АУТОР(И): Marijana Ž. Zeljić, Milana M. Dabić Boričić, Svetlana M. Ilić
Routine word problems are thoroughly described and categorized to combine, change, and compare problems. This paper investigates how 2nd, 4th, and 6th-grade students solve integrated combine and compare problems. We used the integrated combine and compare problems with consistent language (CL) formulation, inconsistent language (IL) formulation, or more complex structure. Our research sample consists of 44 students in 2nd grade, 48 students in 4th grade, and 42 students in 6th grade from schools in Belgrade. The results show that stu- dents are more successful in solving problems with CL than with IL formulation at all levels of education. Students from the 2nd, 4th, and 6th grade are equally successful in solving the CL problem. The surprising result is the nonexistence of a significant difference in the achieve- ment of students in 4th and 6th grade on the IL problem, which could indicate an obstacle in the development of relational term understanding after introducing algebra into mathematical education. Low achievement on the problem with more complex structure showed that students have issues with the modeling process and that they are not eager to use algebraic strategies or graphical representations. These results imply a need for a systematic approach to teaching routine problems after introducing algebra in mathematics education.
word problems, combine problems, compare problems, problem solving strat- egies, mathematical education.
- Blum, Leiss (2007). W. Blum, D. Leiss, How do students and teachers deal with modelling problems, In: C. Haines, P. Galbraith, W. Blum, S. Khan (Eds.), Mathematical modeling: Education, engineering, and economics, Chichester: Horwood, 222–231.
- Boesen, Helenius, Lithner, Bergqvist, Bergqvist, Palm, Palmberg (2014): J. Boesen, O. Helenius, J. Lithner, E. Bergqvist, T. Bergqvist, T. Palm, B. Palmberg, Developing mathematical competence: From the intended to the enacted curriculum, Journal of Math- ematical Behavior, 33, 72–87.
- Boonen, Jolles (2015): A. J. H. Boonen, J. Jolles, Second Grade Elementary School Students’ Differing Performance on Combine, Change and Compare Word Problems, International Journal School and Cognitive Psychology, 2(122), 1–6.
- Briars, Larkin (1984): D. J. Briars, J. H. Larkin, An integrated model of skill in solving elementary word problems, Cognition and Instruction, 1, 245–296.
- Carpenter (1986): T. P. Carpenter, Conceptual knowledge as a foundation for pro- cedural knowledge: Implications from research on the initial learning of arithmetic, In: J. Hiebert (Ed.), Conceptual and procedural knowledge: The case of mathematics, Hillsdale, NJ: Erlbaum, 113–13.
- Carpenter, Hiebert, Moser (1981): T. P. Carpenter, J. Hiebert, J. M. Moser, Prob- lem structure and first-grade children’s solution processes for simple addition and subtrac- tion problems, Journal for Research in Mathematics Education, 12, 27–39.
- Carpenter, Moser (1984): T. P. Carpenter, J. M. Moser, The acquisition of addition and subtraction concepts in grade one through three, Journal of Research in Mathematics Education, 15, 179–202.
- Cummins (1991): D. Cummins, Children’s interpretation of arithmetic word prob- lems, Cognition and Instruction, 8, 261–289.
- Cummins, Kintsch, Reusser, Weimer (1988): D. Cummins, W. Kintsch, K. Reusser, R. Weimer, The role of understanding in solving word problems, Cognitive Psychology, 20, 405–438.
- De Corte, Verschaffel (1986): E. D. De Corte, L. Verschaffel, Eye-Movement Data as Access to Solution Processes of Elementary Addition and Subtraction Problems. Retrieved in January 2022 from https://files.eric.ed.gov/fulltext/ED273450.pdf.
- De Corte, Verschaffel, De Win (1985): E. De Corte, L. Verschaffel, L. De Win, Influence of rewording verbal problems on children’s problem representations and solutions, Journal of Educational Psychology, 77, 460–470.
- De Corte, Verschaffel, Pauwels (1990): E. De Corte, L. Verschaffel, A. Pauwels, Influence of the semantic structure of word problems on second graders’ eye movements, Journal of Educational Psychology, 82, 359–365.
- De Koning, Boonen, Jongerling, Van Wesel, Van der Schoot (2022): B. B. De Koning, A. J. H. Boonen, J. Jongerling, F. Van Wesel, M. Van der Schoot, Model method drawing acts as a double-edged sword for solving inconsistent word problems, Educational Studies in Mathematics, 111, 29–45.
