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We are currently in our “Angles and Triangles” unit and one part of the chapter that I enjoy is using the Angle Sum Theorem and Exterior Angle Theorem. We are just wrapping up both theorems and I can see the four stages of mastery within my students as I work with them.
Unconsciously Incompetent: We started with “What do you notice? What do you wonder?” on the first day (which I used as a previous example in the last module). I pasted three congruent triangles and rotated them to form a straight line, thus the three interior angles of a triangle add to 180 degrees. In the end, not one student in this particular class made any connections with the straight line formed by the three angles – chalk it up to plain ignorance. They did not know what they did not know.
Consciously Incompetent: Now we are to the stage where one angle measure is missing and the students are asked to find the missing angle measure knowing the other two angle measures. What took some time for the students is to understand how the three angle sums are a simple equation when equated to 180 degrees. It was even more apparent when variables were added. Many came to a standstill.
Consciously Competent: As of today, most students are at the Consciously Competent stage. They are able to write an equation and solve for an unknown. It is a good place to be but I noticed a disconnect still which is why I want them at the unconsciously competent stage.
Unconsciously Competent: Here is an example of a math problem that separates this stage from the last stage. The angle measures are 3x, 5x – 10, and 20. More than one student raised their hand today and said “What do I do? There are negatives now.” To be honest it is a little soulcrushing that a few negative numbers are a roadblock. However, a student who is Unconsciously Competent does not let that get in the way – it is business as usual.
Lastly, a Consciously Masterful student understands how writing an equation is an actual problemsolving strategy that can be used under a wide range of scenarios. Would not that be a nice place to be?