Investigation of the ability of task design of elementary mathematics teachers about addition and subtraction

Teachers need a variety of knowledge and competencies to teach mathematics well, and examining teachers' status of these skills and competencies can be a good area of research. One of the competencies needed by teachers is the competence of designing mathematical tasks. Mathematics task plays a key role in students' mathematics learning, and " what students learn is largely defined by the tasks they are given." There are various definitions of mathematics task in research literature. Watson and Sullivan (2008) define mathematics task as information that serves as the prompt for student work, presented to them as questions, situations, and instructions that are both the starting point and context for their learning. Various classifications for mathematics tasks have been presented so far. One of them is the classification of tasks according to the type of their function in the teaching process. In this type of classification, tasks are classified to 1- Warm up task; 2- learning task; 3- review task; 4- Practice task; 5. Assessment tasks. Assessment tasks are tasks that teachers use to evaluate students' performance, and because of the role and importance of assessment in the mathematics teaching and learning process, can play an important role in teaching and learning process. With considering the importance of the ability to design task, one of the appropriate situations that can study teachers' ability in this case by that, is the mathematical problem posing situation. Silver (1994) defines the mathematical problem posing as producing of a new problem from a situation or experience, new formulating of problems. Stoyanova and Ellerton (1996) classified problem-posing situations according to their structure into three categories: 1) free; 2) semi-structured; and 3) structured. A review of research literature revealed that one of areas deceived by researchers is examining relationship between teachers' task design ability and teacher' problem posing ability, and evaluating teachers' task design ability using problem posing situations. Accordingly, the present study aimed to investigate task design ability of new elementary Mathematics books’ provincial instructors based on ‎problem posing situations about addition and subtraction operations. The reason for this study of the two operations of addition and subtraction is the role and importance of these two operations in individuals' mathematical learning. Addition and subtraction is the foundation of school mathematics and has an impact on mathematical content across different grades. A descriptive survey method was used to conduct this study. The statistical population of the study consisted of the new elementary Mathematics books’ provincial instructors. 151 of whom participated in one of State Mathematics Supply Courses New Mathematics Textbooks for Elementary School were selected by convenience sampling. To collect the data. A researcher-made test was used in which their' assessment task design ability for two addition and subtraction operations was evaluated through the free problem posing situation. The posed problems by the participants were analyzed on two basis: 1) addition and subtraction and 2) task design. For the analysis of problems based on addition and subtraction operation axis, closed-ended symbolic problems were analyzed using Van de Walle et al. (2016) model and closed-ended verbal problems were analyzed using Riley et al. (1983) model. To analyze problems based on task design axis, symbolic problems analyzed regarding their open-ended and closed-ended responses and verbal issues analyzed based on 1- open-ended and closed-ended questions, 2- Shao (2018) model and 3- Shimizu et al. (2010) model. The results of the posed problems based on the addition and subtraction operations axis showed the problems that the teachers posed symbolically were more limited to the two forms of "unknown result (addition)" and "unknown result (subtraction)" and the problems presented in others formats were low (less than 10% each). The analysis of the verbal problems revealed that the problems were mostly in two modes of change and composition and percentage of problems in comparison form was very low (7%). It was also found that the verbal problems wasn’t not in the form of "Compare 6". The problems analysis base on the task design axis also revealed that very few of the posed problems (4% symbolic and almost 0% of verbal ones) were open-ended and most of them were closed-ended. The results of the verbal problems analysis revealed that although most of them had real-life contexts (89%) but there was no task designed to engage students in solving practical problems. The results also revealed that among the designed tasks, there was no task that was completely "authentic", "rich" and "complex", and although most of them involved students at the “application” level, they were tasks that link them With real life and application limited to mentioning an object or an event from the real world and not really contextual. On this basis, it can be concluded that most of them were low-level tasks. Given the impact of mathematics textbooks’ content on teachers' task design ability, it is suggested to improve the mathematics textbooks design capability so that the content of them can be modified to have necessary diversity generally in all topics and  specifically in both addition and subtraction operations. Another suggestion is enhancing teachers' task design ability, provide appropriate instructional content on mathematics tasks, and conduct empowerment courses and workshops based in these content. This study was conducted using convenient sampling method, so the results cannot be generalized to the statistical population. Accordingly, it is recommended that a similar study be conducted with appropriate sampling methods that can be generalized to the whole population. Another limitation of this study was that teachers were asked to design tasks for assessment. This can lead to limiting them in design of diverse tasks with regard to teachers' perception of assessment as final assessment. Accordingly, different studies are suggested to investigate ability of teachers to design task based on different tasks in mathematics teaching process.