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How Humans Develop the Abstract Geometric Knowledge?

How humans develop and perfect the abstract geometric thinking? A study performed by researchers from Harvard University and Beijing Normal University has been published on PNAS recently, addressing with how four-years-old children use different geometric concepts and giving possible explanation on the origin of humans’ geometric thinking.

“We investigated what geometric information will be used when four-year-old children read maps, and we hope to reveal when humans began to abstractly think about geometric questions, what geometric knowledge would be invoked.” Said the first author, Moira Dillon from Department of Psychology in Harvard University. Dillon and her co-workers noted that four-year-old children could flexibly use the two core geometric representations—angle and distance—into the map reading, but still, they could not compile the two together.

There were 45 four-year-old children participated in the study. First, they completed two non-symbolic test (Figure 1): in the reorientation test, children needed to find the target in three rectangular environments with different length to width ratios. While, in visual analysis test, they had to find one pattern with different geometric characteristics from six patterns. In these two tests, children invoked completely different geometric knowledge.


Figure 1: (A) Reorientation test. The subjects firstly saw the researcher hide a sticker under one angle of a rectangular and they were blindfold and circling, then they were asked to look for the sticker. The children could use the relationship between the wall length and direction to adjust their positions, however, as the length to width ratio decreased, it became more difficult for them to adjust using the distance relation. In the rectangular with a length to width ratio of 6:7, their performance did not surpass random level. (B) Visual analysis test. Among the 16 geometric presentations, the children successfully distinguished 11 distinct patterns while they only showed random guessing level for the 5 patterns with a red frame. In the 5 presentations, two of them were associated with mirror image distinguishing ability, two of them were related with symmetry sensitivity, and the other one is linked with the distinguishing of difference on relative lengths and angles. Image source: Dillon MR, Huang Y, Spelke ES. (2013) Core foundations of abstract geometry. PNAS.

In the following navigation test, children would use maps to find target item. The maps were associated with two environment that differed in their geometric layout: one of the triangle’s side with no corners while the other one had the triangles’ corners with no sides. (Figure 2) “Using map to navigate is an ability obtained in early human development.” Dillon said:” To understand the maps, we need to refine the geometric relationships on the maps to understand the geometric information in the surroundings.” In the distance array (with sides, without corners), the children successfully found out all of the six items in various locations; in the angle array (with corner, without sides), they only showed behavior better than the random level at three of the locations.


Figure 2: Map navigation test. (A) The maps used in the two navigation tests, in which the green dots marked the target item location. Maps in (B) and (C) suggested the top view of the actual triangular environment that were associated with the maps. Generally, the subjects could successfully find the target items in these two environments. Image source: Dillon MR, Huang Y, Spelke ES. (2013) Core foundations of abstract geometry. PNAS.

The researchers pointed out that children’s behavior in reorientation test and the distance array depends on the understanding of distance while in visual analysis test and the angle array, their performance was dependent on the understanding of items’ geometric shapes. When reading the maps, the four-year-old children can’t integrate the information of distance and angle, but they were already able to use these information separately.

“Humans’ unique ability in abstract geometric thinking originated from the mutual geometric knowledge between humans and zebra fish, bumblebee as well as other animals. Only we can use these knowledge in a more flexible way.” Dillon said, they expected to fully understand the influence of developed geometric knowledge on humans’ further formed, more abstract geometric intuition. “Children will be able to relate information of distance and angle when they are 10 to 12 years old. In the following study, we plan to target our research on the development of children from 4 to 10-12 years old.”


Image sourceShutterstock

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