Reflecting on the Foundations of Embodied Learning (Nathan, M. 2021)

Summary: In these pages, Nathan (2021) reflects on the present realities of classroom education, aiming to describe “a coherent, evidence-based framework for how people actually learn” (p. 4). Critiquing the restrictiveness of classrooms, Nathan (2021) describes how students naturally use their bodies to act out early algebraic equations. He draws on numerous metaphors to illustrate how embodied learning permeates everyday educational experiences. One notable example is the use of a balance in algebra, which teachers often employ to depict relationships between sets of quantities. Nathan (2021) identifies two major obstacles to improving education: misguided educational practices and policies, and the scientific fields that inform these practices and policies (p. 6). He rejects the promotion of dualism, or a dichotomous view of education that separates the mind and body.

Later, Nathan (2021) hones in on mathematics education, drawing on Lakoff and Núñez’s (2000) proposition that mathematical ideas rely primarily on a small number of grounding metaphors (p. 147). One example of a grounding metaphor is the relationship between quantities and collecting objects; for instance, the number 2 is internalized through lived experiences of encountering pairs or sets of two objects. Another commonly referenced type is the conceptual metaphor, such as the number line, where numbers placed to the left and right represent locations along a path. The balance metaphor mentioned earlier is also a conceptual metaphor. Nathan (2021) suggests that embodied learning experiences can foster mathematical reasoning through grounding-as-connection making (p. 151).

1) One moment that particularly stopped me was on page 6, where Nathan (2021) describes how children invent and correctly apply iterative guess-and-check methods to work through algebraic constraints. These intuitive methods, however, are often internalized by students as non-legitimate mathematics and deemed unacceptable in school settings. This reminded me of our class discussion and what Kabula shared from her teaching experiences. We discussed how mathematics can feel intuitive at younger grades and early secondary levels, where students are able to “feel” and correctly guess solutions to relatively simple algebraic equations (for example, one- or two-step equations). As students progress, however, the operations become more complex. Kabula noted that with topics such as exponents, students can no longer rely solely on earlier intuitive methods. She also spoke about how she would “plant the seed” of mathematical operations so that students understand there are underlying steps and procedures behind solving algebraic problems.

What stood out most to me was Nathan’s (2021) observation that students internalize the idea that sensible, action-based methods are not legitimate mathematics (p. 6). Reflecting on our class conversations and my own teaching practice, I began to question whether I, too, regard these intuitive approaches as non-mathematical. On one hand, as mathematics becomes increasingly abstract—such as in calculus or university-level mathematics—intuitive methods alone may no longer suffice. On the other hand, most of my students will not be engaging with advanced mathematics regularly. Instead, they will rely on intuitive mathematics in everyday contexts, such as moving a decimal point to calculate 10% when tipping.

Question 1: This leads me to wonder: until what point are intuitive methods considered non-mathematical, and what ultimately determines whether a method is deemed mathematical or not? Whose definitions are we relying on when making these assumptions and categorizations? Are considerations of daily-life applicability taken into account, or are these judgments grounded in deeply rooted Eurocentric definitions of mathematics?

1) Another moment that gave me pause was on page 149, where Nathan (2021) discusses the mental number line. He explains how the conceptual metaphor of numbers as places along a path is often learned at home and within the community. This metaphor may be shared collectively within a culture or taught explicitly; however, this is not necessarily the case for children from different cultural and community backgrounds. Nathan (2021) draws on a case involving Portuguese immigrant children in Toronto who struggled with basic arithmetic because they lacked foundational arithmetic metaphors, including the number line (pp. 148–149). These children had not been exposed to the number line metaphor, whereas many Western cultures reinforce it through games and everyday activities in the home, supporting its internalization from an early age.

From my prior experiences, I strongly embraced the idea that mathematics is a universal language and leaned on this belief as part of my personal motivation to enter this profession. Reading this section, however, made me reflect on how many mathematical metaphors I take for granted as universal. The number line, in particular, felt like a shared experience—something I assumed every student simply “knew.” Yet, when I consider Indigenous ways of knowing and other non-Western epistemologies, this assumption begins to unravel. Who is to say that a number line must be linear at all? Why could it not be circular, rectangular, or even three-dimensional?

This realization has important implications for my teaching practice. If mathematical understanding is grounded in culturally mediated metaphors, then teaching mathematics cannot assume a single, universal starting point. Rather than using the number line as a neutral or obvious assumption, perhaps it would be more meaningful to ask students how they visualize numbers. By doing so, I can work on validating alternative modes of representation that emerge from students' lived experiences. I imagine an activity around this would be having students draw and represent numbers spatially in various ways, ideally revealing spatial relationships that are important the students' cultural contexts. 

This re-orientation challenges deficit-based interpretations of mathematical abilities. When students struggle with concepts such as arithmetic or number sense, the issue may not be a lack of ability, but a mismatch between the metaphors privileged in school mathematics and those developed at home or in the community. Recognizing this shifts my role as an educator from correcting misunderstandings to expanding students’ repertoires of mathematical meaning-making. In doing so, mathematics classrooms can become spaces that honor diverse ways of knowing while still supporting students in navigating dominant mathematical representations they will encounter in schooling and beyond. 

Question 2: What other deeply rooted mathematical methods are assumed to be universal, and how might these assumptions shape whose ways of knowing are valued in mathematics classrooms? How might our classrooms look if we treated representations like the number line not as universal truths, but as culturally situated metaphors? Honestly, I find this possibility unsettling! Would it be more difficult to teach mathematics without enforcing a single, standardized foundational metaphor? This tension destabilizes my deeply internalized assumptions about the role of the teacher and challenges my sense of control within the classroom.