- De Koning, Boonen, Van der Schoot (2017): B. B. De Koning, A. J. H. Boonen, M.Van der Schoot, The consistency effect in word problem solving is effectively reduced through verbal instruction, Contemporary Educational Psychology, 49, 121–129.
- Fuson (1992): K. C. Fuson, Research on whole number addition and subtraction, In: D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning, New York: Macmillian, 243–275.
- Hegarty, Mayer, Green (1992): M. Hegarty, R. E. Mayer, C. E. Green, Compre- hension of arithmetic word problems: Evidence from students’ eye fixations, Journal of Educational Psychology, 84, 76–84.
- Hegarty, Mayer, Monk (1995): M. Hegarty, R. E. Mayer, C. A. Monk, Comprehen- sion of arithmetic word problems: A comparison of successful and unsuccessful problem solvers, Journal of Educational Psychology, 87(1), 18–32.
- Khng, Lee, (2009). K. H. Khng, K. Lee, Inhibition interference from prior knowl- edge: Arithmetic intrusion in algebra word problem solving, Learning and Individual Dif- ferences, 19, 262‒268.
- Kintsch (1988): W. Kintsch, The role of knowledge in discourse comprehension: A construction–integration model, Psychological Review, 95, 163–182.
- Kintsch, Greeno (1985): W. Kintsch, J. G. Greeno, Understanding and solving word arithmetic problems, Psychological Review, 92, 109–129.
- Koedinger, Nathan (2004): K. R. Koedinger, M. J. Nathan, The real story behind story problems: Effects of representations on quantitative reasoning, Journal of the Learn- ing Sciences, 13(2), 129–164.
- Lewis (1989): A. B. Lewis, Training students to represent arithmetic word problems, Journal of Educational Psychology, 81, 521–531.
- Lewis, Mayer (1987): A. Lewis, R. Mayer, Students’ miscomprehension of relation- al statements in arithmetic word problems, Journal of Educational Psychology, 74, 199–216. Lithner, (2008): J. Lithner, A research framework for creative and imitative reasoning, Educational Studies in Mathematics, 67(3), 255–276.
- Littlefield, Rieser (1993): J. Littlefield, J. J. Rieser, Semantic features of similarity and children’s strategies for identification of relevant information in mathematical story problems, Cognition and Instruction, 11, 133–188.
- Marzocchi, Lucangeli, De Meo, Fini, Cornoldi (2002): G. M. Marzocchi, D. Lu- cangeli, T. De Meo, F. Fini, C. Cornoldi, The Disturbing Effect of Irrelevant Information on Arithmetic Problem Solving in Inattentive Children, Developmental neuropsychology, 21(1), 73–92.
- Morales, Shute, Pellegrino (1985): R. V. Morales, V. J. Shute, J. M. Pellegrino, Developmental differences in undersanding and solving simple word problems, Cognition and Instruction, 2, 41–57.
- Mwangi, Sweller (1998): W. Mwangi, J. Sweller, Learning to solve compare word problems: The effect of example format and generating self-explanations, Cognition and Instruction, 16, 173–199.
- Nesher, Greeno, Riley (1982): P. Nesher, J. G. Greeno, M. S. Riley, The Development of Semantic Categories for Addition and Subtraction, Educational Studies in Mathematics, 13(4), 373–394.
- Nesher, Hershkovitz, Novotna (2003): P. Nesher, S. Hershkovitz, J. Novotna, Situ- ation Model, Text Base and What Else? Factors Affecting Problem Solving, Educational Studies in Mathematics, 52(2), 151–176.
- Okamoto (1996): Y. Okamoto, Modelling children’s understanding of quantitative relations in texts: A developmental perspectives, Cognition and Instruction, 14, 409–440.
- Okamoto, Case (1996): Y. Okamoto, R. Case, Exploring the microstructure of children’s central conceptual structures in the domain of number, Monographs of the Society for Research in Child Development, 246 (61), 27–58.
- Pape (2003): S. J. Pape, Compare word problems: Consistency hypothesis revisited, Contemporary Educational Psychology, 28, 396–421.
- Powell, Fuchs, Fuchs, Cirino, Fletcher (2009): S. R. Powell, L. S. Fuchs, D. Fuchs, P. T. Cirino, J. M. Fletcher, Do word problem features differentially affect problem diffi- culty as a function of students’ mathematics difficulty with and without reading difficulty?, Journal of Learning Disabilities, 42, 99–110.
