Challenging Systems: My 2021 Research Statement

I wrote this research narrative as part of my portfolio for promotion to professor at Bridgewater State University in 2021. I’m sharing it here in case it helps the interested reader of this blog to better understand my somewhat scattershot research agenda (at least since 2014, as far back as my portfolio needed to go).

Disclaimer: Everyone’s research narrative is different, not least because of the context in which it is written. Mine is reflective of both (a) my position as a faculty member at a teaching-focused primarily-undergraduate institution and the ways this shapes expectations of scholarly activity and productivity; and (b) my necessity to marry my scholarship to my large service profile, since I have always held simultaneous administrative roles at the university. Your needs and contexts will vary! And… if I had had more time, I probably would have written a shorter version.

My Research Narrative

It is in the nature of the Teacher-Scholar that their best research often begins in the classroom, and their best teaching often begins in the field.

My scholarship has included examples of both kinds throughout my tenure at Bridgewater. Nearly all of my publications, professional presentations, and workshops have either been inspired and informed by, or directly a collaborative product of, working with undergraduate students. I have always enjoyed mathematics as a subject, a discipline, and an intellectual pursuit. Mathematician Francis Su writes in his 2019 book Mathematics for Human Flourishing that the practice of mathematical thought embodies many of the qualities that most nourish the human spirit: the desires for play, for beauty, for truth, love, and justice. Su’s book has found tremendous popular resonance with mathematicians, math teachers, and students in part because, I believe, it directly confronts the fact that math is too often seen, taught, and experienced in ways that are divorced from its human aspects – or worse, are inhumane. It is a reminder that math is neither human nor inhuman except insofar as we choose to make it so: for ourselves and, more importantly, for the students we serve. For my part, however much I have delighted in the intellectual practices of mathematics for my sake and for its own, I have always been far more satisfied when that work has had an immediate opportunity to intersect with the lives of people. That includes my students, but also college students of all stripes, as well as faculty, staff, and administrators across the educational system.

Math is neither human nor inhuman except insofar as we choose to make it so.

Because the teaching mission looms so large in my professional identity, it is both affirming and natural that that mission is reflected in my research. My goal as a scholar is to provide others with the resources and understanding to improve the experience of teaching and learning mathematical and quantitative content in their classrooms and across their institutions. 

This is crucial work for the future of our educational system in the United States in particular. For example, while Massachusetts’ high school students continue to outperform all other U.S. states in measures of mathematics performance, the U.S. as a whole performs well below the average for developed countries, and many countries’ national averages, including Canada and Switzerland in addition to Singapore, Japan, Korea, and China, far exceed even Massachusetts’. In a world economy that at the macro-level increasingly runs on sophisticated mathematical and statistical algorithms and machine-learning; and at the micro-level increasingly requires every firm and agency to collect, analyze, understand, and act upon large sets of data, from a purely capitalist perspective the U.S.’s economic position is jeopardized by the poor mathematical skill outcomes of our next generation.

Beyond these capitalist implications, the uniquely American strain of ideological anti-intellectualism prevalent in much of our political discourse throughout our country’s short history continues to find particular expression as a cultural dismissal of mathematics as an important skill set for life. These messages may originate outside the walls of the academy, but they are also frequently reinforced within them. Even individual math teachers who couch their feedback to students after a poor performance with well-meaning messages like “it’s okay, not everyone can be good at math” only compound the problem, demotivating students by underscoring the almost ubiquitous (and false) belief that mathematical skill is a fixed, innate capacity instead of one that everyone can develop if given the right strategies, supports, and most of all, significant rationale for doing so.

Beyond individual educators, though, our larger educational systems and curricula are also complicit in perpetuating cultural messages about the unimportance of math. Many students (and teachers!) first notice the role that curriculum plays in demotivating math students during the sequence of high-school courses beyond the first year of algebra. Even the most skilled teacher in an Algebra 2, precalculus, or trigonometry course in high school must frequently answer to students’ questions of “why do we need to learn this?” And “when are we ever going to use this?” There are many valid responses to these questions that speak to the value of the advanced skills and techniques of these courses for students who will take more advanced math and science courses later in their educational journey. And, there is a case to be made that mathematical learning, regardless of its direct applicability, expands students’ broader analytical and intellectual capacity: this case has been made at least since the time of Plato who wrote “any one who has studied geometry is infinitely quicker at learning other subjects than one who has not.”

Neither rationale is necessarily incorrect. But, I question their relevance to all students.

Our larger educational systems and curricula are also complicit in perpetuating cultural messages about the unimportance of math.

