Mathematics faculty and educational researchers are increasingly recognizing the value of the history of mathematics as a support to student learning. The expanding body of literature in this area includes recent special issues of *Science & Education* and *Problems, Resources and Issues in Undergraduate Mathematics Education* (*PRIMUS*), both of which include direct calls for the use of primary historical sources in teaching mathematics. Sessions on the use of primary historical sources in mathematics teaching at venues such as the Joint Mathematics Meetings regularly draw large audiences, and the History of Mathematics Special Interest Group of the Mathematical Association of America (HOMSIGMAA) is one of the largest of the Association’s twelve special interest groups. In this blog post, which is adapted from a recent grant proposal, we explore the rationale for implementing original sources into the teaching and learning of undergraduate mathematics, and then describe in detail one method by which faculty may do so, namely through the use of Primary Source Projects (PSPs).

Teaching from primary sources has long been common practice in the humanities and social sciences [11, 17]. Reading texts in which individuals first communicated their thinking offers an effective means of becoming mathematically educated in the broad sense of understanding both traditional and modern methods of the discipline [9, 18]. The use of original sources in the classroom promotes an enriched understanding of the subject and its genesis for instructors as well as students.

In contrast to many textbook expositions, which often present mathematical ideas in a distilled form far removed from the questions that motivated their development, original sources place these ideas in the context of the problem the author wished to solve and the setting in which the work occurred. Problems and the motivations for solving them are more apparent and natural in primary sources, and the works of these thinkers are more compelling than traditional textbook expositions. Exposing the original motivations behind the development of “esoteric” mathematical concepts may be especially critical for placing the subject “within the larger mathematical world,” thereby making it more accessible to students [22]. Further, primary texts seldom contain specialized vocabulary (which comes with later formalism), thereby promoting access to the ideas by students with varied backgrounds.

Precisely because they give students the opportunity to interpret results as they were originally presented and then reformulate them in modern terms, original source readings encourage robust understanding of mathematics. Engagement with the original problems from which concepts arose has been observed to invite questions of a different nature than students generally pose in more traditional approaches as it provides students with a basis for making their own reflections and developing their own judgment, thereby helping them to see how to develop ideas and reason with them on their own [12, 13]. Drawing on Sfard’s work on “mathematical discourse” and “commognition” [23], Kjeldsen and Blomhøj have further suggested that the reading and interpretation of original sources may even be essential for raising students’ awareness of what constitutes a proper definition, or deciding whether a solution is correct and complete [16], a key feature of success in the study of advanced mathematics. Jankvist further proposes that original source materials may help with the “transition problem” between educational levels [14].

Finally, the reading of original texts provides a means of responding to calls within the Science, Technology, Engineering, and Mathematics (STEM) education community to engage undergraduate students in authentic research at the frontiers of the field. While the importance of such engagement for socializing and retaining students within the disciplinary community has been noted by Project Kaleidoscope Director J. Narum [20] and others, the present research frontier in mathematics is generally too far removed from the undergraduate experience to make this possible. Engaging students with texts that represent the state of mathematical knowledge at an earlier stage of development can, however, provide undergraduates an opportunity to experience research at a frontier stage as it was practiced by some of the greatest mathematical minds throughout time.

Despite the benefits of primary source materials detailed above, and granting the wide availability of such materials via published collections and web resources [6, 8], there are significant challenges to incorporating primary sources directly into the classroom. Using secondary historical sources, such as [15], may suffice to reap some of the benefits of the original works; however, use of such sources carries its own difficulties, including the risk of placing too much emphasis on learning the history of mathematics per se, as opposed to using that history to support the learning of mathematics.

One approach to addressing these issues is through PSPs, which are curricular modules designed to teach core mathematical topics from primary historical sources rather than from standard textbooks. Each PSP is designed to cover its topic in about the same number of course days as classes would otherwise. With PSPs, rather than learning a set of ideas, definitions, and theorems from a modern textbook, students learn directly from mathematicians such as Leonhard Euler, Augustin-Louis Cauchy, or Georg Cantor. This distinction is crucial to PSPs: they are not designed to teach history; rather, they use history as a tool to better teach mathematics.

