Attention mechanism has been regarded as an advanced technique to capture long-range feature interactions and to boost the representation capability for convolutional neural networks. However, we found two ignored problems in current attentional activations-based models: the approximation problem and the insufficient capacity problem of the attention maps. To solve the two problems together, we initially propose an attention module for convolutional neural networks by developing an AW-convolution, where the shape of attention maps matches that of the weights rather than the activations. Our proposed attention module is a complementary method to previous attention-based schemes, such as those that apply the attention mechanism to explore the relationship between channel-wise and spatial features. Experiments on several datasets for image classification and object detection tasks show the effectiveness of our proposed attention module. In particular, our proposed attention module achieves 1.00 % Top-1 accuracy improvement on ImageNet classification over a ResNet101 baseline and 0.63 COCO-style Average Precision improvement on the COCO object detection on top of a Faster R-CNN baseline with the backbone of ResNet101-FPN. When integrating with the previous attentional activations-based models, our proposed attention module can further increase their Top-1 accuracy on ImageNet classification by up to 0.57 % and COCO-style Average Precision on the COCO object detection by up to 0.45. Code and pre-trained models will be publicly available.