- Resnick (1983): L. B. Resnick, A developmental theory of number understanding, In: H. P. Ginsburg (Ed.), The development of mathematical thinking, New York: Academic, 109–151.
- Riley, Greeno (1988): M. S. Riley, J. G. Greeno, Developmental analysis of lan- guage about quantities and of solving problems, Cognition and Instruction, 5, 49–101.
- Riley, Greeno, Heller (1983): M. S. Riley, J. G. Greeno, J. H. Heller, Development of childrens problem solving ability in arithmetic, In: H. P. Ginsburg (Ed.), The develop- ment of mathematical thinking, New York: Academic Press, 153–196.
- Schroeder, Lester (1989): T. L. Schroeder, F. K. J. Lester, Developing understand- ing in mathematics via problem solving, In: P. R. Trafton (Ed.), New directions for elemen- tary school mathematics, Reston, VA: NCTM, 31–42.
- Schumacher, Fuchs (2012): R. F. Schumacher, L. S. Fuchs, Does understanding relational terminology mediate effects of intervention on compare word problem?, Journal of Experimental Child Psychology, 111, 607–628.
- Schwarzkopf (2007): R. Schwarzkopf, Elementary modelling in mathematics les- sons: The interplay between “real-world” knowledge and “mathematical structures”, In: W. Blum, P. L. Galbraith, H. W. Henn, M. Niss (Eds.), Modelling and applications in mathematics education. The 14th ICMI Study, New York: Springer, 209–216.
- Stern (1993): E. Stern, What makes certain arithmetic word problems involving the comparison of sets so difficult for children, Journal of Educational Psychology, 85, 7–23. Stigler, Lee, Stevenson (1990): J. W. Stigler, S.-Y. Lee, H. W. Stevenson, Math- ematical knowledge of Japanese, Chinese, and American elementary school children, Reston, VA: National Council of Teachers of Mathematics.
- Van der Schoot, Bakker Arkema, Horsley, Van Lieshout (2009): M. Van der Schoot, A. H. Bakker Arkema, T. M. Horsley, E. D. C. M. Van Lieshout, The consistency ef- fect depends on markedness in less successful but not successful problem solvers: An eye movement study in primary school children, Contemporary Educational Psychology, 34(1), 58–66.
- Van Dooren, De Bock, Vleugels, Verschaffel (2010): W. Van Dooren, D. De Bock, K. Vleugels, L. Verschaffel, Just answering … or thinking? Contrasting pupils’ solutions and classifications of missing-value word problems, Mathematical Thinking and Learning, 12(1), 20–35.
- Verschaffel (1994): L. Verschaffel, Using retelling data to study elementary school children’s representations and solutions of compare problems, Journal of Research in Math- ematics Education, 25, 141–165.
- Verschaffel, Van Dooren, Greer, Mukhopadhyay (2010): L. Verschaffel, W. Van Dooren, B. Greer, S. Mukhopadhyay, Reconceptualising Word Problems as Exercises in Mathematical Modelling, Journal für Mathematik–Didaktik, 31(1), 9–29.
- Verschaffel, De Corte, Pauwels (1992): L. Verschaffel, De Corte, A. Pauwels, Solv- ing compare problems: An eye movement test of Lewis and Mayer’s consistency hypothesis, Journal of Educational Psychology, 84, 85–94.
- Verschaffel, Depaepe, Van Dooren (2014): L. Verschaffel, F. Depaepe, W. Van Dooren, Word problems in mathematics education, In: S. Lerman (Ed.), Encyclopedia of mathematics education, Dordrecht, the Netherlands: Springer, 641–645.
- Wassenberg, Hurks, Hendriksen, Feron, Meijs, Vles, Jolles (2008): R. Wassenberg,P. P. Hurks, J. G. Hendriksen, F. J. Feron, C. J. Meijs, J. S. Vles, J. Jolles, Age-related improvement in complex language comprehension: results of a cross-sectional study with 361 children aged 5 to 15, Journal of Clinical and Experimental Neuropsychology, 30(4), 435–48.
- Willis, Fuson (1988): G. B. Willis, K. C. Fuson, Teaching children to use schematic drawings to solve addition and subtraction word problems, Journal of Educational Psychol- ogy, 80, 192–201.
- Zeljić, Dabić Boričić, Maričić, (2021): M. Zeljić, M. Dabić Boričić, S. Maričić, Problem Solving in Realistic, Arithmetic/algebraic and Geometric Context, Education and