Yıl 2019, Cilt 10 , Sayı 2, Sayfalar 476 - 500 2019-09-10

A General Analytical Model for Problem Solving Teaching: BoS

Ali Kürşat Erümit [1] , Vasif Nabiyev [2] , Temel Kösa [3] , Mehmet Kokoç [4] , Ayşegül Aksoy [5]


In this study, a general analytical model called Bag of Solution (BOS) was developed to help students understand and solve mathematical problems. The model is based on graph theory, a topic under discrete mathematics. The types of problems to be modelled for BoS were determined by looking at densities of the problems in the central placement examinations and exam preparation books. As a result, three types of problems were selected; namely Mixture, Worker and Motion problems. In order to develop a common model for solution of the three types of problems, a total of 1509 mixture, worker and movement problems were examined. After the analysis, the problem types were taken together, and variable relations were determined, and a common graph model was created. Since it is an algorithmic model, it allows solving problems both by paper and pencil and computer. This study proves that different types of problems (with different scenarios, objects and object relations) can be solved using a single model. It is expected that the BoS developed in this study will offer two benefits. It is hoped to both provide an algorithmic basis for computer-aided instructional materials, adaptive systems and intelligent tutoring systems to be developed for problem solving and also help students to develop a new understanding of the problem-solving process. A common graph structure that can covers the entirety of a problem can allow students to construct their own learning while solving the problem step by step.

