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# Deborah Loewenberg Ball: Teaching/Learning Mathematics Teaching

Communicated by *Notices* Associate Editor William McCallum

To its credit, *Notices* has been publishing a series of profiles celebrating the work and careers of several accomplished women in mathematics. Deborah Loewenberg Ball might, at first consideration, seem an unlikely entry into that august company, in that neither her education nor her early life interests exhibited particular mathematical inclinations. Indeed, her first serious engagement with a challenging math problem came after she had graduated college, where she had majored in French. How she addressed that problem is an important part of her story. I write from the perspective of a colleague, friend, and collaborator.

## 1. Family and Early Education

Let’s start from the beginning.

Deborah’s family heritage is part of a long line of German Jewish intellectuals. Those who survived Nazi persecution fled to the United States and other countries. Her great-great-uncle, Ernst Cassirer, was a prominent twentieth-century philosopher, who embraced both sides of the Neo-Kantian division between the natural and the human sciences, a sensibility also present in Deborah’s work. Her father, Gerhard Loewenberg, was an eminent political scientist, who developed the field of comparative study of legislatures, especially in postwar Europe. He was also a much-appreciated professor and academic leader. Deborah’s mother, Ina, is broadly talented—a scholar in philosophy, a certified public accountant, a published poet, an accomplished photographer, and a polyglot. Deborah’s one sibling, Michael, is a professor of chemical engineering at Yale. Her husband of more than 50 years, Richard, has also been a teacher, and their three children, Sarah, Joshua, and Jacob, all educators in different contexts, have been central to her life and her learning.

Deborah was born in March of 1954 just as her father began his 16 years on the faculty of Mount Holyoke College. Her early education was in South Hadley, Massachusetts, where she was one of only two Jewish children in her elementary school. She read fluently by age four. Her first-grade teacher was unsure how to deal with her and awkwardly arranged for her to read current events materials aloud to the sixth-grade class. Also when she was four, her father teasingly challenged her to prove that the two of them were related. Her “proof” can be seen in Figure 1, in which she even appeals to her birth certificate.

During her father’s sabbatical in Germany, to study the evolution of the Bundestag, Deborah, then age 7, attended public school and became fluent in German. In the late 1960s, her father accepted a faculty position at the University of Iowa, so Deborah finished high school in Iowa City. While in high school she also spent a term in Zurich on a student exchange program. Her only science-related courses were Algebra I and biology. Instead, she immersed herself in the humanities—English and languages, studying French, Spanish, and German.

Having completed two years of college work in high school, it took Deborah only two more to earn a BA at Michigan State University (MSU) in 1976, majoring in French and elementary education. Because of her love of language, she studied Spanish and German as well as French. Realizing that she didn’t want to become a translator, she decided to pursue teaching, choosing elementary education because elementary teachers teach all subjects, and because of Michigan State’s excellent teacher education program.

Having always felt herself to be somehow “different,” Deborah was deeply interested in people—in how others think and in their experiences and perspectives. In her travels she met people of different identities and cultures, becoming aware both of enormous gaps of experience and opportunity and of the histories and social forces that shaped inequality and injustice. Across time, she had also held a range of jobs: as a cocktail waitress in a bowling alley; as an accounts tabulator at Kmart; and in a bakery, eventually as a cake decorator. Baking became a continuing métier for her, from which her students and friends continue to benefit. In each work environment, she noticed that, though the educational opportunities of her coworkers were often less than hers, they skillfully performed complex work involving rapid mental arithmetic calculations, substantial memory retrieval, complex relational work, and complicated multitasking, both physical and mental. It struck her how accomplished, yet how little noticed or appreciated, was their everyday skilled performance. This deepened a sensitivity to equity that has infused all of her life’s work, especially teaching.

