The GROWTH Framework™ for Math Teacher Development

“What is this?” Realizations are everything that gives meaning to a mathematical concept for a learner.  The term realizations is borrowed from Sfard (2008) and is defined as the associations that a learner (teacher and student) might draw on and connect to make sense of a mathematical concept (Davis & Renert, 2014). 

Possible realizations for a mathematical concept include formal definitions, algorithmic knowledge, metaphors, images, computational applications, and physical gestures. As evidenced in other concept study research (Berkopes, 2014; Davis, 2011, 2012; Davis & Renert, 2009, 2012, 2014), teachers investigating their realizations for mathematical concepts provide an environment where the development of their knowledge of the concept is possible. 

Simply asking what subtraction is is an example of a realization technique. An individual’s realizations of mathematical concepts evolve and can be shared by many or are unique to an individual in a collaborative group. It is important to note “the assertion and assumption here is not that any particular realization is right, wrong, adequate, or insufficient” but rather that the process of making realizations will allow for further development of what teachers know about the mathematical concept under development (Davis & Renert, 2013, p. 253). As teachers progress towards becoming master teachers, their realizations of mathematical concepts should change as a reflection of their growing sophistication. 

“Where do I see this and why?” Landscapes are organizing and comparing lists of realizations for a particular mathematical concept to determine where and why they emerge. They are the studying of mathematical concepts as they evolve based on context and application for students. This type of activity for teachers produces a landscape, defined as a macro-level view (Berkopes, 2014; Davis, 2011, 2012; Davis & Renert, 2014) of a mathematical concept. 

Davis and colleagues noted a difference in the utility of various realizations for a mathematical concept. Some realizations for a mathematical concept remain viable in most mathematical contexts where teachers encounter the concept, while other realizations are relatively situation-specific or learner-specific. For example, the realization of division as “repeated subtraction” varies in viability depending on the number sets to which it is applied. 

Other researchers (Shulman, 1986; Ball, Thames, Hill, 2008) refer to this as curricular or horizon knowledge– the scope and sequence of why and when certain mathematical concepts arise when working with children.  Mathematical concepts are not introduced at certain times in the K-12 setting by tradition but rather when concepts “hold together and fall apart in different contexts and circumstances” (Davis & Renert, 2014, p. 43). Studying a mathematical concept through the lens of when it is viable and when it is not– builds a robust understanding of the concept. Through modeling and shared experience, this work also develops the mathematical habits of mind (MHM) that embodied mathematics requires. 

“What is the consequence of my realization?”  “Entailments” are the tracking and scrutinizing the consequences of any one realization of a mathematical concept. Davis and Renert (2014) described each realization of a mathematical concept as carrying a series of nested consequences due to the nested and axiomatic system that builds contemporary mathematical curriculum. Davis and Renert also described the entailments emphasis as a process where teachers gain access to fresh and innovative approaches to a concept, enabling them to move through cycles toward a master educator. 

MTI provides this through a series of mathematical topics followed by a shared open interrogation of that topic in various contexts. These are not mathematics lessons, nor are they lectures. The goal is to model the productive struggle to interrogate something that one thought was well defined by carefully providing contradictions to assumed facts. Care must be placed on the uncomfortable feeling that arises from this pursuit, and this feeling is meant to build empathy for students learning mathematics. 

“How are these used in combination?” The activity of creating conceptual blends and collapsing diverse realizations of a mathematical concept is defined as “blends” (Davis & Renert, 2014).

The first three principles aim to develop distinctions between realizations and the consequences of those realizations for a mathematical concept. Put simply, the first three realizations are about broadening definitions of mathematics and the teaching of mathematics. 

The blending emphasis is categorically different: teachers are asked to seek out “meta-level coherences by exploring the deep connections among identified realizations and assembling those realizations into a more encompassing interpretation” (Davis, 2012, p. 12). For example, the realization of fraction multiplication as “parts of parts” could be blended with the realization of fraction multiplication as the “shrinking of a number line.” The blends of these realizations focus the collaborative effort on defining the deep connections between the two realizations, resulting in potentially new emergent possibilities for the concept. Davis and Renert (2014) describe blends as a deliberate shift in emphasis from “multiple (and potentially disjointed) meanings toward coherent and encompassing definitions” (p. 71). 

