Math problems are a nearly unavoidable feature of school programs ranging from kindergarten to college. In my experience, the process of learning math usually takes on a familiar form, regardless of the type of problem being learned. Carolyn Bufford and colleagues’ (2014) study, entitled “The Psychophysics of Algebra Expertise,” provides evidence in support of the idea that alternative paradigms of mathematics instruction could give rise to increased proficiency in the domain of algebraic problem solving.
Under the traditional method of mathematics teaching, instructors walk students through an example problem step-by-step and explicitly tell their students “rules” or “principles” for solving a certain class of math problem. For example, in algebra classes much attention is paid to solving polynomial equations for zero (often called “finding the roots”). In order to assist learners in solving a polynomial equation, such as x2 + 6x = –8, a teacher would give students several rules to follow in order to solve the equation for zero. They would probably tell students to “make one side equal to zero by adding eight to both sides” and “factor out the common terms.” Eventually, the students would be expected to use a mixture of these rules to arrive at the factored solution: (x + 2)(x + 4) = 0. This is generally where instruction ends in the contemporary math classroom. Students then practice what they were taught by completing problem sets.
As someone who struggled with finding the roots of many polynomials in middle and high school, I always had trouble decoding the structure of math problems, even after receiving this rule-based instruction. It seemed as though my teachers and fellow students had an intuition for solving math problems that I fundamentally lacked at the time. Fortunately, recent research in cognitive science has developed some theories for why I may have felt that way. In particular, Carolyn Bufford and colleagues describe a possible way for helping students gain mathematical fluency through a short and interactive computer program.
Bufford et al.’s study is based on the premise that this explicit rule-based approach to mathematical problem solving is inadequate for developing mathematical expertise. In their view, learning to solve math problems is akin to learning a different language. When you read the words on this blog, you are not consciously thinking of the sounds and meanings associated with each word. Rather, this process happens automatically and without much effort for most native English speakers.
“Perceptual learning, the improvement of information pickup due to experience, is a key part of expertise in virtually every domain, including math. Mathematicians look at a problem or equation and immediately see what concepts, facts, and procedures apply to it or what transformations they can apply to it, but novice math students don’t see problems and equations the same way because novices’ perception hasn’t been tuned to math objects the way the experts’ vision has.”
– Carolyn Bufford on the benefits of perceptual learning interventions
Research has shown that expertise in many domains including math requires a similar form of fluency. In order to look at the aspects of mathematic problem-solving that you cannot just memorize, Bufford et al. used an online learning program called the Algebraic Transformations Perceptual and Adaptive Learning Module (PALM) developed at UCLA. This intervention focuses on helping students to learn see math better – to quickly and accurately pick up the relationships between different equations, increasing fluency with equations, rather than teaching explicit rules for problem-solving.
This module consists of a series of computer-delivered algebraic equations. On the screen, one target equation was shown with four possible answers below. Each of these four answers would take on the form of another equation, but only one of these equations would be an algebraically legal transformation of the target equation.
In Bufford et al.’s study, this Algebraic Transformations PALM was tested on college undergraduates. Participants were first assessed for their pre-existing knowledge of algebra. After this pre-test and a handful of practice problems, half of the participants (n = 26) were randomly assigned to the control condition and were sent home for the day.
The other half of the students (n = 25) were assigned to the intervention, which Bufford et al. call the “learning condition.” These students continued on by engaging in the perceptual and adaptive learning module. The length of this training session was determined by however long it took until the participant met a certain criterion threshold, indicating that they were proficient in solving problems like the one shown above. On average, the perceptual learning group practiced for 30 minutes and solved 142 practice problems.
The next day after the first experiment session, both the control and perceptual learning intervention group were invited back to complete a test of their ability. Participants were tasked with determining if two algebraic equations were the same with only a short exposure to the problem (between 200 milliseconds and 1593 milliseconds). Importantly, these equations were different from the math problems presented the previous day. Importantly, this task was different from the transformations task on the previous day, because the assessment equations were simpler, and the assessment task was only about how the equations looked, not mathematical transformations
As shown in the figure above, results from the experiment revealed that the perceptual learning intervention improved participants’ ability to solve these problems more accurately under time pressure compared to the control group. As one might predict, this finding only applied to those who scored lower on the algebra pre-test, indicating that advanced students may have previously acquired the pattern-recognition abilities gained by the lower-performing students during the perceptual learning task.
I personally enjoyed reading and thinking about this study, because it puts forth a novel way of improving student’s math performance with only a relatively short 30-minute computer-guided learning program. Math need not always be learned in a solely rule-based question–solution style. Rather, educators must consider the unconscious and implicit processes that underlie the solving of complex problems.
If you find this research interesting, the benefits of perceptual and adaptive learning modules have also been found in other studies of math, medical, and language learning by Bufford’s colleagues in UCLA’s Human Perception Lab. The Algebraic Transformation PALM and other PALMs including SAT math are available on the Insight Learning Technology website.
Images: (1) Photo by Kaboompics.com on Pexels.com. (2) Figure from Bufford et al., 2014, p. 11. (3) Figure from Bufford et al., 2014, p. 13.