## 2. Early Teaching and Teacher Mathematical Knowledge

For 13 years (1975–1988) Deborah taught public elementary school, grades 1–5 in East Lansing, Michigan. Her husband, Richard Ball, was a middle school teacher. Deborah and Richard found that their experiences as parents infused their work as teachers and reciprocally. Deborah loved teaching and felt generally satisfied with her progress in developing as a teacher, with the exception of one subject: mathematics. She was puzzled and frustrated that things the children seemed to understand on Friday were often forgotten by Monday. She began to suspect that there were important things about teaching mathematics that she did not understand. Inspired by educational philosophers like John Dewey, Joseph Schwab, and Jerome Bruner,Footnote^{1} she realized that this must begin (but not end) with a more robust knowledge of mathematics itself, something thus far underdeveloped in her education. So, she took and performed well in a four-term calculus sequence, the first two at Lansing Community College, the final two at MSU, plus a number theory course that she loved. This study enabled her to see situations that came up in her teaching in a new light. For example, one day in her first-grade class, she had the children try to measure the area of their hands. They traced their hands on graph paper and then tried to count the grid squares within the outline, with some improvised procedures for squares on the boundary. One child suggested getting the graph paper that the fifth graders used, with smaller squares, so that they might be able to get a more accurate measurement. At that time, Deborah was taking integral calculus, so she recognized that the child was noticing something that was a significant mathematical insight. How should she respond? Not by giving a mini-discourse on calculus, or even on the subtleties in defining area. She responded, “That’s a really interesting idea. Why don’t you get the fifth-grade graph paper, and let’s try it?”

^{1}

Deborah was inspired by Bruner’s famous declaration, “We begin with the hypothesis that any subject can be taught effectively in some intellectually honest form to any child at any stage of development.”

During part of these years of teaching elementary students, Deborah was a doctoral student in the MSU College of Education, majoring in curriculum, teaching, teacher education and professional development, and policy with a minor in mathematics education. For her 1988 doctoral dissertation, titled “Knowledge and reasoning in mathematical pedagogy: Examining what prospective teachers bring to teacher education,” she developed an interview protocol that she used to explore the mathematical understanding, as well as the ideas about learning, teaching, and students, that preservice teachers held before even beginning their professional training.Footnote^{2} Here is an item from a survey, used in a study of content knowledge for teaching, that she later developed from an open-ended interview in her dissertation:

^{2}

Her interview protocol was later used by Liping Ma in a comparison of mathematical knowledge of Chinese and American teachers.

Which of the following story problems can be used to represent the meaning of divided by ?

- a)
You want to split pies evenly between two families. How much should each family get?

- b)
You have $1.25 and may soon double your money. How much money would you end up with?

- c)
You are making some homemade taffy and the recipe calls for cups of butter. How many sticks of butter (each stick cup) will you need?

Few of the study participants selected the correct answer. Most surprising was that even the mathematics majors were unable to explain the meaning of division by .

## 3. Making Teaching Practice Accessible to Study

Next, as an assistant professor at MSU, Deborah continued to teach elementary school every day, then just as the math teacher in Sylvia Rundquist’s third grade class. Together with her colleague, Magdalene Lampert, who was also teaching fifth grade in Thom Dye’s class, they set out to conduct a major study of “the work of teaching,” with a special focus on mathematics teaching. Often, the scientific study of a phenomenon proceeds first by close and systematic empirical observation of the phenomenon of interest, with detailed data gathering. A characteristic feature of classroom teaching presents an obstacle to this: it is typically private, carried out behind closed doors. A few accomplished teachers are sometimes featured as exemplary, to be emulated, much as listening to operas might equip you to sing arias. This is unlike most other skilled professions—like medicine, nursing, architecture, art, acting, aircraft piloting, plumbing—in which extensive supervised actual or simulated performance is the norm as part of professional training. Teaching, instead, is often positioned as something one can just learn on one’s own, through experience. Deborah argued that even more important is that teaching be carefully taught, practiced, and coached. Ball and Lampert set out to design ways for others to engage with the work of teaching, with its complexity and dilemmas, through investigations of primary records of practice.

What Deborah and Magdalene proposed to do was to make their actual practice public, not as a model of exemplary teaching, but to produce a corpus of primary data with which to empirically study the work of teaching. They proposed to the NSF to fully document an entire academic year of their teaching, Deborah of third grade, Magdalene of fifth. The school in which they were both teaching had a diverse student body and teaching faculty, including Black, Latine, and white children, of many different cultures and different home languages. The records they proposed to collect comprised videos with transcripts of every class, student notebooks, teacher journals, homework and quizzes, and interviews with the children. The complexity and scale, innovative use of then available technology, and concomitant expense of such an undertaking were unprecedented. It is a tribute to the vision of the NSF program directors that they decided to fund this. Data were generated during the 1989–90 academic year, and the outcome more than fulfilled its promise. It provided a primary data source for Magdalene Lampert’s seminal book, *Teaching Problems and the Problems in Teaching* (2001, Yale University Press) as well as the materials on which Deborah drew as she sought to study and analyze the mathematical work of teaching.

