An fMRI study of the Interplay of Symbolic and Visuo-spatial Systems inMathematical Reasoning

The purpose of this fMRI study was to provide evidence for the mathematician’s belief that mathematical thinking emerges from the interplay between symbolic and visuospatial systems. Twelve participants were given algebra word problems and depicted the quantitative relations on a mental number line or made parts of an equation. The regions activated in depicting the picture were also recruited to make an equation. Mathematics is a language. Many scientists say that mathematics is a language to describe the nature of phenomena they are looking at. It is well known that Nicolas Burubaki, a group of mathematicians, stressed the crucial role of formal symbol systems in mathematics. On the other hand, many mathematicians and physicists emphasize the role of visuo-spatial reasoning in mathematics, which recruits qualitative, languageindependent representations. For example, Albert Einstein stated “Words and language, whether written or spoken, do not seem to play any part in my thought process.” As Dehaene, Spelke, Pinel, Stanescu, and Tsivkn (1999) suggested, mathematical thinking may emerge from the interplay between symbolic and visuo-spatial systems. In this fMRI study, we approach this problem and provide evidence for this kind of mathematical thinking. Psychological studies have revealed that if a problem apparently looks like a pure symbolic task, it can require students to have some visuo-spatial representations. For example, Griffin, Case and Siegler (1994) showed that the mental “number line”, a qualitative representation of the number system, is crucial readiness for early arithmetic. Lewis used a number-line-like diagram to train undergraduate students having difficulty to solve “compare” word problems (problems containing more-than or less-than relations), and succeeded in improving their performance. Paige and Simon (1966) proposed that solving word problems is not a simple translation of problem sentences into equations, as Bobrow’s (1966) STUDENT did, but needs “physical cues,” a visuo-spatial representation. The 6 grade students who used our “Picture Algebra” strategy (Koedinger & Terao, 2002) to solve the compare word problem showed relatively high performance. We expect that using this strategy may better prepare students to learn formal algebra. Functional magnetic resonance imaging (fMRI) gives us a new source of information about the mental representations used in mathematics. Dehaene et al. (1999) showed two different mental representations are used for different tasks. Exact calculation (e.g., 4+5=9) elicited left-lateralized activation in the left inferior frontal lobe, together with left angular gyrus and left anterior cingulate. This pattern was interpreted as suggesting that the participants recruited their symbolic systems and did language dependent encoding. Approximation (e.g., 4+5 is closer to 8 than 6), on the contrary, elicited bilateral parietal lobes activation. This pattern was interpreted as suggesting that the participants recruited visuo-spatial systems and did language independent encoding. Dehaene, Piazza, Pinel and Cohen (2003) reviewed neuro-imaging and neuropsychological evidence concerning various numerical tasks and proposed a hypothesis that three parietal circuits are related to number processing. The horizontal segment of the intraparietal sulcus (HIPS) appears to be a core quantity system, analogous to a mental number line. This area seems to be supplemented by two other circuits. One is the bilateral posterior superior parietal lobule (PSPL), which is considered to be involved in attention orientation on the mental number line. The other is the left angular gyrus, which is likely to support manipulations of numbers in a symbolic form (e.g., exact calculation). Dehaene et al. (1999, 2003) suspected that mathematical thinking may emerge from the interplay between symbolic and visuo-spatial systems but did not provide direct evidence for this idea. For example, exact calculation and approximation mainly depend on the symbolic system and the visuo-spatial system, respectively, not necessarily a collaboration between these two systems. In this study, we try to provide direct evidence for such a collaboration. We decided to use algebra word problems for three reasons. First, previous psychological research suggests that visuo-spatial reasoning plays a crucial role in solving these problems while they explicitly require students to use symbols (i.e., equations). This kind of problem is expected to show the interplay between symbolic and visuo-spatial systems. Second, algebra word problems are widely used in school mathematics curriculum, so that we can say the observed interplay is a prevailing form of reasoning, not a special form isolated to a very specific task. Third, there are plenty of studies using algebra word problem, the accumulated findings help us in valid reasoning from our results. If algebra word problems recruit the visuo-spatial system as well as the symbolic system, we should see activation of some visuo-spatial areas when students try to make a correct equation for a problem. To find visuo-spatial areas, we asked our participants to make a pictorial representation of the problem in one condition. This task should activate visuo-spatial areas and most of these areas should also be activated when we ask the participants to make an equation of the same problem in another condition. We especially expect that the two hypothesized parietal visuo-spatial systems, HIPS and PSPL, show activation in both conditions.