Perspective Chapter: Mathematics for Teaching – It’s Not (Just) Pedagogy

Abstract

The specialized mathematics content knowledge that is required for effective teaching is multidisciplinary and non-trivial, and the subject of multiple, typically nonspecific interpretations. The purpose of this chapter is twofold: to provide a scalable definition of mathematics knowledge for teaching (and learning), and posit a new conceptual framework, informed by the constructs, assumptions, expectations, beliefs, and theories that differentiate the distinctly mathematical aspects of this type of teacher knowledge. To do so, we build on Shulman’s original conception of pedagogical content knowledge. By using a lens drawn from our own critical practice as mathematics educators, we articulate a working definition which is clear enough to catalyze the development of re-imagined and robust mathematics teacher education standards, programs and courses. By extension, we call upon post-secondary institutions to provide interdisciplinary environments in which “translational” skills and processes are nurtured and developed. We adapt the term “translational” from healthcare to describe an integrated teacher education program that would aim to train a new generation of “hybrid” mathematician/educators to be effective translators of mathematics; mathematics education research; and their intersection across audiences. By experiencing how classroom-focused mathematics and cognitive/pedagogical sciences intersect, the graduates of translational mathematics education programs will be uniquely equipped to deliver improved instruction resulting in increased student achievement.

1. Introduction

This chapter is offered as part of the on-going efforts in teacher education to determine teachers’ needs related to their mathematical understanding as needed for effective teaching. In our own work, we have encountered conflicting views on what background and course work are needed in the preparation of effective elementary and secondary teachers of mathematics, and confusion as to what aspects are seen as “mathematical” and what aspects are “pedagogical,” “cognitive,” “instructional” or “developmental.” And if indeed, as is now generally recognized, that teachers’ understandings of mathematics for the purposes of teaching are indeed specialized, how can such aspects be more precisely described and defined, other than by saying that content knowledge is special or distinct and requires focused preparation programs centered around specific knowledge and competencies. What are these mathematical competencies?

2. Rationale

Our reasons for focusing on a re-examination of the mathematics and mathematical processes (such as reasoning and representation) inherent in Kindergarten to Grade 12 mathematics curricula; how these should be understood by teachers; and, the implications for teacher education programs, stem from one local factor and one broader, international phenomenon.

As of February 1, 2025, all teacher applicants and internationally educated teacher applicants seeking certification in our Canadian province of Ontario will be required to pass a Math Proficiency Test (MPT) in order to apply for licensure through the Ontario College of Teachers (OCT)¹. Any individual seeking certification to teach in Primary (K-Grade 3), Junior (Grades 4–6), Intermediate (Grades 7–9) or Senior (Grades 10–12) curricular divisions (including those with ‘teachable’ subjects which do not include mathematics), will need to pass this test of fundamental mathematics skills (including content up to Grade 9) and curricular components (instruction, differentiation and assessment). The motivation for the mandatory mathematics test for new teachers is not only to improve Grade 3, 6 and 9 students’ achievement on annual standards-based mathematics evaluations, but also because all certified teachers in Ontario, regardless of their teacher education program (Primary/Junior, Junior/Intermediate or Intermediate/Senior) could possibly be assigned to teach mathematics to students in Grade 6 or below and in some cases, assigned to teach Grade 7 to 12 mathematics [1]². While we agree with the principles of credentialling mathematics teachers, echoing the sentiment of Deborah Ball who once said “teachers must know the subject they teach. Indeed, there may be nothing more foundational to teacher competency,” [2] we, like others in agreement, have been harshly criticized, finding ourselves amid enormous controversy surrounding this new policy [3]³. The vocal backlash and harsh criticism arguing against the test has been, most notably, from teacher candidates, teacher unions and Faculties of Education. Loud support for the test has been voiced by other stakeholders (including parents and the Ministry of Education⁴) looking for greater public accountability and teacher professionalism. As provincial assessments by the Education, Quality and Accountability Office (EQAO)⁵ continue to report unsatisfactory student achievement scores in meeting the provincial standard over the last few years (about 60, 50 and 54% of Grade 3, 6 and 9 students, respectively), multiple invested parties have demanded swift action and scalable solutions to improve student achievement. Curricula at multiple grade levels are being hastily rewritten and mandated. In other words, new learning for teachers and students, predicated on immediate reforms to pre-service and in-service teacher education programmes to provide timely and relevant teacher preparation for the new content is required.