These records of practice were deployed as a kind of empirical “text” for others to study the work of teaching as it relates to various kinds of research questions. Specifically, Deborah wanted to understand the nature and form of mathematical knowledge that teaching mathematics entails. Of course, this would depend in part on the mathematical topic and the level of the students. But there should be some mathematical practices, ways of thinking, and sensibilities that apply across these different contexts. Deborah was aware of the ways that teachers and other educators commonly thought about these questions. But, going back to the influence of Dewey, Schwab, and Bruner, she felt it important to incorporate the perspectives of mathematicians as the disciplinary experts. Moreover, rather than ask for their disengaged opinions and reflections on what mathematical knowledge is needed for teaching children, she was now in a position to invite them to examine primary data (especially video) of teaching, and to ask, more concretely, “What mathematical issues do you see in these classroom episodes?” Of course, the problems occur in contexts that involve more than just the mathematics, as do the solutions. These questions could not be adequately answered with intellectual speculation and reflection, or based on university-level mathematics. But, she believed, mathematicians might see important mathematical aspects of events that might remain invisible to others.

To this end, Deborah enlisted several mathematicians, of whom I was one, to examine these records. Others included, for example, Peter Hilton, Herb Clemens, and Phil Kutzko. I was asked first to look at an interesting episode from the January data. Recall that the data ranged from September until May, so January was midyear. I was quite excited by the rich mathematical thinking and interactions of the children. I thought that these children showed amazing mathematical curiosity and engagement. I later more fully appreciated the importance of the many subtle teacher moves that supported the children’s thinking and discourse. The longitudinality of the data allowed me to also “visit” the first classes, in September, when I saw the mathematical tasks and the teacher’s mathematical prompts to which the students were just beginning to respond. It was somehow the teaching that supported these September-children to grow into the January-children, and the comprehensive data allowed me to analyze the pedagogy that supported that transformation.

I continued this study of the 1989–90 data, recorded through annotations of the transcripts, from my office in the Columbia Mathematics Department. Deborah moved from MSU to the University of Michigan (U-M) in 1996, and I spent a spring 1997 sabbatical at U-M to work with Deborah’s research group. After returning, I found it difficult to continue this work remotely, away from the research group, so I moved to U-M in 1999, with a joint appointment in the Department of Mathematics and the School of Education.

## 4. Mathematical Knowledge for Teaching

Over the next several years, Deborah led her group in the development of a theory of “mathematical knowledge for teaching,” (MKT). Unlike other theories that claim such a title, the knowledge base here was an in-depth empirical study of teaching practice. The theory was framed as a refinement of Lee Shulman’s seminal work on “pedagogical content knowledge” (PCK) and was represented in the “egg” diagram in Figure 3.Footnote^{3}

^{3}

From D. L. Ball, M. H. Thames, and G. Phelps, *Content knowledge for teaching: What makes it special?*, Journal of Teacher Education **59** (2008), no. 5, 389–407, http://dx.doi.org/10.1177/0022487108324554.

The right half is PCK, as a kind of hybrid pedagogical and content knowledge developed by Shulman and his colleagues at Stanford (Shulman, 1986; Wilson, Shulman, & Richert, 1987). The left half presents a new decomposition of “pure” content knowledge, which was a unitary component in Shulman’s framework. A novel feature is the prominent appearance of “specialized content (in our case, mathematics) knowledge” (SCK). This new construct designated a knowledge of mathematics not involving pedagogy that is needed for teaching but is not typically needed for, or known by, other professionals, including mathematicians. Its discovery posed new questions about mathematics content courses taught in mathematics departments for future teachers. Where, and how, would SCK be taught?

In order for the theory of MKT to have some bearing on instruction and on student learning, it was important to have measures of MKT and of desired outcomes. To this end, Deborah and Heather Hill, together with Mark Hoover, and several talented doctoral students, led the development of measures of MKT, and another of “the mathematical quality of instruction” (MQI). For student learning outcomes standard measures were used. Working with colleagues Brian Rowan and Stephen Schilling who, along with Heather Hill, brought extensive quantitative analytic expertise, the team was able to show that teachers’ MKT, as they conceived it, is related to student learning outcomes.

## 5. Teacher Education: The Work of Teaching

Given this line of research, a new question emerged: Where and how could teachers develop this kind of mathematical knowledge for teaching? The teaching population in the US numbers close to four million. It is America’s largest occupation. This scale made clear to Deborah that this could not be an isolated agenda. It had to be integrated into a fundamental transformation of US teacher education and professional development—something perhaps analogous to the reform of US medical education, inspired by the Flexner Report in the early 1900s. Her approach was to ground this in an in-depth and fine-grained analysis of the work of teaching, something that was so far, perhaps unsurprisingly, largely understudied. This launched the next phase of Deborah’s work.

