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Using Variation Theory to Design Tasks to Support Students' Understanding Using variation theory to design tasks to support students’ understanding of logarithms Annmarie O’neil, Helen Doerr To cite this version: Annmarie O’neil, Helen Doerr. Using variation theory to design tasks to support students’ understand- ing of logarithms. CERME 9 - Ninth Congress of the European Society for Research in Mathematics Education, Charles University in Prague, Faculty of Education; ERME, Feb 2015, Prague, Czech Republic. pp.433-439. hal-01286925 HAL Id: hal-01286925 https://hal.archives-ouvertes.fr/hal-01286925 Submitted on 11 Mar 2016 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Using variation theory to design tasks to support students’ understanding of logarithms AnnMarie H. O’Neil and Helen M. Doerr Syracuse University, Syracuse, USA, [email protected] In this paper, we discuss three implementations of a is a dearth of research literature involving student task in which students were asked to generate examples learning of logarithmic functions, in general, and the of logarithm expressions equal to a given value. We sit- algebra of logarithms, in particular. Of the literature uate the design of the task in variation theory and in available, much of the focus is on suggested mathe- research on learner generated examples, which describe matical and pedagogical approaches to logarithms learning as developing students’ ways of seeing, particu- with no empirical data related to students’ learning larly in regards to the dimensions of variation and the using these approaches. Weber (2002) concluded that range of permissible change. The analysis of the three many students do not have a process understanding implementations reveals students’ understanding of of exponentiation and logarithms and suggested nu- logarithms, as well as what is possible to learn given the merical approaches to encourage the development of task-as-implemented, or the enacted object of learning. these understandings, but not for logarithms directly. We claim that using variation theory in task design can Wood (2005) suggested verbal explanations from stu- support students in developing important capabilities dents about the meaning of logarithmic expressions for reasoning about logarithms in powerful ways. such as in order to build an understanding of this expression as a numerical value. Confrey and Smith Keywords: Logarithms, variation theory, learner generated (1995) and Panagiotou (2011) both suggest drawing on examples. the historical development of logarithms as a basis for teaching about logarithms. Confrey and Smith fur- ther claim that building the isomorphism between the While exponential functions have been emphasized counting, or additive, world and the splitting, or mul- as a key mathematical understanding in secondary tiplicative, world is building the rules of logarithms. school (Confrey & Smith, 1995), their inverses, loga- The purpose of this study is to address this research rithmic functions, have received very little attention gap by exploring a task design that develops students’ in the research literature. Besides the significance understanding of logarithms. In the recent ICMI 22 of logarithms for their relationship to exponential Study, Margolinas (2013) indicated the importance of functions, the applications of logarithms to various tasks for generating mathematical activity that afford phenomena, such as sound, earthquakes, and human students the opportunity to encounter concepts and growth (Wood, 2005), are important in their own right. strategies. This study focuses on three iterations of a task designed to elicit students’ current understand- According to the Common Core State Standards ing of logarithms, as well as lead them to generaliza- (National Governors Association Center for Best tions related to the properties of logarithms. Practices & Council of Chief State School Officers, 2010), recently adopted in the United States, the emphasis The task design and analysis is situated within both on the algebra of logarithms in a typical Algebra II variation theory, developed by Marton and colleagues course, as well as subsequent courses in mathematics (Marton, Runesson, & Tsui, 2004), and research relat- (e.g., precalculus), means that upwards of 2.5 million ed to learner generated examples, LGEs, (e.g. Watson American students will be expected to engage with & Mason, 2005). Variation theory is most concerned and develop an understanding of these ideas each with the object of learning, comprised of three as- year (National Center for Education Statistics, 2014; pects: (1) the intended, (2) the enacted, and (3) the lived National Science Board, 2010). Despite this, there (Marton et al., 2004). The intended object of learning CERME9 (2015) – TWG03 433 Using variation theory to design tasks to support students’ understanding of logarithms (AnnMarie H. O’Neil and Helen M. Doerr) is what the teacher intends the students to learn at iant while changing the base, which then determined the outset or in the planning of a lesson. The enacted the input. Marton et al. contend that systematically object of learning is what was actually made possible varying certain aspects, while keeping other aspects for students to learn in the implementation of a les- invariant, can prepare students for various other sit- son. The lived object of learning is what the students uations related to the capability in question. Fusion, actually did learn at the completion of the lesson, the last pattern of variation, is the experiencing of and beyond. Marton and colleagues (2004) defines all of the critical aspects simultaneously. Through learning as the development of capabilities, where fusion, learners develop the ability to make gener- a capability is described as seeing, experiencing, or alizations that link the critical aspects of an object of understanding something in a certain way. In order to learning (Holmqvist, 2011). For instance, discerning develop a particular capability (way of seeing, experi- the relationship between the base of a logarithm and encing, or understanding), one must simultaneously the input, in order to hold the value of the logarithm focus on the critical features of the particular object invariant, is a result of fusion and experiencing si- of learning. A variation theory perspective claims multaneous changes in both the base and the input that we can only focus on that which we discern; we of the logarithm. can only discern what we experience to vary; we can only experience variation if we have experienced dif- Drawing on Marton’s work, Watson and Mason (2005) ferent instances previously and can juxtapose our suggest that LGEs are an appropriate way to introduce previous experiences with our current experience new concepts in mathematics. The use of LGEs, howev- simultaneously. er, may appear to be in conflict with variation theory, as the task designer/instructor concedes control of Marton and colleagues (2004) contend that we can the presentation of specific examples to the students. only learn that which we experience to vary. In or- Variation theory, however, does not suggest particular der to ascertain the enacted object of learning, it is ways of arranging for learning, only that variation necessary to be concerned with what varies and what must be present for discernment. Rather, Marton remains invariant in a learning situation. Marton et and colleagues (2004) claim that the particular way al. describe what varies and what remains invariant of arranging for learning is dependent upon the thing as a pattern of variation and identify four of these: (1) to be learned, and research can be undertaken to de- contrast, (2) generalization, (3) separation, and (4) fu- termine the most conducive arrangement for student sion. The first pattern of variation, contrast, refers to learning of that particular thing. In this sense, then, the comparison between what something is and what there is no tension between variation theory and the it is not. For example, in the context of logarithms, use of LGEs, as LGEs, when used in conjunction with log 100 = 2 can be contrasted with log2 8 = 3 in order collaboration, can create the variation in features to discern which aspects of a logarithm statement necessary for discernment. can vary. In order to understand what it means for a logarithm to have base ten, students need to expe- Watson and Shipman (2008) found that the use of rience logarithms that are not base ten. The second LGEs, in a supportive classroom atmosphere, can suc- pattern of variation is generalization, which refers to cessfully introduce new concepts for both advanced experiencing the varying appearances of an object of and low-achieving learners. Watson and Shipman learning in order to separate it from irrelevant fea- suggest that while the discernment of critical fea- tures. For instance, seeing log2 8, log 1000, log3 27, and tures of a concept (Marton’s dimensions of variation) ln e3 as equivalent log expressions that all equal three, through a set of examples may reveal the structure of can help students to generalize what it means for a the concept, learning through exemplification occurs logarithm to be equal to three. The base of the loga- through discerning the generalization of relation- rithm is an irrelevant aspect here; but the power of ships across the dimensions of variation. Drawing on the input in terms of the base is significant. The third this work, the task used in this study was developed pattern of variation, separation, involves varying a with the intended object of learning as generaliza- particular aspect of an object of learning while hold- tions related to the properties of logarithms, such as x ing the other aspects invariant.
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