<aside> 📌 We've identified that "modeling" might be a generative frame for framing the intellectual activity and backdrop of projects which might offer rich opportunities to engage skills and perspectives traditionally associated with Algebra I. This document attempts to define what that means and describe some of the ramifications for the PBLL's design and our design process.
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There is a long and rich history of cognitive scientists, educational researchers, and philosophers of science looking to modeling as a way of understanding the way that we understand things as people and the way that we develop understanding and knowledge within fields. The reason that these fields intersect because the general, professional work of a group like scientists shares a great deal in common with the specific, intellectual work of someone learning something: i.e. you are looking at things you don't understand and trying to figure out how they work.
Richard Feynman analogizing physics to understanding a game of chess
Richard Feynman analogizing physics to understanding a game of chess
But this leaves out an important detail: both when you are iteratively developing your model of the world, and when others have developed models of the world, you are often explicitly interacting with your and others' representations in an attempt to hew your mental model more closely to reality. Hestenes depicts this as three, interrelated worlds:
via Hestenes, "Fig. 5. — Constructivist epistemology maintains that knowledge of the Physical World is achieved by constructing models of physical phenomena. The objective conceptual models constructed by the cooperative activity of scientists are distinguished from mental models constructed in the minds of individuals. Labels on the interactions between the three Worlds are meant to be suggestive rather than definitive."
With this perspective in hand, it is interesting to begin to look at . One way of seeing Common Core through this lens is to observe that shape ("Geometry") and quantity ("Number & Quantity") are two, fundamental characteristics of our world. Generalized representations ("Algebra") let us abstract these, and mappings between these representations ("Functions") let us articulate relationships between and amongst shape and quantity. In the real world, nothing is ever precise or certain, and so we develop tools to deal with error and uncertainty ("Statistics & Probability").
This suggests that when developing projects, the design process should begin with what a domain where many phenomena:
One of the primary virtues of the Common Core is its emphasis on:
focus ⇒ "focus strongly where the standards focus", i.e. "declutter" youth experiences
coherence ⇒ "think across grades, and link to major topics in each grade", i.e. think in progressions, not standards, ensuring ideas are connected and framed in a unified way, appropriate to individuals' background and aptitude at that point and time
and rigor ⇒ "in major topics, pursue with equal intensity":
Many attempts at "project-based learning" think of tackling these dimension in order (i.e. "First discuss and teach the ideas, then practice the skills, then apply them to projects and problems." Jeff Robin unpacks a few of the issues with this beautifully in "What Project-Based Learning Isn't":