Graph theory, problem modeling, problem solving, bag of solution
  • Allahverdi, N. (2002). Expert systems: The artificial intelligence application. Ankara: Atlas Publishing.
  • Aqda, M. F., Hamidi, F., & Rahimi, M. (2011). The comparative effect of computer- aided instruction and traditional teaching on student’s creativity in math classes. Procedia Computer Science, 3, 266-270.
  • Anzelmo-Skelton, N. (2006). Learning style, strategy use, personalization of mathematical word problems and responses of students with learning disabilities. International Journal of Special Education, 21(1), 1-10.
  • Arnau, D., Arevalillo-Herraez, M., Puig, L., & Gonzalez-Calero, J. A. (2013). Fundamentals of the design and the operation of an intelligent tutoring system for the learning of the arithmetical and algebraic way of solving word problems. Computers & Education, 63, 119-130.
  • Baki, A. (2008). Kuramdan uygulamaya matematik eğitimi. Ankara: Harf Eğitim Yayıncılık.
  • Ben-Hur, M. (2006). Concept-rich mathematics instruction: Building a strong foundation for reasoning and problem solving. Alexandria, VA.: Association for Supervision and Curriculum Development.
  • Beqiri, E., & Tahiri, M. (2014). An effective use of information and communication technology in education systems of countries in south-east europe. Academic Journal of Interdisciplinary Studies, 3(2), 91-102.
  • Berry, J., & Houston, K. (1995). Mathematical modelling. Bristol: J. W. Arrowsmith Ltd.
  • Blatchford, P., Bassett, P., & Brown, P. (2011). Examining the effect of class size on classroom engagement and teacher–pupil interaction: Differences in relation to pupil prior attainment and primary vs. secondary schools. Learning and Instruction, 21(6), 715-730.
  • Blum, W. (2011). Can modeling be taught and learnt? Some answers from empirical research. In G. Kaiser, W. Blum, R. BorromeoFerri, & G. Stillman (Eds.), Trends in teachingand learning of mathematical modeling (pp. 15-30). New York: Springer.
  • Cai, J. (2003). Singaporean students mathematical thinking in problem solving and problem posing: An exploratory study. International Journal of Mathematical Education in Science and Technology, 34(5), 719-737.
  • Chang, K. E., Sung, Y. T., & Lin, S. F. (2006). Computer-assisted learning for mathematical problem solving. Computers & Education, 46, 140-151.
  • Chen, T., Mdyunus, A., Ali, W. Z. W., & Bakar, A. (2008). Utilization of intelligent tutoring system in mathematics learning. International Journal of Education and Development Using Information and Communication Technology, 4(4), 50-63.
  • Chingos, M. M. (2012). The impact of a universal class-size reduction policy: Evidence from Florida’s statewide mandate. Economics of Education Review, 31, 543-562.
  • Chiu, M. M., & Klassen R. M. (2010). Relations of mathematics self-concept and its calibration with mathematics achievement: Cultural differences among fifteenyear olds in 34 countries. Learning and Instruction, 20(1), 2-17.
  • Cho, H., Glewwe, P., & Whitler, M. (2012). Do reductions in class size raise students’ test scores? Evidence from population variation in Minnesota’s elementary schools. Economics of Education Review, 31, 77-95.
  • Crippen, J. K., & Earl, B. Y. (2007). The impact of web-based worked examples and self-explanation on performance, problem solving, and self-efficacy. Computers & Education, 49, 809-821.
  • Çelik, D., & Güler, M. (2013). Examination of realistic problem solving skills of sixth grade students. Dicle University Ziya Gökalp Education Faculty Journal, 20, 180-195.
  • De Corte, E., & Masui, C. (2004). The CLIA-model: A framework for designing powerful leaming environments for thinking and problem solving. European Journal of Psychology of Education, 19(4), 365-384.
  • Dede, A. T. (2017). Examination of the relationship between modelling competencies and class level and mathematics achievement. Elementary Education Online, 16(3), 1201-1219.
  • Dharwadker, A., & Pirzada, S. (2011). Graph theory. India: Amazon.
  • Doruk, B. K., & Umay, A. (2011). The effect of mathematical modeling on transferring mathematics into daily life. Hacettepe University Journal of Education, 41(41), 124-135.
  • Elia, I., van den Heuvel-Panhuizen, M., & Kolovou, A. (2009). Exploring strategy use and strategy flexibility in non-routine problem solving by primary school high achievers in mathematics. ZDM Mathematics Education, 41, 605-618.
  • English, L. D., & Gainsburg, J. (2016). Problem solving in a 21st century mathematics curriculum. In L. D. English, & D. Kirshner (Eds.), Handbook of international research in mathematics education (pp. 313-335, 3rd ed.). New York: Taylor and Francis.
  • Erdem, Z. Ç., Doğan, M. F., Gürbüz, R., & Şahin, S. (2017). The reflections of mathematical modeling in teaching tools: Textbook analysis. Adıyaman University Education Sciences Journal, 7(1), 61-86.
  • Erdoğan, A. (2015). Turkish primary school students’ strategies in solving a non-routine mathematical problem and some implications for the curriculum design and implementation. International Journal for Mathematics Teaching and Learning, 2, 1-27.
  • Erümit, K., Karal, H., & Nabiyev, V. (2012). Proposing a model for computed analyzing of the motion problems without parameter. Education Sciences, 7(2), 565-573.
  • Garcia-Santillán, A., Flores-Zambada, V., Escalera-Chávez, M. E., Chong-González, I. S., & Lopez-Morales, J. S. (2012). Students, computers and mathematics: How do they interact in the teaching-learning process? (An empirical study on accounting, management and marketing undergraduate students). International Journal of Learning & Development, 2(2), 178-200.
