*How can I use ProofBlocks to help provide accommodations for mainstreamed special education students?*

First off, just making theorems visual and physical helps immensely. It has been our observation that most of our students don’t organize multistep problems very easily or efficiently in their heads. Having to recall each theorem and hold that in mind while planning a proof is not a trivial exercise. For our special education students, then, we require that they demonstrate on a quiz that they have memorized the details of each theorem and definition. (That quiz, given to the rest of the students as well, does not ask them to apply the concepts, but only show that they know what each theorem is.) Once they have the theorems memorized, they have our permission to use a set of ProofBlocks on the end of unit test to help plan and organize their proofs. Providing this accommodation is easy, discreet, and effective while still in no way decreasing the level of geometric knowledge required to earn the grade.

ProofBlocks have also been used in at least one special education class, and the reported results were tremendous. These students mastered the basic triangle congruence proofs (including Reflexive property, basic definitions, and CPCTC) in essentially the same amount of time as the students in the general education classes. Despite having never taught Geometry before, this teacher acknowledged the relative ease with which she instructed (and the students grasped) proof as compared with all the previous material in the course.

*How do I use ProofBlocks to support the language acquisition of my EL students?*

One of the beautiful things about ProofBlocks is how effortlessly it isolates the skill of logical reasoning, providing scaffolding instead of roadblocks for those skills that would normally deny students access to proof. In the case of English learners, the blocks help separate the meaning of the theorems from the formal mathematical language. Instead of having to wade through sentence after sentence of translation and decoding for each early proof, students are free to just focus on the geometry of the problem and what it means to make a logical argument.

Of course, the process of designing a block itself can also be a very valuable activity. While we normally do not go through this process with the triangle congruence theorems, we do emphasize it for the definitions. In one lesson, the students are given the formal definition of a midpoint, for instance, and are asked to fill in the general format of the information that should go on the inputs and outputs of a blank block. (This activity can be made easier if students are provided with the definitions in “if and only if” form instead of regular textbook-speak, though we usually use our textbook itself as it will be the resource available to them when they go home). It seems only fair to warn that this particular activity is frequently very difficult for students for at least two reasons: (1) the textbook language is not easy for students to decipher (EL or not), and (2) most students do not understand the definitions nearly as well as their teachers believe they do. (We’ve heard the second comment over and over from enough teachers that we feel it is worth passing on.) Students usually need a great deal of practice with the definitions of midpoints, bisectors, and perpendicular lines not just before, but also right in the middle of the unit on triangle congruence.