In trying to formulate arguments to support measurements of mathematics teacher competency, we have found ourselves recurrently struggling to articulate or find satisfactory definitions of what is meant by knowing ‘mathematics’ in the context of teacher education and classroom practice. In terms of course design, we have argued that mathematics courses for teachers must contain more than standard mathematics courses. But what is the “more?” Mathematician Hyman Bass, who by his own admission, has become significantly engaged in school mathematics, states that the “more” is strictly mathematical knowledge (not about students or about pedagogy) that proficient teachers need and use, yet is distinct and thus not known by many other mathematically trained professionals, for example, research mathematicians [4]. Like us, Bass argues that this uniquely distinct type of knowledge required by teachers is not something likely to be part of the instruction in content courses for teachers situated in mathematics departments. For example, multiple representations and their associated reasoning is mathematics, and yet most research mathematicians would dismiss such content as being “school math,” not “deep math,” and thus better left to the pedagogues.

These false impressions and egregious simplifications underscore the fundamental, yet elusive distinction between the mathematical knowledge needed for teaching and the mathematical knowledge needed for other occupations or professions in which mathematics is used. The goal of academic mathematics courses taught by research mathematicians seems to be immediate and insular, i.e., individuals in the class need to “know” the content solely for their own understanding and its applications to other domains in science, technology, engineering and beyond. In sharp contrast, the goal of mathematics courses for teachers is for the candidates not only to “know” mathematics for themselves, but to be nimble in situating and sequencing the content along a K-Grade 12 continuum (i.e., what concepts came before and what comes after); agile in deconstructing the content in order to present it to students in multiple ways using age- and mathematical/developmentally-appropriate approaches and representations together with the associated reasoning; and, anticipate (with the goal of pre-empting) well-documented student misconceptions by calling upon firm foundations in how learning happens. Bass reminds us that novice mathematics teachers’ “self-talk” must focus on mathematics, responding to classroom discourse [4], by first asking oneself “What is the significant mathematics happening in my classroom right now?” “What should I be looking for, and be sensitive to, mathematically, in this situation?” and “What are the instructional moves I need to make to shepherd this discussion back to the important mathematics that is the goal of this lesson?” It goes without saying that the elephant in the room is that this specialized, multi- and interdisciplinary content knowledge requires both a highly specialized teaching context and instructor—one who knows and can create experiential opportunities to concretize the intersection of mathematics content, cognition, curriculum, pedagogy and developmental trajectories—and ones unlikely to be found in or the result of current pre-service (or in-service) education programs.

Our second concern arises from the ever-widening net that is used to capture all that falls under mathematics “curriculum.” With the explicit inclusion of arguably important issues such as diversity and inclusion; topics such as culturally-relevant mathematics; socio-emotional learning (e.g., foci on mindset, perseverance, risk-taking, relationships, and attitude); and, rapidly-changing recommendations around how to teach (e.g., intentional instruction versus inquiry-based learning), we note two distinctly negative trends: the diminishing focus on and elimination of mathematics content itself, and the failure to differentiate teaching mathematics from teaching in general. We argue that teaching mathematics is distinct from teaching in other subject areas, and that student learning of mathematics cannot be achieved unless mathematics content, developmental and pedagogical principles, and research-based practices in mathematics teaching are inseparable in the learning trajectory of mathematics from Kindergarten to Grade 12. By stretching the curriculum to include financial, data and information literacy, coding and other tangential topics, less time, in an already over-crowded curriculum, can be dedicated to the types of experiences that could improve both the teaching and learning of mathematics.

To address the need for a common understanding of what it means to be a “mathematically qualified educator,” we offer an over-arching model for the cumulate body of mathematics knowledge uniquely needed for teaching the K-Grade 12 curriculum. By extension, we argue that by developing an easily interpretable and implementable definition, the need for significant, specific and rigorous learning goals (including standards) for mathematics teacher education programs may lead, not only to more deliberate and appropriate mathematics teacher education course design and implementation, but also to improved outcomes for students along the elementary to secondary continuum, a connection that is well-established in the literature [5].

Both our definition and our recommendations for mathematics teacher education programs are drawn from our own critical practice with individuals—teachers, parents, mathematics educators, mathematicians, administrators, researchers in fields beyond mathematics and students—representing a wide range of perspectives on what change is needed and why. We are advocating for dramatic changes to what is considered to be mathematics knowledge unique to teaching as well as teacher preparation. These proposed changes are not abstract, but rooted in the intersection of our own work with and research involving multiple stakeholders who are impacted by current classrooms and contexts. We propose that the ideas proffered in this chapter are the outcomes of genuine critical practice based on the analysis of our own communities of practice and research. By synthesizing the perspectives of a diverse network who have been trying to transform mathematics education practically, but separately, we integrate many voices collectively.

3. Background

We are mathematics educators, one with a secondary mathematics classroom teaching background and one whose focus has been on elementary mathematics education, respectively. We bring disparate, yet rich and formal knowledge of mathematics content, mathematics education research and a shared belief that higher expectations bring higher achievement, regardless of the age of the learner. Separately, in the context of multiple studies and professional self-assessments, we have amassed data in the form of qualitative and quantitative surveys, individual and group interviews, paper and pencil tests, and annotated observations, of over 1000 prospective teachers as they work to understand the mathematics that we believe they will need for effective teaching.

