During the Song and Yuan Dynasties, mathematicians represented by Yang Hui and Zhu Shijie created various simple multiplication and division methods and "Song Formula" in practical mathematics, which became an important part of abacus calculation. At the same time, the abacus has been widely used by the people through the file. Therefore, abacus calculation in this period, whether it is a calculation tool or an arithmetic algorithm, has been further developed.

When did the modern beam-piercing abacus originate? Due to the lack of literature, the researchers of abacus history have different opinions. Based on the inferences of various schools, there are mainly theories of Han Dynasty, Tang Dynasty, Song Dynasty, Yuan Dynasty and Ming Dynasty.

There are mainly Mei in Qing Dynasty and Zhu Yongmao in modern times who hold the theory of Han Dynasty. They are all based on Shu Shu Ji Yi, but there is no other proof. Whether Shu Shu Ji Yi is a fake or not is still inconclusive. Most researchers think that the abacus tool in Shu Shu Ji Yi can only be called the predecessor of modern abacus, even though the abacus in Shu Shu Ji Yi Yi.

Yu Jieshi, Li, Yin Changsheng and other scholars hold the view of prosperous Tang Dynasty. The most representative figure is Professor Li, who proposed that an algorithmic climax in the middle of the Tang Dynasty was due to the emergence of an abacus with a beam passing through a file, which laid a theoretical foundation for "the founding abacus of the Tang Dynasty" and thus created the academic viewpoint of "the founding abacus of the Tang Dynasty".

Holding the theory of the end of Yuan Dynasty and the beginning of Ming Dynasty is mainly based on the reference to abacus in literature and opera works in the end and middle period of Yuan Dynasty. For example, in the 16th year of Yuan Shizu Zhiyuan (AD 1279), Liu Yin wrote a poem about abacus in the Collected Works of Mr. Jing Xiu. In "The Record of Dropping Out of Farming", Tao degraded the maids and servants into abacus beads, and they had to be dialed before moving. In Yuan Qu Xuan, "Pang Ju mistakenly released the afterlife debt" mentioned "I went to the abacus to dial the age", and so on. In literature and drama, using abacus beads as a metaphor to say that abacus has become more popular also shows that it is a relatively new thing. Therefore, it is considered that abacus appeared in the middle of Yuan Dynasty and was widely used in the late Yuan Dynasty and early Ming Dynasty.