectly. Students are wrong in counting maths operations such as adding, subtracting, multiplying, and dividing wrongly.
Conclusion Error	Students are determining conclusions incorrectly Students do not write conclusions

Indicator	Name of the error
Indicator 1	Finding the unknown side length in a right-angled triangle
Indicator 2	Determine the hypotenuse of a right-angled triangle
Indicator 3	Determining the lengths of the sides in a right-angled triangle (45º -45º- 90º)
Indicator 4	Determining the lengths of the sides in a right-angled triangle (30º -60º - 90º)
Indicator 5	Distinguishing a right-angled triangle from other triangles
Indicator 6	The geometric meaning of the Pythagorean theorem
Indicator 7	Verify that a triple is a Pythagorean triple
Indicator 8	Correct understanding of the Pythagorean theorem based on the given information
Indicator 9	Finding the trigonometric ratios of a right-angled triangle (sin, cos, tan, cot, sec, csc)
Indicator 10	Finding the value of the angle using the inverse of the trigonometric functions (arcsin, arccos, arctan)
Indicator 11	Finding the area of a right-angled triangle

ID	Question	Mathematical Objective / Concept
01	Find the area of square B in the below figure. (justify your answer).	The geometric meaning of the Pythagorean theorem.
02	Which of these sides form a right-angle triangle? (justify your answer). (a) 5, √8 33 (b) 8, 15, 17 (c) √2, √2, 2 (d) 4, 8, √32	Verify that the triple is a Pythagorean triple.
03	The diagram below shows a trapezium ABCD, find the length of hypotenuse of triangle DCE to the nearest tenth. (a) 16.2 m; (b) 8.5 m; (c) 13.7 m; (d) 10.8	Finding the unknown side length in a right-angled triangle. Determine the hypotenuse of a right-angled triangle.
04	Find the unknown side length in the right-angle triangle to the nearest whole num-ber? (justify your answer). (a) 2√2; (b) 4; (c) 4√2 ; (d) 2	Determining the lengths of the sides in a right-angled triangle (45 - 45 - 90).
05	What is the value of the length y in the right-angle triangle? (a) 6√3; (b) 2√3; (c) 6; (d) 12	Determining the lengths of the sides in a right-angled triangle (30 - 60 - 90).
06	The high of the triangle is 17 cm less than the length of its base, and the length of the hypotenuse is 25 cm, find the area of this triangle. (justify your answer).	Correct understanding of the Pythago-rean theorem based on the given in-formation. Finding the area of the right-angled triangle.
07	Find the measure of < B to the nearest degree. (a) 44 m; (b) 55 m; (c) 46 m; (d) 35 m	Finding the trigonometric ratios of a right-angled triangle (sin, cos, tan, cot, sec, csc). Finding the value of the angle using the inverse of the angle.
08	The sides of a figure have 12 cm, 18 cm, and 20 cm, what type of figure is it. (justify your answer). (a) Right triangle (b) Acute triangle (c) Obtuse triangle (d) Not triangle.	Distinguishing a right-angled triangle from other triangles.

	Experimental Group		Control Group
Variable	Mean	SD	Mean	SD
Q1	0.88	1.0	0.67	0.92
Q2	0.79	0.84	1.33	0.85
Q3	0.68	0.77	1	0.90
Q4	0.32	0.53	0.12	0.33
Q5	0.47	0.56	0.27	0.45
Q6	0.03	0.17	0.09	0.29
Q7	0.47	0.51	0.58	0.50
Q8	0.24	0.43	0.12	0.41
Percent average	0.49	0.60	0.52	0.58

	Experimental Group						Control Group
Variable	Data	Concept Error	Strategy Error	Calculation Error	Conclusion Error	Careless Error	Data Error	Concept Error	Strategy Error	Calculation Error	Conclusion Error	Careless Error
Q1	13 38.2%	23 67.6%	8 23.5%	6 17.6%	0 0%	2 5.9%	20 60.6%	31 93.9%	16 48.5%	4 12.1%	0 0%	1 3%
Q2	15 44.1%	28 82.4%	2 5.9%	0 0%	8 23.5%	2 5.9%	2 6.1%	24 72.7%	0 0%	5 15.2%	3 9.1%	5 15.2%
Q3	19 55.9%	15 44.1%	7 20.6%	4 11.8%	0 0%	9 26.5%	8 24.2%	19 57.6%	8 24.2%	3 9.1%	0 0%	9 27.3%
Q4	8 23.5%	13 38.2%	4 11.8%	2 5.9%	0 0%	20 58.8%	15 45.5%	18 54.5%	10 30.3%	1 3%	1 3%	15 45.5%
Q5	10 29.4%	22 64.7%	3 8.8%	5 14.7%	3 8.8%	12 35.3%	10 30.3%	15 45.5%	4 12.1%	1 3%	2 6.1%	17 51.5%
Q6	21 61.8%	16 47.1%	8 23.5%	5 14.7%	0 0%	11 32.4%	22 66.7%	16 48.5%	4 12.1%	10 30.3%	0 0%	11 33.3%
Q7	3 8.8%	18 52.9%	1 2.9%	0 0%	2 5.9%	15 44.1%	3 9.1%	19 57.6%	0 0%	0 0%	0 0%	13 39.4%
Q8	10 29.4%	14 41.2%	0 0%	0 0%	6 17.6%	18 52.9%	5 15.2%	20 60.6%	0 0%	1 3%	4 12.1%	12 36.4%
Percent Average	36.4%	54.8%	12.1%	8.1%	7%	32.7%	32.2%	61.4%	15.9%	9.5%	3.8%	31.5%

[TFN1] Source: Author elaboration(2023)

Examining the errors made by high school students while applying the Pythagorean Theorem

Analisando os erros cometidos por estudantes do ensino médio ao aplicar o Teorema de Pitágoras

Abstract