Automatic Assessment of Student Answers for Geometric Construction Questions

Automatic Assessment of
Student Answers for Geometric
Construction Questions
BUDDHIMA WIJEWEERA, GIHAN DIAS, SURANGIKA RANATHUNGA
DEPARTMENT OF COMPUTER SCIENCE AND ENGINEERING
UNIVERSITY OF MORATUWA, SRI LANKA

Outline
Introduction
Motivation
Problem
Objectives
Literature Survey
System Functionality
System Evaluation
Demonstrating the System

Introduction
“A Geometric Construction is an accurate drawing done by a compass and a
straightedge following a set instructions to obtain a desired outcome.”

Motivation
“Question 9, which was from Geometry, was the least attempted
question. Only 12% of the candidates had attempted this question”
H. Silva and B. Banneheka, “Factors Associated with Mathematics Results at GCE (OL) Examination,”
Vidyodaya J. of Humanities and Social Sciences, vol. 3, pp. 23–43, 2011.

Motivation
The percentage of candidates that
have obtained marks in the range
0 - 2 is 41%
The percentage of candidates that
have obtained marks in the range
8 - 10 is 15%
(from candidates who attempted
the question)
Source: G.C.E (OL) Examination Evaluation Report - 2011 Mathematics

Problem
Students should be provided more opportunities to practice geometric
constructions
Giving close attention to geometric constructions done by each student
is a time consuming task
Teachers have to attend to the weak points of every student during a
limited time period

Objectives
More engaging and interactive personal learning experience to the
student
Automatically assess student answers and provide marks similar to
Ordinary Level (Sri Lanka) examination
Provide feedback to improve geometric skills

Literature Survey
Building a geometric construction tool
◦ Facts need to consider when creating a geometric construction tool
Software for geometric constructions
◦ Current implementations of geometric construction tools and evaluation
Geometric construction representations
◦ Proposed representations for describing a geometric construction
Geometric construction validation
◦ Validating a geometric construction

Software for Geometric
Constructions
Existing software:
Geogebra
Cinderella2
C. a. R (Compass-and-Ruler)
Geometria

Software for Geometric
Constructions
All of them contains high-level construction tools
Some are desktop-based and platform-specific
Some are not open-source
None of them provide platform for creating a geometric construction
learning environment

Geometric Construction
Representations
Geometric Construction is different from a drawing
Presenting a geometric construction that other software can
understand
◦ Predrag Janičić ’s [25] perspective on a geometric construction language
◦ Pedro Quaresma et al. [26] work on creating a common geometric
construction language
◦ The Intergeo Consortium’s effort to build I2G standard [28][29]

Geometric Construction
Validation
Gabriel and Andreas’s [32] introduction to “Compatibility Criterion”
◦ Compatibility Criterion: geometric construction should not violate the
restrictions of constructing
In CleverPHL, Christian and Ulrich [33] have implemented the idea of
‘replay constructions’, to validate Geometric Constructions
Most of the above methods are semi-automatic

Solution
Composition of the system

Representing Marking Scheme
Marking scheme (rubric) consists the solution and distribution of marks
for the question
To represent the separation of sub-questions, alternative answers, a
common format is used
To represent the geometric construction, I2G standard is used

I2G Format
Elements specify basic entities : Point, Line, Circle
Constraints specify the relation between elements:
◦ free_point
◦ line_through_two_points
◦ line_perpendicular_to_line_through_point
◦ point_intersection_of_two_lines
◦ circle_by_center_and_point etc.

Representing Student Answer
The student’s answer is represented using SVG format
Answer can be constructed using any tool that outputs SVG
representation

Assessment System
Infers the student’s answer
Validates student’s answer against rubric
Calculates the best-possible marks distribution
Generates feedback for the answer

Flow of the Assessment system
Rubric String, Answer String
Parse rubric string Parse answer string
Validates Answer
Result String

Inferring the Student’s Answer

Validation : Matching Steps
Validation logic goes through each and every step mention in the rubric
file
Pick the relevant, identifies elements from the answer (which are left
from previous steps)
Identified elements in one step, help to identify elements in next steps
Complex constructions use other elements such as complex-arcs
Always goes through alternative answers (mark-sets), and selects the
best one

Validation : Identify Elements
Identification of Perpendicular Bisectors
Step Description
1 If already found during inferring, return it
2 Identify the possible perpendicular line in
diagram by checking perpendicularity
3 Identify Complex-Arcs which are having centers
similar to given 2 points
4 Among those, select Complex-Arcs which are
intersecting and having same radius

