Self-supervised learning (SSL) has emerged as a powerful technique for learning rich representations from unlabeled data. The data representations are able to capture many underlying attributes of data, and be useful in downstream prediction tasks. In real-world settings, spurious correlations between some attributes (e.g. race, gender and age) and labels for downstream tasks often exist, e.g. cancer is usually more prevalent among elderly patients. In this paper, we investigate the learning dynamics of SSL and observe that the learning is slower for samples that conflict with such correlations (e.g. elder patients without cancer). Motivated by these findings, we propose a learning-speed aware SSL (LA-SSL) approach, in which we sample each training data with a probability that is inversely related to its learning speed. We evaluate LA-SSL on three datasets that exhibit spurious correlations between different attributes, demonstrating that it improves the robustness of pretrained representations on downstream classification tasks.