*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 manipulatives?*

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.

*Is ProofBlocks meant as a replacement for two-column proof?*

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).

*When do students begin the transition to two-column proof?*

As students gain comfort integrating definitions into their proofs, we usually start the process of transitioning to the two-column format.

*If ProofBlocks are so cool, why would students want to switch to two-column proof?*

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 work.

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 will happen.

*What would a sample pacing plan look like for teaching triangle congruence using ProofBlocks?*

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.