Solving Quadratic Equations with the Right Side of the Brain.
Our brain is comprised of two lobes that develop unique functions and quite different ways to process information about the world around us. They usually work together helping us learn and make sense of our experiences. But formal education privileges the functions of the left hemisphere and treats the functions of the right hemisphere as if they interfere with our ability to learn. Is is possible to organize didactic content to make them work together again? Even for abstract content?
The ground-breaking work by Roger Sperry during the 70s, and a host of neuroscientists that came after him, showed with quite clarity that our cerebral cortex is comprised of two lobes that develop exclusive functions and quite different ways to process information about the world, in what has been called Lateral Specialization. In general terms, the left hemisphere appears verbal and conceptual, abstract, analytical, digital, logical, temporal and possibly the seat of self-consciousness. Meanwhile, the right hemisphere seems non-verbal, concrete, synthesizing, analog, spatial, atemporal and highly attuned to emotional expression.
Not surprisingly, it is the left side of the brain, with its analytical powers and digital communication abilities, the one that is privileged by formal education in the western world, while the right side of the brain, with it’s intuitions and empathetic abilities, seems to distract and get distracted from the universal truths in study.
But this is not really the case. I argue that this distractibility happens only when the imagination of the student, understood as fantasy but also understood as the images that give form to her perceptions, is not being properly handled, which is by definition true every time we teach to the left side of the brain alone.
Let’s imagine a group of students solving some well-formed quadratic equations. By well-formed I mean that they come in the form of an irreducible univariate polynomial of degree two:
\[ax^2 + bx + c = 0\]
[1]
By solving, I mean that the students are able to draw fairly accurately the parabola expressed by the equation. This is usually accomplished by finding and plotting points \((x,y)\) according to:
\[f(x) = ax^2 + bx + c\]
[2]
When it does, it is possible to find where the parabola cuts the abscissa (x-axis) by solving:
\[x = {-b \pm \sqrt{b^2-4ac} \over 2a}\]
[3]
Possibly all this brings some (hopefully not too painful) memories from high school. The expected (analytical) way to solve this is to dutifully replace \(a\), \(b\) and \(c\) by the corresponding numbers in the polynomial and solve the equations for \(y=f(x)\), finding enough points of the parabola to help determine its shape.
Let’s return to the proposed students. This is not immediately evident from the formulas but after drawing a number of them they will realize that the parabolas are symmetrical, which helps tremendously in finding other points and drawing the curve. They will also quickly learn that the parabola cuts the ordinate (y-axis) at \(c\), which means that they already have a point of the parabola without having to make any calculations \((0,c)\). Experience will also show that if \(a\) is positive, the curve opens to the top, and if it’s negative, it opens to the bottom. This is great because by simply looking at the sign of \(a\) we can immediately discard half the possible shapes most students will find while studying this subject. Furthermore, the shape of the curve opens and tends to flatten the closer \(a\) gets to zero. This insight can also help estimate the shape of the parabola. In which quadrant should the parabola be drawn? To make a long story short, experience will show that when \(a\) and \(b\) have the same sign, the vertex will appear to the left of the ordinate (y-axis), and to the right if they are different.
No least important, after drawing enough parabolas the students will have noticed that its curve could either touch the abscissa (x-axis) once, cut it twice or not touch it at all. This is indicated by the bit:
\[\sqrt{b^2-4ac}\]
[4]
... that we find in formula [3]. After enough familiarity with the formula it’s easy to see that if \(b^2\) equals \(4ac\) then the parabola touches the x-axis only once. If \(b^2\) is bigger than \(4ac\) then it cuts the x-axis twice; but if \(b^2\) is smaller than \(4ac\) then the parabola doesn’t touch it at all. And look, no square root calculations needed!
Confronted to the exercise:
“Draw the parabola described by: \(2x^2+2x+2=0\)”
These students will quickly identify that the parabola opens to the top \((a>0)\) and cuts the y-axis at 2. Intuitively they can tell that 2x2 is less than 4(2x2) (even without solving the multiplications) and therefore this parabola doesn’t touch the x-axis. And because \(b>0\), just like \(a\), the vertex sits squarely on the II quadrant, looking more or less like the figure to the left:
Compare our intuitive perception of \(2x^2+2x+2=0\) with the plotted rendition of \(y=2x^2+2x+2\), presented to the right and performed by a graphing calculator. They are strikingly similar.
This allows visualizing the general position of the parabola even before any computation is made. This is fantastic information because it can show errors that could spring in an attempt to solve the equations. For example, if in calculating the vertex, y appears negative or x results positive, there is clearly a mistake, something that would take longer to identify without this mental image.
Please note the recurring concept of “experience” during the previous description. The memorizing of the formulas doesn’t immediately provide information about the shape to be drawn (although it defines it), but playing with the formulas (“play” in its most ludic definition) helps to visualize this information, thanks to patterns that emerge after the repeated effort to plot them. During these re-encounters, the eyes of the face see formulas but the eye of the mind slowly starts seeing parabolas.
Someone could argue that by noticing signs and the range of some numbers in a well formed quadratic equation we are, in effect, making an analysis. But please note that we are deriving conclusions based on a familiarity with the signs and their meaning, created by playing with quadratic equations, and not by applying a systematic logical calculation, which is what we do when we solve the equations. In this sense we are not solving equations, we are reading them and integrating them as a gestalt. This direct experience with the subject of study, and the familiarity it creates, is generating a perceptual ability.
Clearly, a merely intuitive knowledge of the parabola can not be enough to study it and be able to derive all its properties (we have not considered the focus point, the directrix and other useful properties, each with very precise and abstract definitions) but imagine how much easier understanding or even discovering these properties would be if while the eyes are seeing the numbers and letters in the formula, the mind is quite immediately forming images of the parabola they represent. While the abstract left side of the brain deals with \(ax^2+bx+c=0\), the right side of the brain deals with \(2x^2+2x+2=0\) or any other concrete expression, providing the students with a comprehensive perspective that includes the simultaneous analytic examination of the parts and the gestaltic configuration of the whole, in what has been termed Perceptual Thinking.
Now, I am not trying to suggest that this is the way quadratic equations should be taught. What I am saying is: look at how much deeper the understanding of abstract ideas can become when the teaching involves the right side of the brain; look at how much more resourceful, intimate and personal the perception of a subject of study can be when the right side of the brain, and the local and transient experiences it requires, are included as part of the teaching process of our cherished universal truths, not merely as homework, but as an integral part of the learning required to make sense of them.
I hope that the notion of an education that involves the right side of the brain that helps students learn to better perceive a subject of study intrigues you. I further develop these ideas about the psychology of education, and even offer a model to correspondingly organize didactic content, in my book “Master Online Teaching” that will shortly be available at Amazon in print and eBook format: