Many normative theorists want to contribute to making the world a better place. In recent years, it has been suggested that to realise this ambition one must start with an adequate description of real-life practices. To determine what should be done, however, one must also fundamentally criticise existing moral beliefs. The method of reflective equilibrium offers a way of doing both. Yet, its practical usefulness has been doubted and it has been largely ignored in the recent practical turn of normative theorising. This paper offers a complementary methodology to the method of reflective equilibrium, referred to as Rawlsian constructivism, which brings forth its practical merits. With the support of Rawlsian constructivism, the method of reflective equilibrium becomes a tool for public reasoning about practical problems which aims to facilitate shared solutions. The process of reflective scrutiny is used, not in the search of moral truth, but rather to highlight what stands in the way of solutions to problems agents face in different domains of social life. The practical value lies in scrutinising reasons for action that are taken for granted, explicating new rationales for action and highlighting neglected points of agreement. The paper exemplifies this approach with a process of justifying individual obligations to combat climate change. Normative theorists who share the practical agenda have correctly noted the importance of bottom-up investigations of subject domains. This paper argues that the next step should be to utilise this version of the method of reflective equilibrium to explore the potential for morally progressive solutions to these problems.