This section of the chapter views mathematical understanding from an **enactive** perspective. I focused in on 2 passages of the reading: the first regards mathematical thinking as a form of communication:

“…our reasons for communicating are not to establish mathematical objects but rather, mathematical objects are necessary for the conceptual development of our own internal thinking and when interacting with others…we do not start with mathematical objects and then communicate, we communicate and through this dialogic process, mathematical objects come into being” (Sfard, 2004 in Thom, pg. 125)

While the second views mathematical understanding as growth:

“…learners are autopoietic beings who determine what phenomena will be experienced as perturbatory and who specify the ways in which they structure their mathematical thinking…mathematical understanding…comes to be through the structural changes within, between, and among learners and their environments.” (Kieren and Pirie, 1992; in Thom, pg. 129)

The first quote makes me uncomfortable. I can reflect on my past experiences and understand that I saw images in my head when thinking about math, and then communicated what these images “were” to me. I have difficulty grasping Sfard’s concept that we communicate then form objects…I think that we see a problem, and envision it in our minds, and then communicate to solve the problem, or understand the problem, or seek further explanation of the problem.

The second quote, although wordy, makes much more sense to me. Kieren and Pirie’s model of mathematical understanding on page 128 of *Re-rooting* follows with my thinking process as I was trying to understand and sort through Sfard’s theory! In Kieren and Pirie’s model, mathematical growth is structured as follows (primitive knowing as the basis):

- Primitive knowing
- Image Making
- Image Having
- Property Noticing
- Formalising
- Observing
- Structuring
- Inventising

This model is dynamic, and allows for understanding to occur on multiple levels at once; as it incorporates “folding back” to allow for deeper understanding. I really appreciate this model, and will use it when planning activities in my future classroom. I also like how this model seems to be in line with how Jennifer has been teaching our class all year: mathematical understanding develops through interactions with, between and among learners and their respective environments.

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