A smartcard-based authentication and key agreement scheme for telecare medicine information systems enables patients, doctors, nurses and health visitors to use smartcards for secure login to medical information systems. Authorized users can then efficiently access remote services provided by the medicine information systems through public networks. Guo and Chang recently improved the efficiency of a smartcard authentication and key agreement scheme by using chaotic maps. Later, Hao et al. reported that the scheme developed by Guo and Chang had two weaknesses: inability to provide anonymity and inefficient double secrets. Therefore, Hao et al. proposed an authentication scheme for telecare medicine information systems that solved these weaknesses and improved performance. However, a limitation in both schemes is their violation of the contributory property of key agreements. This investigation discusses these weaknesses and proposes a new smartcard-based authentication and key agreement scheme that uses chaotic maps for telecare medicine information systems. Compared to conventional schemes, the proposed scheme provides fewer weaknesses, better security, and more efficiency.