West Asian Mathematics
Once people in West Asia figured out how to write down numbers, about 3500 BC, they quickly began to want to use cuneiform to write down other mathematical ideas. The earliest example of this that we have is from about 2700 BC. It shows a multiplication table to help people figure out the area of a space by multiplying width by length. The first column is the width, the second is the length, and the third column is the area. It uses a system for writing down large numbers in base 60 (the way our clocks work today).
Math book (Babylon, about 2000 BC)
This tablet, from about 2000 BC, was a school math book for teaching kids how to calculate inheritance. The problem asks how much each of seven boys would get when their father died, according to Babylonian law. Apparently the law said they should each get a different proportion, with the oldest getting the most and the younger kids less and less. Whoever did the math worked up from the bottom (which was not normal), and also made a mistake in his or her calculations!
Here's a Babylonian math problem you could try to solve yourself. Are you as smart as Babylonian kids?
By the time of the Babylonians, mathematicians were working out quadratic equations like the Pythagorean Theorem, computing square roots and cube roots, and the factors of sixty (because they often worked in base 60).
After the Assyrians formed their empire, mathematical developments seem to have slowed down after this initial rush. But mathematicians continued to work on new ideas. Assyrian mathematicians, working around 1000 BC, first had the idea of dividing the circle into 360 degrees (still working in base 60, so 360=6x60). During the Persian Empire, around 500 BC, people first began to use the abacus (nobody knows whether the abacus was invented in Persia or China, or both about the same time). An abacus is a way of calculating large numbers quickly by moving pebbles along grooves, or beads along wires.
When Alexander conquered the Persian Empire in 331 BC, Greek mathematicians were able to talk to Persian and Indian mathematicians much more easily than before. A lot of new ideas came out of these conversations. The most important was the development by men like Aristotle and Euclid of the idea of the logical proof: that you could start from a few accepted axioms (basic facts) and work gradually by logical steps to show that more complicated ideas also had to be true (or could not be true).
Under Parthian and Sassanian rule, scholars concentrated on bringing together knowledge from the countries around them - India, China, and the Roman Empire. There doesn't seem to have been much independent thinking going on, but we really don't know much about it. In the 600s AD, the Arabs conquered West Asia and established the Islamic Empire. Around the same time, Indian numbers revolutionized mathematics. You can read about that here.