Non-symbolic division task

To examine the children’s ability to solve non-symbolic division problems, we adapted the procedures and task paradigm used in McCrink and Spelke (2010, 2016). The division experiment comprised three blocks: intro, training, and test. There were two task conditions for division: divide-by-two (Division 2) and divide-by-four (Division 4); these conditions were presented in three blocks each, making a total of six blocks in the task.

Intro Block. The purpose of the intro block was to help participants understand the division task. For example, in the Division 2 condition, participants were initially presented with two blue rectangles that changed in size (i.e., increasing or decreasing) over the course of several seconds. Then, the rectangles stopped changing size and a magic wand appeared on the left side of the screen, sweeping up and down the rectangles with a shimmering magical sound. After several seconds, the two rectangles merged, and the experimenter said: “Look! Our magic dividing wand! Originally, there were two blue rectangles, and now there is one rectangle. It reduced the quantity. The magic wand made the two blue rectangles become one!” After watching this short clip, the participant watched another transformation trial movie. This transformation movie was identical to the previous one, but when the magic wand appeared, the rectangles were obscured by a screen. After the wand had finished waving, the video was paused and the participant was asked to try to guess the number of rectangles behind the screen. If the child gave the correct answer, he/she could advance to the training block. In our study, only one child provided an incorrect answer on the first attempt (he/she then answered correctly on his/her second attempt); all of the other children were able to respond correctly on their first attempt.

Training Block. In the training block, an array of blue rectangles was presented on the left side of the screen. The experimenter pointed to the array and said: “Now, there are many rectangles. There are too many rectangles to count. We will concentrate and use our imagination.” After 5 seconds, the array was obscured, and the magic division wand appeared above it. The experimenter said, “Look! The rectangles are dividing,” At this time, a comparison array comprising pink rectangles appeared on the right side of the screen. When the movie was paused, children were asked to choose the side (left or right) that contained more rectangles (Figure 2). The experimenter sat behind the child, asked neutral questions (e.g., “Which side of the screen do you think has more rectangles?”), and let the participant answer. When the child responded, the experimenter recorded the response and re-played the trial movie. When the video was played back, the obscuring screen was removed, providing feedback. The training block comprised six trials.

Non symbolic division task – Training block (McCrink and Spelke, 2010). Reprinted from McCrink and Spelke (2010) with permission from Elsevier.

As in McCrink and Spelke (2016), we manipulated a distance factor in the division task. We set this factor by setting the relationship (or “distance”) between the comparison array and the transformed array (i.e., the correct outcome) to be relatively disparate or close. For the disparate conditions, the number of rectangles in the two arrays differed by a factor of 2.0, while for the close conditions, they differed by a factor of 1.5. There were four distance conditions, /2.0, *2.0, /1.5, and *1.5, respectively, and these were applied in both division conditions. For example, in the disparate conditions (i.e., /2.0, *2.0), the number of small rectangles in the comparison array was either double (*2) or half (/2) that of the transformed array. In the close conditions (i.e., /1.5, *1.5), the value of the comparison array was either the correct amount *1.5 or the correct amount /1.5. For example, if an initial array comprised 16 rectangles, the correct outcome in the Division 2 condition would be eight rectangles. If the disparate condition was applied to this example, the number of rectangles in the comparison array would be four or 16, while the number would be six or 12 when the close condition was applied. That is, trials featuring the disparate condition were easier than those featuring the close condition, as arrays with a distance factor of 2.0 were more discriminable than arrays with a distance factor of 1.5. As the previous study has already shown the effects of the distance conditions, we used the same manipulation for the distance. For more details, please refer to McCrink and Spelke (2016).

Test Block. The 16 test trials were conducted identically to the training block, with the following three exceptions: (1) neither correct answers nor feedback were provided for children’s responses. Instead, the experimenter provided consistent positive feedback (e.g., “Good job!” and “Let us try another one!”), (2) in the training trials, the stimuli in the comparison array were all small rectangles of unified size; however, in the test block, the stimuli in the comparison arrays were rectangles and, for all distance conditions, each array occupied the same area and had the same contour length (Figure 3). Thus, if a child made numerical comparisons simply based on the area and/or length of the array, he/she would perform below chance level in the test block. For a child to successfully solve a problem, he/she needed to consider the exact change in the number of rectangles, not the perceptual variables that changed depending on the value of the array, and (3) if participants seemed to count the numbers using their fingers or by moving their lips, the experimenter intervened in the test block. Each trial lasted until the participant made their choice.

Non symbolic division task – Test block (McCrink and Spelke, 2010). Reprinted from McCrink and Spelke (2010) with permission from Elsevier.