Positivist and constructivist are two forms of philosophy, both of which have been applied to theoretical and practical elements of crime. Broadly speaking, the positivist theory argues that criminality is primarily a biological or psychological feature. Constructivist theories concentrate less on why crime happens and more on who decides what actions should be considered a crime.
Most positivist theories on crime are based on physical and mental attributes being the main reason why some people engage in criminal behaviour and others do not. This covers a wide range of specific explanations. Some variations suggest mental abnormalities are to blame, while others point to genetic factors making people predisposed to crime. In some cases, a positivist theory might relate to specific types of crime, such as testosterone levels affecting the likelihood of violent crime.
Not all positivist theories place as much importance on the biology of the individual in causing criminality. A rival school of thought says environmental factors such as experience of crime being "normal" or social depravation may play a big role. This variation is at the heart of the "nature vs nurture" debate. It can also be a political debate as highlighted by politician Tony Blair's slogan of being "tough on crime, tough on the causes of crime."
Constructivist theories (also called constructionist theories) do not attempt to explain why people commit crime: indeed, they do not view this as the most important factor. Instead constructivists prefer to explore how and why society decides that some actions are considered acceptable and others should be criminal. In its purest form, a constructivist theory holds that no action is inherently criminal and that this is solely a label constructed by society.
Positivist and constructivist views are not simply alternative answers to the same question. In fact they seek to answer different questions: positivist theories are a wide range of answers to why people commit crime, while constructivist theories answer why something is a crime in the first place. The two sets of theories aren't necessarily mutually exclusive. For example, whether you believe criminality is caused more by biological or social factors is linked to whether you believe a particular action should be considered a crime, such as using marijuana.