None of us were born knowing math concepts. Each of us needs to grow knowledge structures for ourselves, in our own minds. How does that happen? Our senses allow us to absorb information and knowledge from the world around us. The more fully we can engage our senses with a topic the more deeply and fully we absorb it.
The activities on this website are based on an understanding of how we grow mental images for mathematics. The images that come to the forefront of our minds as we engage with mathematics are grown over time. They contain portions of our experiences from the past, intertwined with the present, and affecting how we will view and understand things in the future.
Much of what we know as adults we take for granted, not recognizing the multiple experiences that allowed us to grow this understanding. It takes time to grow images. Through interactions with our environment, our images begin to grow. Over time we gradually transition to no longer needing the interaction, but instead, we are able to mentally manipulate these images in our minds to aid in thinking and understanding. Often it is productive to go back and forth between the stages of Image Making and Image Having before the mental ideas become solid in our minds.
We grow images through our senses. Spatial reasoning type activities are very rich in sensory input. And as has been well established in research, spatial reasoning and mathematical thinking are intimately linked. An important feature of spatial reasoning is action based activities, this can include drawing, moving objects, making gestures and/or using your imagination.
My understanding of the nature of images and growth of mathematical understanding
I am deliberate in using the term growing, both because it is a dynamic process that is in a constant state of change, but also because building does not fit my understanding of what is occurring in our minds. My understanding of images is more like an organic entity; the term building does not seem to fit. Building seems to suggest that we take in information in chunks — taking a separate piece of material and attaching it to another separate piece. They do not meld into one another and become one, but remain as attached distinct pieces, and the growth that has occurred is only that which is in proportion to the size of the piece added.
I believe our images to be more organic and in constant motion. Our images stretch out to absorb new perceptions, which stimulates a growth that is not necessarily in proportion with the new material that is received. The perception may stimulate a growth well beyond the mass that has been absorbed. This growth may cause the image to attach or be intertwined with another image that has already been established, causing a massive growth. Or alternatively, this perception may just float to the base of the structure only to help in solidifying or thickening the foundation. This is demonstrated by student behaviour as they will sometimes continue on with what seems like minimal growth then all of a sudden have a massive leap in their understanding. Growth in mathematical understanding is not as we might think — a linear sequential process, but rather a complex dynamic act of continual reforming and often sporadic growth.
Pirie-Kieren Dynamical Theory.
It is this organic process of growth within images that is foundational to the Pirie-Kieren (1994) Dynamical Theory for the Growth in Mathematical Understanding. Pirie and Kieren (1994) have developed a model for describing this growth of mathematical understanding as a “whole, dynamic, leveled but non-linear, transcendently recursive process” (p. 166). It involves moving back and forth between different ways of knowing, represented within the model below as a series of nested layers or levels.