Generally, there seems to be a tendency in our mathematical curricula to encourage the inertial, linear, and algorithmic-heavy approach to learning and assessment. It is imperative that, as educators, we break the cycle of these mind-numbing approaches and disturb them with various pedagogical tools that offer students the chance to let their resistive, non-linear, creative selves shine through.
One such method (aka PCP, read on) would be to insist that students every now and then invent their own mathematical functions, expressions, equations, or real-life application (word) problems that address specific concepts on one hand and allow the students to demonstrate their true conceptual grasp of those concepts on the other hand.
I conceived and started using the Personalized Conceptual Problem (PCP) idea to address some of the shortcomings of routine-laden problems that students typically encounter in most textbooks. These PCPs are personalized and tailored to each student's individual conceptualization of the concepts at hand, open-ended, and still do not sacrifice any procedural or algorithmic processes that are equally important in our students' progress in their mastery of mathematics. Following are two examples of such PCPs that deal with algebra topics.
PCP 1: Invent your own absolute-value equation that uses one of your initials as its variable (unknown) and would have no solutions at all. Show the detailed process of solving your equation for the given variable, graph the solution on a number line, and share how you conceived your equation in the first place. (EC: How about inventing your own inequality that has no solutions at all?)
PCP 2: Write your own math application problem (word problem) whose solution process must eventually involve the use of the equation 7x + 13 = 41, solving it, showing the detailed solution process, and providing the real life meaning of each of the equation's constants.
One such method (aka PCP, read on) would be to insist that students every now and then invent their own mathematical functions, expressions, equations, or real-life application (word) problems that address specific concepts on one hand and allow the students to demonstrate their true conceptual grasp of those concepts on the other hand.
I conceived and started using the Personalized Conceptual Problem (PCP) idea to address some of the shortcomings of routine-laden problems that students typically encounter in most textbooks. These PCPs are personalized and tailored to each student's individual conceptualization of the concepts at hand, open-ended, and still do not sacrifice any procedural or algorithmic processes that are equally important in our students' progress in their mastery of mathematics. Following are two examples of such PCPs that deal with algebra topics.
PCP 1: Invent your own absolute-value equation that uses one of your initials as its variable (unknown) and would have no solutions at all. Show the detailed process of solving your equation for the given variable, graph the solution on a number line, and share how you conceived your equation in the first place. (EC: How about inventing your own inequality that has no solutions at all?)
PCP 2: Write your own math application problem (word problem) whose solution process must eventually involve the use of the equation 7x + 13 = 41, solving it, showing the detailed solution process, and providing the real life meaning of each of the equation's constants.
In the comments area, share your thoughts about the PCP assessment approach, your variation on it, and sample problems of your own. Thank you
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