In developing effective mathematics teaching, one essential voice often goes unheard—the student's. While the classroom may be filled with sound, authentic listening demands attention to the vibration of mathematical thinking and reasoning that emerges from students’ mistakes, questions, and actions. Doing this well transforms teaching from a monologue into a collaborative energy exchange, a dialogue. I know what some of you may be thinking right now—You haven’t heard the word vomit of middle schoolers all day. I assure you that I have. I have a (delightful) child of that age in my home now, and I have also spent many years of my career working with them. There is a lot of wasted oxygen there, but that is part of listening. It is part of the co-creation of learning.
“Children will listen to you after they feel listened to.” ~Jane Nelsen
Have you ever heard of Aboriginal Songlines? This rich oral tradition will guide this writing, along with understanding listening as the art and music of vibrations between people. Aboriginal Songlines (Dreaming Tracks) are integral to Indigenous Australian culture and spirituality. They are paths across the land (or sometimes the sky) that mark the routes followed by the Ancestors as they created the world. These Songlines connect places, animals, plants, and cultural stories, creating a rich web of geography, memory, and knowledge. Aboriginal Songlines embody an oral tradition of passing down knowledge, cultural values, and landscape. I think mathematics, too, can be seen as a cultural map—a modern form of storytelling and exploration. Aboriginal Songlines represent an ancient way of engaging with embodied learning, deeply connected to context and highly participatory. Bringing students into mathematics education should be seen similarly: a journey of co-creation where students are active navigators of their learning, empowered to discover and construct knowledge collaboratively.
Mistakes in learning are often feared, particularly in mathematics. Dr. Jo Boaler’s work (Boaler, 2008, 2015, 2024) has shown that this fear is a significant obstacle to student engagement. Students who develop a fixed mindset believe their abilities are static, and mistakes reflect their inherent limitations. They start to build an identity as a mathematics learner from narratives built on false premises and contradictions. However, this trajectory is not the only way—and it can be reversed. Education research (Robertson & Gerber, 2000) discusses how identity is not a static trait but constantly evolving as children engage with the world around them. With that framework, you can see how formal education plays a significant role in shaping a child's identity.
Aboriginal Songlines fascinate me in this context. They use the lyrical term kinship lineage to convey that the ancestral people are the custodians of a songline. It was their duty to uphold the obligation of passing the song on in perfect form to the next generation. This was done to educate the next generation about the people's cultural heritage and done in what was called “stages of knowledge.” The limitations of oral traditions are apparent. There are no reference documents other than art and memory—both have a shelf life. However, the ancient cultures used this opportunity of mistake as an advantage for deeper internalization of the Songline. Imagine, for instance, a Songline being passed down and a child missteps in recounting part of the movement. Rather than believing they were inherently not gifted at travel or capable of learning their heritage, the mistakes were used to spark dialogue that developed a greater understanding of the land and story. Listening in this context is traveling with a novice and reacting to the novice’s embodiment of the story in connection to the journey involving artifacts, stars, and land. This act refreshes the story for the elder and reinforces the novice's learning, entirely in line with how we see teaching as the best way to learn mathematics. When students make errors in mathematics, their missteps reveal their current conceptualization, providing an opportunity to guide them through a deeper exploration of the material and deepen the teacher’s mathematics-for-teaching “memory” as well.
“As you think, you vibrate. As you vibrate, you attract.” – Abraham Hicks
Music and songs, like the Aboriginal Songline, are often associated with vibrations. The strings on the guitar, the clarinet's air column, and the drum's head are examples of vibrating systems. These systems on most musical instruments comprise two or more vibrating systems working together to produce sounds loud enough to be heard by the human ear. To connect the metaphor, we, as educators, must recognize that these “vibrations”—systems of listening to mistakes, questions, and insights—are the vibrating systems working together so that a master teacher can hear the learning journey in our student’s minds.
A highly tuned teacher is a master listener; this is both an applied scientific and creative skillset. The science is akin to an engineer practicing vibration analysis, the diagnostic action of listening to vibrations to prevent mechanical failure. This is vital to ensuring that a mechanical system doesn’t fail and impede production. Teachers listen to these vibrations not to prevent mistakes but rather to use them to enhance learning. The creative part is cultivating the mistake into positive vibrations rather than fear, an invitation to explore and reflect.
A classroom thrives when students’ voices are valued and they are given the space to contribute meaningfully to the learning process. That learning benefits both the teacher and the students and co-creates mutual growth. To co-create, there has to be mutual regard. In the Zulu culture, they call this sawubona, which translates to “I see you.” More than politeness, the translation means I see you, and by seeing you, I bring you into being. This is part of Ubuntu, a set of closely related Bantu African-origin value systems that emphasize the interconnectedness of individuals with their surrounding societal and physical worlds. Hindu cultures use namaste, which translates to “I honor the Spirit in you, which is also in me.” Namaste is humbling. It is intended to recognize that we are all on an equal standing—all of us are children of divinity and, therefore, of one. To meet children at their level and have an equal standing in learning is to have the confidence to be humble and listen. Their unique perspectives and lived experiences bring depth and richness to the subject matter, and by listening deeply to these voices, teachers can transform their classrooms into spaces of co-creation.
