User:Numberslogicacquisition/sandbox

Cognitive Conflict Approach in Teaching Algebra:Notes

Determination of Learning or Knowledge Conceived. And Whether the knowledge is procedural of Structurally Oriented:

1. Someone with a procedural understanding will speak of a process and steps that need to be done, and provide a list of those steps as an explanation of why they produced an answer. A person with structural understanding will describe the concept as an object, or compare the equivalency of objects, often supported by visual image when asked to explain why they produced an answer.

2. Written opinions of a solution rather than their ability to solve a question.

3. Someone with a procedural understanding will list the steps of a procedure when asked to explain why they produced an answer, whereas someone with structural understanding will describe the concept as an object, explain how a process also represents an object, compare objects, or compare the equivalency of two structures, often supporting their explanation with a diagram.

Justification

Problem Solving

The following explains the combination of procedural and structural understanding when solving a problem and the correct and incorrect way to go about it.

When analyzing a problem analyze the relationships then decide between various operations to perform. Check the validity of the transformations by using a quick example. <--- Not sure what this exactly means. Could mean taking an algebraic equation and converting over into a graph and flipping or turning an image from an original equation rather than just applying an algorithm or formula to solve the problem.

You should be able to justify your decisions beyond just reciting a list of movements that were done while transforming formal expressions. In doing so you could add this as step 4 for the steps provided above in proving structural knowledge over just memorizing a procedure. And finally concluding that structural understanding of a concept if, when faced with an unfamiliar problem involving the concept, they analyze the structure of the problem rather than automatically applying a familiar procedure, and justify their transformations as the solve it.

Control

Initially analyze the problem and constantly monitor and assess their solutions as you progress. Assess if each step taken is valid and if it will help you progress to a solution. Argue with yourself as you work along. Selecting and pursuing the right approaches, recovering from inappropriate choices, and in general monitoring and overseeing the entire problem-solving process is a major determinant of problem solving success or failure.

If someone with a structural understanding comes to a point where the objects are not equivalent they will look for a structural clue and they will persevere and try to determine why the object they ended up with is not equivalent. With procedural understanding, the operations and transformations are steps in an algorithm to be followed; there is no way to check the correctness except with a formal checking method like substituting the eventual answer back into the original question. As someone who has procedural understanding completes a process they do not know if they have made an error. All they have to go by is if x=7 or x>7 as what would act as a "halting signal", so the students will leave an answer that is wrong incomplete when they reach this stage.