Paralleling the work of Lee Shulman, Suzanne Wilson, and Pam Grossman, Deborah noted differences in the preparation of practitioners in other skilled lines of work—for example, surgeons, nurses, airplane pilots, and plumbers. This typically involved decomposition of the work into distinct, more directly learnable practices, and then also learning fluent integrated performance through both close observation of practice (think of the surgical theater) and coached rehearsals and simulations.

During this period Deborah was appointed dean of the University of Michigan School of Education (2005–2016), and the redesign of its teacher education program was a major part of her agenda. To this end, she began by enlisting the faculty in a two-year analysis of the basic everyday practices of teaching. A list of close to 100 such practices of various grain sizes and complexity was identified. This diverse list was clearly impractical as a foundation for teacher education. So, for elementary teacher education, the next phase was a process of distillation and consolidation to arrive at a short list of what they termed “high leverage” teaching practices, and the criteria for choosing them. The resulting, and still evolving, list of nineteen high leverage practices (HLPs)Footnote^{4} included things like:

^{4}✖

(HLP 1) leading group discussions;

(HLP 2) explaining and modeling content;

(HLP 3) eliciting and interpreting individual students’ thinking; and

(HLP 11) communicating with families.

Of course, “content” refers traditionally to school subjects like language arts, mathematics, science, social studies, etc. To organize the teacher education program around these high-leverage practices required reconceptualizing the courses. Each content area course would address a small subset of the HLPs, grounded in that specific content.

To more broadly and enduringly amplify this reconceptualization of teacher education, Deborah founded and directs TeachingWorks,Footnote^{5} an organization housed at U-M that designs resources and practice-based approaches to teacher education that support educators to enact equitable teaching practice that nurtures young people’s learning and actively disrupts patterns of injustice.

Though Deborah’s innovative scholarship remains prolific, she already received, in 2017, the Felix Klein Medal for lifetime achievement in mathematics education research from the International Commission on Mathematics Instruction (ICMI). This is the highest honor worldwide in mathematics education research. The three most highly cited mathematics education scholars in the world are Deborah Ball (73585); Paul Cobb (51794); and Alan Schoenfeld (45544), each of them also a Klein Medalist. And Deborah co-authored the two most highly cited journal articles in mathematics education research, both related to MKT.

But, as has already been indicated, Deborah’s interests in education are broader than mathematics education. In 2017–18 she was president of the (25,000 member) American Educational Research Association (AERA). In her presidential address,Footnote^{6} she introduced and vividly illustrated the notion of “discretionary spaces” in teaching. Drawing on the work of political scientist Michael Lipsky, this idea illuminated how policies and norms, no matter how prescriptive and specified, inevitably leave room for interpretation and (conscious or unconscious) consequential choices by on-the-ground actors that ultimately shape the enactment of the policy or practice. These discretionary actions by teachers can not only profoundly affect student learning, but also either reinforce or disrupt racism and other forms of oppression that infiltrate classrooms from the ambient society, culture, and institutions.

## 6. Deborah the Teacher

If there is one thing that best characterizes Deborah as a person, a scholar, and a leader, it is that she is quintessentially a teacher. She started with 17 years teaching elementary grades in East Lansing schools, partly during her doctoral studies and postdoctoral work at MSU. She has since taught a range of university courses, such as methods courses for teacher candidates, foundations and policy courses in the graduate program, and courses for students across the university with interests in education. Her courses draw many students and she earns high ratings for her teaching and her support of students. She regularly leads professional development workshops around the country. Further, for two decades now, she has taught 10-year-old children each summer in the Elementary Math Lab (EML). I say more about this below.

Teaching (and learning) for Deborah is a calling, a life practice, her way to interact with the world, to nurture and empower people of all ages, stations, and identities, and to disrupt institutions and practices that oppress them. She assumes that *all* children bring substantial knowledge and skills, and so considers it a core task of the teacher to elicit that and use it as a foundation for what new material the children are ready to learn. This contrasts with the negative orientations of much teaching that makes deficit judgments of children’s competence, and with testing that tests not what children know, but what they don’t.

