, it generally refers to the way a new technology has revolutionized the way we’ve come to know things to work. Ride-sharing apps, for example, have upended the taxicab business.
But disruptive ideas have their place in the classroom too. Cognitive theorists use a term coined by Jean Piaget to refer to what disruptive ideas do for learning. Disequilibrium is a state of mind caused by an imbalance between what we previously understood and what we are learning that produces a desire to know more. Often, constructs are redefined in our minds as a result of synthesizing old and new.
In mathematics, disequilibrium occurs in scenarios that perhaps should be called “Lies my teacher told me.” In preschool, children are taught to recognize rectangles and squares as two distinct shapes. The understanding that one is a special case of the other comes later. Young children are often first introduced to the counting numbers. This number set fails, however, when these children are confronted with how to share three sandwiches among four friends. The door is then opened to fractions. Focusing on common misconceptions, like “two doesn’t go into five,” is a way teachers can capitalize on disequilibrium in the classroom.
The following work shows one area where previously learned algebra material fails. This radical equation is solved by using inverse operations and factoring.
√2x – 5 = x – 4
(√2x – 5)2 = (x – 4)2
2x – 5 = x2 – 8x +16
0 = x2 – 10x + 21
0 = (x-7) (x-3)
None of the steps are wrong, but don’t forget to check the answers in the original equation. The solution x=3 yields √2(3)-5 = (3 ) – 4 ⇒ √1= -1, which is untrue. The act of squaring both sides in the second line is the source of the extraneous solution, and using a graphing approach to solving this equation should illuminate the issue.
Previously learned material from other disciplines can cause disequilibrium as well. Beginning Geometry students learn that the area of a triangle can be found by taking half the product of the measures of a triangle’s base and height, the segments labeled in the triangle on the left. Since in the English language, base refers to the bottom, a student may fail to see that the area can be found for the triangle at right.
In fact, the math definitions for base and height are purely based on their relationship to each other (that being perpendicular) and not in any way connected to how the triangle is oriented. The bigger picture understanding is that shapes maintain their properties despite any rigid motions. Geometry software is a great tool for expanding upon this understanding.
Speaking of the language of mathematics, the simple terms “and” and “or” can create their own share of disequilibrium when students try to harmonize their math meanings with their English connotations.
Many, if not most, students benefit from learning through moments of disequilibrium. For, these moments prompt the questioning of why something once believed to be correct, is actually not accurate. Educational publishers, therefore, have the responsibility to try and support teachers in creating these moments.