Conference Agenda

The proposed paper deals with the question how students learn from instructional explanations (in business teaching), in particular, to what extent they rely on the presented concrete example and/or on the explicated general rule when solving application and transfer tasks.

For concept learning, Tennyson and Cocchiarella (1986) state – referring to prototype learning research – that general definitions are helpful but that examples play a predominant role for understanding. Several experiments investigating the acquisition of psychological concepts substantiate the claim that giving definitions and examples leads to better learning than giving definitions only (Balch 2005; Rawson/Ruthann/Jacoby 2015), and suggest that students make more use of known examples than of general definitions when classifying new cases (Zamary/Rawson 2018).

Similarly, in teaching principles that enable students to deal with certain types of tasks/problems, the best strategy seems to be to explicate the general principle and to illustrate and elaborate it using a variety of examples (Fortmüller 1997). Several experimental studies document that abstract and concrete information complement each other in principle-learning and, thus, the combination of rule and example training fosters transfer (Chen/Daehler 2000; Cheng et al. 1986; Fong/Krantz/Nisbett 1986). Theories of analogical reasoning generally assume that concrete examples are easier to understand than abstract principles, and that generated understanding can be transferred to novel situations (Gentner/Loewenstein/Thompson 2003). However, the relation of abstract information and example details in mental representations as well as their role in problem solving is questionable (Reeves/Weisberg 1994). Based on structure-mapping theory (Gentner, 1983, 1989) and pragmatic schema theory (Gick/Holyoak 1983; Holyoak/Koh 1987) on the one hand and exemplar theories on the other hand, Ross (1987) formulates two contrasting hypotheses: According to the principle-cuing hypothesis, learners use examples to access the abstract principle, which is then – once understood – directly applied to solve a novel problem. According to the example-analogy hypothesis, a principle is only understood in terms of an example, such that learners need the example in order to be able to apply the principle to the problem at hand. Empirical findings mostly support the example-analogy view (Chen/Daehler 2000; Holyoak/Koh 1987; Ross 1987). Ross and Kilbane (1997) show that when an abstract principle is explained and afterwards illustrated by an example, students primarily rely on the example. They argue that it might be preferable to employ an embedded-principle method in order to tie the explanation to the example.

In order to investigate the impact of concrete examples and general rules in instructional explanations in business teaching on students’ understanding and achievement, Schopf (2021) compared several versions of a teacher explanation (example-rule, rule-only, example-only) on the topic “break-even point” in an experimental study with second year business academy students. The hypothesis that a teacher explanation with an example-rule pattern is perceived as more understandable and results in superior achievement, than a teacher explanation which only contains the general rule, was confirmed. The finding that in the application tasks mean performance of the example-rule and the example-only groups was significantly better than mean performance of the rule-only group, while mean performance in the transfer task was generally very poor and did not significantly differ between groups, may lead to the conclusion that students from the example-rule and the example-only groups were able to solve the application tasks by using the concrete example as an analogue without having fully understood the underlying general principle. However, students’ notes on the work sheets did not provide sufficient evidence to support this assumption.

Thus, in a follow-up project the author wants to dive deeper and analyse students’ thinking processes.

Problem solving is reported as an essential criterion for the appropriation of mathematical skills. According to mathematicians, psychologists, and mathematics educators, it allows in particular to access the meaning of mathematical notions (e.g. Schoenfeld, 1985).

With this respect, Problem Posing has been recognized as a valuable activity in mathematics education (e.g. Ellerton, 1986; Singer, Ellerton & Cai., 2013 ; Cai et al., 2015). However, researchers (e.g. Zhang & Cai, 2021) have shown the complex nature of teaching mathematics through problem posing. This paper would like to contribute to this research trend.

In this paper, we will focus on multiplicative problem posing. Our purpose is to provide a first understanding of a practice of using problem posing to teach mathematics over a long duration. Our research is part of a larger one, namely the ACE research (Arithmetic and Comprehension in Elementary School). This research proposes a mathematical program in first, second and third grade.

In ACE, a prominent emphasis is put on public representations of mathematical relations seen as “any configuration of characters, images, or concrete objects that symbolizes an abstract idea” (Goldin & Kaput, 1996) and may include manipulative materials, pictures or diagrams, spoken language, or written symbols.

Our early studies concerning one-step additive problems focused on semantic features (Riley, Greeno & Heller, 1983). An extension of our study focus on multiplicative word problems in grades 2 et 3. Our research is organized in a Cooperative Engineering (Sensevy & Bloor, 2020), where teachers and researchers have progressively built a curriculum, in the way of lesson studies (Miyakawa & Winsløw, 2009) with some representational tools. These representations have been worked out at the same time as means for exploring numbers and as problem-posing tools. In this research, the teachers belong to the research team, and are considered as teachers-researchers.

We focus on the fundamental fact, in the teaching-learning process, that students learn by relying on a previous set of meanings, that we may call an already-there (CDpE, 2019). In order to teach, a teacher has to gain a deep understanding of this already-there. For doing that, we argue that a promising avenue consists of organizing the teaching-learning process on the basis of specific examples of the way problems are posed and solved. These emblematic examples of the practice can be seen progressively as exemplar, in Kuhn's sense (Kuhn, 1977). In that way, we will see how a system of exemplars is designed to help students to pose multiplicative problems on their own. In this paper, we thus investigate the following research question: How can a teacher organize problem posing tasks by giving habits of representations of mathematical relationships?