*Deborah as leader*: Carla O’Connor, in introducing Deborah’s AERA Presidential Address, said:

Her teacher identity also permeates how she works as a leader and as a public servant. As dean of the U-M School of Education, where I had the privilege of serving as her associate dean for four years, I witnessed how Deborah interpreted every meeting and every convening as an opportunity to cultivate communities of learning and action. She established the goals of the convening, assessing who would be in the room—the identities, knowledge, and resources they brought. And, in cooperation with her cofacilitators, she would determine which instructional tools and strategies could be most powerfully leveraged, when and by whom, to support the stated objective. And all of this was outlined clearly in her written “lesson plan.”

I can testify to this. Deborah always honors and takes responsibility for the quality of the time she asks her colleagues/students to meet. Her well-attended faculty meetings were edifying and engaging, on themes from the school budget, to review and reform of our instructional programs, to equitable practices, and more, often enlisting outside expertise.

*Deborah and STEM*: These skills enriched every environment she entered, even those whose culture and knowledge base were remote from her own. For example, even through the so called “Math Wars” and beyond, Deborah remains one of the most highly respected mathematics educators by the research mathematics community. She participated in the “Reaching for Common Ground…” effort (AMS *Notices*, Oct. 2005). She is a fellow of the AMS. More pointedly, Deborah has twice been appointed to the National Science Board (NSB), a presidential appointment. The NSB is the governing board of the National Science Foundation. It covers the full range of sciences and typically has only one education member (and at most one mathematician). An educator has a limited role in its planned agenda, and often a little-heeded voice otherwise. Deborah’s reappointment was welcomed by its members because of the clarity and focus she brought to their deliberations. Even on topics for which she had little technical expertise, she demonstrated great listening, learning, and enabling skills, these being qualities not always highly developed in the other members’ environments.

Let me next offer some vignettes of Deborah’s classroom teaching.

*Teaching teaching place value*: This is a methods class for intern elementary teachers, in which Deborah is introducing them to the teaching of place value in the early grades. I will describe this in what may seem like excessive detail, but there is a point to this. My description draws from a video recording of one such class that I also observed firsthand.

To simulate a lesson, Deborah as teacher (T) and a dozen of the teacher candidates in the role of children (C) sit cross-legged in a circle on the floor. Another dozen or so teacher candidates sit at tables around them to observe and take notes on the floor dialogue and actions. A box of more than two thousand popsicle sticks is poured into the middle of the circle. T asks, “How many do you think there are?” Various Cs offer (widely varying) guesses. T: “How can we find out?” A few Cs say, hesitantly, “Count them?” T: “Ok, let’s do that, … together.” Then the Cs begin eagerly gathering individual sticks and counting them. It quickly becomes apparent that they are running out of room for the sticks and finding it hard to keep track of the counting. Some of the Cs say they could collect the sticks into bundles. But another C asks, “How can we hold the sticks in a bundle together? T says, “We can use rubber bands,” and she presents a box of them to distribute to the Cs. At this point, T “exits” from the lesson simulation and asks the teacher candidates on the floor, “What do you think the (real) Cs are going to do when they get the rubber bands?” There is some discussion of the Cs using them as slingshots, and related behavioral issues. They then return to the simulation, and T demonstrates how to use a rubber band to bundle several sticks. But how many sticks should be in a bundle? It becomes clear that the bundles of each C should be of the same size, thinking of eventually combining their collections. But what size? This engenders some lively discussion: Two would be too small (too many bundles); but also we don’t want them too big. Five is a good possibility. Then T urges 10, but emphasizes that this is a choice, and it could as well be otherwise. Then the counting resumes, but now with bundling. During this activity, T asks a variety of questions. T holds up a bundle and three sticks and asks, “How many is this?” A C answers, “13.” T: “How do you know?’ T does not accept “10 + 3 = 13,” but asks for a direct proof—unbundling the 10 and counting all the loose sticks. T also asks questions like, “What are different ways to show me 24 sticks?” All of this happens while the counting goes on. After a while, the same crowding problem as before emerges; too many bundles! This leads to “super-bundles.” But how many bundles in a super-bundle? Again, there is a *choice* to be made, and T proposes for consistency to make it 10. There ensues a sequence of now more complex representation questions: “How many sticks in two super-bundles, three bundles, and four sticks?” “Show me 302 sticks.”

