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Synthesis: In contrast to analysis (i.e., taking apart), at the synthesis level students put things back together. Given the pieces, there might be more than one way to do this. In terms of mathematics, students might take the pieces they’ve learned, and put them together to solve problems not yet encountered in the actual classroom setting. Synthesis is involved when creating something new. Advanced students might be asked to create a new theory. Synthesis is tested via major projects, for example, which might be long term involving creativity and application of all that students have learned on a topic.
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Analysis: At this level, application is taken a step further. Students must be able to take a situation apart, diagnose its pieces, and decide for themselves what tools (e.g., graph, calculation, formula, etc.) to apply to solve the problem at hand. Rather than just understanding and applying individual concepts, students understand the relationship among concepts. Case studies in business, for example, fit this level. The level of difficulty can be controlled for novices to experts by the number of issues presented in the cases requiring analysis. Likewise, this process to control difficulty can be used for any mathematics problem-solving scenario based on level of expertise of learners. For example, at elementary levels, students are introduced to analysis when a few extraneous facts are included in a problem, which are not needed to solve it. At an analysis level, students are able to appreciate that some problems do not have a unique solution and there is more than one way to defend a position or solution method, as in a case study.