The section, in one of the best algebra books, for Linear Equations in One Variable consists of ten pages. Important definitions (concepts) are presented on about a page, there are 9 examples of solving equations, and 111 exercises. The examples and exercises are all numerical and are really the same except for different numbers. The concepts are not discussed nor are there any questions about those concepts.
The final step of each example (and expected for exercises) is a check to determine if the solution makes the equations true. This ignores the fact that all the equations generated by the additions and multiplication properties are equivalent. An excellent opportunity to teach the power of deductive reasoning is wasted.
Following is the normal pattern for “teaching” this material.
- Teacher shows PowerPoint slide of each of the examples.
- Why – they are in the book
- Diligent students copy numerical information from the slides.
- Important conceptual comments are ignored
- Teacher assigns all the odd numbered problems.
- Why so many – math is not a motor skill
- Most students defer working problems until time to cram for a test.
- Because passing the test is all that counts
- Next day the routine is repeated for the next section from the book.
- Completely ignores students’ comprehension
- When the chapter has been completed, a chapter test is administered.
- Too infrequent
- The chapter test consists of 10 problems chosen from the list of assigned exercises.
- Arbitrary and ignores content
- Checking for correct answers constitutes grading.
- Grading unimportant skill
If your point of reference is the following list, then you will see nothing wrong with the above eight steps.
- How you were taught mathematics
- How you were taught to teach mathematics
- How your mentors and colleagues teach mathematics
- How textbooks are designed and written
However, if your point of reference is recent discoveries in cognitive science, then it will be plain that each of the eight steps is terribly wrong and ineffective.
It seems to me that we have refined massed practice into a streamlined practice which “feels” good but is effective. We have known it to be ineffective since 1885. Educators learn slowly.
I recently learned that many/most teachers consider each section of a chapter to constitute a lesson. All the math textbook authors I know use books, chapters, sections, and units as a means of displaying the natural hierarchical structure of the subject. Those subdivisions were not designed with lessons in mind. The teacher who views each section as a lesson will probably not teach the important hierarchical structure of the subject. If that structure is absent, mathematics becomes a hodgepodge collection of random ideas which are nearly impossible to learn.