The structure of the typical middle grades math textbook follows this basic outline:

1. Vocabulary review

2. Brief explanation of concept.

3. 2-5 very specific examples of the concept in isolation.

4. 1-2 very specific examples of the concept in a problem solving situation.

5. 20-50 problems that mirror the examples.

The problem with this structure is that if teachers simply "taught the text(test)" students would have almost no opportunities to see the interconnectedness of mathematics concepts. Nor would they have the opportunity to think creatively to solve any problems since they would just be parroting back a technique rather than applying said technique in new or novel situations.

A better lesson/textbook structure would be:

1. Teacher presents students with a rigorous and relevant problem that they can think about but cannot solve without specific mathematics concepts that the teacher will introduce later.

2. Discussion should take place among the students about the best possible strategy to solve the problem. Teachers should participate in the discussion and ask questions that guide students toward the realization that they need certain mathematics concepts to achieve their goal.

3. The students and teacher work together to come up with a solution to the problem.

4. After the problem is solved through exploration the teacher bridges the gap between the students' problem solving process and the specific skills/goals the teacher had in mind when the lesson began. Essentially, the teacher begins teaching only after the students have been given a change to struggle with the problems at hand.

The benefit to my process is that it gives students a chance to be interested in the concept first. It also gives the students a need for the mathematics that the teacher plans to introduce. Teaching based on curiosity and need is harder to do and requires more planning, more time for students to make mistakes, and more time for questions...but students learn from questions...they don't learn from lectures (or textbooks).

The only example I can think of where this is taking place in any organized fashion ( because it is hardly organized when I do it! Like I said, it's pretty difficult to pull off and often takes months to get the students used to learning like this) is at Dy/Dan with his WCYDWT? ( what can you do with this ? ) series.

All of this being open-source text has been scrapped and I'm starting from somehow constrain what is essentially experiential/collaborative/free form learning to a .pdf file. Wish me luck.

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