Textbook

Our textbook is Algebra I, Concepts and Skills, by McDougal Littell. Textbook work is only a part of this class. The textbook helps establish the course outline, provides explanations, examples, and practice exercises, and ensures with state-mandated standards.

Quality textbooks are a good resource for the following reasons:
1. They provide a reasonable course outline, which can be further modify or expanded.
2. They help ensure compliance with state standards.
3. They provide sheets of practice problems.
4. They teach students how to use a textbook, in particular the table of contents, index, and glossary.
5. They help reassure parents who may have concerns that an alternative program--no matter how worthwhile--may not be preparing students for mainstream high school.
6. They provide an excellent example of what's wrong with mainstream teaching. This item probably needs more explanation. Read on...

Textbooks and State Standards

State standards set an inappropriately low bar because they focus almost exclusively on the memorization of an overly broad range of facts and methods. This often lead to "drill and kill" instruction which may appear to meet the standards in the short term, but rarely meets the real needs of students.

Given the state requirement to memorize a vast quantity of shallowly understood factoids, there is often too little time for the in-depth study and exploration which alone leads to deeper mathematical thinking and insights. Perhaps an example of the difference between textbook memorization and inspired mathematical thinking is helpful.

Teaching how to convert between Fahrenheit and Celsius

Typical Textbook Style Instrucgtion: Memorization

1. The formula is presented and students are told they will learn to use it.
2. Example problems are provided demonstrating how the formula is used. Students read these examples and memorize the process. (Note that the only cognitive challenge for students is to stay interested long enough to memorize the formula and method. No mathematical thinking is required.)
3. A page or two of practice problems are provided, often as homework. Once students show they can successfully perform the memorized procedure on  a set of virtually identical problems, the lesson is "learned" and the class rushes on to the next topic.
4. 'Advanced' students are often given more challenging problems that may require some creative and mathematical thinking. Students who are considered 'Basic' or 'Below basic' are often given sheets of simpler problems on the faint hope that more "drill and kill" will "make it stick."

Teaching Through Thinking and Inspiration

Please Note: A prerequisite for success with the following approach is student enthusiasm for learning. Such approaches can not be dropped out of context into any classroom. They are an integral part of a complete educational approach, such as those developed by such seminal thinkers as Rudolf Steiner, Paolo Freire and Henry Giroux.

1. A challenging problem is presented. It is given in a context that is relevant to the students and their world. For example, the teacher might say, We have some visitors from Mexico in our class today. To measure temperature, they along with almost all other regions of the world use the Celsius system while we in the US still use the Fahrenheit system. We want to study and measure temperature today, so we need to find a way to translate between these two systems. Can we develop a formula to easily convert between Fahrenheit and Celsius?
2. The teachers then leads the class through the process of inventing the formula. Leaving as much as possible for students to develop on their own. The general process, which models the scientific method is as follows:
3. Gather relevant facts and observations. For example, we notice that: 
       32° F  =    0° C
              and that
      212° F = 100° C
4. Given these two statements (or linear equations) we can use algebraic methods, such as linear substitution to solve this system of equations. From here we can derive two formulas, one for converting from Faranheit into Celcius and one for converting from Celcius into Faranheit. We can also explore how these formulas are mathematically identical although they help us in different ways due to variations in their form.
5. By inventing/discovering the formulas themselves, students are more likely to remember them. They are also more likely to see important underlying patterns, such as the fact that this line of reasoning now enables them to 'invent' formulas for any two linear scales. We can now "play" (in the best sense of the word) with this idea by converting between the above temperature scales and others, such as Kelvin or even completely new scales that we create ourselves.
6. Students will have their own ideas about how to extend this lesson. They may try the method on other scales having nothing to do with temperature, such as centimeters and inches, pounds and liters, dogs and cats, etc. Some ideas will 'work' others will not. The patterns that result from this research are a deep well of potential discovery.
7. If time allows, the teacher can present the important topics of scientific certainty and formal proof.
      - "How do we know that our formulas are always accurate?"
      - "Can we find situations in which the formula fails?"
      - "How can we prove with absolute certainty that our formula is always accurate?"
      - "Can we explain our proof to others in a way that is scientifically and mathematically valid?"
8. For key topics, main lesson book pages can be created in which students demonstrate their process for inventing formulas and proofs.
9. After all this deep and original thinking, students are far more likely to remember this lesson for the rest of their lives. They may in time forget specific details of the formulas, but they will always know that they have the capacity to discover such formulas themselves. They will carry with them a greater confidence in their ability to do original mathematical thinking, and will not feel that they must forever rely only upon memorizing poorly understood methods invented by others.

Many textbooks are poor reference materials. They tend to be heavy, superficial, error-prone, filled with distracting pictures, and contain constant infomercials for the publisher's website. Few recent textbooks contain complete indexes and glossaries. This is because such essential elements are not a part of the state standards. For suggestions on better reference books, see my list of recommended books.