Helping students know'further'-increasing the flexibility of students' knowledge using symbolic invention tasks

Helping Students Know ‘Further’ – Increasing the Flexibility of Students’ Knowledge Using Symbolic Invention Tasks Ido Roll (idoroll@cmu.edu) Human Computer Interaction Institute, Carnegie Mellon University 5000 Forbes Avenue, Pittsburgh, PA 15213, USA Vincent Aleven (aleven@cs.cmu.edu) Human Computer Interaction Institute, Carnegie Mellon University, 5000 Forbes Avenue, Pittsburgh, PA 15213, USA Kenneth R. Koedinger (koedinger@cmu.edu) Human Computer Interaction Institute and Department of Psychology, Carnegie Mellon University, 5000 Forbes Avenue, Pittsburgh, PA 15213, USA Abstract evaluate a set of examples with regard to one aspect of the data. Figure 1 shows an example of such a task, in which students are asked to invent a method for comparing the variability of two datasets, in order to choose the more consistent one (i.e., where data is “closer together”). Following the invention attempt, students receive direct instruction on canonical methods and practice them. For example, following the task detailed in Figure 1, students receive instruction on Mean Absolute Deviation and practice applying it. While students often fail to invent general valid methods, research suggests that this experience prepares them to better learn independently from learning opportunities that follow the instruction (Schwartz & Martin, 2004; Kapur, 2008). IPL tasks use contrasting cases to direct students' attention to deep features of the domain. Rather than analyzing a single dataset, as commonly done in show- and-practice problems, IPL tasks ask students to compare two or more sets of data that vary along a single deep feature. For example, the two sets in Figure 1 have the same average and sample size but differ in their range. The contrasting cases also give students a baseline against which to evaluate their inventions, since their intuitive comparison of the cases is often clear and correct (Schwartz, Sears, & Chang, 2007). The invention activity itself, prior to instruction, can be divided into two parts. First, students analyze the contrasting cases and rank them intuitively according to the target construct (e.g., variability). We refer to this stage as comparative reasoning, since students reason about the task by comparing the different cases. The second part is the design of mathematical methods, in which students attempt to invent general valid methods that rank the cases in the same way as their intuitive ranking. We refer to this stage as generative reasoning, since students generate symbolic methods to quantitatively compare the contrasting cases. Table 1 shows a summary of the IPL process. In the current study, we begin to unpack the IPL process and its effects. Our first research question evaluates the different roles of comparative vs. generative Invention as Preparation for Learning (IPL) is a teaching strategy in which students attempt to develop novel solutions prior to receiving instruction (Schwartz & Taylor, 2004). This method was previously shown to prepare students to learn independently from future learning opportunities that build upon the materials learned in class. We began unpacking the IPL process by identifying its components and evaluating the contribution of generative reasoning (in the form of symbolic invention) on top of comparative reasoning (in the form of ranking alternatives). An in-vivo study in 6 middle-school classes with 105 students found that generative reasoning is an essential component of IPL. Furthermore, we found that students who attempted to invent symbolic models during the IPL process (generative reasoning) were able to invent new strategies during the post-test. At the same time, students who completed the IPL process without designing symbolic methods were in need for worked-out examples in order to solve new-strategy problems in the post-test. We propose a mechanism that explains how invention leads to the observed increased flexibility in students’ knowledge. Keywords: Invention as Preparation for Learning; Preparation for Future learning; Transfer; Comfort Zone; Generative Reasoning. Introduction Invention as Preparation for Learning (IPL) is a teaching strategy that uses constructivist instructional methods and direct instruction in a complementary fashion (Schwartz & Martin, 2004). First, students are asked to invent general methods (and their mathematical expressions) to The Bouncers Trampoline Company tests their trampolines by dropping a 100 lb weight from 15 feet. They measure how many feet the weight bounces back into the air. They do several trials for each trampoline. Here are the results for two of their trampolines: Trampoline A: {1 3 5 7 9} Trampoline B: {3 4 5 6 7} Which trampoline is more consistent, that is, its test results are closer together? Figure 1: The trampoline IPL task.