Advocating for required coursework has been a career-long endeavor for the first author in particular, having begun teaching mathematics courses for prospective elementary teachers in 1989— in a mathematics department—and continuing to do so to the present time in a faculty of education. Similarly, in her first year as a Primary/Junior instructor at a Faculty of Education—1998—the second author was allotted 12 course hours in which to “prepare” teacher candidates to teach and assess students using the mathematics curriculum policy document required for Kindergarten to Grade 6. With her influence, by 2021, the course was increased to 60 hours over three academic terms. As professional colleagues, we have long reflected on and discussed the benefits and detriments of mathematics “education” courses, when some are offered by mathematics departments in Faculties of Arts and Science, and others, by Faculties of Education [6].

With courses commonly offered in two different faculties by different categories of scholars, i.e., research mathematicians versus education staff (ranging from tenured faculty members to graduate teaching fellows with no classroom experience and no mathematics education specialization), it is not surprising that there has been and continues to be confusion at the institutional level about who “owns” mathematics education, decisions around what it should comprise, and why. Moreover, there is no agreement on precisely what constitutes “mathematics education” in the context of teacher preparation and professional development over the long course of educators’ careers. Over many years and in multiple contexts, including local, national and international mathematics education conferences, when the topic of mathematics for teaching arises, we have encountered a common barrier: there is a pervasive perception that the mathematical knowledge, problem solving, reasoning, perspectives, and resources that we, as mathematics educators with strong mathematics backgrounds, deem to be most important for mathematics teacher education courses, are looked down upon and dismissed as mere ‘instructional pedagogy topics’ by both institutional administrators and research mathematicians (even some who purport to have an interest in mathematics education). The content of the highly specialized mathematics course work foundational to and essential for mathematics teacher preparation is not recognized as an aspect of mathematics, valued for being rich and multidisciplinary, nor considered to be complex and interdisciplinary to teach. Furthermore, the defining qualities of the rare instructor who can teach such courses effectively, continue to be unspecified by hiring teams in Education and Arts & Science, resulting in increasing numbers of new mathematics education faculty who have a targeted interest in topics like children’s literature with a mathematics theme, “unplugged coding,” or diversity, (the list goes on), but lack mathematics, developmental psychology, cognitive science and pedagogy, or are research mathematicians who “dabble in education” but lack the strong interdisciplinary background needed to prepare and support teachers.

Our goal here then, is to adapt the Translational Institute in Medicine (TIME) model. The mission of TIME is to enhance collaboration and optimize communication by sharing expertise. This is achieved by offering a curriculum that interweaves rigorous research with authentic “clinical” experiences in a multidisciplinary environment that crosses departments and disciplines.

Notwithstanding the important contribution of the rich “mathematics for teaching” framework described by Ball and colleagues [2], our focus is to drill down deeper into particularly the specialized content knowledge aspect of the Ball et al. model, using the original Shulman conception of pedagogical content knowledge [7] as the framework to posit a Translational Institute in Mathematics Education curriculum for mathematics teacher education.

4. Framework

Perhaps the most oft-cited model of mathematics for teaching was created by Ball and colleagues [2]. These authors identify a number of sub-domains that fall under two main headings, i.e., subject matter knowledge and pedagogical content knowledge. While we appreciate the model’s comprehensiveness, the two major domains and their respective subdivisions can be overwhelming and may contribute to a lack of clarity in designing mathematics courses for educators. For example, the “subject matter knowledge” domain is further distributed across three subdomains, namely common content knowledge, specialized content knowledge, and horizon knowledge. Of particular interest to us is the “specialized content knowledge” piece of the model. While we realize that the other category in the model, “pedagogical content knowledge” (to use its terminology) still draws heavily on mathematics in lesson design, responses to students and so on, our goal is to add clarity to what content mathematics courses for teachers, framed in the subject matter knowledge part of the model, should constitute, and hence we draw particularly from the specialized content knowledge category, and attempt to further articulate its description.

According to Shulman, the mathematics content a teacher should know includes:

1. The most useful forms of representation of the concepts and topics.

2. The most powerful analogies, illustrations, examples, explanations, and demonstrations of those topics and concepts.

3. The ways of representing and formulating the content that make it comprehensible to others [7].

Shulman continues by saying that “since there are no single most powerful forms of representation, the teacher must have at hand a veritable armamentarium of alternative forms of representation” ([7], p. 9). Importantly, the reference to the key roles played by mathematical representation appears over and over in the Shulman description above, and in fact, throughout his 1986 paper. Indeed, the knowledge of representations and their associated mathematical reasoning is a particularly important, but often weak, area of understanding for prospective teachers [8].

A challenge we have encountered repeatedly, is the (mis)interpretation of the mathematical ideas of representation and reasoning, and the (mis)perception that these ideas are really ‘pedagogy,’ and should thus be housed in education curriculum and instruction courses for teachers. We fundamentally disagree.

