Joe Rosenstein: The Art of Being Discrete

It’s our pleasure to welcome Professor Joseph G. Rosenstein to the Math Blog. He is Distinguished  Emeritus Professor of mathematics at Rutgers University.  His biography is extensive and you may read about his accomplishments on his web page .

Michael Paul Goldenberg: Welcome, Joe. It’s a pleasure to be speaking with you again. I know you from attending a summer program for high school teachers in applied graph theory at DIMACS in the summer of 2001. You were running a parallel program for middle and elementary school teachers. You were quite a legend among the high school teachers who were returning from previous years. I was told to be  on the lookout for someone with a big beard, some sort of splashy tie, and shorts. Do you still dress like that during your summer work? Any reasons beyond a personal taste for the “Joe Rosenstein Look”?

Joe Rosenstein: Yes, I still look much the same. I still have a beard – my beard is now almost 52 years old – but, curiously, it is now much whiter than in the past. I always wear a tie, year-round, and whenever they’re out of style (which is often), they can be described as “splashy;” I started wearing ties over 50 years ago, and have quite a collection. I also wear shorts in the late spring and summer every year; indeed, with global warming, this year I wore shorts through the end of October. There is no mathematical, pedagogical, spiritual, or political reason for the way I look. It’s just who I am. 

MPG:  How did you first become interested in mathematics as a serious pursuit?  

JR: I can’t point to a specific moment or incident when I decided that mathematics would be the direction in which I would go. As a high school student, I would go to the main public library in Rochester, New York and devour whatever math books they had; I scored 100 on all the math Regents exams given in New York State (including Solid Geometry). It was natural for me to continue to focus on mathematics when I went to college. Initially, I was very discouraged, because most of the 75 students in the honors calculus course in which I was enrolled in Columbia had gone to high schools like Bronx Science and were much better prepared than me, but I stuck it out, and I was one of only six students who survived that two-year honors course. So clearly math and I were meant for each other. Therefore, it was clear that I would go to graduate school in mathematics.

I went to Cornell because while at Columbia I became interested in mathematical logic and Cornell was then (and still is) a center for mathematical logic, which indeed became my focus in graduate school. My mentors and thesis advisors at Cornell were Anil Nerode and  Gerald Sacks. One day I was told that I would be going to the University of Minnesota as an Assistant Professor, and that is what I did. That was September 1969.

I continued doing mathematical research for about 15 years and then decided to write a book called “Linear Orderings” which included much of the research that I had done, but was also intended to organize and put in one place all that was known about linear orderings; that book was published in 1983 in Academic Press’ Series on Pure and Applied Mathematics. By 1987, however, my focus had shifted from mathematical research to mathematics education.

When I am asked on occasion why I chose to go into mathematics, and to become an academic mathematician, I respond that I was good at math and never asked the question of what I really wanted to spend my life doing … so I just continued what I was doing successfully. Sixty years ago, young people often did not ask questions about their future, but rather took the path that was somehow laid out for them. As time passed, I learned that I had other interests and talents, and have pursued them as well. But I have no regrets about spending my life with mathematics..

MPG: What led you to a focus on discrete mathematics? 

JR: Before 1989 I had not focused on discrete mathematics, although in retrospect it had become clear to me by then that much of my research would now be called either theoretical computer science or infinite combinatorics. However, in that year, a proposal was submitted to the National Science Foundation to create a “Science and Technology Center” based at Rutgers whose focus was on discrete mathematics. As part of the proposal, applicants had to describe how their center would reach out to precollege teachers and students. By then I was already quite involved in running programs for high school teachers; I had started a well-attended annual “precalculus conference” (the 32nd conference will take place in March 2018!) and an annual week-long summer institute for new math teachers (and a parallel institute for new science teachers). So I think that we could make a convincing case that we could do the outreach to schools that NSF required. Indeed, our proposal was funded by NSF and we were one of the initial half dozen Science and Technology Centers; our center is called DIMACS, the Center for Discrete Mathematics and Theoretical Computer Science, and almost 30 years on, it is a flourishing center with an impressive track record and international reputation.