In the first case, it is undeniably true that advanced algebra, analytic geometry, and calculus are indispensable skill sets for students who will pursue further education in STEM fields in college. This is particularly true in the physical sciences, engineering, and at the advanced levels of social sciences such as economics. And it is true that we ought not have a K-12 math curriculum that a priori restricts the capacity of some students to choose these paths. But, especially in the last several decades, the prominence of engineering and physical science in the world economy that existed during the 20th-century period of industrialization, the Cold War, and the space race has been surpassed by the new 21st-century period of technologization, mass data collection, and artificial intelligence. The new century still needs scientists and engineers, but it arguably will be characterized by an even greater need for data professionals, statisticians, and analysts. And it is not only specialists in these areas who will be needed: data literacy and ethical data analytical skills will be necessary for every sector of the workforce. The 20th-century, calculus-centered K-12 math curriculum is not equipped to meet this need, and we cannot rely upon higher education to be a place of first exposure to these critical skills and literacies for the bare majority of U.S. students who pursue higher education.

As one of my math colleagues sardonically noted regarding the primacy of calculus in today’s curriculum: “[We] didn’t want to lock students out of a field that needs it. So we decided to lock everyone into it.” 

The first signs of change, however slow, have seen high schools begin to offer introductory statistics courses either as alternatives to senior-year calculus, or as additions to it for students whose schedules allow. Many of these are Advanced Placement courses or are only offered on schools’ “honors tracks”, meaning that they do not yet have the ability to reach all students, particularly students who might most benefit from such an alternative but who are marginalized in the system by their prior performances in the advanced-algebra courses intended to prepare them for calculus. Moreover, there is an acute shortage of high-school math teachers with the expertise to teach a statistics course, and this makes scaling up statistics education an even more daunting challenge. Yet this is a need that we at Bridgewater are uniquely positioned to help meet, and have begun to address in the ways we advise pre-service secondary math teachers in our department.

The new century still needs scientists and engineers, but it arguably will be characterized by an even greater need for data professionals, statisticians, and analysts.

To the second rationale, that any mathematical pursuit builds students’ broader analytical and intellectual capacities and increases their uptake of, and success in, other subjects, the evidence to support this claim is equivocal at best. There is research showing that building mathematical reasoning skills does have a benefit to students’ conditional logical reasoning skills — that is, their ability to identify and critically evaluate contingent claims (identifying premises and supposed conclusions, and assessing the validity of an argument that purports to demonstrate the sufficiency of the former for the latter). This is certainly a valuable analytical capacity for students to develop both for their other studies and for their everyday lives and civic engagement. 

But I maintain that even if this were the most important downstream intellectual capacity for us to develop in students, it will only do so for students who actually learn and succeed in those math courses designed to build that capacity. It does students no good to hear that “learning how to complete-the-square will help you be a more successful thinker” if they do not ultimately learn not just how to perform the context-free algebra task for an exam, but moreover the concepts, logical principles, and reasoning behind the process. Only deep mathematical learning can be expected to have these broader intellectual benefits, yet for too many students a focus on “getting the answer”, informed both by structural incentives such as standardized testing and by individual beliefs about the uncreative, instrumental nature of mathematics itself, leads to vanishingly superficial understandings of the subject that evaporate the moment after they are put to paper.

My scholarship is focused on challenging these beliefs and these systems, from the individual level where students’ and teachers’ beliefs about the nature and importance of math are narrowed (including teachers of other subjects across the curriculum), to the organizational level where institutions can develop new systems, networks, and sets of values that support both student success in their mathematical and quantitative courses, and faculty ownership and engagement in the effective teaching of these skills across the curriculum. I have been fortunate to connect through this work with other faculty and administrators who are passionate about being change agents in these areas on their campuses as well, and I have learned much more from these networks of collaboration than I could ever expect to give back in kind.

In the following I describe my scholarly activities during the review period in three main categories of inquiry:

  1. Quantitative reasoning and organizational development;
  2. Peer instruction programs and retention in STEM; and
  3. MATHCALA: a pedagogical game for abstract algebra.

Quantitative Reasoning and Organizational Development

Quantitative reasoning (QR) — defined as the use of quantitative evidence and mathematical techniques to substantiate critical reasoning within an authentic context — is a skill set whose importance to the educational trajectories, career opportunities, and civic participation of all people in the 21st century cannot be overstated. Today’s “data-drenched” workplaces, media landscapes, and public policy debates make critical evaluation and effective communication of quantitative ideas more vital to the health of our societies than ever before.

