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MONTO: A Machine-Readable Ontology for Teaching Word Problems in Mathematics

Introduction The Indian National Curriculum Framework (NCF) (NCF, 2005) and the National Council of Teachers of Mathematics (NCTM, 1980) both focus on development of problem solving in school mathematics. A majority of research work done on problem solving in mathematics has been conducted in the US, and yet, according to Trends in International Mathematics and Science Study (TIMSS-2007), East Asian students perform better than the US students (Gonzales et al., 2008). It is true that, over the past decade, a few Indian students have won gold and silver medals in International Mathematics Olympiads; however, studies of students' mathematical learning conducted in Indian metros show that even students from top schools perform below the international average (EI Reports, 2009). The Program for International Student Assessment (PISA) (PISA, 2012) reported that the mathematical proficiency of students from Indian states such as Tamil Nadu and Himachal Pradesh was lower than the average of countries of the Organization for Economic Co-operation and Development (OECD: http://www.oecd.org/). A problem is defined as "to search consciously for some action appropriate to attain a clearly conceived, but not immediately attainable aim" (Polya, 1981). For this study, the term mathematical problem refers to a problem that is solved by using mathematical models, formulas, mathematical logic, and rules. In the historical review of problem solving, there is a dichotomy between the terms problem solving and doing exercises. The term problem solving refers to the use of various heuristic strategies, pattern searching, and control functions for selecting the appropriate strategy, whereas doing exercises refers to the use of known procedures and methods (Schoenfeld, 1985). For the scope of this paper, we consider mathematical problems at K-12 level. Problem solving is introduced at school level when students learn word problems including a real world scenario. At that point students experience difficulties because they need to understand the real world scenario, connect it to the mathematical language and convert it into the mathematical model to solve. Hence, the paper focuses on teaching word problems. Despite more than seven decades of work in teaching problem solving (Polya, 1981; Polya, 1946; Schoenfeld, 1985; Silver, 1985; Marshall, 1995; Jonassen, 2011), classroom teaching of solving mathematical problems at school level has remained a great challenge. The paper is organized as follows. Initially, the theory of mathematical thinking, knowledge and behavior is discussed, drawing from the literature of mathematics education, to give a theoretical framework for the argument. Secondly, the state of the art for teaching word problem solving is discussed, and the gaps are identified. The next section contains a discussion of the methodology for developing ontology and of the proposed ontology, considering one particular domain with an example. The next section compares the proposed ontology with existing teaching ontologies described in the literature. The final section includes an evaluation of MONTO, the conclusion, and possible future work. State of the art Mathematical thinking, knowledge, understanding Once students are taught mathematical problem solving, they should be able to explore patterns and seek solutions to the problems and not just memorize procedures, and should be able to formulate conjectures and not merely do exercises. The implicit objectives of teaching mathematical problem solving at school level--such as development of mathematical thinking, logical thinking, and critical thinking--are expected to be achieved after some years. While teaching mathematical problem solving, Schoenfeld (1985) conducted a study of students and came up with a framework for the analysis of mathematical thinking and behavior that contains four components (Table 1). A student who has been taught mathematical problem solving is strong in analyzing a large amount of quantitative data, uses mathematics in practical ways, and is analytical both in thinking on her own and in examining the arguments put forward by others (Schoenfeld, 1992). …