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Neural Correlates of Arithmetic Calculation Strategies

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Neural Correlates of Arithmetic Calculation Strategies Cognitive, Affective, & Behavioral Neuroscience 2009, 9 (3), 270-285 doi:10.3758/CABN.9.3.270 Neural correlates of arithmetic calculation strategies MIRIA M ROSENBERG -LEE Stanford University, Palo Alto, California AND MARSHA C. LOVETT AND JOHN R. ANDERSON Carnegie Mellon University, Pittsburgh, Pennsylvania Recent research into math cognition has identified areas of the brain that are involved in number processing (Dehaene, Piazza, Pinel, & Cohen, 2003) and complex problem solving (Anderson, 2007). Much of this research assumes that participants use a single strategy; yet, behavioral research finds that people use a variety of strate- gies (LeFevre et al., 1996; Siegler, 1987; Siegler & Lemaire, 1997). In the present study, we examined cortical activation as a function of two different calculation strategies for mentally solving multidigit multiplication problems. The school strategy, equivalent to long multiplication, involves working from right to left. The expert strategy, used by “lightning” mental calculators (Staszewski, 1988), proceeds from left to right. The two strate- gies require essentially the same calculations, but have different working memory demands (the school strategy incurs greater demands). The school strategy produced significantly greater early activity in areas involved in attentional aspects of number processing (posterior superior parietal lobule, PSPL) and mental representation (posterior parietal cortex, PPC), but not in a numerical magnitude area (horizontal intraparietal sulcus, HIPS) or a semantic memory retrieval area (lateral inferior prefrontal cortex, LIPFC). An ACT–R model of the task successfully predicted BOLD responses in PPC and LIPFC, as well as in PSPL and HIPS. A central feature of human cognition is the ability to strategy use) and theoretical (parceling out the effects of perform complex tasks that go beyond direct stimulus– strategy-specific activity). response mappings. However, the cognitive capacities that Our approach in addressing these challenges was three- enable this competency—selective attention, maintenance fold: (1) to investigate a domain in which much is already and transformation of internal representation, top-down known about the neural substrates, (2) to use a task in sequencing—also result in the potential for more than one which the strategies can be disambiguated behaviorally, legitimate route to the response. People navigate with ref- and (3) to employ computational modeling in order to erence to landmarks or in absolute space; reading involves make specific predictions about the contributions of the direct semantic retrieval, but can also use orthographic strategies. In the following sections, we will expand on information; quantities can be computed exactly or esti- each aspect of our approach. Briefly, building on research mated roughly. Such strategic variation has implications into the neural underpinnings of numerical and arithme- for understanding the neural underpinnings of behavior, tic processing, we investigated two strategies for solving in terms of locus, level, and duration of activity. multidigit multiplication problems. The strategies could Strategic variation, both within and between partici- be distinguished by the timing of participants’ keypresses, pants, has been documented behaviorally in numerous do- enabling us to confidently assess strategy use. Finally, mains (Kwong & Varnhagen, 2005; LeFevre et al., 1996; using the ACT–R cognitive architecture (Anderson, 2007; Rogers, Hertzog, & Fisk, 2000). For methodological rea- Anderson, Qin, Sohn, & Stenger, 2003) with task analyses sons, early functional imaging research focused predomi- of the strategies, we modeled participants’ behavior and nantly on simple short tasks in which the assumption of generated predictions of the BOLD response in regions monolithic processing pathways may be justified. How- of interest (ROIs). ever, as researchers image increasingly complex tasks, the need to consider the effects of strategy grows. The inves- Cortical Areas Involved in Arithmetic Processing tigation of strategies using imaging has the potential to Lesion and functional imaging studies have consis- enrich our understanding of neural functioning, but there tently (although not exclusively) identified the parietal are challenges that are both practical (accurately assessing lobe in arithmetic processing (Cipolotti, Warrington, & M. Rosenberg-Lee, [email protected] © 2009 The Psychonomic Society, Inc. 270 NEURAL CORRELATES OF STRATEGIES 271 Butterworth, 1995; Dehaene, Spelke, Pinel, Stanescu, intraparietal sulcus activity (Delazer et al., 2005). Al- & Tsivkin, 1999; Jackson & Warrington, 1986; Kahn & though