PSPs employ a selection of excerpts from primary historical sources that follows the discovery and evolution of the topic in question. Each PSP contains commentary about the historical author, the problem the author wished to solve, and information about how the subject has evolved over time. Exercises are woven throughout the project, requiring that students actively engage with the mathematics as they read and work through each excerpt. At appropriate junctures, students are also introduced to present-day notations and terminology and are asked to reflect on how modern definitions have evolved to capture key properties of solutions to problems posed in the past. Learning from the PSP via in-class activities and discussions replaces standard lectures and template blackboard calculations.

As an example, the PSP *Networks and Spanning Trees* [19], opens with Arthur Cayley’s discovery of a pattern for the number of (labeled) trees on *n* vertices. Cayley used the term “tree,” without any definition, to describe the logical branching when iterating the basic process of (partial) differentiation. Students are asked to follow in Cayley’s footsteps and arrive at the same observations, hinting at an algebraic pattern for the number of such trees. Students are then asked to find the gaps in Cayley’s 1889 “proof” [7], and to reflect on what would constitute a valid argument. This is followed by Heinz Prüfer’s rigorous counting of (labeled) trees [21], motivated by the problem of enumerating all possible railway networks on *n* (fixed) towns so that (i) the least number of railway segments is used, yet (ii) travel remains possible between any two towns. The module continues with Otakar Borůvka’s 1926 solution to finding the most economical way to connect *n* towns in a rural region to an electrical network. He devised an ingenious algorithm to solve this problem by connecting each town to its nearest neighbor, and then iterating the algorithm on connected components until a connected graph (tree) resulted [4, 5, 10]. After working through these specific applied problems of Prüfer and Borůvka, students are asked to reflect on how the modern definition of a tree captures the proprieties sought by both authors, and what lemmas or theorems have evolved from these historical sources.

Classroom implementation of this and other PSPs is extremely flexible, thereby enhancing their adaptability to a wide variety of institutional settings. Although they work best by utilizing a combination of in-class activities and out-of-class homework, PSPs can be completed individually or in small groups, or assigned as a one-to-two week assignment requiring a written paper addressing all the exercises in the PSP. Other instructors assign a PSP in parts over a four-to-five week time period, and ask students to submit written answers to each exercise in installments. The written portion of a PSP allows students to react to the historical sources, organize their thoughts through mathematical exposition, and rediscover groundbreaking ideas for themselves.

Faculty interested in finding a wide collection of PSPs are encouraged to consult [1, 2, 3].

**References**

[1] J. Barnett, G. Bezhanishvili, H. Leung, J. Lodder, D. Pengelley., I. Pivinka, and D. Ranjan, *Learning Discrete Mathematics and Computer Science via Primary Historical Sources*, http://www.cs.nmsu.edu/historical-projects, accessed on January 4, 2015.

[2] Barnett, G. Bezhanishvili, H. Leung, J. Lodder, D Pengelley and D. Ranjan, *Teaching discrete mathematics via primary historical sources*, http://www.math.nmsu.edu/hist_projects, accessed on January 4, 2015.

[3] J. Barnett, G. Bezhanishvili, H. Leung, J. Lodder, D. Pengelley, I. Pivkina, D. Ranjan, and M. Zack, *Primary Historical Sources in the Classroom: Discrete Mathematics and Computer Science*, Loci: Convergence (July 2013), DOI 10.4169/loci003984, http://www.maa.org/publications/periodicals/convergence/primary-historical-sources-in-the-classroom-discrete-mathematics-and-computer-science, accessed on January 18, 2015.

[4] O. Borůvka, *O jistém problému minimálnim, (On a Certain Minimal Problem)*, Práce Moravské Přdovědecké Spolecnosti v Brně **3 **(1926), 37–58.