Validation : Identify Elements
Other Elements
Identification of Perpendicular to a Line
Identification of Angular Bisectors
Identification of Parallel Line to a Line
◦ Angle-copy method
◦ Rhombus method
◦ Using perpendicular-lines method

Sample Question
Question:
Using only a straightedge with a cm/mm scale and a pair of compasses and showing construction lines clearly,
(i) Construct a line segment AB such that AB = 8 cm, and the angle ABC such that ABC = 60 degrees and BC =
10 cm. Complete ABC triangle.
(ii) Construct the perpendicular bisector of AB.
Question Number Answer Marks
(i) Drawing AB or BC 1
4
60 degree angle construction 1
Completion of triangle 2
(ii) Constructing perpendicular bisector 2 2
6

Sample Answer
Completed within 16 milliseconds
Final results...
Marks for sub-question 1 :: 4.0 out of 4.0
STEP PASSED:: sub-question: 1, markset: 2, step: 1, marks: 1.0
Marks for sub-question 2 :: 0.0 out of 2.0
STEP FAILED:: sub-question: 2, markset: 1, step: 1
STEP ERROR:: There is an incorrect construction for bisector-line through points
A and B
STEP ERROR:: Please refer: http://www.mathopenref.com/constbisectline.html
STEP ERROR:: Place the compass point on A and B, and draw an arc on each side
of the line without changing the compass width
>> Total marks for the question:: 4.0 out of 6.0

Uniqueness
Construction of geometric construction gives close-experience to paper-
based environment
Validates student’s answer considering construction process
Gives partial marks and feedback based on student’s answer
Evaluates a large number of answers within a short period of time
Developed based on platform-independent & widely-accepted
technologies and standards

Data sources used
Source No. of Answer Scripts Purpose
Tuition Class 01 15 Testing
Tuition Class 02 10 Testing
Government School 01 – Grade 11 20 Testing
Government School 04 – Grade 11 7 Tuning
Tuition Class 03 5 Tuning
Government School 05 – Grade 10 10 Tuning

Evaluation Criteria
Evaluating Assessing Module
◦ Answers were manually graded according to a marking scheme
◦ Answers were graded using the system
◦ Two grades (manually and by system) were compared to evaluate the system
performance
Experienced teachers’ instructions were considered to provide feedback
and identify common errors

Answer Grading
Accuracy of the system was 97% out of 100 answer scripts graded
Answer scripts belonged to 5 different questions
Marks distribution :
◦ 17% students have scored 0 – 3 marks,
◦ 21% have scored 4 – 6 marks
◦ 62% have scored 7 – 10 marks.
◦ From these answer scripts, two students have scored zero marks and 43
students scored full marks.

Demo
https://youtu.be/-_tATk_-eGs

Future Work
Extending the system to suite other examinations.
◦ This will require modifications such as support for more constraints in rubric
file, support for more SVG elements (ellipse, polygon) and validating more
construction types.
Feedback generation can be improved
Incorporate student-model, and machine learning techniques
◦ This will help to select questions based on student’s skill level

References
[1]S. Gulwani, V. A. Korthikanti, and A. Tiwari, “Synthesizing geometry constructions,” in ACM SIGPLAN Notices, 2011, vol. 46, no. 6, pp.
50–61.
[2]“The Math Forum.” [Online]. Available: http://mathforum.org/library/drmath/view/61335.html. [Accessed: 01-Nov-2016].
[3]H. Silva and B. Banneheka, “Factors Associated with Mathematics Results at GCE (OL) Examination,” 2015.
[4]H. P. Tam and Y.-L. Chen, “A regional survey of Taiwan students’ performance in geometric construction,” in Proc. 36th Conf. of the Int.
Group for the Psychology of Mathematics Education, 2012, vol. 4, pp. 131–38.
[5]S. Chakraborty, D. Roy, and A. Basu, “Development of knowledge based intelligent tutoring system,” Advanced Knowledge Based
Systems: Model, Applications and Research, vol. 1, pp. 74–100, 2010.
[6]“Geogebra.” [Online]. Available: https://www.geogebra.org/. [Accessed: 01-Nov-2016].
[7]“Cinderella.” [Online]. Available: http://www.cinderella.de/tiki-index.php. [Accessed: 01-Nov-2016].
[8]“C.a.R. - Compass and Ruler.” [Online]. Available: http://zul.rene-grothmann.de/doc_en/. [Accessed: 01-Nov-2016].
[9]B. B. Bantchev, “A brief tour to dynamic geometry software,” CaMSP February 2015 Newsletter, 2010.
[10]M. El-Demerdash and U. Kortenkamp, “The effectiveness of an enrichment program using dynamic geometry software in developing
mathematically gifted students’ geometric creativity in high schools,” 2010.
[11]M. L. Crowley, “The van Hiele model of the development of geometric thought,” Learning and teaching geometry, K-12, pp. 1–16,
1987.

