Teaching with Dignity: Reaching Beyond False Premise and Contradiction

Introduction: How Contradictions Can Breed Hatred

I remember distinctly the hatred I saw in the eyes of an undergraduate in one of my entry-level mathematics courses. And yes, it was hatred, which I recognized as a very intimate emotion. Since I did not know her, the target of her hatred could not have been me. Curious, I approached her after the lecture, at the moment when students were packing up and sprinting for freedom. I asked, “I noticed this course seems to have generated severe reactions for you. Can you help me understand why?” My curiosity paid off (often the case); I got a story. She told me, rather spiritedly, that she wanted to be a therapist for children with special needs and that she had failed a non-credit-bearing college algebra course three times before being forced to retake my course. She said, “I’m not a math person, but I will be a great therapist. Why do I need college algebra to be a therapist? This is stupid!” My response left her puzzled. I said, “I’m not teaching college algebra; I’m teaching an intellectual way of being. This course, to me, is the means to connect with you in a way that, without it, I would otherwise not have the opportunity.” She wasn’t buying it.

This narrative may resonate with you. If so, you’ll be glad to know it has an inspiring ending. After this student and I worked hard to build a relationship based on trust and empathy, she got an A+ in my course. You might ask, what happened next? Well, that is the true inspiration. She got an A+ in her next credit-bearing course, Finite Mathematics (oh, you’ve heard of it!?). She also became one of the most thoughtful tutors ever employed at the Learning Center I designed and managed at the University. I posit that this was true because I could empathize with her experiences and help push her thinking on the contradictions of her statement that she “wasn’t a math person.” This was not her truth.?

“Contradictions do not exist. Whenever you think that you are facing a contradiction, check your premises. You will find that one of them is wrong.”~Ayn Rand

Unwinding this untruth she had told herself changed everything for her. It's time for you and others who have found a contradiction-based relationship with mathematics to jump into this reality. One of your premises is wrong.

Narrating the Premises and Superficialities in the Journey of Learning

In reflecting on how to make this point more transparent and relate its importance to the work of teaching, I kept returning to the themes of maturity and contradictions. As frustrating as it is, our understanding of the world and our ability to learn are sometimes impeded by the depth of our maturity. I’m not talking about the maturity of keeping your room clean or being able to cook a good meal. I’m talking about the intellectual dignity that comes from time, experience, and learning from your mistakes.?

“Maturity is the ability to think, speak and act your feelings within the bounds of dignity. The measure of your maturity is how spiritual you become during the midst of your frustrations.” ~Samuel Ullman

As has been the case with my writing, I turned to literature to help shape the point I was hoping to clarify about the work of teaching. Maturity is the central theme of Samuel Ullman’s poem Youth, which conveys a powerful message about the essence of youthfulness, not as a physical state but as a state of mind and spirit. This couples well with JD Salinger’s classic The Catcher in the Rye. In Catcher in the Rye, Salinger takes readers on a journey through the mind of Holden Caulfield, a young man who is acutely aware of the contradictions and what he sees as superficialities in the world around him. Holden’s struggle to reconcile these observations, his maturity, and his values mirror the challenges that we all face. It’s a seeming contradiction — the necessity of remaining youthful, optimistic, and open to creativity and inspiration while developing maturity and wisdom.

Caulfield questions everything around him; you can feel his enthusiasm and passion on the page. However, he lacks wisdom. Teachers also have youthful enthusiasm; it's part of what attracts and keeps educators in the classroom. Researchers call the impact of a teacher’s approach (including this contagious enthusiasm) entailment (Davis & Renert, 2014). Entailment refers to the natural consequences or implications of adopting a particular perspective, approach, or concept in teaching and learning. For example, teaching a mathematical concept in a specific way will entail certain learning outcomes or student misconceptions. In this context, there’s an elegance to using this term, as it signifies mathematics as a tool for logic and as a tool that guides philosophy.?

Entailments represent key mathematical skills to be carefully applied in mathematics for teaching. In teaching, entailment suggests that inevitable consequences or deeper understandings naturally follow when introducing a concept or teaching method. For example, introducing a topic like subtraction by only using the words “take away” can lead to contradictions when applied to negative numbers. What does it mean to take away a negative quantity, and why do you add a positive quantity when you do this? These might be positive (deepening understanding) or negative (leading to frustration). The goal of a teacher is to generate teaching wisdom by observing these entailments and reflecting on how they shape students' learning trajectories. When telling stories about mathematics, this wisdom helps make the plot coherent. Entailment often implies a direct and logical connection, it tends to operate within structured systems (like mathematics or logical reasoning). While the results in the classroom are unpredictable, a teacher can study and reflect on how she thinks about and represents a concept. This leads to teaching wisdom and maturity, the ability to think, speak, and act about mathematics through teaching with dignity.

The False Premise of Phoniness in Mathematics

Holden Caulfield’s journey is one of observation—he sees the world around him through a youthful lens that prompts revulsion to what he calls phoniness, contradictions, and inconsistencies in the adults around him. As I’ve returned to the book repeatedly over the years, I’ve come to understand that this reflects his youthful immaturity that colors his perception of the world. He is unable to reconcile these complexities with his idealistic views. His black-and-white thinking represents an error in his premises, leading him to oversimplify the world.?