[5] ______ , *Přispěvek k řěsení oťazky ekonomické stavby elektrovodnich sítí (A Contribution to the Solution of a Problem on the Economical Construction of Power Networks)*, Elecktronický obzor **15 **(1926), 153–154.

[6] R. Calinger, *Classics of Mathematics*, 2nd ed., Prentice-Hall, Engelwood Cliffs, New Jersey, 1995.

[7] A. Cayley, A* Theorem on Trees*, Quarterly Journal of Pure and Applied Mathematics **23 **(1889), 376–378.

[8] L. Euler, *The works of Leonhard Euler online*, Available at http://eulerarchive.maa.org, accessed on January 4, 2015.

[9] M. Fried, *Can Mathematics Education and History of Mathematics Coexist?*, Science & Education **10 **(2001), 391–408.

[10] R. L. Graham and P. Hell, *On the History of the Minimum Spanning Tree Problem*, Annals of the History of Computing **7 **(1985), 43–57.

[11] M. de Guzmán, *Enseñanza de las ciencias y la mathemática*, Revista Iberoamericana de Education **043 **(2007), 19–58.

[12] H. N. Jahnke, *The use of original sources in the mathematics classroom*, History in mathematics education: the ICMI study (Fauvel, J. and van Maanen, J., ed.), Kluwer Academic, Dordrecht, 2002, 291–328.

[13] U. T. Jankvist, The use of original sources and its possible relation to the recruitment problem, *Proceedings of the eighth congress for the European society for research in mathematics education* (B. Ubuz, Ç. Haser, and M. A. Mariotti, eds.), Middle East Technical University, Ankara, Turkey, 2013, 1900–1999.

[14] _______, *On the Use of Primary Sources in the Teaching and Learning of Mathematics*, International Handbook of Research in History, Philosophy and Science Teaching, Springer Verlag, New York, 2014, Matthews, M. (editor).

[15] V. Katz, *A History of Mathematics: An Introduction*, 2nd ed., Addison-Wesley, New York,1998.

[16] T. H. Kjeldsen and M. Blomhøj, *Beyond motivation: history as a method for learning meta- discursive rules in mathematics*, Educational Studies in Mathematics **10 **(23 September 2011), DOI 10.1007/s 10649–011–9352–z.

[17] D. Klyve, L. Stemkowski, and E. Tou, *Teaching and Research with Original Sources from the Euler Archive*, Loci: Convergence (April 2011), DOI 10.4169/loci003672.

[18] R. Laubenbacher, D. Pengelley, and M. Siddoway, *Recovering Motivation in Mathematics: Teaching with Original Sources*, UME Trends **6**, DOI 10.1007/s 11191–012–9470–8, Available at the website http://www.math.nmsu.edu/~history/ume.html, accessed January 4, 2015.

[19] J. Lodder, *Networks and spanning trees*, 32 page curricular module available at [1].

[20] J. Narum, *Promising Practices in Undergraduate Science, Technology, Engineering, and Mathematics (STEM) Education*, Evidence on Promising Practices in Undergraduate STEM Education – Commissioned Workshop Papers, National Academy of Sciences, June 30, 2008, Available at the website http://sites.nationalacademies.org/dbasse/bose/dbasse_080106, accessed January 4, 2015.

[21] H. Prüfer, *Neuer Beweis eines Satzes über Permutationen*, Archiv der Mathematik und Physik **3 **(1918), no. 27, 142–144.

[22] N. Scoville, *Georg Cantor at the Dawn of Point-Set Topology*, Loci: Convergence (March 2012), DOI: 10.4169/loci003861.

[23] A. Sfard, *Thinking as communicating: Human Development, the Growth of Discourse, and Mathematizing*, Cambridge University Press, New York, 2008.

