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.