Supporting students’ QR skills has been at the core of my identity as a scholar and faculty development leader during my time at BSU. I embraced that work from within my roles as a mathematics faculty member and Math Services tutoring director, in part as a project to convince colleagues teaching in departments across campus that, much like writing, QR is a cross-disciplinary skill that faculty in all disciplines can be equipped to teach and support — and one need not be a math teacher to do so. This commitment led me first into faculty development leadership at BSU with the newly-founded Quantity Across the Curriculum (QuAC) initiative, and as QuAC grew in activity and my circle of collaborators widened, I had increasing opportunities to give back to the scholarly community from what we were learning in our BSU programs.

This I did in two principal ways. My published writings focused primarily on the organizational-development aspects of building quantitative reasoning programs, informed by how QuAC originated at BSU and gained traction as a faculty-led initiative. Meanwhile, I also remained an active presenter and workshop facilitator at professional conferences on the topic of defining, assessing, and integrating QR across the college curriculum. 

Publications in this area since 2014

Matthew Salomone, Kathryn Bjorge (2016).
Building Effective Quantitative and Math Faculty Development”, http://dx.doi.org/10.5038/9780977674435.ch10 in G. Coulombe, M. O’Neill, M. Schuckers (Eds.) QMaSC: A Handbook for Directors of Quantitative and Mathematical Support Centers, Neck Quill Press, http:// scholarcommons.usf.edu/qmasc_handbook .

Amber Parsons, Matthew Salomone, and Benjamin Smith (2019).
Going Public: What Institutional Moments Bring Everyone to the Table?“, in L. Tunstall, G. Karaali, and V. Piercey (Eds.) Shifting Concepts, Stable Core: Advancing Quantitative Literacy in Higher Education. MAA Notes #88. (Mathematics Association of America, MAA Press). Print ISBN 978-0-88385-198-2. Electronic ISBN 978-1-61444-324-7.

Peer Instruction Programs and Retention in STEM

The trajectories of students into and through educational pathways in science, engineering, technology, and mathematics is frequently compared to a “leaky pipeline”. While this is an evocative metaphor, it carries with it the implication that the pipeline was designed not to leak — an assumption to which the history of educational systems in the U.S. has historically run counter. Equal access to high-quality science and math education for students across all gender, racial, and socioeconomic categories remains an elusive goal. For many students, the existing pipeline was in some ways designed to leak them out.

Students’ transitions from high school to college are known to be one of the most critical junctures in their paths toward earning the STEM degrees that are both needed in today’s workforce and which open the doors for well-paid, stable careers for those students. At many universities, though, large proportions of first-year students intending on pursuing science and math encounter difficulties — both in and around their academic coursework — that lead them to abandon their plans. At Bridgewater, for instance, the 2006 Project Compass project turned up evidence that BSU’s first-generation, low-income, and nonwhite students, who collectively comprise a majority of our student body, faced disproportionately long odds on remaining in the STEM majors they entered college hoping to complete.

For many students, the existing STEM pipeline was in some ways designed to leak them out.

The STREAMS grant, a National Science Foundation grant awarded to BSU in 2010, provided approximately $1 million over five years to support a range of interventions designed to increase retention of our STEM students: not only their retention at the university, but more specifically their retention within STEM majors in the Bartlett College of Science and Mathematics. STREAMS funding supported a pre-first-year STEM summer bridge program, a STEM mentorship program, articulation work with transfer institutions to ensure STEM transfer students arrive with appropriate coursework to meet the requirements of BSU’s majors, and faculty and curriculum development work within the college. 

It also supported the design and implementation of a supplemental learning assistance (SLA) program of peer instruction in five of the College’s major introductory courses (Biology 121, Chemistry 141, Computer Science 151, Math 161, and Physics 243). The peer instruction models designed by faculty in each department were varied to meet the needs of their curriculum: biology, computer science, and mathematics faculty created one-credit corequisite courses for their peer leaders to engage students in small-group problem solving and inquiry; chemistry and physics faculty chose to integrate peer leaders’ work into existing contact time for the courses. Faculty development efforts assisted instructors to make the most effective use of peer instruction, creating as much continuity as possible between students’ interactions with faculty and with peer leaders (initially called “PALs” after the Academic Achievement Center’s related program). 

I was a part of the STREAMS grant’s team of faculty co-investigators, and coordinated the implementation of the SLA program for the College. In addition to working with grant P.I. Dr. Tom Kling (physics) and co-investigator Dr. Ann Brunjes (English) to assist departments in their curriculum designs to accommodate peer instruction, I also was responsible to recruit, hire, provide training for, and supervise the employment of the College’s peer leaders which numbered on average between 15-20 each semester. As the STREAMS grant funding began to end in 2014-2015 and the program was institutionalized through College funding, I passed these responsibilities to other colleagues, first Dr. Bob Cicerone (geology) and then my math colleague Dr. Jackie Anderson, who still coordinates the program as of this writing.

