Think back to your school days. Do you remember a teacher who just made things “click” for you? Or maybe someone who inspired you to stretch yourself beyond your comfort zone? Decades of educational research tell us that these memorable educators generally share two key traits: being BOTH knowledgeable and effective. If those descriptors sound subjective to you, it won’t be surprising to find they’ve been the trigger for decades of “math wars” — and their subsequent (unintended) consequences.
Understanding the problem and coming to a solution requires specificity and objectivity. What exactly does it mean to be a knowledgeable and effective mathematics teacher? As you might imagine, the answer is complex and begins with how mathematics education's knowledge, learning, and purposes are characterized. In this edition of the Mathematics for Teaching blog, we will first unpack the essential elements underpinning the required qualities of being a knowledgeable and effective teacher. Grounded in that objective definition, we’ll move to explain the 6 principles that guide MathTrack Institute’s GROWTH framework and how this framework is situated within the body of research that sparked its innovation and development.
The Evolution of Educational Research: The Shift from Knowledge of Teaching to Content-Specific Teaching Knowledge
Process-Product Research
Research in the 1980s by Thomas Good, Douglas Grouws, and Howard Ebmeier out of the University of Michigan focusing on mathematics teachers found that certain teacher behaviors positively influenced students’ performance on basic skills tasks but not tasks requiring problem-solving skills. This research defined teacher knowledge by non-content-specific teacher behaviors and tracked effectiveness by student performance on standardized assessments (Good, Grouws, & Ebmeier, 1983).
This was much better than previous attempts that made comparisons between teachers in a sample relative to each other. This is the process measures, statistically linked to production outcomes or process-product research for teacher knowledge and effectiveness. This orientation to teaching, called general pedagogy or teacher methods, meant that if the content being taught was held constant, these were the process measures that led to better student outcomes. This makes sense as we think of a teacher’s ability to conduct whole class activities, dissuade unproductive behaviors, initiate positive student behavior, and have high expectations in a relatively relaxed learning space as more productive learning environments, whether it be History and World Languages, Physics or Mathematics. However, this approach drew criticism because a well-organized class with orderly students looks different within a math class than it could look in another content domain.
Production-Function Research
In the 1990s, several separate teams of researchers from various fields, including economists, mathematicians, and mathematics education researchers, at Stanford University, Harvard University, the University of Michigan, National Chi Nan University in Taiwan, and The World Bank, among others developed the next evolution of education research, dubbed production-function research. Here, the focus is on three classes of variables: teaching ability (content knowledge), teachers' motivation (locus of control), and classroom context (where and who was taught). The most important change was teaching ability or teacher knowledge, specifically in regard to their content background (Harbison & Hanushek, 1992; Mullens, Murnane, & Willet, 1996; Rowan, Chiang, & Miller, 1997). This was measured by proxy variables that established measurable teaching ability, such as major, number of content courses taken, and scores on certification exams by prospective or current teachers. To remain consistent, production-function researchers still measured teacher effectiveness on student performance using standardized assessments, usually measuring procedural mathematical knowledge. Imagine these on a spectrum:
Critiques of this paradigm were that the number of math classes you took as an undergraduate didn’t adequately measure what you knew about mathematics or certain teacher content knowledge. For example, when would you learn how the curriculum works and why the K-12 mathematics curriculum has the scope and sequencing that it has? These are intuitively vital to teaching but weren’t measured in either spectrum of research paradigms and, therefore, weren’t considered for teacher effectiveness. Taking math classes like differential equations wasn’t correlating to higher test scores for students… there had to be a reason for that.
Lee Shulman’s Combined Paradigm
By the early 1990s, a new paradigm was introduced by a researcher named Lee Shulman from Stanford. For the first time, it was considered equally important to deeply understand mathematics and effective classroom activity behaviors. This was also the first time that a research group defined the common use knowledge of a subject, like mathematics, as different from the knowledge of that subject that a teacher needed. Put another way, teachers need to know something different about mathematics than what other professions need to know about mathematics (Ball, 1990, 1991; Borko et al., 1992). The spectrum, with the addition of Shulman’s work, would look like the following:
Shulman’s article, in which he first discussed this understanding of teacher knowledge, has been cited more than 38K times in educational research. To say it made an impact is an understatement.