Home to eminent mathematicians such as Paul Erdős, John von Neumann, and George Pólya, Hungary has a long tradition of excellence in mathematics education. In the *Hungarian approach* to learning and teaching, a strong and explicit emphasis is placed on problem solving, mathematical creativity, and communication. Students learn concepts by working on problems with complexity and structure that promote perseverance and deep reflection. These mathematically meaningful problems emphasize procedural fluency, conceptual understanding, logical thinking, and connections between various topics.

For each lesson, a teacher selects problems that embody the mathematical goals of the lesson and provide students with opportunities to struggle productively towards understanding. The teacher carefully sequences the problems to provide focus and coherence to the lesson. These problems do more than provide students with opportunities to learn the mathematical topics of a given lesson. Indeed, the teacher sees the problems she poses as vehicles for fostering students’ reasoning skills, problem solving, and proof writing, just to name a few. An overarching goal of every lesson is for students to learn what it means to engage in mathematics and to feel the excitement of mathematical discovery. Click here for a sample task from a 5th grade classroom at Fazekas Mihály School in Budapest.

Another hallmark of the Hungarian approach is the classwide discussion of approaches to problems. After working on problems individually or in small groups, volunteers come to the front of class to share their solutions. Because of the non-trivial nature of these problems, students learn to communicate their thinking with clarity and precision. When a student gets stuck, others chime in to offer support and suggestions in a friendly manner. The teacher creates a welcoming environment that is conducive to the sharing of students’ mathematical experiences.

In such a classroom, the teacher’s role is that of a motivator and facilitator. He provides encouragement and support as students engage with the task at hand. He offers guidance when a student is stuck and probes when clarification is needed. After the student investigation, the teacher highlights important ideas embedded in a concrete problem, and summarizes and generalizes their findings. In particular, the teacher’s summary makes sense and is meaningful, because students have had the experience of playing around with these ideas on their own before coming together to formalize them as a class.

Moreover, the teacher repeatedly asks, “Did anyone get a different answer?” or “Did anyone use a different method?” to elicit multiple solutions strategies. This highlights the connections between different problems, concepts, and areas of mathematics and helps develop students’ mathematical creativity. Creativity is further fostered through acknowledging “good mistakes.” Students who make an error are often commended for the progress they made and how their work contributed to the discussion and to the collective understanding of the class.

My interest in this approach to teaching has led to my involvement in Budapest Semesters in Mathematics Education (BSME), a new semester-long program in Budapest, Hungary designed to introduce the Hungarian approach to American and Canadian undergraduates and recent graduates. Conceived by the founders of Budapest Semesters in Mathematics (BSM), BSME is specifically intended for students who are not only passionate about mathematics, but also the teaching of mathematics. Participants will immerse themselves in mathematical exploration to experience first-hand learning in the Hungarian approach; then they will investigate how to bring this pedagogy into their own future classrooms. They will observe Hungarian mathematics classrooms and will have the opportunity to plan and teach their own lessons to Hungarian students (in English).

One of the core benefits of the Hungarian approach, and one that I am excited for BSME participants to bring back to the US, is that students acquire the mathematical habits of mind that allow them to think like a mathematician. As Cuoco, Goldenberg, and Mark describe,

Much more important than specific mathematical results are the

habits of mindused by the people who create those results. … The goal is not to train large numbers of high school students to be university mathematicians. Rather, it is to help high school students learn and adopt some of the ways that mathematicians think about problems. … Although it is necessary to infuse courses and curricula with modern content, what is even more important is to give students the tools they will need in order to use, understand, and even make mathematics that does not yet exist.(pp. 375-376)

Given the wide-spread adoption of the Common Core State Standards, as well as the recently published *Mathematical Education of Teachers II* (MET2) report by CBMS and NCTM’s *Principles to Actions*, our teachers are now expected to provide learning experiences that lead to the acquisition and development of students’ mathematical habits of mind, so that “all students learn to become mathematical thinkers and are prepared for any academic career or professional path that they choose.”* Preparing teachers with the knowledge and skill set to cultivate such a learning environment is now an important national need in mathematics education, and the Hungarian approach has the potential to play a critical role in this endeavor.

