Look at the material used for the bow. If the bow is made of all wood, then it was likely made before Bear began to mass produce bows in 1949. Bows made of only wood that have a "Bear Products" stamp on them were made before 1949. A decal that says "Running Bear" on a wooden bow means it was made in 1948.
Search the bow riser for a medallion. If it has a copper coin medallion, then it was made in 1959. An aluminum medallion was then used by Bear in 1960 and 1961. Pewter was used in 1962, brass was used in 1963 to 1970 and a nickel-silver medallion was placed on the riser from 1971 to 1972. Starting at the end of 1972, a raised medallion of gold and chrome was used in all Bear bows and is still used today.
Date your Bear bow by serial number if it was made between 1965 and 1969. Serial numbers were placed on the limbs of most Bear bows, along with the weight and length of the bow. The first number stood for the year of manufacturing. So, a "5" at the beginning meant it was made in 1965, a "6" was made in 1966, a "7" in 1967, an "8" in 1968 and "9" at the beginning meant it was made in 1969.
Search for any type of decal on the bow. A small running bear decal was placed on bows from 1948 to around 1953. Then, a large standing bear with "Glass Powered Bow" written under it was placed on bows from 1953 to 1955.
Visit the Bear Archery website to date newer model bows. Bows made in the last few years will have the model name on them. The Bear Archery website has a list of every model released in recent years and what year it was launched.