| Proposal Type: | Individual Paper |
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| Domain: | Learning and Social Interaction |
| SIG: | Individual Differences in Learning and Instruction |
| Type | Submitted Paper |
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PC and projector |
| Paper Details |
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| Title | Reviewing students’ conceptions of problem solving. In fact, what sort of problems are they talking about? |
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| Abstract | This paper deals with an important source of individual differences often forgotten about in the classrooms. It presents a study about primary school and compulsory secondary school students’ conceptions of mathematical problems and the consequences these conceptions might have for the teaching and the assessment of these students’ mathematical performance by their teachers. A total amount of 60 students aged 8 to 16 from 9 different urban and suburban schools were clinically interviewed in order to access their conceptions of mathematical problems. Two different basic conceptions of mathematical problems were identified through a qualitative content analysis. The results reveal that students’ conceptions are linked in some way to the classroom assessment practices in which they take part. Educational experience (thus age) and (most important) achievement seem to be influencing elements on the way they conceive of mathematical problems, as far as older low-achievers share the same simple conception of mathematical problem as younger students. |
| Summary | (For the sake of anonymity, any reference has been avoided) Context and aims Along the decades of 1980-1990 a lot have been published about students’ conceptions of problem solving. Researchers coincide in pointing at the following conceptions (or beliefs, as some prefer): “a problem is something difficult to solve”; “you should cope with it in 5 minutes or less, if not, you are not able to solve it”; “problem solving is only for clever people”, etc. Interestingly, none of these studies discuss a previous question that, from this paper’s author’s point of view, must be thought of when asking people about problem solving, namely:
Subjects The study presented in this paper was carried out in Research questions It is assumed by the author that the students construct their conception of mathematical problems as they participate in mathematics classes along their school experience. After this premise, the research questions were stated as:
Method As for the method of research, the students were confronted with ten different mathematical tasks, interrogated about their teachers’ usual assessment practices and asked to assume a hypothetical assessors’ role, pretending to assess a classmate’s mathematical learning. This different research tasks allowed the researcher to gather information about the students’ conceptions of mathematics, assessment and problems –last being focus of this paper-. The students’ answers were fully transcribed. A qualitative content analysis was carried out by means of a particular qualitative software package. A first deductive scheme of categories was drawn from literature as a point of departure for the analysis. After several phases of analysis new categories were defined inductively from the data. All this recursive procedure of analysis was carried out by the same research; the software allows the verification of intra-rater reliability. Results The results of the study indicate that there at least two different conceptions of problems held by students while problem solving. In other words: when two students tell as that problems are difficult to solve, or that only clever people are able to solve problems… they might be not talking about the same kind of problems, indeed. The first conception of mathematical problems found among students was defined as a ‘standard-structural’ conception. Students refer to problems as only the standard word problem: a problem must be composed by a starting descriptive statement, which must followed by a direct question, and a response sentence is expected to be given. That is:
The second conception of mathematical problems found among students was defined as a ‘superficial-structural’ conception. Students identify problems as long as some key elements are visible in the statement; for instance, numbers, the money symbol, a question symbol, and key words such as ‘give’, ‘take’, ‘sell’, ‘win’, regardless of any other structural aspect of the task. Since the author assumes that the students construct their conception of mathematical problems along their participation in mathematics classes, it is not surprising that differences could be found with respect to the students’ age: older students conceive of problems more frequently from a standard perspective, whereas younger students stuck with superficial elements. But the answer to the second research question is unexpected, indeed: low-achievers, present more often a superficial conception of a mathematical problem quite independently from their age (up to 16 years!). Conclusions The results of this study lead us to reflect about consequences at two levels: at a theoretical level, the literature about students’ (and probably also teachers’) conceptions of problem solving must be revised, since the basic shared understanding of the term between the researcher and the subjects was always presumed but now this basic intersubjectivity must be challenged. Second, at a more practical level, possible consequences for low-achievers, in terms of a likely discrimination against better achievers, since very frequently teachers tend to avoid using problems (both as learning goals and teaching resources) with the low-achievers. |
| Keywords | At-risk students Beliefs Mathematics education |
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| Authors | ||||||
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| Name | Surname | Institution | Country | EARLI Number | Presenting | |
| Ana | Remesal-Ortiz | Universidad de Barcelona | Spain | aremesal@ub.edu | * | |