“When people talk, listen completely. Most people never listen.” – Ernest Hemingway
In Zen and Buddhist traditions, listening is a mindful act rooted in presence and awareness. To listen deeply is to be fully present, tuning into the speaker's underlying context, emotions, and energy. This is the form of hearing that I’m describing for the classroom, where the goal is not simply to identify the “right” answer but to engage with the diversity of pathways, interpretations, and possibilities. Teachers practicing mindful listening are better positioned to guide students through their discoveries, modeling intellectual curiosity and non-judgmental exploration. Let’s practice together using research (Rushton, 2018) activities that provide examples of typical student mistakes:
Take a look at each of the examples provided in the figure. In each one, a student made a mistake in their methodology for solving the task they were provided. It is helpful to understand that sometimes, we make errors without being motivated by false narratives. Have you ever tripped while walking, despite having decades of experience walking? Of course you have. Beyond the probability of a casual mistake, you can interrogate student written mistakes to understand their thought processes and allow them to interrogate these responses as part of the vibration exercise.
In the mistake of question #6, what types of errors can you interpret from the student’s work without having the opportunity for dialogue with the student? They could have erred in understanding division with integers, a deeper understanding of the inverse relationship for solving equations with elementary operations, or misunderstanding the meaning of a negative sign as a coefficient to a variable expression. What other mistakes could you interpret? The more you can visualize, the less energy you will need to interpret, so you can spend that energy on listening if you have the opportunity for dialogue.
“Listening is the first step in making people feel valued. Mindful listening allows us to do more than take in people’s words; it helps us better understand the how and why of their views.”~Rebecca Z Shafir
Listening and giving value to others takes courage. According to research, teacher reluctance to use error analysis is related to teachers' fear of encountering mistakes by novice students (Rushton, 2018). The fear is related to the idea that exposure to mistakes will result in students adopting those methodologies rather than learning “the right way.” Yet, research suggests that this is not the case and that students retain information longer when mistakes are part of the learning process. Listening and learning from mistakes is part of learning. As I’ve shared, the culture of First Nations and Songlines chose to learn this way to respect the ongoing cycle of engagement, mistake, and rediscovery without judgment. In Zen and Buddhist traditions, listening is a practice of mindfulness and presence and respecting the energy involved between people. In the classroom, when students contribute ideas, whether correct or mistaken, they contribute to the shared journey and energy of mathematical discovery. The teacher validates these contributions and supports a collective meaning-making process by listening deeply and responding with thoughtful questions. Even better, the teacher models the intellectual way of being by handling mistakes without judgment.
Let’s try what it feels like to listen without judgment. Here is a number:
If I were to ask you, how many ones are in this number? You may reach out to your working understanding of place value and answer zero, which I have often observed as being accepted as correct in teachers' classes. But, I observed a young man in one of my teacher’s classes answer 400. His classmates laughed at him, which isn’t out of the realm of possible in a room full of 8- and 9-year-olds. Terrifying. The teacher responded by saying that wasn’t correct and then asked a different question to reword the same question. How many ones are in the ones place? The student said zero, and the class moved on. The reality is that the number 400 does have 400 ones. That is what that number means, and that flexibility in understanding numbers is advantageous. But getting laughed at for a perceived mistake introduced social anxiety and fear that could be circumvented in a classroom of mindful listening. Without correcting the human nature of criticism for mistakes, we lack the connection of namaste, sawubona, and Ubuntu that can be part of the culture of mutual energy exchange in learning.
Teachers act as models of intellectual behavior by demonstrating how to grapple with complexity and engage in thoughtful dialogue. The (H) in GROWTH—Hear Your Students—is the pedagogical problem-solving of a mathematics teacher that “aims to capitalize on the interpretive potentials that arise on the collective level when individual expertise is drawn together around perplexing problems” (Davis, 2012, p. 15). A master teacher, like a custodian of a Songline, does not simply transmit knowledge but navigates alongside the learner, modeling how to question, explore, and build understanding through collective resilience. This involves the value system of Ubuntu, which we train as part of the GROWTH framework. This models courage and understanding that we as human societies benefit more from truly seeing each other and our matching energy, than taking that energy away through judgment and criticism. Here is an excerpt from a speech by President Teddy Roosevelt in 1910 in Paris that I reach back to again and again for inspiration:
“It is not the critic who counts; not the man who points out how the strong man stumbles, or where the doer of deeds could have done them better. The credit belongs to the man who is actually in the arena, whose face is marred by dust and sweat and blood; who strives valiantly; who errs, who comes short again and again, because there is no effort without error and shortcoming; but who does actually strive to do the deeds; who knows the great enthusiasms, the great devotions; who spends himself in a worthy cause; who at the best knows in the end the triumph of high achievement, and who at the worst, if he fails, at least fails while daring greatly, so that his place shall never be with those cold and timid souls who neither know victory nor defeat.”
When a teacher pauses to listen to a student's reasoning, he or she conveys that learning is not about getting it right the first time but diving into the "how" and "why" of a mathematical task. They model courage and work from a namaste framework instead of being critical and judgmental. Criticism requires little empathy, understanding, courage, or intellectual way of being—not a Songline that we want to persist for generations.
I’ve covered a lot of ancient human ways of operating that span culture, time, race, and civilizations. These cultural ways of being represent a conclusion from the Songline of humanity: our existence needs to recognize that we each exist because others exist. As a teacher, practicing mindful listening and tuning into the vibration of learning in the classroom is part of teaching humanity’s cultural heritage. Teachers, in one way or another, are the custodians of this heritage as they are often the elders in our society who have the most influence on our children outside of the home. A learning environment that greets learners with “sawubona” or “I see you” is more than politeness. It is courage that we each can recognize the vibration of another person's worth and dignity, saying, “I see your experiences, passions, pain, strengths, weaknesses, and future.” The vibrations of that future have the potential to produce a beautiful harmony.