Let us go back to the year 1600. This is the time where we should start looking for origins of wasan. At this time Japan was under control of daimyo, or in western terms warlords, who were still fighting about dominancy. Some of them were very powerful and the country was in almost continuous turmoil.   
In 1600, during the famous Sekigahara battle daimyo were defeated by Tokugawa Ieyasu. This was the starting point to a new period in the history of Japan – a period of almost 250 years without wars in Japan. After the battle Tokugawa Ieyasu moved to a small, at this time, provincial town Edo, today’s Tokyo. The country was united and many changes started taking place. 
We have to notice that it was the time when Spanish, Portuguese and Dutch tried to settle down in Japan, and strengthen their trade. At the same time missionaries from these countries wanted to convert as many souls as possible. As we probably guess, the trade with foreigners was not a problem for anybody. However, converting people to Christianity was not very much welcomed by the two main religions in Japan – Shinto and Buddhists. This was in fact the main source of tensions in the country. In order to keep people calm Tokugawa Ieyasu issued an edict ordering Portuguese and Spanish to leave Japan, removing missionaries, to destroy all Christian churches, and forbidding Christianity in Japan. Tokugawa Ieyasu died a few years later, but his grandson Tokugawa Iemitsu finished the task of removing foreigners. In 1641 there were practically no foreigners in Japan. Only a small group of Dutch merchants, members of the East India Company, was left in a small, artificially made 200x70m island in Nagasaki waterfront. All these changes started a new period in Japan, sometimes called sakoku, or a closed country. All exchange of information between Japan and rest of the world was forbidden. Foreigners had no right to enter Japan, and Japanese had no right to travel outside of the country.
In this moment we have to notice that for centuries Japan was under the strong influence of China. There were many things that came to Japan from China: science, scientific and technical innovations, art, and Buddhism. This way came to Japan also many mathematical discoveries. This includes the most famous Chinese mathematics textbook Jiuzhang Suanshu, or The Nine Chapters of Mathematical Art, and of course soroban – a very specific form of abacus. As we may suspect closing the country also stopped any type of exchange with China.
Closing the country had not only negative effects. The most important – it stopped fights both internal as well as those with foreigners. It also forced, and in fact helped, Japanese to develop their own forms of art, and science. The local art, science and culture started developing rapidly. This concerns also mathematics. For the above mentioned reasons this period also has another name genroku, that means renaissance.
Genroku (元禄) was a Japanese era name after Jōkyō and before Hōei. This period spanned the years from 1688 through 1704. The reigning emperor was Higashiyama-tennō (東山天皇). The years of Genroku are generally considered to be the Golden Age of the Edo Period. The previous almost hundred years of peace and seclusion in Japan had created relative economic stability. The arts and architecture flourished.  A sense of optimism is suggested in the era name choice of Genroku meaning "Original happiness". (Source Wikipedia)
If we look closer into Japanese history we will see that during this time they developed famous Kabuki opera, Noh dance, tea ceremonies, the famous garden architecture, flower arranging, several famous painting schools including the famous Katsushika Hokusai an ukiyo-e painter, poetry including the most wonderful type of Japanese poetry haiku, and literature. These are things in the Japanese culture that we in the West appreciate the most.
Here comes a very interesting problem. This is a long period of peace. What to do with the huge crowd of samurai who lost their lords, jobs, and know nothing but how to kill opponents with their long and short swords? This is rather hard question. Just imagine what was expected that a superb man should know? At this time a man should know medicine, art, poetry, tea ceremony, dance and music, arithmetic and of course how to read and write. Therefore it was a huge challenge for a samurai to find a new goal in his life. However, many of them succeeded in this new reality and they became painters, some of them even famous painters, actors, poets, writers, physicians and teachers.  The Western readers of this paper can observe this transformation of samurai in a very famous movie series Musashi (NHK, 2003).
Miyamoto Musashi (宮本 武蔵) also known as Shinmen Takezō, Miyamoto Bennosuke, or by his Buddhist name Niten Dōraku, was a Japanese swordsman and samurai famed for his duels and distinctive style. Musashi, as he was often simply known, became renowned through stories of his excellent swordsmanship in numerous duels, even from a very young age. He was the founder of the Hyōhō Niten Ichi-ryū or Niten-ryū style of swordsmanship and the author of The Book of Five Rings (五輪書 Go Rin No Sho), a book on strategy, tactics, and philosophy that is still studied today (source Wikipedia).
As a result of all these changes many of the samurai started creating schools, known as juku. Most of them were small country schools with few students. Major topics taught there were reading, writing and arithmetic. Who attended these schools? Usually, the students were adults, starting from samurai, craftsmen, and merchants to countrymen. This was the place where there are roots of sangaku and the origins of specific mathematics that we can find on sangaku tablets.

Reconstructed sangaku from the Takemizuke shrine, Nagano prefecture, 450x200cm

How wasan is related to the western mathematics

Looking at the history of Japanese mathematics in this period we can find two kinds of mathematics – the one, very picturesque, on sangaku tablets developed by juku students as well as a more serious mathematics similar, in many aspects, to the mathematics developed at this time in Europe. Surprisingly, mathematicians in Europe and in Japan were working on similar problems. History of Japanese mathematics is not the main topic of this paper; therefore we will mention here only a few major facts. Readers interested in the history of Japanese mathematics can find a good source of information in the history of Japanese mathematics by Smith & Makami.
Now, let us briefly look what Japanese mathematicians did during this time. They developed a theory of determinants even more advanced than the one created by Leibnitz later in Europe. They discovered many geometry theorems that were discovered later in Europe by Casey, Malfatti, or Soddy. They perfected soroban calculations including multiplication and division of numbers. In fact, they were very good in handling arithmetical operations on big numbers, as well as equations of very high orders. They developed a theory that is similar to our integral calculus. They published books on calculating areas and volumes of complicated objects obtained from intersection of solids. Some of the most famous Japanese mathematicians of this period are Seki Kowa or Seki Takakazu, Yoshida Mitsuyoshi, Immamura Tomoaki, Muromatsu Shigekyo and many others. In 1627 Yoshida Mitsuyoshi published the book Jink-ki, which means Big and Small Numbers. Jink-ki was probably the most popular mathematical book in Japan and it had more than 300 editions.