Transformational learning not only involves developing novel meanings, but it also entails coming to see those meanings are more productive and generative than prior meanings.

## Inspiration

When working with teachers, whether prospective or practicing, we must face the insights gained from research on neuroplasticity and the habitualization of meanings/ways of thinking.

## Description

Place yourself in the following situation: in working with a teacher, you intend to support their developing of a particular meaning, *M1*. You determine that he holds meaning *M2* and meaning *M1* and *M2* are disparate; *M2* is not a foundational way of thinking for *M1* and, in fact, can inhibit his construction of and ability to teach for *M1*. This raises the question: how do we engage with the teacher in a way that both honors *M2 *and affords constructing *M1*?

There are numerous ways to approach this problem. Traditional approaches involve using tasks in which *M2* is not relevant, but *M1* is. Similarly, one might use tasks that target *M1* through focused, closed-ended questions and design. Such tasks are useful to engender *M1* and possibly draw connections with *M2*. Yet, such tasks can be so contrived as to feel too disjoint from the classroom for teachers. Or, such tasks leave a teacher perplexed as to the implications of *M1* for their instruction in comparison to *M2*. Relatedly, such tasks only afford *M1* and do not challenge *M2* so that the former becomes their predominant or habitual meaning. With respect to teachers, for the tasks they envision teaching, *M2* remains the predominant meaning because it is more familiar and "does work". A solution to this issue is to simultaneously target the two meanings in order to draw them into competition with each other. Hence, the name competing meanings*.*

In my work with teachers I often want to draw a rate of change meaning, *M1*, into competition with a shape-based, slope as direction or movement meaning, *M2*. In our work, we have found that *M2* does not act as a sufficient springboard for the development and habitualization of *M1*. Said another way, *M2* as derivative of or subordinate to *M1* is more productive and generative than *M1* as derivative of or subordinate to *M2*.

The two graphs presented above provide the source material for a task I've used to draw meanings into competition with each other. Here, I acknowledge that the competing meanings process above is typically cyclical and iterative across a number of tasks, but for the purpose of this study guide we'll keep things linear and simplified. The task is presented in two parts. The left graph is first provided, but *without *the axes-labels “*x*” and “*y*”. The teacher is told a student provided the graph (without labels) as a solution to graphing “*y *= 3*x*”, and we ask them to consider how the student might have been thinking. After the teacher has exhausted the number of ways a student might have been thinking, they are presented the same graph but with variable labels. They are told a student clarified their solution and they are asked to comment on the graph, its accurateness, and how they would respond as a teacher. The right graph is created by most teachers during one of the phases. They create it by rotating the given graph to horizontally orient *x*.

The task incorporates the competing meanings perspective by using the following principles: (a) it sensibly affords *M1 *and *M2*; (b) the use of *M2* likely produces a perturbation, but is still viewed as relevant to the task; (c) if *M2* engenders a perturbation, *M1* is a likely meaning for the student to use or develop; (d) the use of *M1* will reconcile a perturbation stemming from *M2*; (e) due to different outcomes, the teacher can reflectively compare the affordances and constraints of each meaning; and, critically, (f) the teacher is likely to perceive the task and each as relevant to their instruction. Relating these task features to the competing meanings figure above, (a) and (b) involve problematizing an extant meaning; (a), (c), and (d) involve an accommodation via the enactment of an alternative meaning; and (b), (d), (e), and (f) support reflecting on and comparing extant and alternative meanings. The *competing *aspect of competing meanings is also apparent due to *M1* and *M2* resulting in different conclusions. With *M2*, the solution and its rotated version are inaccurate (e.g., the “slope” is wrong in its rise-run or direction). With *M1*, both are accurate (e.g., each is the set of points so that *y *is three times as large as *x* and for any change in *x*, *y* changes by three times that amount). This in-the-moment incompatibility is key to considering which meaning is better viewed as derivative of the other (e.g., slope as an implication of rate of change is more generative and generalizable than rate of change as an implication of slope).

At its foundation, competing meanings is about aiding teachers in separating what is critical to a mathematical idea from what is merely a product of representational practices and conventions. This requires their construction of meanings that enable them to do so. They must be confronted in a way that prompts them to critically analyze meanings in terms of their similarities, differences, affordances, and constraints.

… even though it looks like a negative slope…we call it slope because it’s visual and it’s easy to visualize a negative and positive slope. But that’s only visual on our conventions of how we set it up…slope is rate of change, we can still see that for like equal increases ofxwe have an equal increase ofyof three. And so for equal positive increase of one we have an equal positive increase of three. And so, it is a positive slope.

It’s smart [of a student] to understand that it’s not glued.

Oh. It’s clever. We have a clever kid over here. OK, so it now technically isyequals threex…it’s just not the standard way of doing it…They see the relationship betweenxandy.

## Further Reading

Forthcoming.

Moore, K. C., Silverman, J., Paoletti, T., Liss, D., & Musgrave, S. (2019). Conventions, habits, and U.S. teachers’ meanings for graphs. *The Journal of Mathematical Behavior, 53*, 179-195. https://doi.org/10.1016/j.jmathb.2018.08.002

Moore, K. C., Stevens, I. E., Paoletti, T., Hobson, N. L. F., & Liang, B. (2019). Pre-service teachers’ figurative and operative graphing actions.* The Journal of Mathematical Behavior, 56*. https://doi.org/10.1016/j.jmathb.2019.01.008

Moore, K. C., Liang, B., Stevens, I. E., Tasova, H. I., & Paoletti, T. (2022). Abstracted Quantitative Structures: Using Quantitative Reasoning to Define Concept Construction. In G. Karagöz Akar, İ. Ö. Zembat, S. Arslan, & P. W. Thompson (Eds.), *Quantitative Reasoning in Mathematics and Science Education* (pp. 35-69). Springer International Publishing. https://doi.org/10.1007/978-3-031-14553-7_3

Moore, K. C., Stevens, I. E., Tasova, H. I., & Liang, B. (2024). Operationalizing figurative and operative framings of thought. In P. C. Dawkins, A. J. Hackenberg, & A. Norton (Eds.), *Piaget’s Genetic Epistemology in Mathematics Education Research*. Springer, Cham. https://doi.org/https://doi.org/10.1007/978-3-031-47386-9_4

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