Graph-based semi-supervised learning (GSSL) has long been a research focus. Traditional methods are generally shallow learners, based on the cluster assumption. Recently, graph convolutional networks (GCNs) have become the predominant techniques for their promising performance. However, a critical question remains largely unanswered: why do deep GCNs encounter the oversmoothing problem, while traditional shallow GSSL methods do not, despite both progressing through the graph in a similar iterative manner? In this article, we theoretically discuss the relationship between these two types of methods in a unified optimization framework. One of the most intriguing findings is that, unlike traditional ones, typical GCNs may not effectively incorporate both graph structure and label information at each layer. Motivated by this, we propose three simple but powerful graph convolution methods. The first, optimized simple graph convolution (), is a supervised method, which guides the graph convolution process with labels. The others are two 'no-learning' unsupervised methods: graph structure preserving graph convolution () and its multiscale version GGCM, both aiming to preserve the graph structure information during the convolution process. Finally, we conduct extensive experiments to show the effectiveness of our methods.