5. Goals

To follow we illustrate and describe the rich mathematical nature of the representations repeatedly mentioned by Shulman [7], and the deep (and developmental) reasoning which ensues, using several examples. We further argue, based on a large dataset, that while such mathematics has to be provided during curriculum and instruction courses if no other mathematical course options for teachers are available, that much better results can be obtained if prospective teachers have the opportunity to focus on these particularly mathematical concepts, either prior to or concurrently with curriculum and instruction (informally, ‘methods’) courses. Prospective teachers viewing a sample lesson, for example, who do not have the required knowledge of the representations and models involved, will typically miss many of the important pedagogical aspects in their quest to sort out the mathematics [9]. This contrasts with participants who are already familiar with the mathematical ideas who can focus on the lesson itself at a higher level.

To avoid confusion with existing models, we will term the specialized mathematics concepts needed by teachers for effective teaching as specialized mathematics for teaching, or SMT. To follow, we attempt to problematize, illustrate, and define this term in a useable way. We begin with addressing some preconceived notions and common problems, first by offering a series of vignettes drawn from our research and other experiences.

6. Math for teaching: what’s the problem?

To follow are several scenarios illustrating some of the challenges in describing and understanding SMT, which also illustrate the perceptions of some sample stakeholders. They are offered as a collective, with discussion to follow. In each case, the reader is invited to ponder the problem illustrated. Each vignette has been chosen to illustrate a particular challenge with the conceptualization of the field of specialized mathematical content knowledge for teaching.

6.1 Story one

A research mathematician and mathematics educator are chatting at a conference, looking at new mathematics manipulatives. The manipulatives in question were rectangular prisms made of plastic to represent base ten blocks. The mathematics educator is quite excited by these, as the shapes are exactly proportional – a tray of ten ‘ones’ is used to illustrate 10, and the tens tray is proportional to the unit piece. Similarly, a tray of ten 10’s, the hundreds tray, is proportional to both ones and tens. Commonly used base ten blocks do not have this self-similar property.

The mathematician is unable to understand why the mathematics educator is excited … and after an explanation of the need to represent multidigit numbers in concrete, highly visual, accessible and accurate ways to preservice teachers by the mathematics educator, the mathematician comments “but cannot they just see these things in their head?”

6.2 Story two

An important representation of the operation of multiplication is the area model. The area model can be used in elementary grades to develop whole number products, such as 12 × 13, with flexible understanding. We take it as a given that this model is an example of the kind of specialized mathematics for teaching, SMT, that we are describing here. Having said that, we have observed that teacher candidates with formal post-secondary mathematics backgrounds know a ‘rule’ for simplifying the product of two binomials such as (x + 2)(x + 3), yet are unable to explain developmentally why this method—as well as the other possible simplification methods—make sense and could contribute to children’s computational fluency and calculational flexibility. In fact, many seem to be unaware of the area model for whole number multiplication.

6.3 Story three

This story took place in a grade 2/3 classroom. The teacher was providing examples of division questions on a smart board to the children. All the examples presented by the teacher were of the style “a case of 24 apples is divided equally between 4 charity baskets. How many apples should go in each basket?” The children were given about six such examples, all using this same model of division (sometimes called ‘equal sharing’ or the ‘partitive’ model of division).

This was followed by a worksheet. The first question on the worksheet asked the children to use a drawing of 12 circles to illustrate 12 ÷ 6.

First, the teacher was approached by two boys who had drawn the following (Figure 1):

Figure 1.

The boys’ model drawn to illustrate 12 ÷ 6.

The teacher told the boys they were correct.

Next, a pair of girls showed her their picture, as below: (Figure 2)

Figure 2.

The girls’ model drawn to illustrate 12 ÷ 6.

The teacher ‘corrected’ the girls’ picture to look like the boys’ picture. One of the girls began to cry, in obvious frustration.

6.4 Discussion of the stories

The first story illustrates a situation which is deeply problematic but often disregarded in course development. Understanding the starting point and needs of teacher candidates is crucially important. A “top down” approach in which it is assumed that candidates have experienced their own K-Grade 12 mathematics education in conceptual ways, and can call upon ‘learned’ visualizations illuminating the reasoning behind the ideas to lead the kind of instruction in which children develop mathematical ideas using concrete tools and other representations, is neither realistic nor helpful. Indeed, such an approach can further exacerbate the pervasive sense of elementary teachers that they aren’t good” at mathematics [6]. And even teacher candidates who do have significant mathematical background often are unaware of suitable developmental progressions for learning, or in this case, the uses, benefits, pitfalls, and overall nuances of classroom manipulatives and other representations, and how these connect to, and support, developmental reasoning [10].