To learn more about the Hungarian approach, consult the following articles:

- Andrews, P., & Hatch, G. (2001). Hungary and its characteristic pedagogical flow.
*Proceedings of the British Congress of Mathematics Education*, 21(2). 26-40. - Stockton, J. C. (2010). Education of Mathematically Talented Students in Hungary.
*Journal of Mathematics Education at Teachers College*, 1(2), 1-6.

*National Council of Teachers of Mathematics. (2014). *Principles to Actions: Ensuring Mathematics Success for All*. Reston, VA: Author, p. vii.

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Job opportunities for graduates with degrees in the mathematical sciences have never been better, as the world is being viewed through increasingly quantitative eyes. While standard statistical methods remain the work horse for data analytics, new methods have appeared that help us look for all sorts of hidden patterns in data. Examples include statistical methods inspired by tools from abstract algebra, geometric data analysis based on methods from algebraic topology, and new machine learning methods, such as deep neural nets, combined with novel optimization methods. Most importantly, perhaps, an eye trained for the discovery of patterns can go beyond standard analysis approaches through ad hoc data interrogation. Mathematics can be viewed as the science of (non-obvious) patterns, so it is not surprising that a solid mathematics education makes for excellent training in data analysis. It is now more widely known than ever that mathematics is the key enabling technology for the solution of the most difficult scientific problems facing humankind. Human health is arguably at the top of this list. I will focus here on data analytics in healthcare, a field growing by leaps and bounds, although one can make similar statements about the need for mathematical scientists in many other areas.

The holy grail in health care lies in the integration of three types of data: basic research and clinical models, electronic health records, and population health data, such as health insurance claims data, an important step toward making personalized medicine a reality. For instance, genetic profiles of large populations, combined with their health records, lifestyle information, and insurance claims history can help us develop predictive tools for the attributes of healthy aging. Across these application areas, there are severe shortages of qualified data scientists who are able to go beyond the application of software tools to an understanding of the underlying algorithms and their limitations, developing and implementing modifications or new algorithms.

Regarding basic research and clinical practice, new, so-called next generation sequencing technologies are providing insights into molecular events at the genome level as well as the level of molecular networks, uncovering new approaches to the search for targeted drugs against a host of diseases. Mathematical and statistical models, based on gene sequence and expression data, combined with measurements of proteins and metabolites, provide the tools to distinguish normal cells from cancer cells, for instance. Molecular profiles of patients suffering from schizophrenia, combined with behavioral and clinical data, can point to more targeted drug prescriptions. New data types are being developed at breakneck speed, and data analysis methods are struggling to keep pace. In genomics, for instance, new sequencing technologies, such as atac-seq, allow the detection of so-called epigenetic features that capture the status of chromatin, a “wrapper” of DNA that needs to be unpackaged before a gene can be transcribed, or data that capture information about gene-gene interactions that utilize information about the 3D structure of chromosomes, rather than just linear sequence information.

The use of electronic health records promises to revolutionize the delivery of health services. Here too, challenges arise from the quantity of data, their heterogeneity and, frequently, a lack of appropriate data analytics methods, for instance the development of predictive models for patient response to a certain diabetes drug, given co-morbidities such as hypertension or heart disease. Finally, private and public health insurance providers have large quantities of data to analyze for their policy decisions. A major bottleneck in all these areas is the lack of qualified data analysts.

In my experience, M.S. and Ph.D. level mathematicians have the perfect intellectual skill set to excel at this type of problem. (Not surprisingly, the National Security Agency and hedge funds have recognized this some time ago.) The best background is a solid training in fundamental mathematics, algebra, analysis, topology, etc., combined with programming skills. Equipped with this skill set, acquisition of algorithms from statistics, bioinformatics, machine learning, and topological data analysis, to name a few, is straightforward, together with the needed domain knowledge. The intellectual flexibility of someone with solid mathematics training frees them from the limitation of practitioners with modest mathematics training to go looking for nails that fit their particular hammer.