not an arithmetic task, algebra problems solved Whitaker, 1991; Menon, Rivera, White, Glover, & Reiss, with a verbal strategy had strong prefrontal activity but 2000; Warrington, 1982; Whalen, McCloskey, Lesser, & little parietal activity, whereas the same problems that Gordon, 1997). The triple-code theory (Dehaene, Piazza, were solved using an equation-based strategy showed Pinel, & Cohen, 2003), summarizing imaging and neuro- the opposite pattern (Sohn et al., 2004). In these cases, psychological research into arithmetic and number pro- qualitatively different strategies resulted in different loci cessing, proposes that numbers are not represented in a of cortical activity. single unified manner. Instead, representations are dis- tributed over several different areas that each code for a Behaviorally Distinct Strategies in Mental different aspect of numbers: The horizontal intraparietal Multidigit Multiplication sulcus (HIPS) codes for numerical quantity or magnitude, Multidigit arithmetic involves more than repeated the posterior superior parietal lobule (PSPL) corresponds single- digit operations; instead, complex algorithms to the spatial and attentional aspects of number process- must be executed that interleave the basic operation with ing, and the angular gyrus is involved in the verbal pro- carries or borrows and, in the case of long multiplication cessing of numbers. Under this proposal, only the HIPS and division, addition and subtraction steps, respectively. is a uniquely numerical area; the PSPL is part of general Moreover, the properties of arithmetic (e.g., commutiv- attention system, and the angular gyrus is part of a left- ity, distributivity) afford some flexibility in the order of lateralized perisylvian language network. processing. This reordering produces distinct strategies These functional divisions suggest that the arithme- (where a strategy is a goal-directed procedure [Siegler & tic operations may rely on different areas of the cortical Alibali, 2005] under the deliberate control of the partici- number processing network. For example, multiplication pant [Naus & Ornstein, 1983]). For example, the count- is learned by rote memorization of the “times table” and on strategy, used by children to solve simple addition hence should involve verbal processing, which is sup- problems, involves retrieving the number of fingers of ported by the angular gyrus. Conversely, subtraction is the first addend and then counting up to the second ad- purported to be accomplished through reference to an in- dend to arrive at the final result. A subtle improvement ternal number line that is traversed in reaching an answer. on this strategy, known as the min strategy, is to always Thus, an area such as the horizontal intraparietal sulcus, begin with the larger addend first, regardless of the order which is involved in magnitude computation, should play of presentation (Siegler & Shrager, 1984). Compare a greater role in subtraction. Consistent with this interpre- solving 2 1 21 with the count-on strategy versus with tation, in single-digit tasks, the HIPS is more active during the min strategy: Both strategies involve retrieving and subtraction than during multiplication, whereas the oppo- counting, but the counting is much reduced in the latter site pattern holds in the angular gyrus (Chochon, Cohen, case. Strategies that are distinguishable by the order in van de Moortele, & Dehaene, 1999; Lee, 2000). The PSPL which elements are processed have not been investigated has been shown to be more active in subtraction over mul- using neuroimaging. The primary purpose of the pres- tiplication (Lee, 2000), but has also been implicated in ent study was to establish that such reordered strategies nonmathematical spatial attention tasks (Corbetta, Kin- involve the same neural substrates, but that they can be cade, Ollinger, McAvoy, & Shulman, 2000; Culham & distinguished by their profiles of cortical activity within Kanwisher, 2001; Wojciulik & Kanwisher, 1999). Lesion those substrates. studies support this distinction between verbal and quanti- The domain chosen to investigate these questions was tative aspects of arithmetic processing, finding a dissocia- mental multidigit multiplication (i.e., multiplying a large tion between subtraction and multiplication deficits (De- multiplicand by a multiplier, without external aids). Of haene & Cohen, 1997; Dehaene et al., 2003). Moreover, the various methods of solving multidigit multiplication patients with multiplication impairments also tend to have problems, we selected two strategies that differ in the aphasia, which is consistent with a verbal representation order of the individual single-digit multiplications. The of multiplication facts (but see van Harskamp, Rudge, first strategy involves working from right to left, multi- & Cipolotti, 2002,
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