The second story further illustrates how easy is it to overlook the developmental thread of an idea, when approaching it from a more abstract perspective as well as the idea that post-secondary mathematics courses can replace the need for SMT. Our data consistently illustrate that the conceptual underpinnings of typical procedural calculations are poorly understood by many prospective teachers [11], even those with strong mathematics backgrounds [12], and that exploring with manipulatives is often an important key to progress [9]. The related problem is the difficulty, especially for a more sophisticated mathematical knower, to unpack the levels of understanding and abstraction needed to prompt developmental understanding. For example, we have heard teacher candidates with post-secondary mathematics backgrounds, who are asked to explain an idea conceptually, claim that a higher-order (typically formulaic) algebraic ‘explanation’ is how they ‘understand’ an idea.

Specifically in the case of the mathematics in Story Two, both whole number products, and binomial products, can be understood and connected using the area model. Yet the traditionally-taught algorithm for whole number multiplication begins with the units, for example the ‘2 × 3’ sub-product in an example such as 12 × 13, while the typical North American algorithm for binomials, ‘FOIL’ begins with the left-most quantity, so for the example provided here, (x + 2)(x + 3), it leaves the ‘2 × 3’ calculation to the last. An ‘aha’ moment often comes when teacher candidates realize these methods rely on the same representation, and either one can proceed in any order. Once again, we emphasize that this is a mathematical realization.

Furthermore, teacher candidates subsequently begin to see that allowing students to construct, discuss, and then adapt these models to develop their own computational methods can later result in an understanding of an abstraction of the initial idea (in this case, abstracting the whole number area model to one showing multiplying expressions with variables) as a generalization of something they already understand. Such opportunities may also open the door to respecting that the traditional algorithm used in North America is merely one of many “non-standard” models developed by different cultures to calculate the products of multidigit numbers. Ultimately, the same area model can be further exploited to develop factoring techniques, and even the quadratic formula [13], and these constructs are often “newsflashes” to intermediate–senior teacher candidates taking mathematics courses for prospective high school teachers.

Story three is disheartening, yet sadly representative of similar events played out over and over in classrooms we have observed. The girls’ model was in fact the ‘correct’ representation of 12 ÷ 6, based on the model of the division operation illustrated by the contextual examples provided by the teacher earlier. The examples shown by the teacher used the ‘equal sharing’ or ‘partitive’ model of division, i.e., the idea of evenly dividing an amount into equal groups. The boys, on the other hand, were using a different interpretation of division, sometimes called the measurement or quotative model, in which the second number refers to the size of each group (not the number of groups). We are not clear (unless due to gender bias) as to why the teacher also switched her understanding from partitive to measurement along with the boys—at any rate she seemed totally unaware she had done so. Another explanation is that perhaps the teacher subconsciously assumed that circling two groups of 6 dots was quicker than circling six sets of 2 dots. At any rate, both sets of students’ representations were ‘correct’ but in fact the girls’ model adhered more closely to the provided classroom examples—it certainly wasn’t ‘wrong’ as the teacher indicated to them.

The point here is that SMT is rich, subtle, contextual, developmental, and non-trivial. It needs to be explicitly studied and learned, in order to be used effectively in classroom teaching; moreover, the higher level of mathematical learning an individual has attained, may require more unpacking and purposeful developmental reconstruction of concepts.

7. Discussion of the construct of SMT

The three chosen vignettes illustrate several aspects of SMT. The first is that SMT is fundamentally based on representation and reasoning, as initially posited by Shulman. Often, suitable representations need to be specifically learned and studied, especially in cases where prospective teachers’ backgrounds are very traditional and procedurally-based. As well, teachers often need a range of representations in their toolkit, not all of which they have necessarily seen before.

Secondly, as discussed, SMT is deeply developmental in a mathematical sense. That is, the knowledge of what concepts and representations must already be in place, and how to build on these, is critical. Jumping steps, without attention to the connections and abstractions needed, is not helpful. Nor is using a higher-order abstraction to define a lower order concept [14]. Since multiple representations and various pathways are often possible, teachers need to be aware of these. Seeing the developmental progression from a vantage point of a strong mathematical background adds another layer, in that more and more unpacking is needed, as well as attention to what (mathematical) building blocks are needed.

The third example illustrates just how rich, complex and nuanced such understanding of mathematics must be. While many pedagogical factors might also be drawn from the last example, the fact remains that it was the teacher’s narrow mathematical understanding of division models that derailed the interaction with the girls; sadly, such experiences, felt over and over by students, can not only powerfully sow the seeds of mathematics anxiety, but result in enduring misconceptions.

8. Defining SMT

According to the Purdue University online Writing Lab, we use definitions to avoid misunderstanding with our audience by introducing the term; then stating the class or object to which the term belongs; and, finally, expanding on the differentiating characteristics that distinguish it from all others of its class. As an illustration of this three-step process, they provide this example: Astronomy (term) is a branch of scientific study (class) primarily concerned with celestial objects inside and outside of the earth’s atmosphere (differentiating characteristics).

