By Jennifer Sherman
Proof blocks are a completely new way to teach writing proofs that places primary emphasis on the logical reasoning and geometry involved in proving a concept, instead of on painstaking memorization and recall. The design forces students to approach the problems in a logical manner and allows them to be aware of whether or not their proof is correct and complete.
Each theorem, postulate, or definition is represented by a block and each proof is the assemblage of a selection of these “proof blocks.” As students learn theorems and postulates, they develop a toolkit of blocks that they can use write proofs. Each block has requirements placed on its inputs and outputs which are derived directly from the precise wording of the definition, postulate, or theorem. These requirements make it explicit what each theorem provides with no room for misinterpretation or confusion. With the formats of acceptable inputs and expected outputs clearly delineated, it is intuitive for students to create logical arguments. Assumptions and givens in the problem are represented as blocks with no required input and are the start of the whole chain. The proof is complete when you reach a block with an outgoing statement that matches what you were trying to prove. Consider the following example:
Like a flow proof, a proof with blocks shows connections between information and steps in the proofs. However, unlike flow proofs, the theorems and postulates are the explicit tools which students use in order to build a proof step by step. When just starting out, students keep track of which postulates and theorems are available to them with toolkits of labeled blocks which they can manipulate and which list what kinds of information is needed in order to use them. In addition, breaking away from the numbered two column format also allows students to work forward and backward without having to keep track of their work in their heads. This method takes all the painful bookkeeping out of proofs and frees them to focus on creating a logical argument which is the fun of proofs!
This form of proof also allows teachers more freedom. It is very rare for students to create an illogical argument, because at any step in the process, it is completely explicit what you know and what you can say. For the same reason, I have found that generally students are aware of when their proof is complete and correct without needing teacher affirmation. Not only does this realization empower the students, but it frees the teacher to focus more attention on those students who truly need assistance. Since proof block proofs are just as rigorous as any other type of proof, they can be trivially transformed into two column or paragraph proofs.
Powerpoint slides and handouts are linked at the bottom of the page.
© Jennifer Sherman 2007
proofblocks@gmail.com
| Attachment | Size |
|---|---|
| ProofBlocks Intro.doc | 40 KB |
| ProofBlocks Template.doc | 58 KB |
| CMC-ProofBlocks.ppt | 2.25 MB |
| Logic Activity (p93).doc | 72 KB |
