Conceptual and Procedural Knowledge During Strategy Construction: A Complex Knowledge Systems Perspective

ABSTRACT

This article elaborates a new direction for studying the construction of novel strategies that enables researchers to model the conceptual underpinnings of students' observable strategic actions during episodes of mathematical problem solving. The nature of the relationship between conceptual and procedural knowledge has been persistently debated for decades. Recently, there has been mounting empirical evidence that conceptual and procedural knowledge can develop by mutual bootstrapping in a bidirectional and iterative fashion (e.g., Rittle-Johnson & Schneider, 2014). However, the very constructs of conceptual and procedural knowledge (especially procedural knowledge) have been critiqued for inconsistency in definition and lack of operationalization (Star, 2007). The analysis in this article addresses this critique by modeling the diverse forms of conceptual and procedural knowledge needed to implement a strategy as a complex knowledge system: a strategy system. Furthermore, the strategy system model is used to elaborate processes of mutual bootstrapping between conceptual and procedural knowledge at a moment-by-moment time scale. The strategy system model builds upon the Knowledge in Pieces epistemological perspective (diSessa, 1993), and coordination class theory in particular (diSessa & Wagner, 2005). Both the theoretical notion of a strategy system and the bidirectional model of mutual bootstrapping between conceptual and procedural knowledge are illustrated using data from a case study of a pre-algebra student who iteratively refined a procedure for solving algebra word problems. The strategy system model highlights the complexity of both strategies and concepts and offers a window into what can be learned by students during strategy construction processes.

Acknowledgments

This research grew out of my doctoral dissertation at the University of California, Berkeley. I warmly thank my committee—Alan Schoenfeld, Geoff Saxe, and Andy diSessa—for their intellectual support and encouragement of this research. I especially thank Andy diSessa for his ongoing and genuine engagement with me around the ideas I have developed in this article. The published analysis is better because of generative conversations and helpful suggestions from the Functions and Knowledge Analysis Research Groups at Berkeley. In particular, I thank Leslie Dietiker, Darrell Earnest, Shuly Kapon, Olivia Levrini, Julie Nurnberger-Haag, Orit Parnafes, Geoff Saxe, Bruce Sherin, Jack Smith, and my STaR manuscript group for their close readings and comments on previous drafts of this work. Lastly, thanks are due to Noel Enyedy, for his guidance through the review process and to Jim Greeno and three anonymous reviewers who provided insightful and challenging feedback that shaped the quality of the finished article.

Notes

1 In the case of Wason's famous selection tasks, individuals have a hard time solving the given task when posed in terms of abstract principles to be checked but can readily solve the same task when it is framed in terms of policing a social rule.

2 Siegler and Jenkins (1989) defined a strategy as a procedure that is nonobligatory and goal-oriented.

3 Karmiloff-Smith (2013) provided an excellent history of microgenetic analysis as an empirical methodology, in particular focusing on the work of the strategies group in Geneva.

4 Knowledge analysis (diSessa, Sherin, & Levin, 2016) is a broader suite of analytic approaches with similar aims (the study of the form and content of knowledge) that includes, but is not limited to, analyses based on the study of knowledge-in-transition.

5 diSessa et al. (2016) offered an updated treatment on the terminology and advancements with respect to the coordination class construct. I have noted what is relevant to the analysis here, but refer readers to the expanded treatment.

6 In addition to inferential relations between categories of attention, other knowledge participates in inferential chains that support problem-solving actions. This could include ancillary knowledge like number facts or other knowledge that supports implementation of a strategy.

7 The transcription conventions used throughout the analysis are as follows: “…” denotes voice trailing off. // denote an interruption or overlapping speech. [italic brackets] denote explanatory text used to disambiguate unclear references or add detail about gestures or nonverbal actions. (Pause) marks significant pauses in speech and the duration of the pause is recorded. Italics are used for emphasis.

8 It is worth noting that the assumption of change in input being proportional to change in output is not (necessarily) valid, and that Liam's actions are consistent with this.

9 In a similar problem that Liam solved in a later session, he explained that he knew that consecutive numbers are all about the same amount. In the context of this problem, one can infer that Liam's choice of first guess was informed by thinking that roughly 1/3 of 414 would be a good first guess. He did not articulate this reasoning at this time, but it gave a glimpse into the strategic nature of Liam's approximations, even for his choice of first guesses.

10 This is expressed in this top-level presentation as 1 unit change results in a change of D/d unit increments, but I note that in practice, the individual may have other ways to determine how many unit increments are necessary to solve problems (e.g., imagined simulation of iterated covarying change as opposed to division).