Finally, all the sticks are counted, and the collections of each C are gathered in the center. The assembled individual sticks are consolidated into more bundles, and then the assembled bundles are consolidated into more super-bundles. But now there is an abundance of super-bundles. And, after some discussion, they decide to form “mega-bundles” (comprising 10 super bundles). In the end, they have two mega-bundles, five super-bundles, three bundles, and seven individual sticks. And so, the count of the original pile is captured by these four digits: 2537.

Afterward, the lesson is repeated with the two groups of interns exchanging roles. Two big features of this lesson design are worth noting. Rather than directly presenting the teacher candidates/children with symbolic place value notation (this is the ones place, the tens place, …, names to be memorized) it creates a context in which (1) they organically construct/invent a place value system, and (2) they are viscerally able to appreciate the amazing power of notational compression so achieved. Moreover, the bundling stick model is a faithful physical representation of the mathematics; bundling and unbundling are just ways of organizing the physically invariant collection. Performing addition and subtraction in this model gives direct meaning to the symbolic processes of “carrying” and “borrowing”—or, more properly, “regrouping,” carried out physically with two- and three-digit numbers. This contrasts with the base-ten blocks representation, which involves physical exchange in place of unbundling, with cardinal invariance shown by geometric measurement. Finally, one arrives at symbolic place value notation, with its compactness and computational efficiency, achieved with formal algorithms. The symbolism is abstract and remote from its mathematical meaning, but it arrives with a concrete foundation into which it can be translated when needed.

Deborah’s instruction included systematic analysis of these representation systems, which progressed from making the mathematics transparently and directly meaningful but cumbersome, to the powerfully compressed and efficient but opaque in meaning symbolic representation. When children might encounter difficulties in the symbolic system, the teacher candidates were learning to diagnose them and return to a more transparently understood model to help a child understand the meanings of the symbolic manipulations they were making. The teacher candidates were taught to rehearse such explanations, to talk about this and ask questions that could open insights for young learners.

*The broken calculator lesson*: Deborah was invited to teach a lesson to a third-grade class of mostly Black children in Detroit, whom Deborah had never met. When Deborah showed the regular teacher the lesson plan, she expressed reservations about whether the students could actually handle the lesson.

The class was high energy, with lots of sound and motion. Deborah gave them a picture of a hand calculator and explained that the 2 and the 5 keys were broken. She asked them, “Can we still use this broken calculator?” After eliciting various student opinions, she posed a sequence of addition and subtraction problems, to be performed on the broken calculator. For example, if the 2 and the 5 keys were broken, how might one use the broken calculator to add 32 + 51? The children successfully engaged with these substantial challenges with great enthusiasm.

What mathematics learning was supported by this activity? Given some arithmetic expression to calculate, the children had to construct an equivalent expression (having the same value) that did not involve the digits 2 and 5. For example, 32 + 51 = 33 + 50 = 33 + 40 + 10. More than arithmetic practice, this implicitly involved early algebra skills and equivalence of arithmetic expressions.

With regard to the regular teacher’s expectations, Deborah has always, when meeting students she did not know, chosen to *not* look at previous teachers’ or others’ judgments about the mathematical capabilities or potential behavior issues of the individual students. For her, each child comes with high promise and no presumed deficiencies. This is a crucial stance to take, especially for Black and Brown children who have been historically marginalized and pervasively seen by their white teachers through deficit lenses.

*Proving mathematical impossibility:* I turn next to a third case: Deborah teaching a summer program class (the EML) of racially and linguistically diverse soon-to-be fifth graders. The goal of the program is growing the children’s mathematical identities through a focus on their skill as mathematical thinkers in preparation for fifth grade. The curriculum she developed involved a combination of foundational work on fractions, including their placement on the number line, plus some novel and challenging problem-solving activity. In this regard, the reform mathematics education literature has urged the teaching of practices like reasoning and proving across all grade levels, K–16. But this has proved difficult to achieve even at the undergraduate level. Deborah has developed a complex problem called “The Train Problem” on which they work, often in small groups, over several days. It is a nontrivial problem even for most adults.

A plain mathematical statement of the problem goes as follows. Consider the set of numbers, S = {1, 2, 3, 4, 5}. Note that The Train Problem has two parts. (I) Can every (whole) number . be expressed as the sum of a subset of S? (II) Is there an ordering T = (a, b, c, d, e) of S such that every number 15 is the sum of some consecutive terms in T? In what follows, I describe how Deborah stages this problem for the children and, in particular, how she motivates them to seriously engage with it. I draw from one particular year in which I observed the development of the children’s work.

