Your mathoscope is good when pedagogical content knowledge is in harmony with foundations of math. However, when they are in conflict, your mathoscope is bad, and confusion, rote procedural learning and homework bloat are in your future.

When the lesson plan is based on math foundations it is logical, concise and consistent. It can be stated quickly and concisely. The rules are bound to the concepts. The rules teach the concepts. The procedural rules teach the concept. Practicing the procedural rules in drills teaches the concept.

Self discovery is possible to go the next step or do applications of the concept and its procedures. Students can be led to self discover the rules or concepts in some cases. This may take ingenuity and practice to work out such lesson plans. The polished and concise rules then would be taught as a foundation to move forward. These would be available in the textbook or materials.

When the lessons are in conflict with math foundations, then false impressions and confusions are created in the students’ minds. These don’t go away with practice, applications, or homework. Instead, the students learn the rules procedurally and not conceptually. It is harder for them to retain the material because it contains logical inconsistencies.

PCK and foundations of math need to be harnessed together as twin horses to lead the students on faster than is possible when one is ridden separately and definitely better than when they are not working in harmony.

Math foundations gives us short consistent statements of the correct math. Math foundations avoids bloat. When we remove what is not the explanation, we have shortened the lesson. Math foundations makes sure that what is left is the explanation.

The job of PCK is then to help us teach the math foundations explanation to the students. The math foundations explanations all fit together. There is no way to tie math together except by using math foundations. This is because that is the job of math foundations and it has already in large part done this. Math foundations is the explanation of elementary math and logic.

Bloated lessons that meander around and don’t teach the math foundation concept are our lot if we do not mind the teachings of math foundations. Nor should we expect on the fly to do as good or better than centuries of hard work on math foundations. These are improvements in the explanations of elementary math. If we don’t use them, we are back to Euler 1765 Elements of Algebra forever.

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