This paper proposes two intuitive metrics, skew and stereotype, that quantify and analyse the gender bias present in contextual language models when tackling the WinoBias pronoun resolution task. We find evidence that gender stereotype correlates approximately negatively with gender skew in out-of-the-box models, suggesting that there is a trade-off between these two forms of bias. We investigate two methods to mitigate bias. The first approach is an online method which is effective at removing skew at the expense of stereotype. The second, inspired by previous work on ELMo, involves the fine-tuning of BERT using an augmented gender-balanced dataset. We show that this reduces both skew and stereotype relative to its unaugmented fine-tuned counterpart. However, we find that existing gender bias benchmarks do not fully probe professional bias as pronoun resolution may be obfuscated by cross-correlations from other manifestations of gender prejudice.