cheesemonkey wonders

cheesemonkey wonders

Sunday, November 12, 2023

TEN TRUE STATEMENTS - Using Cognitive Load Theory to Build Toward Mastery of Proofs - Geometry

 I've been reading and thinking a lot about cognitive load theory in Geometry class, thanks to Michael Pershan, Greg Ashman, Dylan Wiliam, and Ilana Horn. 

I've pared back what I ask students to do using a new structure I've been calling "Ten True Statements." It could be twelve or eight or nine, but ten is a nice number. Here's the basic idea.

Students are given a problem that includes a diagram and a statement, but my instructions to them are extremely non-pathway-specific. I ask them to generate at least ten true statements about the situation. I given them a specific amount of time and then I yell, "GO!"

I circulate, but only provide just-enough of a hint to table groups to help them get themselves unstuck. The purpose here is to learn how to ask for help and not just stay stuck.

Here's one of the problems they did on Thursday:

congruent triangle problem

I consider this activity purely generative. Students need practice in brainstorming.

I want them to lose themselves in flow so they can practice using their reference materials to develop as many ideas (aka "true statements") about the figure as they can, together with justification. 

I don't care about the order of statements. I don't care if statements are relevant to a proof pathway. 

The habits of mind I am trying to cultivate are to learn how to brainstorm more gently with their minds without judgment; to use their tools as a memory aid; and to document their thinking process.

My theory of action is this: the more practice they have in generating true statements and in deriving new true statements from previous true statements they have generated, the easier it will be for them to learn how to put their true statements and justifications into order.

I am trying to focus their working memory just on the generation of true statements. 

All four classes are really loving this activity, so I have to go find more suitable problems for the week.


  1. Love this idea. What are some of the things they said were true for this example?

    1. THAT was incredibly revealing for each table group and gave me a ton of formative assessment data on where they were at in their understanding. Some groups started by restating the given information as discrete true statements, such as "BD=DG." Less confident groups often said things like "Triangle BDG is isosceles." That gave me the opening I needed to help them connect their reference materials to what they could identify on the diagram.

      More fluent/confident groups of learners dove into annotating their diagrams with color and other indicators. In general, the more annotations on their diagram, the more mature their understanding was. The groups that had accumulated more statements started spontaneously drawing arrows between and among statements. That told me a lot about their understanding of implication and sequencing.

      This gave me a lot of flexibility to coach the groups who felt really lost, asking them which things in the diagram they could confidently annotate, asking them to state the names of things, etc.

      I got an enormous amount of information on where students are, and how variable it is from one course section to another.