TNL offers six different Thought Exercises to prompt authentic math learning. As their categorization implies, TE's are more than mere routines. They create a know-building community that enhances learning skills and reinforces conceptual understanding: drive.google.com/file/d/1XCS3...

So, how do we convince the masses of the necessity of replacing a public school system that was solely developed for the preparation of labor for one concerned with maintaining a highly informed, critical thinking, positive change affecting, humanity-centered citizenry?"

Here are three ideas for feedback:
(1) Discuss the way in which the learner expressed equality; 1/2 does not equal 3.5/3
(2) Introduce the concept of complex fractions
(3) Express 3.5/3 multiplicatively, 3.5(1/3) or (3 1/2)(1/3), and apply the distributive property: 3(1/3) + 1/2(1/3) = 1 1/6

Learn more about our work here: www.thenumberlab.com

Yes: "...But I teach so they can know the math well enough to score well (not 'test prep')."

Truth. And this is the reason that math PD has to be concerned with mathematics itself, not merely the implementation of a program/curriculum. Teachers can implement with greater fidelity when they are able to identify and make connections between mathematical ideas for themselves.

Yes. Excellent point.

Learners would have the opportunity to make sense of expressions and equations by identifying underlining mathematical ideas. A lesson on "how to" simplify complex expressions becomes conceptualizing addition through a discussion of units and an analysis of terms.
drive.google.com/file/d/1mPRK...

Indeed. Learning is more powerful when it happens in community.

Yes! Meaning-making and sense-making are essentially human tasks. Storing, sorting, and organizing information are essentially technological tasks (one we've designed tools precisely for!). Our brains can certainly do both, but the reason we offload the latter is to facilitate the former.

The teacher is expected to do all of the thinking.

"This might be the most fundamental aspect of constructivist classrooms: adults modeling for children that learning is not merely a schoolhouse collection of correct answers but that the world itself is full of complex problems w/ multiple perspectives...that merit our thoughtful consideration..."