Explicit-Objective Research
For the next 20 years, into the mid-2000s, researchers focused on building on the promise of Shulman and colleagues’ notion and then labeling that knowledge as explicit knowledge for teacher effectiveness (Explicit-Objective Research). This was mostly done by researchers from the University of Michigan, Michigan State University, and Stanford Universities (Ma, 1999; Hill, Rowan, Ball, 2005; Hill, Sleep, Lewis, & Ball, 2008; Izsak, 2008; Izsak & Araujo, 2012; Izsak, Orrill, Cohen, & Brown, 2010).
Other researchers at University of Alberta and University of Calgary concentrated on designing models for developing teacher knowledge, understanding tacit and emergent frameworks for that knowledge, and building from theories to make claims about the nature and development of such knowledge (Adler & Davis, 2006; Davis & Renert, 2009, 2014; Davis & Simmt, 2003, 2006; Davis, 2011, 2012; Simmt, 2011). All the while, the utility of and very notion of the discipline of mathematics was also changing (Lakoff & Nunez, 2000). Technology and the understanding of complexity and embedded systems led to understanding the discipline's utility for far more than simply creating calculators out of students.
MathTrack’s GROWTH Framework Is Built on the Foundation of This Research
For more than a decade, the founders of MathTrack Institute (MTI) have been creating and testing our GROWTH framework to move from theory to beyond proof of concept. Our guiding animus has been to define and efficiently map out how to train mathematics teachers effectively through an institutional approach. MTI defines the understanding required to be a knowledgeable and effective mathematics teacher as mathematics-for-teaching (MFT). We combined Shulman’s work and how it was taken forward within mathematics while simultaneously situating it within a theory for learning and cognition called enactivism that understands tacit and emergent behaviors.
Why Enactivism? Enactivism allows mathematics to be a product of human beings, shaped by our brains and conceptual systems and the concerns of human societies and cultures. This orientation helps overcome one of the largest barriers in society to teaching and learning mathematics—the mindset that you either are a math person or you’re not. If indeed everyone is a math person, we can eliminate this cultural barrier for people and help them see that not only are they innately mathematical but that they can, with proper effort, be very good at math. All people, not just a chosen few. It also has another advantage. Under this frame of understanding, we at MTI believe that learning is best achieved through experience, situated in the work, and evolved by the cycle of try, fail, debug, and try again. If we are all “not yets” in some way, what can we become?
You are correct if it sounds like a lot is involved in becoming a knowledgeable and effective teacher. By formulating both the conceptual understanding and the practical ability to implement training that would enable this ability, we reworked how higher education should approach this study and development. We had to understand that becoming a knowledgeable and effective math teacher happens at the co-implicated levels of the collective (other teachers and students) and at the individual levels. Therefore, we worked further to combine these methods into Concept Studies for Mathematics Teachers that can be enacted within the work of teaching. MTI positions the study of mathematics for teaching as a rigorous applied mathematical study, which, when sufficiently mastered, merits a bachelor’s degree in applied mathematics.
Situated within the individual and collective, the model implementation for our work is best represented by embedded systems:
Concept analysis has a rich history of the solitary or textbook-based pursuit of understanding mathematical topics (Usiskin, Peressini, Marchisotto, & Stanley, 2003). This work was predominantly led by the University of Chicago School Mathematics Project and a mixture of mathematical historians, philosophers, and those oriented to the evolution or humanistic approach to mathematics. Lesson study is about teachers, situated in their work, testing and designing lesson ideas together and sharing their failures and successes (Chokshi & Fernandez, 2004; Fernandez & Yoshida, 2004). Started as a framework in Japan, ethnographic researchers looked to bring Lesson Study as an operational approach to teacher training in the US. The GROWTH framework synthesizes the power of both into one super-charged approach. For the first time, you can major in these frameworks as an undergraduate and study them to receive your teacher license or gain continuous learning through professional development. We hope this dismantles the notion that a relatively cranky George Bernard Shaw wrote more than a century ago in "Maxims for Revolutionists," an appendix to his play Man and Superman. "He who can, does. He who cannot, teaches.”
