Mathematical Reasoning Cannot Be Taught: A Cautionary Tale

“How idiotic to require education students to take this class!” spat professor Guerzhoy, “This type of reasoning cannot be taught. Either you are a mathematician or you aren’t.” The four other education majors in the class seemed to shrink further into themselves, a pitiful island of students huddled against the far left wall of the classroom. I tried not to blink. I always sat in a neutral space mid-room, straddling the fence between the education students and the math Master’s students who filled the right. Between that and the fact that I occasionally got glimpses of understanding regarding the abstract algebra Professor Guerzhoy scrawled across the board with his nicotine stained fingers, I figured I still had him fooled. He hadn’t figured out that I was just a wanna-be high school math teacher and not a real mathematician. Not yet anyway.

Even when I utilized his office hours to bring him my incomplete proofs, pages of my desperate attempts at explaining concepts I only partially understood in a language of signs and symbols not my native tongue, I never let on. He would dart his eyes back and forth from my proofs to my face, peering darkly at me, and muttering things like, “Unconventional…not at all what I would have…..and yet…,” then grunt and affirm that what I had written was indeed true and to keep thinking about the problem. Each office session would end in a barrage of at-least-you-can-think-logically-not-like-those-education-majors and mathematical-reasoning-can’t-be-taught. I still never let on. I let Dr. Guerzhoy assume what he wanted to assume. Yet all the while, I felt dirty, like I was betraying the other education majors, like I was living a lie.

Finally, about a month into the course, and after another bitter classroom tirade in his almost unintelligible Russian accent about the general stupidity pervading the left hand side of the classroom, I lost my cool and decided to do something. Something had to be done! I was going to be a teacher; did I or did I not believe that math could be taught? I decided that I did believe it, with all of my heart. So, after class I ducked out quickly to lurk in the hall, waiting for that humbled group of education majors. When they heard my proposal, they seemed stunned. I don’t know if they even realized I was one of them before that point, but they all assented to my plan. “Meet me in the cafeteria after class and I will teach you. I will read that book, I will figure it out, and I will teach you!” O, and it worked. I have always understood things best as I tried to explain them. Just the process of trying to simplify and organize a concept well enough in order to teach it solidifies the information in my mind like nothing else. We would sit there in the cafeteria after class working through proofs and discussing abstract algebra and I felt like a teacher. A teacher! Then one day, as we sat there, I happened to glance up, and there, a few tables away stood Professor Guerzhoy, mouth agape and eyebrows knitted. He caught my eyes and his expression shouted disapproval and worse, betrayal. He had thought I was one of his. He had believed me to be a math major. My little charade was up. He never spoke directly to me again, in or out of class, and I never bothered to drag myself into his dark little office for help again either.


That’s the end of my sad little tale. Yet, it’s not the end. Dr. Guerzhoy may be a striking example but he’s not the only professional educator perpetuating the “it-can’t-be taught” lie. He’s not the only one of us buying into the easy out of “they-just-can’t-learn-this.” I have spent the last five years tutoring students with disabilities. They were mostly your students, professors. They were taking math 082, 092 and 122. They’ve been enrolled in Psych classes, English classes, and Chemistry classes. They’ve been nursing students, machining students, and criminal justice majors. I’ve worked with hundreds of students with dozens of disabilities from hundreds of your classes. What’s the one thing they all had in common? They are the students you couldn’t or wouldn’t teach. They are the ones you said “couldn’t be taught.”

Now I realize that tutoring is a whole different world than classroom teaching. I understand that it’s impossible to personalize your instruction for each student who walks into your classroom. Nobody understands that better than a GED instructor. We teach four years of material across four broad subjects over the course of a semester to a group of students whose test scores vary from a second grade equivalency to a twelfth. That’s not what I’m talking about.

I’m talking about the dedicated ex-con who I tutored through his GED. I helped him cope with the dyslexia that danced numbers across the page by uncovering one digit at a time to add fractions. He had to write out his 6, 7, 8 and 9’s times tables before every test because he couldn’t memorize them but he would not give up. He came back to me devastated after a meeting with an academic advisor who told him, “Based on your test scores, you aren’t capable of college math. Perhaps you ought to consider airplane mechanics instead of criminal justice.” That old sense of injustice kicked right in, and I told him he could do whatever he damned well please and that academic advisor had no idea what he was capable of. He went on to pass his college math classes and without any further tutoring, I might add.

I’m talking about the culinary certificate seeking student with ADHD and LD who fell in love with math as we worked together on his 092 class and decided he wanted to change his degree to engineering. His counselor told him he had no aptitude and shouldn’t pursue it. I assured him that no matter what the tests said, he had the spark, I had seen it. He’s well on his way now to his Master’s in biomedical engineering.

I’m talking about the sweet, dedicated young woman with mild mental retardation and a passion for medicine who was told that the most she should ever aspire to was to be a home caregiver or a CNA. She doggedly applied and reapplied to, then finally completed her radiology program even though she had to take physics three times.

I’m talking about the dozens of students who have come to me saying that they went to your offices during office hours and you couldn’t or wouldn’t help them. You told them you already explained it in class. You told them they were too far behind and they should just withdraw. You asked them to consider other career options and majors. I’m talking about the ones who reported that they couldn’t ask questions during class anymore because you mocked them or belittled them. I’m talking about the ones who came to you with their accommodations and, despite the fact that you are legally obliged to grant them, you decided they either didn’t really need them, were being lazy, or you just didn’t have the time.

I have made it my life’s work to teach these students: the low ones, the slow ones, and the ones for whom it never comes easy. I will admit that they don’t always do well with traditional teaching. I will admit that sometimes I have to explain the same concept 10 times in ten different ways before stumbling upon an explanation that makes sense to a student. I will admit that many of them don’t succeed or give up too quickly. However, this is what I want you to know: they all can be taught. Every student. Any concept. I beg you, teach them. And to those of you, and I know there are many, who have gone home and rethought your teaching style, who have stopped in the middle of a lecture to reteach, who have spent countless office hours and personal hours working tirelessly with my students who often seem to show so little progress despite your best efforts, thank you. You are truly teachers.

Note: This was originally published on another of my blogs, P3 to Mickey D’s: Non-Traditional Teaching Tales on 10/2/15.
Photo Credit: Maths by AJC1 on Flickr; used under Creative Commons License.

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