Defining the “term” specialized mathematics knowledge for teaching does not fall neatly into the three-step method because although the term is familiar to most mathematics educators and many mathematicians, it is opaque because of subjective interpretations and evaluations, and in fact, many of the personal definitions that have been posited have only served to increase misinterpretations about what SMT is.

The first step in defining specialized mathematics knowledge for teaching is to establish that it is a distinct and multifactorial body of mathematical knowledge that stands alone. Although the term contains the words mathematics and teaching, it involves many more elements than the two italicized words, which in combination bring us closer to a shared understanding—the true purpose of a definition. If we follow the writing centre examples above, we will say that SMT is a branch of scientific study that encapsulates the mathematical, pedagogical, cognitive and developmental knowledge required by educators in order for students to receive a mathematics education that upholds the integrity of both mathematics and the learner.

SMT begins by valuing and knowing how mathematics understanding develops beginning with quantity and counting, the underlying structures of numbers, through to operations, and their physical and mental images. Beginning with our youngest learners, and continuing throughout the school years including the secondary grades, SMT prioritizes knowledge of appropriate representations and models, and their associated reasoning, as a fundamental factor in SMT. Similarly, while young children perceive shape and space from the world around them, it is the role of the teacher is to advance those perceptions by modeling and providing opportunities for students to mathematize and formalize their earlier knowledge and communicate it with precision. In order for students to transfer lower-order mathematical ideas from one context to another and recognize the connections/relationships that result in a “bigger picture,” they must build upon carefully orchestrated prior knowledge then begin gradually to construct the higher order ideas we call mathematical generalizations. This is what is meant by “mathematically developmental.” The teacher must carefully negotiate the trajectory towards higher-order mathematical constructs, by calling upon and offering concrete models, illustrations and related explanations that bridge to more abstract representations, and lead, ultimately to a generalization. The reasoning involved must rely only on constructs students already are comfortable with, not, as is often unwittingly done in mathematics, by drawing upon unfamiliar higher-order ideas. Without knowing how, when, and with what representations mathematics learning happens in the students’ minds, the introduction of new mathematical content/concepts is futile.

The selection and use of representations is much more than the mere drawing of an outcome—these skills and processes are part of the tangible and cognitive toolkit learners acquire, then use, to explore, reason, and abstract. Teachers must be deeply aware of the suitability of a range of such tools for a given task, as well as the mathematical appropriateness and possible uses and pitfalls. Often teachers need variety of representations, including drawings and manipulatives, and an awareness of the characteristics of each for different purposes. For example, why is the Soroban (Japanese Abacus) a powerful extension of finger-counting, and five- and ten-frames? Why is the Soroban a powerful tool for concretizing place value beyond base ten blocks? When is the number line more useful in exploring integer operations and when are integer counters a safer choice? What are the benefits of using different fraction manipulatives (ranging from fraction strips to grid paper) for exploring different operations? When are concrete algebra tiles most useful and when are online options helpful?

Teachers also need an understanding of the various interpretations and models of the fundamental operations (for example, the partitive versus the measurement model of division) and how and where these interpretations are relevant, as well as which of these models must be in place before proceeding with a new idea. As an example, teachers interested in having students explore the results of dividing by a unit fraction such as in 2 ÷ 1/4 need to ensure students already are familiar with the measurement model of division (“I have to measure two cups of flour but only have a 1/4 cup scoop, how many scoopfuls do I need?”), while exploring a calculation such as 1/4 ÷ 2 (“I want to share the remaining 1/4 of a pizza with my friend”), requires familiarity with the partitive model. Similarly, exploring (−6)÷(−2) is very straightforward if one is familiar with the measurement model of division but nearly impenetrable without it.

This knowledge is rich, complex, nuanced, connected, flexible, deeply mathematical, and non-trivial, usually contextualized in the business of teaching, yet learnable as a discipline in its own right. While such knowledge enables good pedagogy, it is deeply mathematical.

Knowledge of possible connections should extend up to as well as beyond the relevant grade level, as some students may be working at differing levels of generality and formality of a mathematical concept. It is also important that teachers have a sense of what is to come in terms of how some ideas are approached. For example, exploring division by zero as a process is much more productive when the student is at an appropriate developmental age for contemplating the complexity and has not, at a younger age, been “brushed off” by brusque ‘rules’ such as, “You cannot do it. It’s undefined.” Such deep knowledge of specialized mathematics for teaching includes not only an understanding of the related horizon knowledge, but also the mathematical underpinnings upon which understanding is constructed. Thus, horizon knowledge should extend to grades previously as well as those to come.

Knowledge of the mathematically conceptual building blocks of an idea is a critical component of SMT. Yet again we emphasize that this is mathematical knowledge, but of a distinct nature, although certainly fundamentally important for good pedagogy.

