This section is designed to help K-12 teachers and mathematics resource teachers develop and strengthen their own mathematics problem-solving skills, as well as aiding them in fostering these skills among the school students they teach.

As is true for this ADEPT course as a whole, but is worth emphasizing for this section in particular, problems given in all ADEPT courses, and in Conceptual Algebra in particular, range in level. Not only are some problems appropriate for lower-level K-12 classes and students while others are higher-level K-12, but in addition, some problems are, by designed structured to challenge K-12 teachers themselves.

In addition to the mathematical level of an exercise, there is also the dimension of how directly connected it is to the K-12 curriculum: ADEPT problems range from "directly applicable to the K-12 classroom" to "mental push-ups for teachers" exercises which you are encouraged, and supported, towards modifying and customizing them to be appropriate to the level, and needs, of your own students.

Pattern problems play two roles in this part of the course: First, they are almost ideally suited for serving as self-contained exercises in mathematical modeling, problem-solving, and practicing mathematical generalization; secondly, patterns are the foundation for the Language of Algebra. Algebra has been called "generalized arithmetic" and while it is not the only "hat" Algebra wears, it is one of the principal ones. Generalized arithmetic patterns and properties are the basis for algebraic expressions and properties.

For most of the pattern examples, Teachers should refer to Fostering Algebraic Thinking. As an additional reference, an example of a matchstick type problem is found a this Australian guide for example.

The Tips on Problem Solving handout is one which teachers who have taken the ADEPT class have found to be a helpful guide of things to keep in mind as they approach, begin, and proceed forward in the problem-solving process. Just as this Tips handout is modified from those I have used in college-level courses, several teachers who have taken the ADEPT Conceptual Algebra have reported they have themselves, successfully modified it, or modeled their own handouts on it, to create a handout or list for their own K-12 students to keep in mind as they engage in problem-solving.

The Metacognition handout, which is entitled Pedagogical Issues to Keep in Mind During and After Problem-Solving, on the other hand, is essentially a framework which teachers and other educators can use:

1. as a guide in thinking about their own strengthening of their mathematical abilities and power and

2. for thinking about how a given problem may (or may not) apply to this or that K-12 classroom situation, and how it might best be modified if necessary, to be more applicable. As in the case with the preceding handout, it applies across the mathematical curriculum, beyond algebra, and like the preceding handout, it can also be modified by teachers to serve another purpose, namely

3. creating materials, worksheets, or simply a short list of Guiding Questions -- or even a "diary" assignment -- which encourages K-12 students to reflect upon what lessons they have learned while solving a mathematical problem: which skills are most applicable to which types of problems; which general strategies help; and so forth, and also to place it mentally in relation to existing mathematical knowledge (how does this problem relate to other types I have seen before?) and to foreshadow and link to, future mathematical learnings (how can this problem be extended or generalized?) and so forth.

Another part of Conceptual Algebra concerns what I informally call "algebrization" or the process of going from "I see the pattern..but I don't know how to get the formula!" to finding a correct formula. This integral part of the course cannot be done full justice in this Curricular Resource Binder (CRB): Sign up for the course