Beyond Display: From Dynamic Representation to Generative Representational Activity with Large Language Models

Mathematics has always been a language of pictures, symbols, and stories. For centuries, students learned to translate a problem written in words into a drawing, or a drawing into a set of numbers, and back again. These translations were hard work, requiring the mind to hold different versions of the same idea in place at once. In recent decades, computers changed this landscape. They introduced tools that could draw graphs instantly or solve equations without a human hand moving a pencil. These tools did not invent new kinds of math, but they changed how people interacted with the old ones. They made it possible to see how a shape changes when you pull on a corner, or how a graph shifts when you tweak a number, creating a dynamic relationship between the user and the math. Now, a new kind of computer program, known as a large language model, has entered the classroom. These are the systems that can hold a conversation, answer questions, and write text that looks like it was written by a person. The big question for educators and researchers is whether these talking machines are just another tool for doing math, or if they have fundamentally changed the nature of mathematical representation itself.

A researcher at East Carolina University, Kwaku Adu-Gyamfi, set out to answer this question by looking closely at what these new machines actually do. The paper argues that while large language models are powerful, they do not create a new type of mathematical object. You cannot ask a computer to draw a graph, and the result is not a new kind of graph; it is still a graph. Similarly, an equation generated by a machine is not a new kind of equation. The real change lies in how the math is specified and how the work is divided between the human and the machine. The study suggests that we should stop thinking of these models as simply the next step in a line of dynamic tools, like graphing calculators or geometry software. Instead, they introduce a distinct way of working called "generative representational activity." This is a mouthful, but in plain terms, it means a process where a person describes a mathematical idea in ordinary conversation, the computer builds a candidate version of that idea, and then the person can talk about that version to change it, refine it, or ask for a different one, all without ever having to learn a special code or command language.

The paper carefully rules out several ideas that might seem obvious but are not quite right. It is not new for computers to generate answers from natural language; systems that have existed for over a decade can take a sentence like "show me the growth of a population" and produce a chart. It is also not new for computers to create mathematical objects; graphing calculators and geometry programs have been doing that for years. What is new is the specific combination of features found in these conversational environments. In the past, if a student wanted to change a graph, they often had to go back to the original command or use a slider to adjust a number. With a large language model, the student can simply say, "keep the starting point the same, but make the second line go down instead of up," and the system will produce a new version based on that description. The key is that the conversation itself becomes the tool for specifying the math. The student does not need to know the exact formal rules to make the change; they just need to describe the relationship they want to see.

However, the paper warns that this ease of generation comes with a hidden trap. When a computer produces a perfect-looking graph or a correct-looking table in response to a conversation, it can be very tempting to assume the student understands the math behind it. The researcher argues that this is a dangerous assumption. Just because the machine successfully built the object does not mean the student coordinated the different parts of the idea in their own mind. In a traditional setting, if a student draws a graph and then writes the matching equation, the act of drawing proves they connected the two ideas. In a generative setting, the machine does the connecting. The student might have asked for the right thing, but they might not understand why the machine's answer is right, or they might not notice if the machine made a subtle mistake. The paper points out that a correct-looking product is not proof of understanding. It is merely proof that the machine followed instructions.

To illustrate this, the paper looks at how students might work with a geometry problem. Imagine a student asking a computer to create a specific shape, like a four-sided figure with certain properties. In a traditional dynamic geometry program, the student would build the shape by dragging points, and the computer would ensure the shape stayed true to the rules, like keeping opposite sides parallel. If the student dragged a point and the shape broke, they would know they had violated a rule. With a large language model, the student might describe the shape in words, and the computer would generate a picture. If the computer makes a mistake and the sides aren't actually parallel, the student might not notice unless they are actively checking the math. The machine has done the work of construction, but the student has not necessarily done the work of understanding the constraints. The paper suggests that the real learning happens when the student has to look at the machine's output, check if it matches their mental image, and then describe how to fix it if it is wrong.

The research also highlights that this new way of working changes what counts as a skill. In the past, a student's ability to draw a graph or solve an equation was a direct measure of their skill. Now, if a student can get the right answer by talking to a computer, we cannot be sure if they learned the skill or just learned how to ask the right questions. The paper proposes that we need to look for different evidence of learning. We need to see if students can explain why the computer's answer is correct, or if they can spot when the computer is wrong. We need to see if they can take a machine-generated example and use it to understand a deeper rule. The ability to prompt a computer is a new skill, but it is not the same as the ability to do the math. The paper argues that the most important part of the process is no longer the production of the answer, but the coordination of the ideas. The student must still be the one to hold the different versions of the math together in their head and verify that they make sense.

Ultimately, the paper concludes that large language models are not a magic wand that solves the problems of math education, nor are they a threat that replaces the need for human understanding. They are a new kind of environment that shifts the balance of work. They take over the heavy lifting of drawing and calculating, freeing up the human mind to focus on describing, critiquing, and refining ideas. But this shift requires a new kind of vigilance. Teachers and students must be careful not to confuse a machine's ability to generate a correct answer with a human's ability to understand the math. The paper suggests that the future of math education with these tools depends on keeping the focus on the relationships between ideas, rather than just the final product. The goal is to use the machine to generate possibilities, but to keep the human in the loop as the one who decides what is true, what is consistent, and what is worth learning. The machine can build the house, but the human must still understand the blueprint.