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Acquiring the principles of mathematics and science

Exploring how students develop conceptual understanding through cognitive change, conflict, and discovery learning.

AIt has been pointed out that learning mathematics and science is not so much learning facts as learning ways of thinking[1]. It has also been emphasised that in order to learn science, people often have to change the way they think in ordinary situations[2]. For example, in order to understand even simple concepts such as heat and temperature, ways of thinking[1] of temperature as a measure of heat must be abandoned and a distinction between ‘temperature’ and ‘heat’ must be learned. These changes in ways of thinking[1] are often referred to as conceptual changes[3]. But how do conceptual changes[3] happen? How do young people change their ways of thinking[1] as they develop and as they learn in school?

BTraditional instruction[4] based on telling students how modern scientists think does not seem to be very successful. Students may learn the definitions, the formulae, the terminology, and yet still maintain their previous conceptions. This difficulty has been illustrated many times, for example, when instructed students are interviewed about heat and temperature. It is often identified by teachers as a difficulty in applying the concepts learned in the classroom; students may be able to repeat a formula but fail to use the concept represented by the formula when they explain observed events[5].

CThe psychologist Piaget suggested an interesting hypothesis relating to the process of cognitive change[6] in children. Cognitive change[6] was expected to result from the pupils’ own intellectual activity[7]. When confronted with a result that challenges their thinking - that is, when faced with conflict - pupils realise that they need to think again about their own ways of solving problems, regardless of whether the problem is one in mathematics or in science. He hypothesised that conflict brings about disequilibrium, and then triggers equilibration processes that ultimately produce cognitive change[6]. For this reason, according to Piaget and his colleagues, in order for pupils to progress in their thinking they need to be actively engaged in solving problems that will challenge their current mode of reasoning[8]. However, Piaget also pointed out that young children do not always discard their ideas in the face of contradictory evidence[9]. They may actually discard the evidence and keep their theory.

DPiaget’s hypothesis about how cognitive change[6] occurs was later translated into an educational approach[10] which is now termed ‘discovery learning[11]’. Discovery learning[11] initially took what is now considered the Tone learner’ route. The role of the teacher was to select situations that challenged the pupils’ reasoning; and the pupils’ peers had no real role in this process. However, it was subsequently proposed that interpersonal conflict[12], especially with peers, might play an important role in promoting cognitive change[6]. This hypothesis, originally advanced by Perret-Clermont (1980) and Doise and Mugny (1984), has been investigated in many recent studies of science teaching and learning.

EChristine Howe and her colleagues, for example, have compared children’s progress in understanding several types of science concepts when they are given the opportunity to observe relevant events. In one study, Howe compared the progress of 8 to 12-year-old children in understanding what influences motion down a slope. In order to ascertain the role of conflict in group work[13], they created two kinds of groups according to a pre-test: one in which the children had dissimilar views[14], and a second in which the children had similar views. They found support for the idea that children in the groups with dissimilar views[14] progressed more after their training sessions[15] than those who had been placed in groups with similar views. However, they found no evidence to support the idea that the children worked out their new conceptions during their group discussions[16], because progress was not actually observed in a post-test immediately after the sessions of group work[13], but rather in a second test given around four weeks after the group work[13].

FIn another study, Howe set out to investigate whether the progress obtained through pair work[17] could be a function of the exchange of ideas[18]. They investigated the progress made by 12-15-year-old pupils in understanding the path of falling objects[19], a topic that usually involves conceptual difficulties[20]. In order to create pairs of pupils with varying levels of dissimilarity in their initial conceptions[21], the pupils’ predictions and explanations of the path of falling objects[19] were assessed before they were engaged in pair work[17]. The work sessions involved solving computer-presented problems, again about predicting and explaining the paths of falling objects[19]. A post-test, given to individuals, assessed the progress made by pupils in their conceptions of what influenced the path of falling objects[19].

Collocations & key phrases

  • [1]ways of thinking — Particular methods, perspectives, or habits of reasoning.
  • [2]ordinary situations — Normal, everyday conditions or circumstances.
  • [3]conceptual changes — Transformations in fundamental beliefs or mental frameworks.
  • [4]traditional instruction — Conventional teaching methods centered on direct teacher explanation and rote learning.
  • [5]observed events — Physical phenomena or happenings that can be directly seen or experienced.
  • [6]cognitive change — A shift or development in the way a person mentally processes information or understands concepts.
  • [7]intellectual activity — Mental effort involving reasoning, thinking, and problem-solving.
  • [8]mode of reasoning — A specific pattern or way of logical thinking.
  • [9]contradictory evidence — Facts or observations that conflict with a pre-existing belief or hypothesis.
  • [10]educational approach — A specific method or strategy used to facilitate teaching and learning.
  • [11]discovery learning — An educational strategy in which students acquire knowledge by investigating problems and finding solutions themselves.
  • [12]interpersonal conflict — A disagreement or clash of opinions between two or more people.
  • [13]group work — A collaborative learning task performed by several students working together.
  • [14]dissimilar views — Opinions, understandings, or perspectives that differ from one another.
  • [15]training sessions — Organized periods of structured learning or guided activity.
  • [16]group discussions — Conversations where members of a group share ideas to solve a task.
  • [17]pair work — An activity where two students collaborate on a shared task.
  • [18]exchange of ideas — The mutual sharing of thoughts, theories, and explanations between people.
  • [19]falling objects — Physical items dropping downward under the pull of gravity.
  • [20]conceptual difficulties — Challenges or misunderstandings experienced when attempting to grasp complex concepts.
  • [21]initial conceptions — The starting beliefs or understandings an individual holds before receiving new information.

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