Relation between procedural and conceptual knowledge

An additional possibility how a potential impact of our inductions might become visible is the integration of procedural and conceptual knowledge. Although conceptual knowledge might not have profited directly from the interventions tested here, it is pos-sible that one or more of the inductions serve to integrate both kinds of knowledge, which might indicate an increasingly abstract concept of commutativity. The parameters hit rate, false alarm rate and sensitivity (d´) were included as indicators of conceptual knowledge, whereas the number of solved problems separately in each subset of the computation task as well as the resulting Difference value were put into the correlation matrix as procedural measures. An integrated concept of commutativity should be indi-cated by a significant correlation between the difference score and the sensitivity score d’. Table 28 presents the correlations for the three experimental groups separately, as well as collapsed across all conditions. As can be seen, in all experimental conditions there was no significant correlation between the difference score and d’. Even when

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collapsing the three conditions in order to increase power, no statistically significant correlation between d´ and the difference score was detected.

Table 28. Experiment 3: Integration of procedures and concepts: Correlation coef-ficients between procedural and conceptual knowledge in Experiment 3

judgment

Our results provide strong hints that symbolic approximation is more suitable to induce an arithmetic principle like commutativity for a later exact context than nonsymbolic approximation. In our study, children who encountered an induction with nonsymbolic approximate arithmetic containing commutative trials showed no advantage in an exact symbolic task also interspersed with commuted problems afterwards. After symbolic approximation on the other hand, children actually showed a commutativity effect that was of comparable size as the effect after a direct instruction of the principle. So it seems that also symbolic approximation can be used to foster children´s skills in exact arithmetic. Hyde et al. (2014) found in their sample of first graders nonsymbolic arith-metic to be a suitable induction for exact symbolic addition problems. We extended this research question to arithmetic principles and found that in second graders, an approxi-mation training with interspersed problems that allow the use of that principle could even induce the exploitation of the additive law of commutativity. This was only the case for symbolic approximation, though. However, neither of the inductions nor an explicit explanation of the principle led to a specific advantage in increasing conceptual commutativity knowledge. This is in line with several studies that also failed to

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strate the effectiveness of diverse kinds of inductions to increase conceptual knowledge (Fyfe, Rittle-Johnson, & DeCaro, 2012; Matthews & Rittle-Johnson, 2009; Sherman &

Bisanz, 2009).

The commutativity effects found in this experiment were mostly of only medium size.

One explanation might be that second graders already are so proficient in solving addi-tion problems that there was little need for them to exploit shortcut possibilities. Calcu-lating a problem anew might not mean a notable disadvantage in solution time to them.

So in the following experimental series (Experiments 4 a – 4 c), we tested different age ranges with the symbolic approximate arithmetic induction, starting with children at the beginning of first grade. We have demonstrated that at this age, children already per-form above chance in a symbolic approximate arithmetic task (Studies 1 and 2, see also Gilmore et al., 2010), and consistently found no detrimental influence of SES on this measure. Thus, it seems promising to test it also at the beginning of formal education as an induction for principles that children already have some precursory knowledge of.

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13 Experiment 4

We replicated Experiment 3 in order to further investigate the effect of symbolic ap-proximation on procedural and conceptual knowledge of commutativity. Experiment 3 had indicated that symbolic approximation can trigger at least the procedural exploita-tion of an arithmetic principle. We thus repeated our testing with the symbolic approxi-mate arithmetic induction in three age groups to narrow the age range in which symbol-ic approximation might be a suitable means of teaching¹¹.

To that aim, in Experiment 4 a we tested children who had just started school with slightly simplified versions of the symbolic approximate arithmetic induction as well as the computation task used in Experiment 3. In this experiment, we did not include the conceptual measure (the judgment task) because we did not want to overburden partici-pants and furthermore assumed that in children who had just started school there would be no sufficient basic understanding to be fostered by an indirect induction. Neverthe-less extending the testing of Experiment 3, we realized two conditions, one starting with the induction (the symbolic approximation task), and the other with the computation task. Thereby, we were able to test if the positive influence of symbolic approximate addition problems presenting the principle of commutativity is actually a unidirectional one, or if exact commutative problems might in turn also have a fostering effect on sub-sequent approximation.

In Experiment 4 b, we tested children at the end of first grade and used the same tasks as in Experiment 3, including the measure of conceptual commutativity knowledge.

Besides to the two conditions already described for Experiment 4 a, we also instantiated an additional control condition which did not receive exact commutativity problems at all during the course of the experiment. This enabled us to disentangle the effects of the symbolic approximation task and exact commutativity problems on conceptual knowledge (see Chapter 13.2.1 for the more detailed description of the procedure).

Experiment 4 c was identical to 4 b except for the age group tested and some adjusted time limits. This time our participants were third graders, in order to on the one hand test if our symbolic approximation induction is also ꞌworkingꞌ with children who had sufficient opportunity to practice the principle in question, and on the other hand to

11 The experimental series described in this chapter has been prepublished with the permission of the Dean of Research of the University Cologne (see Hansen et al., 2015).

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plore if conceptual knowledge might be fostered by our induction in children who should already have a more stable foundation of conceptual commutativity understand-ing.

13.1 Experiment 4 a: Start of first grade

The main goal of Experiment 4 a was to investigate whether children who had not yet received any formal instruction about the commutativity principle would benefit from symbolic approximate arithmetic problems with respect to spontaneously spotting and applying commutativity-based shortcut options in exact arithmetic problems. For this purpose, we investigated first graders who had attended school for approximately four months and had not yet learned about commutativity in school. Half of the children started with the symbolic approximation task and then received the exact arithmetic problems (approximation-first group). The remaining children were administered to the reversed order of tasks; that is, they solved the exact arithmetic problems (computation task) first and then worked through symbolic approximation task (computation-first group). If the symbolic approximation task triggers the exploitation of commutativity in the exact arithmetic problems like we found in Experiment 3, children in the approxima-tion-first group should show a larger commutativity benefit than the computaapproxima-tion-first group.