9. Assessment

We often hear that such specialized knowledge of mathematics would be difficult to assess, particularly in large-scale tests. Ball and her colleagues have created a wealth of multiple-choice items mainly for elementary level teachers, some of which attempt to measure aspects of what we are terming SMT. We have, over the years, crafted a wealth of such items as well (e.g. [9, 11]), both open-ended and multiple choice, spanning a range of grades. For illustrative purposes in the current discussion, a set of examples of assessment items is provided to follow, crafted to span a range of grades and question styles.

9.1 Examples of assessment items

The figures that follow are simply illustrative, offered to dispel arguments that SMT is too “hard” to assess. Further items have been used and studied in our other work (Figures 3–5) [9, 11].

Figure 3.

Sample of a multiple choice assessment item to assess SMT.

Figure 4.

Sample short response assessment item to assess SMT.

Figure 5.

Sample extended response item to assess SMT.

We have, in our work [9] seen the connection between deeper understanding of SMT as measured by items such as the samples provided above, and higher-quality mathematics teaching. Hence we suggest that SMT can both be improved with suitably focused coursework, which is critically necessary for all prospective teachers of mathematics including those with formal mathematics backgrounds, and it can also be measured and assessed with reasonable effectiveness.

10. Implications for courses for teachers

In 1986, Shulman queried educational researchers about why it was that “no one asked how subject matter was transformed from the knowledge of the teacher into the content of instruction” ([7], p. 6). Now, almost 40 years later, that question remains unanswered and the absence of subject matter in the study of teaching and learning remains what he labeled “the missing paradigm” ([7], p. 6).

Shulman reminds us that the educator must know the “most useful forms of representation of those ideas, the most powerful analogies, illustrations, examples, explanations, and demonstrations—in a word, the ways of representing and formulating the subject that make it comprehensible to others” ([7], p. 6).

There is abundant research to prove that an undergraduate degree is an inadequate pre-requisite for the classroom [15]. Were that true, then every graduate with a mathematics degree should be able to demonstrate the highest levels of subject matter competence, argues Shulman, which is, to teach the subject [7]. The truth is that a post-secondary education in mathematics is about the subject of mathematics, not education.

As a profession, education, specifically mathematics education, must then deliver a program that bridges the chasm between knowing mathematics oneself and knowing it for teaching: holding as its standard the fundamental principle that the defining characteristic of pedagogical accomplishment is knowledge of content [7]. This will require a major pivot because the culture of most mathematics education courses is not to concentrate on mathematics content—numbers and operations (including place value and fractions), algebra, geometry, and data and probability. While true in our home province of Ontario, the phenomenon is widespread. In fact, two studies by the National Council on Teacher Quality (a US body) found

…that just 13 percent of the 860 undergraduate elementary teacher preparation programs reviewed covered critical math topics, including numbers and operations, algebra, geometry, and data and probability [16]. And in 2018, a similar review of graduate programs by NCTQ found that just 1 percent of 201 programs covered these topics [17]

Much of what we have defined as specialized mathematics content knowledge is what Dan Lortie called the crucial “backstage” skill set that teachers possess and apply on the “frontstage” known as the classroom [18]. By this, Lortie meant having the knowledge and schemata that transform a person from being a subject matter “knower” to a subject matter “teacher.”

If we agree with educational psychologists like David Berliner, then we must acknowledge that domain-specific knowledge is a characteristic of an expert in any field [19], including the mathematics teaching profession. The development of expertise requires time, appropriate mentorship, and the integration of disciplines and “laboratories” across which and in which knowledge translation occurs. This last sentence clearly describes the rich preparatory program that already exists in the innovative Translational Institute in Medicine (TIME) model, and one upon which the criteria proffered by Shulman, Berliner, Ball, Bass and others could be actualized given its interdisciplinary nature of linking coursework from multiple specializations with periods of formal observation to enhance the development of professional thinking and action.

Although the TIME model is currently aimed at graduate (M.Sc. and Ph.D. students), the program of 12 courses (three of which are mandatory and three of which are elective), could be adapted to a 3-or 4 year Translational Institute in Mathematics Education program that “packages” courses that would include mathematicians, pedagogues, psychologists, cognitive scientists, neuroscientists, qualitative researchers and experienced mathematics classroom teachers. By so doing, pre-service students would develop, over a rigorous, research-based and holistic preparatory experience, the foundational knowledge and practical experiences that are pre-requisite for the mathematics classroom.