  • Gooding, S. (2009). Children's difficulties with mathematical word problems. Proceedings of the British Society for Research into Learning Mathematics, 29(3), 31-36.
  • Hoffman, B., & Spatariu, A. (2008). The influence of self-efficacy and metacognitive prompting on math problem-solving efficiency. Contemporary Educational Psychology, 33, 875-893.
  • Huang, T. H., Liu, Y. C., & Chang, H. C. (2012). Learning achievement in solving word-based mathematical questions through a computer-assisted learning system. Educational Technology & Society, 15(1), 248–259.
  • Hwang, G. J., & Wu, P. H. (2012). Advancements and trends in digital game‐based learning research: A review of publications in selected journals from 2001 to 2010. British Journal of Educational Technology, 43(1), E6-E10. doi: 10.1111/j.1467-8535.2011.01242.x
  • Jacobse, A. E., & Harskamp, E. G. (2009). Student-controlled metacognitive training for solving word problems in primary school mathematics. Educational Research and Evaluation, 15(5), 447-463.
  • Jeremic, Z., Jovanovic, J., & Gasevic, D. (2012). Student modeling and assessment in intelligent tutoring of software patterns. Expert Systems with Application, 39, 210-222.
  • Jonassen, D. H. (2011). Learning to solve problems: A handbook for designing problem solving learning environments. New York: Routledge.
  • Kaiser, G., & Schwarz, B. (2006). Mathematical modeling as bridge between school and university. Zentralblattfür Didaktik der Mathematik-ZDM, 38, 196-208.
  • Kilpatrick, J. (2013). A retrospective account of the past 25 years of research on teaching mathematical problem solving. E. A. Silver (Ed.), Teaching and learning mathematical problem solving: Multiple research perspectives (pp. 1-15). New York: Routledge.
  • Lesh, R. A., & Doerr, H. M. (2003). Beyond constructivism: Models and modeling perspectives on mathematics problem solving, learning, and teaching. New Jersey: Routledge.
  • Li, Q., & Ma, X. (2010). A meta-analysis of the effects of computer technology on school students’ mathematics learning. Educational Psychology Review, 22(3), 215-244.
  • Lopez-Morteo, G., & López, G. (2007). Computer support for learning mathematics: A learning environment based on recreational learning objects. Computers & Education, 48(4), 618-641.
  • Mohamedi, H., Bensebaa, T., & Trigano, P. (2012). Developing adaptive intelligent tutoring system based on item response theory and metrics. International Journal of Advanced Science and Technology, 43, 1-14.
  • Nabiyev, V. V. (2012). Yapay zeka (5th ed.). Ankara: Seçkin Yayıncılık.
  • Olkun, S., Şahin, Ö., Akkurt, Z., Dikkartin, F. T., & Gülbağcı, H. (2009). Problem solving and generalization through modeling: A study on elementary school students. Education and Science, 34(151), 65-73.
  • Öz, E., & Baykoç, O. F. (2004). Decision theory supported expert system approach to supplier selection problem. Gazi Üniversitesi Mühendislik-Mimarlık Fakültesi Dergisi, 19(3), 275-286.
  • Polya, G. (1957). How to solve It. New Jersey: Princeton University.
  • Popper, K. R. (1999). All life is problem solving (P. Carmiller, Trans.). London: Routledge.
  • Rich, E. (1983). Artificial intelligence. The University of Texas at Austin: McGraw-Hill Inc.
  • Singh, P., & Lokotsch, K. (2005). Effects of transformational leadership on human resource management in primary schools. South African Journal of Education, 25(4), 279-286.
  • Skemp, R. (1986). The psychology of mathematics learning. Suffolk: Penguin Books.
  • Smithers, D. B. (2005). Graph theory for the secondary school classroom (Unpublished doctoral dissertation). East Tennessee State University, Johnson City.
  • Schoenfeld, A. H. (2017). Learning to think mathematically: Problem solving, metacognition, and sense making in mathematics. Journal of Education, 196(2), 1-38.
  • Soylu, Y., & Soylu, C. (2006). The role of problem solving in mathematics lessons for success. Journal of İnönü University Faculty of Education, 7(11), 97–111.
  • Sümersan-Seyhanlı, S. (2007). The effect on the student achievement of graph theory in teaching of the unit of “probability” of primary school 8th (Unpusblished master’s thesis). Balıkesir University, Balıkesir.
  • Sweller, J., Clark, R., & Kirschner, P. (2010). Teaching general problem-solving skills is not a substitute for, or a viable addition to, teaching mathematics. Notices of the American Mathematical Society, 57(10), 1303-1304.
  • Thomas, J. L. (2012). Combination classes and educational achievement. Economics of Education Review, 31, 1058-1066.
  • Vicente, S., Orrantia, J., & Verschaffel, L. (2007). Influence of situational and conceptual rewording on word problem solving. British Journal of Educational Psychology, 77(4), 829-848.
  • Xin, Z., Lin, C., Zhang, L., & Yan, R. (2007). The performance of Chinese primary school students on realistic arithmetic word problems. Educational Psychology in Practice, 23, 145–159.
  • Yavuz-Mumcu, H., & Baki, A. (2017). The interpretation of mathematical modelling skills of high school students in the activities of using mathematics. Ondokuz Mayıs University Journal of Faculty of Education, 36(1), 7-33.
  • Yen, J. C., & Chen, M. P. (2008). Patterns of reflection for problem-solving in a mobile learning environment. International Journal of Education and Information Technologies, 2(2), 121-124.
Birincil Dil en
Konular Sosyal, Eğitim, Eğitim Araştırmaları
Bölüm Araştırma Makaleleri
Yazarlar