We offer a rebuke to Shaw’s statement, stating:
“Those who know mathematics, teach.”
The 6 Principles of the GROWTH Framework
The principles of the GROWTH Framework are specific emphases of the work of becoming a more knowledgeable and effective teacher. We specifically chose “emphasize” because it denotes a way of thinking about teaching that embraces and encompasses the vast breadth of mathematics. We readily acknowledge that we cannot cover all the topics a teacher will encounter, nor would we want to. The key to GROWTH is empowerment, the ability to apply these principles to an educator's work as it unfolds in the complex work environments that teachers inhabit. This methodology has merit whether an educator is a parent, a teacher, or anyone seeking to impact children who are learning mathematics positively. The GROWTH Framework synthesizes 40 years of mathematics education research reviewed above into a coherent framework for math teacher preparation.
- Grasp the meaning
- Reveal the horizon
- Observe the implications
- Weave together concepts
- Teach with peers
- Hear your students
Principle 1: Grasp the meaning
“What is this?” We have chosen “meaning” to encompass deep understanding versus define or definition, which denotes memorization. This may seem strange, as much of mathematics is about having clear postulates and definitions, of which you can surmise later truths or contradictions.
We also use the word realizations, which implies an evolving coming-to-know of a topic built through ever more sophisticated lived experiences. Defining something on your first day of teaching will have little utility to a seasoned veteran teacher with many layers of sophistication for how students learn. Realizations are everything that gives meaning to a mathematical concept for a learner.
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. It is important to note that “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 toward becoming master teachers, their realizations of concepts should change as a reflection of their growing sophistication. They will know more about subtraction so that they can, in turn, share the process of unpacking subtraction with their students and have empathy for the struggle it will require.
Principle 2: Reveal the horizon
When considering a concept like subtraction, a teacher can ask, “Where do I see this and why?” A horizon calls to mind a landscape, which invokes an accurate understanding of what we are trying to accomplish with educators. It encompasses all of the visible features of a land or countryside and invites consideration of their aesthetic appeal. Mathematical horizons, when well understood, have a similar beauty. They are remarkable accomplishments of humanity's capabilities to discover and invent.
These mathematics horizons are about organizing and comparing lists of realizations for a particular mathematical concept to determine where and why they emerge. Think of it as a macro-level view of a mathematical concept like subtraction. When does the notion of subtraction change? What number sets, when applied, do initial realizations for the concept need to evolve? Are there ways of talking about subtraction differently than “taking away”? For example, what does subtraction mean in a quadratic equation? What does it mean when you subtract an expression?
Principle 3: Observe the implications
When considering a mathematical topic in the context of teaching, it is insightful to ask directly, “What is the implication of my realization of this concept?” The implication of an ever-evolving realization is best defined as an entailment or a fundamental byproduct of the realization. Take the realization to its limit and discover why it may not work. This means we need to track and scrutinize the implications, or consequences, of any one realization of a mathematical concept. It also means that we can think differently about that kid in your class who always gives contradictory ideas about the topic being discovered. That isn’t an annoyance; it is, when well directed, an emergence of incredible mathematical capabilities.
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. This will allow even incredibly sophisticated educators to gain access to fresh and innovative approaches to a concept, enabling them to move through cycles toward even greater mastery of their craft (Berkopes, 2014). For example, there are nested consequences of defining multiplication as repeated addition. Is it useful when applying multiplication to specific permutation problems that deal with the order and combination of things? For example, counting the number of outfits that can be made from a given number of shirts, pants, and shoes without considering the order falls under combination problems. Is that different from when order matters? Is it repeated addition that you are manifesting, or is there something different about the nature of the multiplication operation?
Principle 4: Weave together concepts
As educators develop multiple realizations for topics, the logical next step is to ask, “How are these used in combination?” Or when is it appropriate to use them in combination? Creating conceptual blends and collapsing diverse realizations of a mathematical concept is defined as “blends” (Davis & Renert, 2014). The first three principles of GROWTH aim to develop distinctions, whereas blends create coherence. Coherence and pattern recognition are difficult but essential mathematical habits of mind that teachers can build on their own and then empathetically share with their students.