“How do I co-create with peers?”  “Participation” is defined as the planned effort to engage other teachers with the post-formalist orientation to the discipline of mathematics.

If MathTrack Institute is to scale our approach to quality mathematics instruction, then our educators, parents, and students need positive experiences with mathematics as a culturally created body of knowledge. Teachers are “vital participants in the creation of mathematics, principally through the selection of and preferential emphasis given to particular interpretations over others” (Davis & Renert, 2013, p. 251). 

MathTrack’s programs of study are designed to provide the opportunity for educators to develop an awareness of the complex nature of realizations for even elementary mathematical concepts. Teachers, parents, and students need positive experiences with mathematics and understanding that no “realization is right, wrong, adequate, or insufficient” (Davis & Renert, 2013, p. 253). 

“Why do I need this?” “Pedagogical Problem Solving” is an explicit practical link between the concept study environments and the learning environments of the teacher, parent, or peer. Every parent or teacher has encountered a student of mathematics asking—why? Such questions concern the nature of the mathematics students and teachers co-create in the classroom or parent and child co-create in the home. The question of why is a constant part of creating new mathematics with novice students and is the best part of teaching mathematics, but can be the most frustrating for unprepared educators. 

Experiencing this is especially important because MathTrack Institute frames mathematics as a product of human beings and is shaped by our brains and conceptual systems. Often, these questions—for example, “Is a fraction a number?”—are neither innocuous nor straightforward. The various realizations for fractions shared by Lamon (2007), among others, show that fractions can be considered numbers dependent upon the mathematical context. 

Pedagogical problem solving “aims to capitalize on the interpretive potentials that arise on the collective level when individual expertise is drawn together around perplexing problems” of teaching mathematics (Davis, 2012, p. 15). With this emphasis, brief activities during residency or apprenticeship requirements can be tried and experimented with by the teacher, parent, or peer student engaged in teaching mathematics in diverse ways.

MathTrack’s GROWTH Framework™ combines two distinct and significant research paradigms for training mathematics teachers - Concept Analysis and Lesson Study -  situated within a theory for learning and cognition called Enactivism.  Following the publication of The Embodied Mind (Varela, Thompson, & Rosch, 1991), enactivism grew in awareness as a theory of learning in mathematics education research (Ernest, 2006).  This theory positions mathematics as a product of humans that is shaped by experience, cognition, and practical concerns of society and culture.  Our cognition has evolved to the world we know which has a profound impact on how we know and teach mathematics.  

Enactivism defines knowledge as adequate or viable action within the context of the lived experience (Proulx, 2004).  If perception of the world guides our actions in the world, then defining a clear understanding of the word perception and its meaning is important.  Within the framework of Enactivism, perception of one’s environment is considered an active process of categorization made possible by previous interactions with the lived world.  As we receive information from unfolding events, we apply our perceptual and conceptual filters to generate insights, to which we then apply in the real world to generate experiential data (Boisot, 1998).  This loop is considered “knowledge”, which is distinct from understanding, insights, data, and information.   

Thus, Enactivism renders perception and action inseparable in lived cognition (Varela et al., 1991). Two critical elements of enactivism for the work of MathTrack Institute are the individual cognizing agent (student/learner) and the environment that is co-implicated with that agent.  Through the GROWTH Framework™, educators are trained to understand how to foster classroom learning environments with their students. The GROWTH Framework™ is provided through an online format and is job embedded to be situated within their class environment. What educators derive from GROWTH, therefore, is based on the perceptions for viable action within the environments in which they teach. 