For decades as an educator, I’ve encountered countless observations of Mathematics and the black-and-white thinking that leads to an oversimplification and an inevitable rigid focus on “the right way to teach/learn things.”? This robs the discipline I love of the art and music I believe it represents. Inevitably, what is lost is the benefit that can come from hearing its notes.?

"Did you know that the human voice is the only pure instrument? That it has notes no other instrument has? It's like being between the keys of a piano. The notes are there, you can sing them, but they can't be found on any instrument. That's like me. I live in between this. I live in both worlds, the black and white world." ~Nina Simone

In my journey, I’ve had plenty of reasons to lose my youthful enthusiasm for the discipline and been hardened by truths I wasn’t ready to admit at times. But my youthfulness returned when I learned the humble truth (which you might call wisdom) that there are no contradictions—only false premises. Early in their learning, students tend to view mathematical concepts as fixed and rigid. This often comes from similar views modeled by parents, guardians, and teachers alike, who often have carried these views into their adult lives. Regretfully, if you are never exposed to alternative narratives, then this is mathematics for you. Just as Holden struggles to understand the perceived contradictions he sees in people, students may struggle with mathematical concepts that don’t fit neatly into the black-and-white framework they’re presented.?

We may need a concrete example to better apply this to entailment and the development of educator wisdom. A student might initially understand multiplication only as repeated addition. Memorizing these facts and the algorithms for obtaining “the correct answers.” This is akin to Holden’s view that people should be consistent and straightforward—anything beyond this narrow view is frustrating and incomprehensible. When students encounter more advanced concepts, such as multiplying fractions or understanding abstract algebraic structures, their limited view of multiplication becomes insufficient. They may react with confusion or frustration, much like Holden does when faced with the complexities of adulthood. There is no phoniness to multiplication or adulthood but rather a limited/false premise that leads to a seeming contradiction. This is a teacher skill that can be learned, a wisdom that is best generated through reflective practice while engaged in the art of teaching.

Connecting the GROWTH Framework: Observe the Implications (O)

In The Catcher in the Rye, Holden Caulfield’s resistance to growing up and his longing to protect the innocence of children is central to his character. He believes that by shielding children from the inevitable corruption of adulthood, he can preserve their purity and prevent them from experiencing the pain that he struggles to face. However, by the novel’s end, Holden experiences a profound realization when he watches his younger sister Phoebe on the carousel. He understands, perhaps for the first time, that he cannot stop the process of growing up—neither for himself nor for others. This moment of acceptance is crucial for Holden, as it allows him to release some of the fear and grief that have kept him emotionally stagnant.

Similarly, in mathematics education, students and teachers often experience fear, pain, and emotional barriers from past negative experiences with learning math. These feelings of isolation, alienation, and a lack of confidence are not unlike Holden’s struggle with growing up. These emotions create a natural and pervasive fear of mathematics, like a fear of adulthood for Holden. There is a profound contradiction in this, that needs to be labeled. The very fear that has stagnated some adults also drove them into the classroom.?

“I can be changed by what happens to me. But I refuse to be reduced by it.”~Maya Angelou

Many have recounted (Smith & Pantana, 2009) that their reason for becoming a teacher is to prevent students from experiencing the same horrors, to catch them, and to protect them from a negative learning experience. Without the ability to interrogate these premises, they can linger and propagate unintended damage to the very children they strive to protect.

The “O” in the GROWTH framework—” Observe the Implications”—provides a mechanism for overcoming this fear and interrogating the consequences of how we teach mathematics concepts. Observation in this context involves considering the broader implications of a mathematical concept, its evolution, and how it connects to other ideas. By observing the implications (the process of entailment), educators and students can move beyond their varying expertise of any concept and explore deeper connections throughout the broader mathematical structures. Entailment as a process allows us to feel safe in approaching a concept because no matter where you are with your understanding of that concept, there is always room for deeper understanding. Further, it allows teachers to interrogate the consequences of language and representation and how that may lead to student misconceptions. It’s studying mathematics for the gray, the space in between the keys, and the ability to humanize the learning process. Truly a mastery that only resides in its necessity for the work of teaching or perhaps for any of those helping young people grow in their learning.?

Through the GROWTH framework, teachers can first put on their own oxygen mask before trying to help others. This approach will help students see that mathematics is a journey that will challenge their preconceptions and force them to confront the inconsistencies they once feared. Students only learn how to do this through adult modeling. As students are empowered to observe the implications of their learning, they can begin to see that what once appeared disjointed or contradictory is, in fact, part of a more extensive, more coherent system. That system is based on ways of thinking and tools for analysis, not just getting the right answer in a single context.

Conclusion: A More Mature Understanding for Teachers and Students

Holden’s realization that he cannot stop Phoebe from growing up demonstrates a leap in maturity. So, too, can we free ourselves from the grip of our mathematical insecurities, even as adults. Mathematics is not a field to be feared but one to be observed, questioned, and embraced. The journey toward a deeper understanding of mathematics is, in many ways, a journey toward maturity– for students and educators. My own journey has made me a better friend, partner, and father—benefiting from the mistakes and the removal of many false premises. Just as Holden Caulfield learns that he cannot stop the process of growing up, teachers and students must learn that the fear and grief associated with mathematics can be overcome by observing the broader implications of the subject.?

By letting go of past pain and embracing the complexities of mathematics, educators can build healthier, more supportive learning environments where students are free to explore, question, and grow. In doing so, both teachers and students can move beyond their initial fears and engage with mathematics in a way that fosters confidence, curiosity, and lasting intellectual growth.?