The children worked with Cuisenaire rods. These have dimensions (1 cm) x (1 cm) x (n cm) (from n = 1 (white) to n = 10 (orange)). Deborah had them make “trains” (a sequence of rods placed end-to-end).

She told the children that each rod represented a train car with passenger capacity corresponding to its length, and the passenger capacity of a train is the sum of those of its cars. She set the class up as a collective to be called the “EML Train Company,” a company that specializes in making trains using only the cars: white, red, light green, purple, and yellow, each one at most once.

Deborah asked, “What’s the biggest (most passengers) train you can make?” “The smallest?” After a small calculation, they found 15 to be the largest. For the smallest, the children began to engage in an interesting discussion of whether there could be an “empty train.” Deborah then asked if they could make trains with any number of passengers between 1 and 15. Collectively, they constructed trains for each number, and in some cases several trains for some numbers. Incidentally, this activity involved a lot of practice with basic arithmetic as well as the construction of a table to record their findings. At this point, the children felt pretty confident and fluent at train building, and Deborah praised them as being a really smart train company.

She then announced that there was a “customer” who had heard of their high reputation and wanted to hire them to build a special train. This excited the children. In fact, the customer was an adultFootnote^{7} recruited by Deborah, who appeared before the class and announced, “I’ve heard that you’re a really smart train company. I would like you to build me a special train that uses each of the five different size cars, each one only once. I also want to be able to break apart the special train to make smaller subtrains. But I want these smaller trains to be formed of cars that are next to each other in the special train. And I want to be able to make such a subtrain for every number of passengers between 1 (just the white car) to 15 (all five cars).”

After he left, Deborah worked with the children to unpack the several complex conditions of the problem. Pairs of children constructed various trains to see if they worked. For each train, they had 15 numbers to check, but they soon recognized that 1, 2, 3, 4, 5 (single cars) and 15 (all cars), come for “free,” for any train. So, they only had to worry about 6, 7, 8, 9, 10, 11, 12, 13, 14. The children tested these, starting up from 6. For each train, they were able to find, after significant effort, that some numbers could not be made. This work stretched over a couple of days amidst other mathematical work. There was a risk of the children becoming discouraged, especially when the customer visited and asked, “When do you think you’ll have my train?”

While this is a finite problem, therefore, in principle answered by examining all cases, it is a BIG problem: There are 5! = 120 possible trains, and each train requires testing for nine different passenger sizes (6–14 passengers). After some intense work by the group, Deborah contributed a clue. She simulated publicly a phone call from an assistant of the customer, suggesting that it might be important to begin by trying to make trains that would work for the big numbers, such as 13 or 14, instead of just always beginning from the lower end, at 6, 7, etc. She asked whether the children wanted to try this. After further exploration and effort, some of the children realized that the only way to get 14 was to remove 1 (the white car), and that this would be possible only if white is on the end; otherwise, its removal would disconnect what’s left. Similar reasoning shows that the only way to get 13 is to remove the red car. And so, the only possible trains must have red and white on the two ends. Turning the train around, if necessary, one can assume that red is the left end and white the right end. This is a big step in reducing the scope of the possible solution space of the problem. These red-white-ended trains are then determined by the ordering of the three middle cars: green, purple, and yellow.

Prior to working on the train problem, Deborah had given the task of finding all three-digit numbers they could make with the digits 3, 4, and 5 using each one only once. They found all six possibilities but Deborah pressed them to show (prove) that there couldn’t be any others. The children had produced a range of convincing arguments for this.

The children were now able to retrieve (transfer) this earlier work to determine the six possible three-car trains using the green, purple, and yellow cars, each one only once. Thus, they had reduced the train problem to a manageable size, examining six (not 120) trains! These reasoning steps were dramatic, challenging, and transformative. Deborah had them make the six possible trains and then formed the class into six groups, each group to test if its train worked. Collectively, they reached the conclusion that none of the six trains worked to meet the customer’s order and the constraints of the problem. The problem was impossible to solve!

Mathematicians might judge this to be a mathematical success. But this predicament left some of the children still uncertain. They had failed to make the customer’s train and, given past experience, they worried that they were failures. They were disappointed; maybe they were not as smart as the customer had expected. “Will he be mad?” they asked. Deborah assured them that they had done very important and valid mathematical work. One student wondered, “Do you think that someone else might be able to make the customer’s train?” After extended discussion, they began to realize what they had accomplished: No one, no matter how “smart,” could make the train. In fact, they had done very smart work on this challenging problem.

