| In the past couple of decades,problem solving has always been one of the core issues in the reform and development of education.However,Einstein proposed that posing a problem is often more important than solving a problem,since problem solving is probably just a mathematical or laboratory skills,while to raise new problems,new possibilities,and to regard old problems from a new angle requires creative imagination and marks the real progress in science.With the deepening of world multi-polarity,economic globalization and social informatization,the importance and urgency of training creative talents is becoming increasingly highlighted.To enhance our capability of independent innovation,and build an innovation-oriented country will not be reached without cultivating innovative intellectuals,which depends mainly on creative education,the reform of educational ideas,educational content as well as the methods.The revolution from focusing on problem solving to problem posing as creative activities is the possible path to satisfy the increasingly and personalized educational demands,is the new trail to achieve the modernization of education.Problem posing as an instructional goal and an instructional approach,plays an important role on maximizing the quality and equitable distribution of learning opportunities as well as fostering students’ understanding,creativity,and non-cognitive skills.Nowadays,Problem posing is receiving increased attention in school curricula and instruction around the globe.Even though,we still have to admit that there is a gap between the research an practice,between the advocacy and reality.To some extent,problem posing is often marginalized by mainstream research.We still meet a lot of problems on the way forward in the research and practice of problem posing.The most essential and prominent research space is the process of information processing and dissemination of posing problem.Mathematical communication is the cognitive activity of information processing,dissemination and expressing.There is a large body of research on the topic of mathematical communication in problem solving.However,It is still lack of research on the topic of mathematical communication in problem posing.The inherent mechanism of problem posing is different from problem solving,as the product of its communication is the new problem posed according to the problem situation,from which we can only infer the posers’ thinking from one aspect.Thus,we want to know how to explore more about what they are thinking while posing problems,such as how they understand the context of problem posing and how they pose problems.So,it is necessary to explore and examine the processes,performances and regular patterns of mathematical communication in problem posing.Firstly,this study primarily explores the framework of mathematical communication in problem posing based on the theory of information communication and information processing.We conceptualized the communication in mathematical problem posing as having three stages: a)information input—understanding the meaning of the problemposing tasks,b)information processing—determining and understanding the processes of how problems are posed,and c)information output—representing posed problems.The framework is theoretically conceptualized.Then the framework is illustrated by using data from two case studies based on the same problem-posing task.The usefulness of the framework is discussed to understand the mechanisms involved in students’ understanding of the problem-posing situation and posing and representing mathematical problems.Then,we discuss the performance and difference of students’ mathematical communication under different problem-posing tasks formats based on the framework of mathematical communication in problem posing proposed above.The research problems are: a)under the problem-posing task with or without numbers,how do students understand the task,how do they construct and express the new problems? b)under the problem-posing task with or without context,how do students understand the task,how do they construct and express the new problems? In order to solve these two research problems,we specified investigate students’ inner thinking and strategies,performance and the difference between experts and novices in the three stage of mathematical communication in problem posing.In addition,in order to dig deeper into the reasons for the difference in the performance of experts and novices’ mathematical problem posing,we set up the problem solving tasks using the same context with problem posing tasks to explore the possible relationship between problem-solving and problem-posing.This study mainly uses the paper-pencil test,eye tracking & think aloud and interview research methods to solve the research problems.A total of 700 students in the sixth grade were tested in the paper-pencil test from three primary school in city A.The participants attending the eye tracking and think aloud experiment consisted of two groups,one of which was 66 graduate students major in mathematics education or pure mathematics from a university,and the other was 60 students in grade six(note: they just entered middle school and were recorded as sixth grade students.)The research tool include the problem-posing tasks with and without numbers(PPT-Num;Leung & Silver,1997),the problem-posing tasks with and without context(PPT-Context;Cai et al,2019)and problem solving tasks same with the problem-posing context(PPS).The data collection process is divided into the following phases: a)pilot test,interviewed 10 participants who are professor,post doctor,doctor major in mathematics or mathematics education and primary school students and using the semi-open verbal report to adjust and revise the research tool;b)formal test,the paper-pencil test is divided into two rounds.The first round is problem-posing test.The test tasks are selected from the A/B items in the PPT-Num and PPT-Context tools to form 8 sets tests.They were distributed randomly to 700 students from 16 classes in 3 primary schools;The second round is problem solving test.The tasks are selected from the PPS test.The two rounds tests were separated by on month.The eye tracking & think aloud test is divided into two rounds too.The first round is problem-posing test.The test tasks are selected from the A/B items in the PPT-Num and PPT-Context tools to form 2 sets tests.They were distributed randomly to 66 graduate students and 60 six grade students.The second round is problem-solving test.The task are selected from the PPS test.The two round tests were separated by a week.The research results shows that:1.The impact of task format with or without numbers on the three stages of students’ mathematical communication in problem posing.a)The task format with or without numbers do effect the input stage of students’ mathematical communication in problem-posing,specifically reflected on: Students percept understanding the task with numbers more easily;Students need more cognitive load under task without numbers;Compared to task with numbers,students pay more attention on processing and reprocessing the information about the “variables”.b)The task format with or without numbers do effect the process stage of students’ mathematical communication in problem-posing,specifically reflected on: Compared to task without numbers,students construct significantly more solvable problems under the task with numbers;The structural complexity of problems posed by students is affected by the task format with or without numbers.The task without numbers could provide students with opportunities to construct complicated structural problems.Students need more cognitive load to construct problem under the task without numbers,the difference is showed in their information processing of “events”,“variables”and “value of the variables”.c)The task format with or without