Old McDonald Had a Smart Chicken
The obviously relevance has to do with our natural ability of abstraction we often speak of collection of things as a single entity. “The Detroit Lions,” the House of Representatives,” the army, the Rotary Club, Canadians, the solar system. We often sometimes classify objects, people, and ideas according to common shared properties. This makes it much easier to talk about things in general without having to repeat individual instances over and over.
Ongoing research in this area is crucial. For example, knowledge and skill with what domains and practices of mathematics topics and skills have high leverages for teachers’ capacity to teach mathematics with integrity and effectiveness? How can teachers be helped to learn mathematics in ways that are usable for teaching mathematics? What is the impact of teacher’s knowledge on their effectiveness? How can curriculum materials support teachers learning of mathematics, and, conversely, what do teachers need to learn so that they can use curriculum materials critically and skillfully? Improving the mathematics learning of every child depends on making central the learning opportunities of our teachers. Teachers cannot be expected to know or do what they have not had opportunities to learn. This will require a deliberate and sustained focus on identifying the mathematics knowledge needed for teaching mathematics, on understanding its specific uses in teaching, and the careful development of well-designed and taught courses and workshops, materials and supports. We must study alternative solutions to these issues and compare their effects, at scale, holding the goal of high quality effective mathematics instruction at the center.
I believe the best way to help establish these concepts in a struggling students mind, it’s to take a multi-pronged approach: 1) Visually, using something like Venn diagram
2) Linguistically, using “natural language”
3) Logically, or...
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