The customer came to “receive” his train. Deborah discussed with the class how they might respond. They would have to show not only that they could not make his train, but also that it was impossible. And they would show this mathematically. Deborah formed the class into groups to prepare their presentations of parts of the proof and explanations of the various stages of the complex argument.

Before the customer’s arrival, Deborah asked the children what they would do if he said, “Well, I know some pretty smart mathematicians, and maybe I’ll ask them to make my train.” One of the girls responded confidently, “You’d be wasting your time and money.”

## 7. The Elementary Math Lab (EML)

The Park City Mathematics Institute (PCMI), sponsored by the Institute for Advanced Study, has a “vertically integrated” structure, in that it assembles several dynamic programs each summer in a common physical environment: one for active mathematics research development involving both graduate students and leading world experts; one for undergraduates and undergraduate faculty; one for mathematical enrichment of high school teachers; etc. In 2004, Herb Clemens, then PCMI director, invited Deborah to contribute an elementary education component to this mathematically rich and intense environment.

This was the birth of what has become the EML, which moved to the University of Michigan in 2007 and is now under the sponsorship there of TeachingWorks. The above discussion of the “Train Problem” is based on EML data. Resonant with her early work with Lampert, Deborah conceived the EML to be a “theater of public teaching” as well as a laboratory in which teaching was a domain of empirical study and experimentation.

In the EML, Deborah teaches a diverse class of about 30 rising fifth graders for two weeks, with an ambitious curriculum that combines foundational topics with high-level mathematical thinking and reasoning. The teaching is directly observed and studied by teachers and educators who come from around the country. The days begin early, when Deborah shares with the observers the lesson plan for the day’s (2+ hours) class. She answers questions and invites feedback, including suggested modifications. Then the class is (silently) observed, followed by inspection of the children’s notebooks, and then a debriefing with Deborah. This highly organized and multiply purposed structure has some features in common with Japanese Lesson Study, but it is much more complex. All of this activity is professionally recorded and documented. The afternoons offer several professional development options for the observers. The children are given some tutoring sessions by STEM undergraduates, focused on puzzle solving or other topics, followed by some art-related activity. Beginning in 2019, Deborah was joined by coteacher Darrius Robinson and the two of them have developed a team-teaching approach in working in this special program.

The EML is yet another manifestation of Deborah’s essence as a teacher. But it is also an expression both of her institutional creativity and of her continuing commitment to making teaching accessible to primary observation and study.

## 8. Epilogue

Deborah Loewenberg Ball is an educator—a teacher, a teacher of teachers, and, in particular, a teacher of mathematics. And she is a well-recognized and honored leader, scholar, and practitioner in each of these domains. However, her profile differs from those of other women featured in the *Notices of the American Mathematical Society*.

Mathematicians are frequently critical of school mathematics education in this country, being offended by the perceived distortion of mathematical ideas in various curricula and instruction. And they generally give little attention to mathematics education research. But they do not hesitate to voice strong opinions about what teachers should know and understand to teach mathematics. This typically consists of chosen extractions from the disciplinary canon. These views often find a home in policy and curricular documents.

Deborah made mathematical knowledge for teaching a rigorous research question, grounded in a close empirical study of the work of teaching. The work of her research group on MKT made this into what some have called an area of applied mathematics—that is to say, a complex domain of human practice that makes substantial specialized use of mathematics. It revealed novel kinds of mathematical knowing, thinking, and doing that are essential for teaching mathematics, yet had not previously been clearly identified or measured.

Something mathematicians most appreciate about Deborah is her respect for the integrity of the disciplinary knowledge in whatever domain she teaches, mathematics in particular. Although she has followed a different path in and around mathematics, she reflects a mathematical sensibility and depth of mathematical understanding of all that she teaches, as well as an artistry in pedagogical design that elicits critical thinking and a deep engagement with learning. And she does this with children historically and pervasively marginalized and oppressed in our society.

Finally, I think it is important to say that Deborah is deeply spiritual. A faithfully practicing Jew, she has also joined many of her students of other faith traditions in their religious celebrations. At her core, she considers teaching itself to be a spiritual calling, a challenging practice of cultivating human fulfillment.

## Credits

Opening image and Figures 1–3 are courtesy of Deborah Loewenberg Ball.

Figure 4 is courtesy of American Educational Research Association.

Figures 5, 7, and 8 are courtesy of TeachingWorks.

Figure 6 is courtesy of Hyman Bass.

Photo of Hyman Bass is courtesy of Hyman Bass.