References (cont.)
[12]V. Santos and P. Quaresma, “Integrating DGSs and GATPs in an adaptative and collaborative blended-learning Web-environment,”
arXiv preprint arXiv:1202.4833, 2012.
[13]“Web Geometry Laboratory.” [Online]. Available: http://hilbert.mat.uc.pt/WebGeometryLab/. [Accessed: 01-Nov-2016].
[14]I. A. Elena and P. Manuela, “The Interactive Geometry Software Cinderella.”
[15]V. S. Cisuc and P. Quaresma, “Collaborative environment for geometry,” in Experiment@ International Conference (exp. at’13), 2013
2nd, 2013, pp. 42–46.
[16]G. R. Laigo, A. H. Bhatti, P. Lakshmi Kameswari, and H. M. GebreYohannes, “Revisiting Geometric Construction using Geogebra.”
[17]“Cinderella (software).” [Online]. Available: https://en.wikipedia.org/wiki/Cinderella_(software). [Accessed: 01-Nov-2016].
[18]“Geometria.” [Online]. Available: http://geocentral.net/geometria/. [Accessed: 01-Nov-2016].
[19]“Geometra.” [Online]. Available: https://sourceforge.net/projects/geometra/. [Accessed: 01-Nov-2016].
[20]“GCLC.” [Online]. Available: https://www.emis.de/misc/software/gclc/. [Accessed: 01-Nov-2016].
[21]“Emaths Geometry Toolbox.” [Online]. Available: http://www.emaths.co.uk/images/tutorials/geometrytoolbox.swf. [Accessed: 01-
Nov-2016].
[22]M. F. Inayat and S. N. Hamid, “Integrating New Technologies and Tools in Teaching and Learning of Mathematics: An Overview,”
Journal of Computer and Mathematical Sciences, vol. 7, no. 3, pp. 122–129, 2016.
[23]W3C-SVG-Working-Group, “Scalable Vector Graphics (SVG) 1.1,” 2011.
[24]J. Schiller, “Codedred - SVG Support.” [Online]. Available: http://www.codedread.com/svg-support.php. [Accessed: 01-Nov-2016].

References (cont.)
[25]P. Janicic, “Geometry constructions language,” Journal of Automated Reasoning, vol. 44, no. 1–2, pp. 3–24, 2010.
[26]P. Quaresma, P. Janicic, J. Tomasevic, M. Vujosevic-Janicic, and D. Tosic, “XML-based Format for Descriptions of Geometrical Constructions and Geometrical
Proofs,” Communicating Mathematics in Digital Era, CMDE, pp. 15–18, 2006.
[27]S. Gulwani, “Dimensions in program synthesis,” in Proceedings of the 12th international ACM SIGPLAN symposium on Principles and practice of declarative
programming, 2010, pp. 13–24.
[28]S. Egido, M. Hendriks, Y. Kreis, U. Kortenkamp, and D. Marquès, “i2g Common File Format Final Version.”
[29]Intergeo-Consortium, “Interoperable Interactive Geometry for Europe.” [Online]. Available: http://i2geo.net/. [Accessed: 01-Nov-2016].
[30]M. Hendriks, U. Kortenkamp, Y. Kreis, and D. Marques, “Report on Implementations of the Common File Format,” 2009.
[31]P. Quaresma and N. Baeta, “Current Status of the I2GATP Common Format,” in International Workshop on Automated Deduction in Geometry, 2014, pp. 119–
128.
[32]G. J. Stylianides and A. J. Stylianides, “Validation of solutions of construction problems in dynamic geometry environments,” International Journal of
Computers for Mathematical Learning, vol. 10, no. 1, pp. 31–47, 2005.
[33]C. Spannagel and U. Kortenkamp, “Demonstrating, guiding, and analyzing processes in dynamic geometry systems,” in Proceedings of the 9th International
Conference on Technology in Mathematics Teaching. Metz, France: ICTMT, 2009, vol. 9.
[34]L. J. Blanco, “Errors in the teaching/learning of the basic concepts of geometry,” International Journal for Mathematics Teaching and Learning, vol. 24, 2001.
[35]A. Özerem, “Misconceptions in geometry and suggested solutions for seventh grade students,” Procedia-Social and Behavioral Sciences, vol. 55, pp. 720–729,
2012.
[36]C. Alvin, S. Gulwani, R. Majumdar, and S. Mukhopadhyay, “Synthesis of Geometry Proof Problems,” in AAAI, 2014, pp. 245–252.

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