For example, a blended understanding that subtraction can be considered taking away, cutting off, and comparing two quantities allows much more flexible thinking than just one of these realizations. The blends of these realizations focus the collaborative effort on defining the deep connections between the 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). Coherence is a beautiful word for this activity because it embodies seeking the quality of being logical and consistent.
Principle 5: Teach with peers
Teaching is not a silent film or a solitary sport. Yes, you are often the only adult when working with your students in your classroom. But, your peers are the collective with whom you share the struggle of being an educator and creating mathematics with students. It may sound strange that you are creating mathematics with your students. However, we posit that mathematics is much more than the traditional formalist perspective, which views mathematics as purely based on formal systems and logical deduction from axioms. MTI’s perspective (post-formalist) views mathematics as not solely confined to manipulating symbols according to predefined rules. Rather, it is also influenced by human intuition, creativity, and social interactions. Mathematics is a living and evolving discipline shaped by the cultural, historical, and contextual factors influencing mathematical thinking and problem-solving. It is invented and continues to be invented all over the world.
This is where we bring together the individual and the collective through the shared effort of designing and creating mathematical lessons. How can teachers best co-create with peers? “Participation” in creating lessons must be done within the work of teaching with others. You must have the chance to create, fail, and then share the success and failure with others attempting this difficult work. That is why we built our framework, including apprenticeship-based pathways to degrees, situated within the work of teaching. Teachers and parents 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). You may catch yourself saying in frustration, just do it this way; it is easier! This profoundly impacts what mathematics students gain access to; your choices matter.
Principle 6: Hear your students
The final principle of the GROWTH framework is essential because creating a learning environment does not happen without students. You are not a teacher if you do not have or involve your students. You can talk about mathematics, but you certainly are not teaching it. As many teachers love to share, students always ask, “Why do I need to know this?” Rather than seeing this as an annoyance, it should be the most desired question for teachers to receive from their students. This is a chance to co-create and deepen the understanding of everyone involved.
We think about this phenomenon as “pedagogical problem-solving.” The question of why is a constant part of creating new mathematics with novice students. It is also, in our opinion, one of the most enjoyable aspects of teaching mathematics. However, for the unprepared, it can be incredibly difficult.
For example, a student may ask, “Are fractions numbers?” This is neither innocuous nor straightforward. Dr. Susan Lamon from Marquette University(2007), among others, shared various realizations for fractions that show that fractions can be considered numbers depending on the mathematical context. The deep intellectual work of understanding how to answer these questions warrants a prestigious degree specifically geared toward creating more knowledgeable and effective future and current teachers.
CALL TO COLLABORATE
MathTrack’s GROWTH framework is designed to support new, novice, and master teachers to continue to deepen and enrich their understanding of their craft. This framework synthesizes more than 40 years of research into a practical implementation provided by MathTrack Institute in a single, user-friendly platform in a cost-effective, always accessible way. If you, as a school leader or teacher, want to engage with MathTrack Institute about how the GROWTH framework can become a part of your strategy for mathematics education in your school or district, please contact our team. We would be happy to support you with questions that you may have and feedback about our framework. We seek to solve the issue of access to high-quality mathematics educators, which begins with understanding what it means to create more knowledgeable and effective mathematics teachers.
ABOUT MATHTRACK INSTITUTE
MathTrack Institute (MTI) is an Indiana-based distance learning institution aiming to empower math teaching expertise within every school community. In late 2020, its founding faculty set out to re-think math education. Through iterations of research and practice, MTI concluded that teacher preparation was the cornerstone to student success in math. Deeper research highlighted a concerning trend: a significant downturn in traditional routes to teaching, with mathematics experiencing the sharpest decline—40% since 2012. In response to these challenges, MathTrack introduced its GROWTH Framework, an innovative approach designed to scale math teaching expertise equitably.
The GROWTH Framework was developed to provide educators of all levels access to PhD-level math education concepts with a job-embedded format to empower practical classroom use. The framework is delivered through three work-integrated learning programs to serve all stages of educator development:
- Apprenticeship: For degree-seeking educators
- Transition to Teaching: For certification-seeking educators
- Professional Development: For continuing education and staff development
