Kibbe Melissa M, Feigenson Lisa
Department of Psychological & Brain Sciences, Boston University, 64 Cummington Mall, Boston, MA 02215, United States.
Department of Psychological & Brain Sciences, Johns Hopkins University, 3400 N Charles St, Baltimore, MD 21218, United States.
Cognition. 2017 Mar;160:82-90. doi: 10.1016/j.cognition.2016.12.006. Epub 2017 Jan 6.
Solving for an unknown addend in problems like 5+x=17 is challenging for children. Yet, previous work (Kibbe & Feigenson, 2015) found that even before formal math education, young children, aged 4- to 6-years, succeeded when problems were presented using non-symbolic collections of objects rather than symbolic digits. This reveals that the Approximate Number System (ANS) can support pre-algebraic intuitions. Here, we asked whether children also could intuitively "solve for x" when problems contained arrays of four or fewer objects that encouraged representations of individual objects instead of ANS representations. In Experiment 1, we first confirmed that children could solve for an unknown addend with larger quantities, using the ANS. Next, in Experiment 2a, we presented addend-unknown problems containing arrays of four or fewer objects (e.g., 1+x=3). This time, despite the identical task conditions, children were unable to solve for the unknown addend. In Experiment 2b, we replicated this failure with a new sample of children. Finally, in Experiment 3, we confirmed that children's failures in Experiments 2a and b were not due to lack of motivation to compute with small arrays, or to the discriminability of the quantities used: children succeeded at solving for an unknown sum with arrays containing four or fewer objects. Together, these results suggest that children's ability to intuitively solve for an unknown addend may be limited to problems that can be represented using the ANS.
对于孩子们来说，在诸如5 + x = 17这类问题中求解未知加数是具有挑战性的。然而，先前的研究（基贝和费根森，2015年）发现，即使在接受正规数学教育之前，4至6岁的幼儿在使用非符号化的物体集合而非符号数字呈现问题时也能成功解决。这表明近似数字系统（ANS）可以支持代数前的直觉。在此，我们探讨当问题包含四个或更少物体的阵列时，孩子们是否也能直观地“求解x”，这些阵列鼓励对单个物体进行表征而非ANS表征。在实验1中，我们首先确认孩子们可以使用ANS来求解较大数量的未知加数。接下来，在实验2a中，我们呈现了包含四个或更少物体阵列的未知加数问题（例如，1 + x = 3）。这次，尽管任务条件相同，但孩子们无法求解未知加数。在实验2b中，我们用新的儿童样本重复了这一失败结果。最后，在实验3中，我们确认孩子们在实验2a和2b中的失败并非由于缺乏用小数组进行计算的动机，也不是由于所用数量的可辨别性：孩子们成功地求解了包含四个或更少物体阵列的未知和。总之，这些结果表明，孩子们直观求解未知加数的能力可能仅限于可以用ANS表示的问题。
