How long do students use the manipulatives?
It depends on the
student, the class, and the teacher. Generally, the stronger the
student, the faster they’ll give them up. Most of our honors
students will use them for a day or two (if that), and skip
straight to the representational model, drawing the proofs out on
the whiteboards and referring to the physical blocks only to check
the meanings of the theorems. (A sheet with images of the blocks
for each theorem, definition, postulate, and property on it would
serve the same purpose.)
Average students do not usually need to use the triangle congruence
blocks more than a day or two, but the blocks become important
again when they begin to integrate definitions into their proofs.
At this point, the blocks are as much a crutch for organizing the
proofs as they are for reminding students what exactly the inputs
and outputs of each theorem are.
The length of time students use the manipulatives is affected by
the way that teachers encourage their use. Some teachers encourage
students to place the manipulatives on the whiteboards and draw the
connections between the manipulatives themselves. Other teachers
make a greater effort to encourage students to just draw the
outlines of the blocks onto the boards, using manipulatives as a
reference. These choices are as much a decision about the skills of
the student population as they are about pacing. If students are
using the first method, they may want to hold onto the blocks for
longer than they would otherwise.
How do you
convince students to give up the
Most students are not particularly
attached to them, and this hasn’t really been a problem. Once they
know what each theorem means and success begins to breed
confidence, the majority of students let go within a couple days.
meant as a replacement for two-column
Absolutely not! ProofBlocks was designed as
an introduction to proof, not as a replacement for the more
traditional methods. We designed ProofBlocks as a way to make
visible the mental decisions and processes that we ourselves engage
in when we grapple with a geometric proof. We wanted students to
see the connections between steps. We wanted them to successfully
practice not skipping steps and become comfortable with the
required level of rigor.
All that said, there are many reasons to use two-column proof, not
the least of which is that it is universally recognized. In an
environment where state testing is so important, it would be
irresponsible not to teach the students the skills they need to
succeed on these assessments, and we recognize that familiarity
with two-column proofs is key to this success. Two-column proofs
are also generally cleaner looking, requiring less space, and the
format is better suited to certain kinds of problems than the
ProofBlock format is (algebraic proof, for instance).
students begin the transition to two-column
As students gain comfort integrating
definitions into their proofs, we usually start the process of
transitioning to the two-column format.
are so cool, why would students want to switch to two-column
The key to encouraging the transition lies
in assigning proofs that highlight the strengths of the other
formats, and there are at least two ways to entice students into
making the switch. The first is to assign longer and longer proofs.
Students generally hate having to wrap a ProofBlock style proof
onto a new line because they’ve run out of room. The same proof
written in columns will look much shorter, seeming like less
The other way is to move into proofs that are more algebraic in
nature. This part of the transition we facilitate by teaching the
proofs of parallel lines (with all its angle pair algebra) as the
unit following congruent triangles. It is not natural to use the
Addition Property of Equality and write the statements from left to
right. Students recognize this fact and those comfortable with the
geometry will usually make the switch relatively smoothly. Students
who are less confident in the geometry will tend to cling to what
they know and have found success with in the past. These students
will need more practice and support to make the transition, but it
What would a
sample pacing plan look like for teaching triangle congruence using
Every school and every teacher is different. This pacing plan
for the congruent triangle unit was created (and used) several
years ago at Santa Susana High School. It was designed for a bell
schedule where all classes were an hour except for the nearly two
hour Wednesday/Thursday block. Not all the activities and
worksheets referred to are online. In addition, as teachers have
gained comfort with the format and started seeing different needs
in their student populations, much of it has changed. It is a
work in progress that we are providing solely so you can get an
idea of what our unit has looked like in the past.
The sample pacing plan may be downloaded here.