Soon after NSF funded DIMACS, I submitted proposals to NSF to fund programs for high school teachers and high school students. These proposals were also successful; indeed, I received four consecutive grants from NSF for the Rutgers Young Scholars Program (RYSP) in Discrete Mathematics (extending over 8 years) and three consecutive grants from NSF for the Leadership Program (LP) in Discrete Mathematics for teachers (extending over 11 years). The LP continued for another 8 years with funding from other sources and the RYSP celebrated its 26th iteration this past summer.

MPG: What did you learn from the Leadership Program in Discrete Mathematics? 

JGR: In its initial years, the LP provided programs just for high school teachers. We learned from those programs that

• students were motivated by, and indeed excited by, the focus on applications;
• all students were able to learn the topics in discrete mathematics, since there were few mathematical prerequisites for these topics;
• while being accessible to all students, the topics could provide challenges to mathematically talented students;
• discrete math provided an opportunity for a “new start” in mathematics to students who had previously been unsuccessful;
• discrete math also provided an opportunity for a “new start” for teachers, in that techniques (such as focusing on problem solving and understanding, using group work, using questioning techniques) that they had avoided when teaching traditional topics, they were able to introduce successfully when teaching discrete math and were often able to transfer their new strategies into teaching traditional topics although we had initially assumed that discrete math would work only for high school students, the teachers in the LP told us that they were able to introduce discrete math topics successfully to middle school students as well.

As a result, when we applied for a second grant from the NSF, the reach was expanded to include middle school teachers as well. And the third, five-year, grant from the NSF was for K-8 teachers, since many of the topics in discrete math were accessible to elementary students as well, although the LP participants of course had to modify the curriculum and instruction to be suitable for their grade levels. 

MPG: You’ve spent a large part of your career promoting discrete mathematics as part of K-12 curricula on a state and national level. When I first came to the University of Michigan to do graduate work in mathematics education in 1992, I discovered that the State of Michigan already had a discrete mathematics strand embedded throughout its curriculum framework for math. While I was field supervisor of student teachers in secondary mathematics for U-M, I met a local high school teacher who was an enthusiastic booster of discrete math and did presentations for teachers at various conferences. I subsequently arranged to have him speak to my student teachers every semester and eventually was able to get a couple of them placed in his classroom. In my conversations with him, I realized that, outside of his classroom and those of a handful of other teachers around the state, the discrete mathematics strand was being honored far more in the breach than in the observance. The issue, he told me, was that there were absolutely no discrete mathematics items on the annual state assessments. How does this compare to your experiences in New Jersey? 

JR: In the early 1990s it became clear to the leadership of the Mathematical Sciences Education Board, part of the National Academy of Sciences, that improvement in mathematics education should be state-based. Influenced and encouraged by the MSEB, I created the New Jersey Mathematics Coalition in 1991, bringing together leaders from colleges, schools, industry, government, and the public sector in order to improve the mathematics education of New Jersey students. (Similar coalitions were created in subsequent years in many other states, as well as a national organization of such coalitions.) The coalitions advocated for the adoption of state standards.

As the idea of state standards gained momentum, the New Jersey Mathematics Coalition, in partnership with the NJ State Department of Education, was able to get a grant from the US Department of Education to create mathematics standards and a mathematics framework in New Jersey, an effort that I directed, that involved hundreds of New Jersey educators, together with people from the other sectors, and that produced the standards that were adopted by the State Board of Education in 1996 and that produced the New Jersey Mathematics Curriculum Framework to assist teachers and schools in implementing the standards.

These standards included a standard in discrete mathematics. Thus, when the state assessments were developed, they included items on discrete mathematics. Since the effort to develop the 1996 standards was not envisioned as providing specifications for statewide assessments, the standards were revised in 2002 in order to better align the revised state standards with revised state assessments.

As a result, New Jersey’s state assessments have, since soon after 1996 and until about 2008, included questions on discrete mathematics. I cannot avoid mentioning that, during this period, the scores of New Jersey students on the National Assessment of Educational Progress (NAEP) were among the highest in the nation.