Orcid: 0000-0003-4910-4989
Yazar: Ali Kürşat Erümit (Sorumlu Yazar)
Kurum: TRABZON ÜNİVERSİTESİ, FATİH EĞİTİM FAKÜLTESİ
Ülke: Turkey


Yazar: Vasif Nabiyev
Kurum: Karadeniz Technical University
Ülke: Turkey


Yazar: Temel Kösa
Kurum: TRABZON ÜNİVERSİTESİ

Yazar: Mehmet Kokoç
Kurum: TRABZON ÜNİVERSİTESİ

Yazar: Ayşegül Aksoy
Kurum: TRABZON ÜNİVERSİTESİ

Tarihler

Yayımlanma Tarihi : 10 Eylül 2019

Bibtex @araştırma makalesi { turkbilmat486084, journal = {Turkish Journal of Computer and Mathematics Education (TURCOMAT)}, issn = {}, eissn = {1309-4653}, address = {}, publisher = {Türkbilmat Eğitim Hizmetleri}, year = {2019}, volume = {10}, pages = {476 - 500}, doi = {10.16949/turkbilmat.486084}, title = {A General Analytical Model for Problem Solving Teaching: BoS}, key = {cite}, author = {Erümit, Ali Kürşat and Nabiyev, Vasif and Kösa, Temel and Kokoç, Mehmet and Aksoy, Ayşegül} }
APA Erümit, A , Nabiyev, V , Kösa, T , Kokoç, M , Aksoy, A . (2019). A General Analytical Model for Problem Solving Teaching: BoS. Turkish Journal of Computer and Mathematics Education (TURCOMAT) , 10 (2) , 476-500 . DOI: 10.16949/turkbilmat.486084
MLA Erümit, A , Nabiyev, V , Kösa, T , Kokoç, M , Aksoy, A . "A General Analytical Model for Problem Solving Teaching: BoS". Turkish Journal of Computer and Mathematics Education (TURCOMAT) 10 (2019 ): 476-500 <https://dergipark.org.tr/tr/pub/turkbilmat/issue/48621/486084>
Chicago Erümit, A , Nabiyev, V , Kösa, T , Kokoç, M , Aksoy, A . "A General Analytical Model for Problem Solving Teaching: BoS". Turkish Journal of Computer and Mathematics Education (TURCOMAT) 10 (2019 ): 476-500
RIS TY - JOUR T1 - A General Analytical Model for Problem Solving Teaching: BoS AU - Ali Kürşat Erümit , Vasif Nabiyev , Temel Kösa , Mehmet Kokoç , Ayşegül Aksoy Y1 - 2019 PY - 2019 N1 - doi: 10.16949/turkbilmat.486084 DO - 10.16949/turkbilmat.486084 T2 - Turkish Journal of Computer and Mathematics Education (TURCOMAT) JF - Journal JO - JOR SP - 476 EP - 500 VL - 10 IS - 2 SN - -1309-4653 M3 - doi: 10.16949/turkbilmat.486084 UR - https://doi.org/10.16949/turkbilmat.486084 Y2 - 2019 ER -
EndNote %0 Türk Bilgisayar ve Matematik Eğitimi Dergisi A General Analytical Model for Problem Solving Teaching: BoS %A Ali Kürşat Erümit , Vasif Nabiyev , Temel Kösa , Mehmet Kokoç , Ayşegül Aksoy %T A General Analytical Model for Problem Solving Teaching: BoS %D 2019 %J Turkish Journal of Computer and Mathematics Education (TURCOMAT) %P -1309-4653 %V 10 %N 2 %R doi: 10.16949/turkbilmat.486084 %U 10.16949/turkbilmat.486084
ISNAD Erümit, Ali Kürşat , Nabiyev, Vasif , Kösa, Temel , Kokoç, Mehmet , Aksoy, Ayşegül . "A General Analytical Model for Problem Solving Teaching: BoS". Turkish Journal of Computer and Mathematics Education (TURCOMAT) 10 / 2 (Eylül 2019): 476-500 . https://doi.org/10.16949/turkbilmat.486084
AMA Erümit A , Nabiyev V , Kösa T , Kokoç M , Aksoy A . A General Analytical Model for Problem Solving Teaching: BoS. Turkish Journal of Computer and Mathematics Education (TURCOMAT). 2019; 10(2): 476-500.
Vancouver Erümit A , Nabiyev V , Kösa T , Kokoç M , Aksoy A . A General Analytical Model for Problem Solving Teaching: BoS. Turkish Journal of Computer and Mathematics Education (TURCOMAT). 2019; 10(2): 500-476.