Dialog State Tracking (DST) is a core component in task-oriented dialog systems. Existing approaches for DST usually fall into two categories, i.e, the picklist-based and span-based. From one hand, the picklist-based methods perform classifications for each slot over a candidate-value list, under the condition that a pre-defined ontology is accessible. However, it is impractical in industry since it is hard to get full access to the ontology. On the other hand, the span-based methods track values for each slot through finding text spans in the dialog context. However, due to the diversity of value descriptions, it is hard to find a particular string in the dialog context. To mitigate these issues, this paper proposes a Dual Strategy for DST (DS-DST) to borrow advantages from both the picklist-based and span-based methods, by classifying over a picklist or finding values from a slot span. Empirical results show that DS-DST achieves the state-of-the-art scores in terms of joint accuracy, i.e., 51.2% on the MultiWOZ 2.1 dataset, and 53.3% when the full ontology is accessible.