By A. Gwinn Royal, Ivy Tech Community College of Indiana
Currently, I am focusing on mitigating “learned helplessness” with respect to the study of mathematics. According to an article on the APA website (https://www.apa.org/monitor/2009/10/helplessness.aspx), newer research on learned helplessness suggests that the real issue is (lack of) control. This leads me to believe that by affording my students greater control over their own learning (within the bounds of mandated curriculum and instruction), I can deliver them from helplessness to a place where they acquire a keen sense of agency in their academic endeavors. Many of the students I teach are in my courses because somewhere along the way, their study of mathematics has primarily concerned learning to fail. I teach them how to fall.
Teaching Students How to Fall
On an ice-skating outing, a parent of a toddler wants the child to enjoy the experience. There are several approaches to this scenario: the parent can just let the toddler have free reign on the ice, the parent can hold hands with the child, or the child can use a skate trainer. Suppose the toddler is free to explore. This could be dangerous, as there may be no safe place for the toddler to learn to skate by trial and error. Now, suppose the parent and child hold hands to skate. Put yourself in the position of the toddler for a moment—you’re doing your best to keep up with someone whose strides are far longer, smoother, and faster than yours, you’ve got to keep one arm up at an uncomfortably steep angle with the other one frantically waving around, and losing your balance means you’ll just get dragged along. This is less than ideal. Enter the skate trainer: this solves a lot of problems for the toddler because it now becomes a situation within which our inexperienced skater has some measure of control (slower speed, ability to take breaks when frustrated or fatigued, the separation of balancing skills from skating skills). The use of the skate trainer reminds me of Amanda Serenevy’s description for the Traditional Math approach, which includes heavy scaffolding. The kind of helplessness that often results is one of dependency; math students who are almost completely reliant on the instructor to provide hints, cues, and prodding aren’t going to make much headway toward increasingly bigger ideas if they are not given the opportunity to become more metacognitive and confident in their ability to teach themselves how to learn. Moreover, the skate trainer has limited usefulness; the skill of skating is still yet to be fully exploited—the more fun and interesting maneuvers, such as jumping and spinning, would be hampered by the use of extra equipment. If the skate trainer is used moderately, tapered off, and given up during childhood, the child will learn to be resilient (i.e. comfortable with falling and getting back up) as the skills of skating emerge. However, if the child grows into an adult who is still dependent on the skate trainer, it’s much more difficult to separate the two. Adults have an affective filter that tends to inhibit the necessary risk-taking behavior that paves the way for learning. I mitigate this kind of lack of control by incorporating a different approach.
With the Conceptual Math approach, the learner is required to exert a measure of control over certain aspects of their experiences so that deeper understanding can be cultivated. For example, students may be expected to start with a lot more of their own thinking and they are encouraged to explore their own ideas and approaches as they stumble along the path of learning. This is also where I, the instructor, can practice the art of “be[ing] less helpful,” as proposed by Dan Meyer (https://www.ted.com/talks/dan_meyer_math_curriculum_makeover#t-613428).
For example, in solving a multi-step linear equation, such as 5x+24=2x+36, I enlist the help of the class in presenting the solving process. I take suggestions, and execute the orders of my students, whether it involves a mathematically legal move or not, and whether the move makes the problem easier or more difficult to solve. A student recalls the multiplication property of equality and advises that we divide both sides by 5 to rid ourselves of the 5 in the 5x term so that the variable is isolated. Initially, I try to cheat the system by dividing only the 5x term by 5, and the students call me on it right away. I praise them for catching this “error” and I continue with the division; we end up with x+24/5=2x/5+36/5. A few students look uncomfortable; this is clearly not what they expected, but they seem content to move forward. I solicit advice once more. Someone wants to move the constant term to the right-hand side. I attempt to follow the addition property of equality literally by adding 24/5 to both sides (it’s called the addition property, so I always add, right?); another student interrupts and states that, because the term is already positive, we have to subtract it on both sides instead. I (pretend to) protest this because it’s not called the subtraction property. A brave soul haltingly posits that we are really adding the opposite. Several students concur, and I concede the point. I take a moment to encourage the students to speak up even when they’re uncertain, as I expect the entire class to be supportive and helpful throughout our learning activities. We then have the following: x=2x/5+12/5. Someone declares that the problem is solved because the variable has been isolated on the left, but a counter-argument emerges, as there’s still another instance of a variable term on the right. After a few moments of constructive debate—usually, for this situation of variables on both sides of the equation, someone makes an analogy of using a word to define itself—the student who claimed that the problem was finished retracts the proclamation and insists that we continue solving. But we’re stuck; there is no apparent way to get the x-terms to combine. There seems to be the consensus that it is impossible to combine a plain x with a fractional x, and I intentionally allow this misconception to persist until we go back to analyze the problem after it’s completed. As we ponder a strategy, a student expresses consternation that the problem seems harder than it should be and suggests that we get rid of the fractions by multiplying the entire equation by the LCD. Most students nod in agreement, so we arrive at this: 5x=2x+12. Pleased to be free from the dreadful fractions, simultaneously two students suggest moving a variable term to the other side. I ask them if it matters which one I move, but I’m met with silence. I ask, “Can I just pick a term to move then?” I see more nodding, so I proceed to move the 5x to the right-hand side: 0=2x+12-5x. I “forgot” to combine like terms, and I’m promptly reprimanded for that oversight. I correct it to this: 0=-3x+12. A few students who have been quietly following along contribute that I should have moved the 2x to the left-hand side. I feign distress and slowly reach for the eraser to go back, but I’m told that I can just keep going by moving the -3x instead. I feign relief at being let off the hook for my wrong turn, and I arrive at 3x=12. From there, we find the solution x=4. We check the proposed solution by evaluating the original equation at 4, and we find that the solution checks out. Before we leave the problem, I step the class back through our process and suggest things to consider; sometimes, we re-work the problem using alternative strategies.
We learn to strike the balance between launching out into the unknown and making tentative plans to accomplish the learning task. We use our book and notes for basic information, but we allow ourselves to fall (and subsequently get back up) when we are actively engaged in learning. Our follow-up discussion includes labeling the places where we fell so that we can learn to recognize traps and create strategies to deal with them. For instance, our first decision led to falling awkwardly in our problem-solving technique because we got ahead of ourselves. In our re-work, we knew to hold off on the division by 5 until after like terms had been combined on both sides; we used our previous falling spells to think more critically and to make better decisions.
I am not a math major per se; I hold a master’s degree in Education (and I’m proud to wear baby blue). I stumbled into teaching mathematics after having spent many years tutoring students in various disciplines. Math by far was the most hated subject, and I was dismayed that so many people weren’t able to do the most basic calculations without experiencing anxiety on the level of PTSD. I thought that perhaps there was something I could do to help; I continued my studies by adding graduate math courses to my credentials, so here I am. I’m not a math genius (some areas of math are hard for me, too); I share my tales of struggle with my students to let them know that learning new things does not always come easily, and that it is OK to wrestle with a problem. I want them to become critical thinkers willing to ask questions that lead to interesting problems, and to confront those problems once they arise. How do I know they’ve changed? When I hear things such as, “It’s not as bad as I thought,” “I can help my kids with their homework now,” or “I can use this stuff.” I consider that a victory.