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One of the most difficult topics to teach in geometry is the concept of proof. This article describes ProofBlocks as an innovative approach to helping students develop proficiency in proof. ProofBlocks is an intuitive approach to teaching proof that makes proof accessible to students without sacrificing the rigor of traditional approaches. When using ProofBlocks, students find it natural to work from both ends of their attempted proof, breaking it into smaller pieces and searching for the information they need to make logical connections. ProofBlocks is an inherently visual approach that lends itself to the use of manipulatives, thereby addressing the needs of visual and kinesthetic learners, who are often neglected when other forms of proof are taught. This technique provides a wide range of benefits over more traditional forms of proof as taught in the high school classroom. It separates geometric knowledge from logical reasoning, providing a more manageable structure within which students can explore geometry while developing their proof intuition. (Contains 8 figures.)

Abstract Background The most common method of identifying groups of functionally related genes in microarray data is to apply a clustering algorithm. However, it is impossible to determine which clustering algorithm is most appropriate to apply, and it is difficult to verify the results of any algorithm due to the lack of a gold-standard. Appropriate data visualization tools can aid this analysis process, but existing visualization methods do not specifically address this issue. Results We present several visualization techniques that incorporate meaningful statistics that are noise-robust for the purpose of analyzing the results of clustering algorithms on microarray data. This includes a rank-based visualization method that is more robust to noise, a difference display method to aid assessments of cluster quality and detection of outliers, and a projection of high dimensional data into a three dimensional space in order to examine relationships between clusters. Our methods are interactive and are dynamically linked together for comprehensive analysis. Further, our approach applies to both protein and gene expression microarrays, and our architecture is scalable for use on both desktop/laptop screens and large-scale display devices. This methodology is implemented in GeneVAnD (Genomic Visual ANalysis of Datasets) and is available at http://function.princeton.edu/GeneVAnD. Conclusion Incorporating relevant statistical information into data visualizations is key for analysis of large biological datasets, particularly because of high levels of noise and the lack of a gold-standard for comparisons. We developed several new visualization techniques and demonstrated their effectiveness for evaluating cluster quality and relationships between clusters.