MPG: What about on the national level? What happened to discrete mathematics with NCTM and with the Common Core Content Standards for mathematics? 

JR: With the advent of the Common Core standards, discrete mathematics has been essentially absent from the national math standards and, of course, from the New Jersey math standards. The one exception is that combinatorics has been incorporated into the standard relating to probability at the 8th grade level, although systematic listing and counting should be introduced at the elementary level.

MPG: Why do you think there has been so much resistance and inertia when it comes to discrete math in American K-12 mathematics education? 

JR: There has been a traditional and systemic resistance on the part of mathematicians against discrete mathematics, including a reluctance to consider it mathematics. Even Euler, the founder of graph theory, considered his solution to the question of whether a given graph has an Euler circuit as reasoning, not as mathematics. If summations, integrals, partial derivatives, or complex numbers are not present, then it’s not real mathematics. This perspective filters down to the K-12 curriculum into the view that all of mathematics should be preparation for calculus, a view that is echoed by teachers who never learned discrete mathematics in preparing for their teaching career.

Indeed, although the national standards were always intended to include the mathematics that prepare students for college, careers, and citizenship, the Common Core has hijacked the math standards so that it became preparation for calculus. That is very unfortunate, for not all students need to prepare themselves for calculus.

Unfortunately, most teachers don’t know discrete math or its value for their students and most college mathematicians who teach prospective teachers do not know how valuable it would be for their students our future K-12 teachers.

 MPG: Why do you think the Common Core focused on preparing students for calculus.

JGR: Three important reasons that I think led to the Common Core’s focus on preparing students for calculus were (a) a concern about the STEM pipeline, (b) a concern about US students performance on international assessments, and (c) a concern about the number of students who come to college with inadequate preparation for college math courses. From a superficial perspective, each of these supports a calculus-based curriculum.

With respect to (a), our research suggests that it is not that the STEM pipeline is too small (if shortages indeed exist), but rather that it is too leaky; a substantial number of students who already appear to be in the STEM pipeline are provided little encouragement to pursue STEM-based careers and drop out of math after taking AP calculus. 

With respect to (b), if the gap between US students and international students is indeed a problem (of which I am not convinced), then it’s not clear that narrowing our curriculum is an effective way to solve the problem. (In theory, if we spend 100% of our class time on the topics covered in the international assessments, then our students will do better than if we spend 90% of our class time on those topics and 10% of our time on other topics.) Our students’ scores may rise, but their overall understanding and experience with mathematics will suffer.

With respect to (c), very few of the students who come to college unprepared for college mathematics are going to end up taking, let alone succeeding, in calculus. They would be better prepared for college mathematics if they had a stronger experience with problem solving and reasoning.

Looking at the three issues more closely, we see that what all of these students need is not a narrower curriculum, but a broader curriculum, one that focuses more on problem solving and reasoning, as is the case when one incorporates discrete mathematics in the curriculum.

These ideas are discussed in more detail in my article “The Absence of Discrete Mathematics from Primary and Secondary Education in the United States … and Why that is Counterproductive” that will soon appear in the ICMI-13 Monograph published by Springer and entitled “Teaching and Learning Discrete Mathematics Worldwide: Curriculum and Research.”

MPG: I taught math in an alternative high school from 1998 – 2000. Virtually all of my students were testing at a 4th to 5th grade level in literacy and mathematics, and few of them had earned any high school credits in mathematics. During my second year there, after stumbling around trying to find something that would be accessible and interesting to students who feared and loathed mathematics and were very weak in basic arithmetic, to the extent that trying to teach them algebra was essentially futile. I stumbled via the Core-Plus curriculum into a unit on graph theory. While many of them floundered with Euler circuits and paths, something I thought would work for them, I was thrilled to finally find a topic that a more than a few of them liked and with which some of the weakest and most resistant students were extremely successful: graph coloring. What are your thoughts on that as a way to engage students who have not been doing well previously? 