GROWTH Framework™ as a Concept Study

Brent Davis and his colleagues provided insight into answering questions of access, development, and study of this form of collective Mathematics For Teaching (M4T) through their work with teachers in collective mathematical environments called Concept Study (Davis, 2008a, 2008b; Davis, 2012; Davis et al., 2009; Davis & Renert, 2009; Davis & Simmt, 2006; Davis & Sumara, 2007, 2008; Simmt, 2011).  A concept study combines the collaborative work of Lesson Study (Chokshi & Fernandez, 2004; Fernandez & Yoshida, 2004) with the mathematical disciplinary knowledge focus of Concept Analysis (Usiskin, Peressini, Marchisotto, & Stanley, 2003). This combines collectives working together to study mathematics for teaching and learning (pedagogy and content).

Concept Analysis

Concept analysis. (Lakoff & Nunez, 2000; Leinhardt, Putnam, & Hattrup, 1992; Usiskin, Peressini, Marchisotto, & Stanley, 2003) is a fine-grained interrogation of individual mathematical concepts. In MathTrack’s training and blog, these activities are written as an interrogation of a specific concept/topic, like subtraction.  Davis and Renert (2013) paraphrased Usiskin et al.’s (2003) description of concept analysis as examining the historical origins, common usages and applications, and the representations and definitions of a mathematical concept. 

In MathTrack’s program of study, educators explore the many dimensions of that mathematical topic, labeling its complexity and discussing how it develops or is exhibited (curriculum) with students. Through this, educators are intended to develop their cultural approach to mathematics to include empathy as a model for the shared struggle to learn new mathematics.  Lakoff and Nunez (2000) described a concept analysis as “ a mathematical idea analysis, framed in terms of cognitive mechanisms, of what is required to understand—really understand” a mathematical equation or concept (p. 384). MathTrack Institute positions concept analysis as a rigorous applied mathematical study, which, when sufficiently mastered, warrants a bachelor’s degree in applied mathematics and beyond. 

Lesson Study 

As MathTrack aims to scale mathematics expertise at large, the required shift to culture and pedagogy must be acknowledged.  Research suggests to scale sufficiently a methodology that involves a cultural and pedagogical shift (Fernandez & Yoshida, 2004; Stigler & Hiebert, 1999) requires a unique activity where teachers plan, observe, and discuss lessons collaboratively with their peer teachers. This is the collective of various sizes of the mathematics teacher. The key component of lesson study success, as described in Stigler and Hiebert (1999), is that teachers are given opportunities to work collaboratively, situated in the shared work of being a mathematics or other content teacher.  An example of this within MathTrack’s programs of study includes discussion posts where educators review each other's perceptions and ideas.  

Concept Study 

With the development required to become an expert mathematics teacher, it is intuitive and natural to combine both concept analysis and lesson study into MTI’s training.  A teacher’s work involves unpacking mathematical concepts (Ball, Hill, & Bass, 2005) so that students can access the culturally created thought processes and ideas that the concepts represent.  Davis and colleagues take this notion further, as unpacking pries apart a mathematical concept for teaching, while substructing (Davis & Renert, 2014) is how the various parts of the concept unite or conflict to formulate a profound understanding of the concept. 

These two mathematical actions—unpacking and then substructing—are the intended purposes of our professional development's implicitly and explicitly applied emphases (PD). A concept study takes the form of six emphases that emerged from initial pilots (Davis, 2011, 2012; Davis & Renert, 2009, 2013, 2014). The emphases are designed around capturing the potential of learning and developing while doing– specifically within the framework of teaching mathematics as an educator, parent, or peer. Thus, MTI’s content is virtually accessible so that educators and stakeholders can continue to return to them as always-present potentialities. Simply put, we cannot control when the information is most valuable for viable action in the education process, but we can empower access to that information by making it always available. 

Each of these emphases have been incorporated within the GROWTH Framework’s™ Six Principles to make viable action with these concepts possible.  These Principles are designed to progress from interrogation and coherence to the practical application of mathematical concepts and mathematics teaching. MathTrack has designed this evolution of proficiency that, when mastered, will provide evidence of a proficient and highly capable mathematics educator.