numbers do not effect the output stage of students’ mathematical communication in problem-posing.2.The impact of task format with or without context on the three stages of students’ mathematical communication in problem posing.a)The task format with or without context do effect the input stage of students’ mathematical communication in problem-posing,specifically reflected on: Students percept understanding the task with context more easily.However,there is no consistent and stable difference in students’ understanding the task with or without context.b)The task format with or without context do effect the process stage of students’ mathematical communication in problem-posing,specifically reflected on: Compared to task without context,students construct significantly more solvable problems under the task with context;Compared to task without context,the categories of problem structure constructed by students under the task with context are more abundant.Students need more cognitive load to construct problem under the task without numbersc)The task format with or without numbers do effect the output stage of students’ mathematical communication in problem-posing.3.The cognitive behaviors and strategies for three stage of students’ mathematical communication in problem posingBy analyzing 126 posers’ think aloud protocols in the input stage of problem posing,we found the cognitive action of “translation” and “integration” discussed in the previous literature and the cognitive action of “questioning” and “evaluation” were emerged in the data.The “questioning” means the posers express their thinking obstacles or the cognitive conflict with existing cognitive experience encountered during understanding the tasks.The“evaluation”is the posers’ perception of understanding the task on a certain value orientation.The main behaviors during the translation stage are reading(or re-reading)tasks,describing,interpreting and summarizing the meaning of each sentence.The thinking strategies during the translation stage used include inference,summarize(extract)the key variables of each sentence,and interpret their extensional meaning.The thinking strategies during the integration stage used include segmenting or overall the general idea and concatenating information for calculation.During the processing stage of problem posing,we found the cognitive action of“selection”and “organization” discussed in the previous literature.The cognitive action of “selection” appear as selective restatements of sentences or supplementary information outside of the context.The thinking strategies include selecting the existing information in the context,obtaining new information inferred from the information in the context,filtering specific contextual information and supplying information outside of the context.The cognitive action of “organization” appear as describing and explaining the relationship between the conditional elements and target elements.The thinking strategies mainly include using the known relationship of the context,obtaining the new relationship inferred from the existing relationship of the context and adding the relationship outside of the context.During the expressing stage of problem posing,we found the cognitive action of “statement”,“evaluation”and “correction” discussed in the previous literature.The cognitive action of “making the statement” are expressed as organizing the language to descript questions involved with the strategies of independent expression and integrated in reasoning.The cognitive action of “evaluation” is expressed by evaluating certain characteristics of the questions posed according to their own psychological standards involved with the orientations of the complexity and solvability.The cognitive action of “correction” is the description and explanation of the conditions or goals in the problem been posed involved with the strategies of correcting the condition elements and goal elements.After expressing the problems,some poser will solve the problems posed by themselves.In addition,The sequence of problems posed by the poser includes four categories of “parallel migration from the same information”,“vertical migration from the previous problem”,“according to the distribution of the key information in the context”and “change the elements or relationships from the previous problem”.4.The difference and causes of the performance of the subjects with high and low ability in the three stages of mathematical communication in problem posing.From the performance and eye tracking data analysis of the 126 participants who attended problem posing activities on 4 tasks,it showed that the high ability group with a wide range of knowledge took longer to process information at each stage of the problem posing,but the amount of problems been posed was less than that of the low ability group.The quality of the problems posed by the high ability group was slightly higher than that of the low ability group,but there was no significant difference.By analyzing the paper-pencil test data(from 669 participants)of problem posing and problem solving,eye tracking data(from 126 participants)of problem posing and problem solving and the interview data(from 6 participants),we explored that the reasons for the differences in the performance of two groups in the three stages of mathematical communication in problem posing can be summarized as follows:a)In the same context,students’ correctness of problem solving is weakly related to their performance of the problem posing.Therefore,it could not be directly concluded that the high ability group with higher education level and outstanding ability to solve mathematical problems will performance better in problem posing than low ability group.b)Students who can not(or partly)solve problems have the capability to pose solvable problems as much as possible.Therefore,taking account of the learning opportunities,it can not be concluded that the subjects with stronger ability of problem solving and broader knowledge on mathematics will performance better on problem posing than that of less educated subjects.c)The high ability participants are more familiar with problem solving.They just used the knowledge learnt in the primary and middle school to pose problems.Therefore,taking account of their knowledge been used and experience,the high ability subjects did not take the potential advantages in problem posing.The main innovation of this research is to conceptualize the framework of mathematical communication in problem posing as having three stages.The usefulness of the framework is discussed to understand the mechanisms involved in students’ understanding of the problem-posing situation and posing and representing mathematical problems.Therefore,it plays an important role as providing the theoretical support for deeper exploration of the inherent process of problem posing,and laying the foundation for researchers to know what posers think.Secondly,the target data of the input stage and process stage is distinguished by setting the partition in Eprime code.The last is to use the eye tracking technology to assist in verifying the results of students’ performance on mathematical communication in problem posing and reveal the regular pattern of the internal posing processing,so that to improve the effectiveness and precision of assessment.The shortcoming of the research is that there are few think-aloud protocols that can be further analyzed in the stage of understanding the tasks(90% of the participants tend to understand the tasks by reading them or silent reading them).Secondly,the participants of low ability group attending the eye tracking experiments were selected from a good quality school(although a random sample is chosed within the school),it was difficult for us to distinguish the experts and novice in problem posing. |
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