JR: That is exactly right. When I teach courses for prospective K-8 teachers, I start with map coloring. More specifically I provide each group of 4-5 students with a map of the continental United States in which all the states are colored white and an envelope full of paper chips of various colors and ask them to color the states so that bordering states have different colors. I don’t say anything about the number of colors. Of course, each group colors the whole map in one or two minutes, and I then ask them whether they can eliminate one color, then whether they can eliminate another color and so on, and every group always is able to reduce the number of colors to four.

This activity, which we also used to start off the Leadership Program, has the advantage that it doesn’t appear to be mathematics, and therefore does not arouse all of the students’ and teachers’ residual fears about mathematics, the negative experiences they have had with math in the past, and their lack of confidence in their mathematical abilities. Through this activity, they find out that they can be successful in mathematics, and this initial successful encounter – and those that follow – enables them to continue to succeed. This activity, as well as the other activities in the LP, were developed by myself and Valerie DeBellis, who served as LP Associate Director.

My book, “Problem Solving and Reasoning with Discrete Mathematics,”  [note: reviewed this month] also begins with map coloring, although the activity described above does not work the same way with readers of a textbook as is does in a classroom setting. From map coloring, we go to vertex-edge graphs and graph coloring, and then to applications of graph coloring, and then to systematic construction of graphs. This book is designed both for a course for high school students and for a course for prospective K-8 teachers. But it is also appropriate for those who are mathematically curious.

MPG: Do you see any promising approaches to getting discrete mathematics accepted as an option for students in K-12 who may not be ready for or interested in calculus? 

JR: As it becomes clearer that the Common Core standards are the wrong standards, that they are inappropriate for a substantial number of students, the question of which standards are appropriate will be answered on a state-by-state basis, following the conception formulated by the MSEB almost 30 years ago. Perhaps having national standards is a good idea but, given the disastrous standards that were produced in this round, the national standards route will not be taken for many decades. That will make it possible for those math teachers who know about discrete mathematics to attempt to convince their states of its value.

In order for that to happen, those teachers need to work now to institute courses that can serve as models for their colleagues. For the past ten years, their hands have been tied, as schools and districts insist on spending class time exclusively on what’s in the standards. But in the coming decade, I anticipate that those restrictions will be lifted and teachers will have more freedom to explore other mathematical topics, including discrete math.

MPG: What are you working on now? 

JR: High school math teachers will have to convince their supervisors and principals that discrete mathematics is valuable for their students. So I am working on producing a video (actually a series of videos) that are designed for that purpose – emphasizing the importance of problem-solving and reasoning and how discrete math promotes that, emphasizing the importance of seeing how math can be used to solve real-world problems (which in a calculus-based curriculum doesn’t happen very much until calculus), and emphasizing that all students can benefit from learning about these topics. I hope that these videos will convince teachers of the value of discrete mathematics and that teachers will encourage their supervisors and principals to watch the videos and also come to see the value of discrete mathematics. I hope that many will, as a result, introduce discrete mathematics into their schools on an experimental basis and, when successful, will expand its availability to all of their students.

In order for this plan to work, and in order for teachers to realize that my book, “Problem Solving and Reasoning with Discrete Mathematics,” and other books are available, I will also have to develop strategies for reaching teachers, including through social networks – to do which I hope to recruit many other educators to promote discrete mathematics. I hope that some of those who read this interview will join me in this enterprise.

As you may know, we just elected a new governor in New Jersey. I hope that he will be amenable to developing new math standards for New Jersey, and I am prepared to be actively involved in that process. Such an effort may lead to bringing discrete math back into the New Jersey curriculum.

MPG: Anything else you’d like to share with us? 

 JR: The website for my book on discrete mathematics is new-math-text.com. I also have another website for other books I have written on Jewish themes, and particularly Jewish prayerbooks – that is newsiddur.org.

I recently retired after 48 years as Rutgers, and am getting used to adding “emeritus” to my title of “Distinguished Professor of Mathematics.” My Rutgers website is https://dimacs.rutgers.edu/~joer/joer.html

Finally, my wife and I have been married for 48 years, and are very proud of our five daughters, five sons-in-law, and eleven grandchildren.

MPG: Thanks so much for taking the time to speak with us, Joe, and for sharing your passion for mathematics.