The successful institutionalization (and indeed, expansion to several additional departments and courses in the College) was made possible by the demonstrated success of the SLA program in increasing retention of our students in STEM. The following publications were the STREAMS grant team’s efforts to disseminate both the design and organizational development processes that led to the successful, near-simultaneous launch of the SLA program across five different STEM departments in 2010-2011, and subsequently the data analysis model and results that testified to the benefits of the program in increasing STEM retention in our college. Crucially for our institutional context, the latter provides evidence that the mechanism by which student retention was increased is that the SLA program contributed to erasing differences in students’ academic readiness (as measured by their SAT scores) that were previously associated with risk of poor outcomes in the introductory course in their major, which in turn drove attrition from STEM.

Publications in this area since 2014

Thomas Kling, Matthew Salomone. (2015). “Creating a peer-led cooperative learning program to improve STEM retention.” Change: The Magazine of Higher Learning, 47(6), 42-50.

Matthew Salomone, Thomas Kling (2017). “Required peer-cooperative learning improves retention of STEM majors.” International Journal of STEM Education, 4(1), 1-12.

MATHCALA: A Pedagogical Game for Abstract Algebra

Many students of mathematics are also attracted to games of strategy and logic. Most famously, for example, chess, Go, and Set are games whose relatively easy-to-learn rule sets belie very intricate and mathematically sophisticated game-theoretic properties that make the systematic search for winning strategies very challenging. I myself have never identified as one of these kinds of math students: my enjoyment of a game is usually in inverse proportion to the amount of mental effort I expend thinking about strategies!

But conversely, games of strategy and logic have also been shown — including my BSU colleague Dr. Polina Sabinin — to be valuable teaching tools to develop math students’ capacity for abstract reasoning. And this is the promise that ultimately led me to get involved with games: the notion that it could help improve my teaching and help my students better grasp an abstract mathematical idea.

This research project originated in my BSU classroom in Spring 2018. I was teaching one of the fundamental requirements in the upper-level of the math major, MATH 301 Abstract Algebra I, and preparing a lesson on one of the grandest results in the course, the so-called fundamental theorem of finite abelian groups. This result is an example of a “structure theorem”: once we grasp it, we are rewarded with a complete classification of all possible finite abelian groups of a given order. It is analogous to a biologist somehow being able to say “these are the only types of river fishes that have ever existed or can ever exist, and here’s exactly how to tell them all apart.” A powerful result!

In the previous semester, however, my efforts to convey the theorem’s result, its power, and elements of its proof to my students had fallen flat. So the second time around, I decided to design and use a simple chip-stacking game to teach the theorem by analogy.

The game, which I came to call MATHCALA for its similarity to the African bead game known as Mancala (or Ouril in Cape Verde), was an instant success with my students. By removing the baggage of mathematical formalism and notation and allowing them to focus on the game’s simple rule set and piece-moving strategies, they first developed an understanding of the abstract underlying structure underneath the theorem I intended to teach them. And, they enjoyed the process of playing the game, which involved both some cooperative goal-seeking and some friendly competition.

So when it was time for me to teach them the theorem, all that was left for me to do was to “land the metaphor” by relating the game’s pieces and game board back to the mathematical objects they represented (prime-order factors and direct products), and the game’s rule set back to the conclusion of the theorem (concerning how direct products may be rearranged without meaningfully altering their structure). These conclusions which last semester’s students had struggled so much to absorb fell almost naturally into place — and I saw the evidence of improved understanding in students’ work afterward, including observing students using thumbnail sketches of the game to help structure their thoughts before writing a proof.

This work has led to two parallel research articles currently in progress (my goal is to submit them at the same time, as they reference one another), to which I dedicated a considerable amount of time during my Fall 2019 sabbatical leave to researching. One article focuses on the pedagogical value of MATHCALA for students’ learning of abstract algebra, for which I hope to work with colleagues to collect some student feedback data this coming academic year. The other focuses on the mathematical properties per se of the MATHCALA game. This work led me directly to some challenging combinatorics problems for which no closed-form solutions yet exist, but I have been able to develop some partial results using the theory of multiset partitions.

Articles currently in progress on this topic

Matthew Salomone. (In progress.) “MATHCALA: A Combinatorial Game to Teach Classification of Finite Abelian Groups.”

Jeremy Alm, Matthew Salomone. (In progress.) “Partitions of Minimally-Distinguishable and Minimally-Indistinguishable Multisets.”

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