Although humans are not the only animals that communicate among themselves, we are likely the only ones that use language as a tool for thinking—at least as far as we can tell. The use of complex language is viewed as a hallmark of being human—a marvel of human evolution. It has been proposed that language gives rise to the ability for abstract thought by providing a medium for alternative representations (Bloom & Keil, 2001). And so, abstract thought is fundamental to mathematical thinking.
Language: A Tool for Making Sense and Thinking
Once we acquire language, we can talk to ourselves. The most obvious example is engaging in cognitive tasks that are language-dependent, such as imagining how someone else might make sense of a shared experience. Key to our survival as a species, our ability for “theory of mind” is supported by language. The more we hear others talk, the better we can guess what they are thinking.
Furthermore, causal reasoning and social cognition require the support of language. Of course, non-linguistic creatures are also capable of understanding causal relationships; for example, dogs are notorious for jumping up and down in excitement when their owner takes the leash or even walks in the direction of the leash, as they understand, “we are going out.” However, it seems that only humans can distinguish between true and false intentions in others. And this superpower is supported by language. It is believed that this uniquely human ability to recognize lies and false pretenses is only possible through encoding situations into language (Bloom & Keil, 2001).
In the domain of mathematics, the idea that language is a tool for making sense and thinking, holds significant merit. Dehaene (1999) argued that the ability to reason about larger quantities is impossible without some form of language. For instance, reasoning that if you remove two objects from twenty, eighteen will remain, would not be possible in the absence of some form of human language. The relationship between mathematical thinking and language is further supported by studies of children who have suffered brain injuries in certain areas of the brain that process language, and subsequently develop difficulties with mathematical calculations (Dehaene, 1999).
Language also plays an important role in memory formation. Particular to learning mathematics, it is the ability to make connections between mathematical ideas (i.e., conceptual understanding). In other words, integrating newly-acquired mathematical knowledge with prior and existing knowledge is the product of narrative structures which are impossible to imagine outside language (Fivush & Schwarzmueller, 1999).
Various research findings challenge the view that mathematics relies less on language compared to other school subjects. For decades, some educators perceived mathematics as being less language-dependent than other subjects, believing it could be learned with minimal attention to language. This perception is fueled by teaching approaches that prioritized memorizing and executing procedures with little emphasis on understanding (Moschkovich, 2010, 74)
Today, in my experience, remnants of this view persist, though they are held by a diminishing minority of math educators. The idea that communication and mathematical discourse are essential components of understanding complex mathematical concepts is now widely established and accepted. It is explicitly stated in state standards across the nation and embedded in the most popular curriculum materials. While this represents a significant step forward, there is still work to be done – a shared vision is needed.
Academic Language: A Definition and an Understanding
Academic language can be defined as the language of academic settings—specifically, the language used in schools to study and mediate the disciplines. We contrast academic language with ‘everyday language’, the language of daily life. It is a common mistake to think of academic language as being superior to everyday language (Uccelli & Phillips Galloway, 2017).
We recognize that mathematical ideas can be complex, abstract, and difficult to relate to, as they are sometimes detached from routine experiences. However, it is important to remember that life itself can also be very challenging. Navigating the complexities of human social and hierarchical structures can be an overwhelming experience, and we all use language to help us navigate our daily lives. There is no doubt that everyday language is both useful and cognitively demanding.
Nevertheless, there is a difference between academic and everyday language—not necessarily in complexity, but in vocabulary, syntax, pragmatics, and other language features. It is precisely because it is different that, as math educators, we need to own the responsibility of teaching and embracing mathematical language—not because it is superior, but because it is essential in the math classroom.
The necessity to support learners in developing the mathematical register becomes evident when we consider the abstract nature of its discipline-specific vocabulary (e.g., proportionality, function, probability), which math teachers tend to address. However, it becomes even more evident when we consider how such vocabulary functions within the ecosystem of mathematical discourse (e.g., infer, explain, justify), an area that, in my experience, math teachers tend to pay less attention to. Mathematical language has its own features, such as oral and written discourse, along with pragmatics, syntax, and others. Math teachers hold a unique vantage point for teaching the language of their discipline.
Discourse To Develop Understanding and Agency
The use of discourse as a means to understand complex mathematical ideas is well established within the community of math educators and supported by extensive research. Students enter the math classroom equipped with resources rooted in “everyday language,” which they rely on to make sense of the mathematics they encounter (Sfard, 2008). In addition to linguistic resources, learners draw upon their common sense and prior experiences about how the world works, as expressed through gestures, body postures, kinesthetic actions, artifacts, and signs in general (Radford & Barwell, 2016, p. 289).
The genesis of conceptual understanding, using all the sense-making resources available to the learner, becomes a foundation for continued cognitive growth. These abilities are then leveraged to identify patterns, recognize rules, and make predictions. Math teachers are likely to notice emerging skills that manifest in the development of new linguistic resources, bridging everyday language and academic mathematical discourse. These resources arise from the need for more advanced tools to think about and communicate increasingly complex ideas.
Discourse in the math classroom does more than facilitate understanding. As learners develop additional linguistic resources, they also cultivate a voice and a broader range of choices. This emerging agency becomes evident in how communication unfolds in the classroom, revealing power dynamics through who is doing the talking and the thinking. Is it the teacher? The learner? Is the teacher a thought partner to the learners? Is learning mathematics a shared experience, or is it primarily imparted by the teacher? These nuances and power structures are made visible through the discourse in the math classroom.
A Vision
The challenge of underperformance in mathematics, particularly among Black and Brown students experiencing poverty, can be alleviated through the gradual transfer of voice and choice from the teacher to the learner—since only the learner can author their own sense-making. Creating an environment that nurtures agency in the learning process is likely to foster lifelong skills in thinking and reasoning through mathematical discourse. Therefore, the goal is not discourse solely for the proper transmission of knowledge, values, and norms from teacher to student. Instead, it is about discourse that engages learners, enabling them to construct new meanings and understandings of mathematics for themselves (Chapman, 2009).
We propose a vision of mathematical discourse that elicits learners’ ideas and creates a space for peer-to-peer interactions, transcending hierarchical structures in the math classroom. This vision positions discourse as a tool for empowerment, enabling learners not only to answer questions but also to gain the confidence to formulate them. It is a vision where discourse is used to build a shared understanding of complex mathematical ideas and the ways we reason with them. This is not merely the vision of this author; it is a value shared by many (National Council of Teachers of Mathematics, 2024).
References
Bloom, P., & Keil, F. C. (2001). Thinking Through Language. Mind & Language, 16(4), 351-367.
Chapman, O. (2009, January). Discourse to Empower “self” in the Learning of Mathematics. Proceedings of PME, 33(2), 297-304.
Dehaene, S. (1999). The Number Sense: How the Mind Creates Mathematics. Oxford University Press.
Fivush, R., & Schwarzmueller, A. (1999, January 6). Children remember childhood: implications for childhood amnesia. Applied Cognitive Psychology, 12(5), 455-473.
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