Lieven Verschaffel

Open real-world problems play an important role in the teaching and learning of mathematics. Research on open problems comprises a variety of research methodologies, theoretical conceptualizations, and constructs. This systematic literature review aims to synthesize and structure these various approaches by examining the current state of research on open problems in mathematics education. With a full-text analysis of 67 peer-reviewed journal articles, we investigated the descriptive characteristics of studies in this field, the conceptual frameworks used to define the openness of real world mathematical problems, and the structure of problems in terms of their initial, intermediate, and goal states. Additionally, we analyzed the constructs employed in the studies. We identified four sets of conceptual frameworks: (1) psychological frameworks of problem-solving, (2) mathematical problem-solving frameworks, (3) mathematical modelling frameworks, and (4) suspension of sense-making frameworks. The findings demonstrated that most studies have focused on problems with open initial and intermediate states, whereas problems with open goal states have received less attention. Furthermore, cognitive constructs were addressed more frequently than affective-motivational ones. The findings of this review reveal the need for a theoretically grounded conceptualization of the notion of openness in empirical studies. Important future research directions include investigating problems with open goal states, analyzing instruction to improve student performance in solving open problems, and addressing affective-motivational constructs in studies with open problems. This review contributes to a deeper understanding of openness as an important task characteristic and provides a comprehensive synthesis of the available research on open real-world mathematical problems.

Despite extensive research indicating that students often fail to apply real-world knowledge and common sense when solving word problems, the underlying causes remain underexplored. Teacher behavior and instructional methods are potential factors contributing to students' tendency to provide unrealistic answers to such problems. The current study aims to address this gap by examining the cognitive processes and perspectives of pre-service teachers when solving problematic word problems. A group of 146 pre-service teachers (97 females, 49 males) in Iran participated in the study, which consisted of two phases. In the first phase, participants were given three problematic word problems to solve and were subsequently asked to evaluate four different student responses. A significant correlation was found between the participants' responses in the initial test and their evaluations in the second phase. In the second phase, the study employed a phenomenographic approach to explore the thinking processes and perspectives of the pre-service teachers while solving the problems. The analysis of interview data led to the identification of two primary categories of unrealistic problem-solving: "inattention" and "ignoring." In the "inattention" category, the problem solver fails to recognize the relevance of real-world knowledge, while in the "ignoring" category, the solver acknowledges real-world factors but deliberately chooses not to integrate them into the solution. In the end, a model of unrealistic problem-solving is proposed and discussed, with implications for teacher training and pedagogical practices.

This 5-year longitudinal study examined whether high mathematics achievers in primary school had cognitive advantages before entering formal education. High mathematics achievement was defined as performing above Pc 90 in Grades 1 and 3. The predominantly White sample (M age in preschool: 64 months) included 31 high achievers (12 girls) and 114 average achievers (63 girls). We measured children's early numerical abilities, complex mathematical abilities, and general cognitive abilities in preschool (2017). High mathematics achievers had advantages on most tasks in preschool (ds > 0.62). Number order, numeral recognition, and proportional reasoning were unique predictors of belonging to the high-achieving group in primary school. This study shows that the cognitive advantages of high mathematics achievement are already observed in preschool.

Early detection of and relevant information on children’s mathematical difficulties is important to initiate targeted teaching and intervention. This study investigated the extent to which strategy use in single-digit addition provides additional predictive information about 61 grade one children’s (6-year-old) mathematical achievement 3 years later that is not available from a standardised mathematics achievement test. Four predictors available in year one (arithmetic strategy use, mathematical achievement, non-verbal reasoning skills and sex) explained 54% of the variation in grade four mathematics achievement. Arithmetic strategy use was the most important single predictor of year four mathematics achievement (R² = 30%) and explained an additional 12% variation if added to a model comprised by the three other year one predictors. This result suggests that systematically obtained measures of how young children solve single-digit arithmetic problems might provide useful information about their foundational number knowledge, which in turn may reveal how well they achieve later in school.