As with the TIME program in medicine, over an extended professional program, in the mathematics education derivative program, novice teachers will find themselves in focused seminars with expert academics from multiple fields whose research illuminates mathematics as well as its teaching and learning. They will then be tasked with integrating constructs, theories and content by reporting back and reflecting on the immersive observational opportunities where they experienced expert teachers engaged in the complex exercise of “teaching” while novice elementary and secondary school-aged students endeavor to “learn.” Unlike many current teacher education programs, in which learning to teach is the cursory “add on” after graduating with a general undergraduate degree, in a Translational Institute in Mathematics Education program, teacher candidates’ learning would be unified and purposeful from the start. All classes, whether in geometry or child development, will be conducted with complementarity⁶, not in parallel stream, so that future educators learn not only about mathematical content but experience, from the “other side of the desk,” the mathematical trajectories, advantages and pitfalls of lessons taught with concrete materials during authentic activities and lessons with peers and school-aged students. They will experience and in turn, acquire, techniques to formulate questions that will prompt mathematizing of content. They will “play” with the “thinker tools” commonly called manipulatives to see how mathematical explanations using multiple representations move from concrete to abstract. Through careful monitoring and questioning by the instructor, and in-depth sharing with peers, both the developmental progression of the mathematical ideas, and the alternate routes for these to emerge, will be explored and unpacked. Through this process, candidates will encounter their own misconceptions or gaps in understanding the mathematical development process and apply their problem solving through the hard task of promoting students’ understanding, learning to orchestrate their way through children’s misconstructions and remediating accordingly.

Such outcomes are not achievable given the typical requirement of one or two courses in mathematics education offered in our province’s consecutive (post-degree) teacher certification program. We argue, that in their current format of generic “education” courses, “concurrent” education programs, running in a totally disconnected stream from Arts & Science or Engineering degree requirements, for example, are also largely inadequate.

We have had good results from offering at least one course in mathematics for teachers (using our SMT definition) prior to explicitly focusing on more typical pedagogical topics such as lesson design and assessment. We have found [9], that even short content-rich, representation-focused experiences are effective in influencing the values of prospective teachers [11] in terms of what is important for students to know, and how they might begin to nurture such student understanding. Imagine, then, the impact of an extended, immersive TIME program.

11. Conclusions

Given that teachers’ knowledge of the conceptual underpinnings of mathematical concepts has a direct influence on student achievement [5], a concerted effort to define more clearly and articulate SMT has been the purpose of this chapter. Unclear articulation and ongoing (mis)understandings of prospective teachers’ mathematical needs has been a problematic theme throughout both of our careers and continues virulently in our region. Inadequate teacher preparation in mathematics can only result in unsatisfactory mathematical learning by students, ultimately giving rise to what is commonly referred to as “math anxiety” in both students, and eventually their teachers. In our experience, prospective teachers who define themselves as “anxious” about mathematics are adamant and rational about the issue: they know they do not understand mathematics well enough to teach it, and they are fully aware of this problem yet are prepared to head into a classroom where they will be solely responsible for teaching mathematics, after the most cursory preparation. Fortunately, when supported in developing a fuller understanding of mathematics (at least of a few overarching concepts), such conceptions of themselves can change greatly. Indeed, the positive transformation, both mathematically and in terms of self-efficacy, can be quite remarkable, but is unfortunately limited to only a few teacher education faculties who have SMT as a focus and thus, beneficial to only a handful of graduates. The “anxious” students become the new “anxious” in-take in education programs, i.e., the ones who ask, “I’m going to be in your mathematics education course starting in September. We’re just going to learn how to teach math, right? We’re not going to have to do math are we?” Sadly, it is not an exaggeration to say that we both keep tissue boxes on our desks.

We can break the cycle but only if we give teachers the confidence to assume their roles in the mathematics classroom—in other words, beginning with the end in mind, which is, in our opinion, a teacher whose SMT is not in question. A teacher who understands how representations—including manipulatives, drawings, other models, virtual environments, or even at times, movement or hand motions—can be used to support the development of particular mathematical ideas. A teacher who understands how mathematical ideas themselves evolve and require different tools across the grades, yet build upon each other in increasingly sophisticated and ultimately abstract ways. And a teacher who understands that supporting the learning of this content requires new types of problem-solving strategies and reasoning.

We offer our conception of SMT to define the “vision of the mathematics teacher,” by describing the skills and knowledge teachers should embody and exemplify on graduation from a mathematics education program. We believe that this will be useful in the promotion and creation of better program design, and course design, particularly mathematics for teachers’ course content, all of which would ultimately result in improved mathematical understanding for students from Kindergarten to Grade 12.

Based on earlier concepts of Shulman, and further articulated by Bass and colleagues, we have described specialized mathematical content for teaching as a distinct mathematical domain, with unique requirements, which are tied directly to the student learning process and informed by multiple fields of academic study.

It’s about TIME (literally and figuratively) for SMT to be an important requirement in mathematics teacher education. As we have argued above, such teacher knowledge is a critical component of classroom mathematics learning for students. If the academic and “clinical” standards for and preparation to be a mathematics teacher—a highly specialized profession regardless of the grade and age of the students—are to shape positive change in the post-secondary education lecture hall or Grade 3 classroom, there is no time to waste.