What does this mean for the training of mathematics graduates to put them in a position to take advantage of this new “golden age?” As mentioned, I believe that a solid education in “pure” mathematics is best, together with some other skills. This should be complemented by hands-on experience in data analysis, ideally as part of an ongoing analytics project. Needed specific skills can be learned as part of “on-the-job” training. This, of course, requires that mathematics graduate programs partner with organizations such as medical schools, research institutes, companies, or state agencies to provide access to data projects. While this seems like a simple task, it can sometimes pose formidable obstacles. Nonetheless, existing M.S. and Ph.D. programs in mathematics need to make only relatively minor adjustments to their curriculum to train graduates that will be highly sought after in a broad range of healthcare-related organizations. It is worth emphasizing that, even though this contribution is focused on graduate education, many of my comments also apply to undergraduate mathematics and statistics majors.

Of course, many mathematics departments already have new or established activities in data analytics, ranging from entire degree programs, such as a professional M.S. degree at Georgetown University in Washington, DC, complete with industrial internships, to formal course offerings, such as a 1-year course sequence on data analytics at SUNY Albany. (It is generally difficult to glean such information from Departmental websites, and I would be grateful for any information about ongoing or contemplated efforts.) My main hypothesis in this contribution is that there is much that can be done with relatively minor administrative effort or restructuring of the curriculum. Most departments have appropriately generic course offerings on the books that can be used if formal credit is needed. And opportunities for hands-on training are plentiful and can be handled quite informally. The main requirement is probably one or two committed faculty members.

Based on my experience, there is a great willingness on the part of healthcare and biomedical research organizations to provide initial training to mathematicians who might not know the first thing about electronic health records or next generation sequencing, but come equipped with curiosity and some communications skills across fields. Almost all universities and colleges already offer relevant communications training that can be leveraged by a department. Many students are eager to combine their love for mathematics with a desire to solve real-life problems but, in my experience, many of them do not know how they can use their training for careers in “non-standard” settings. While biomedicine and healthcare typically do not offer the high salaries of the financial industry, they do offer a plethora of problems that can be solved by someone with mathematical training, whose positive impact on people’s lives can be clearly seen, providing strong motivation.

In response to questions about the usefulness of mathematics, students are sometimes told by their professors (including me, when I taught mathematics courses) that with mathematical training one can do “anything.” My experience in the life science and healthcare fields has taught me that there is a lot of truth to that assertion.

]]>Somehow, over the last 600 years or so, mathematics has moved from the core of the liberal arts disciplines to entirely outside. We’re all used to this; a “liberal arts math” course is understood to serve non-STEM majors, for example. The reasons for this shift are interesting to ponder (see [1] and [2]), but in this post I suggest that we consider some of its unfortunate present-day implications. It’s also worth considering the broader aim of a liberal arts approach, which transcends disciplinary boundaries.

A well-known exposition of the liberal arts ideal appears in a fifteenth-century treatise of Vergerius, in which he advocates studies “worthy of a free man.” While Vergerius lays out specific areas of study, including the “mathematical arts” –the quadrivium of arithmetic, geometry, astronomy, and music — he opens with the importance of a liberal education to character development. From early on, then, the liberal arts ideal goes beyond eschewing the vocational; it values sustained engagement with abstract concepts as central to the capacity to live a good life.

Although I have bristled at a smorgasbord interpretation of liberal arts, which misses the point in its focus on breadth with little attention to depth, equally concerning is the suggestion that some areas of study are more worthy of inclusion. In a recent report of the Association of American Colleges and Universities, which calls itself “A Voice and Force for Liberal Education in the 21^{st} Century,” “[t]he term ‘liberal arts’ is used … as a description for majors in the humanities, arts, and social sciences.” The common misperception of mathematics as solely practical may have contributed to its exile; it certainly adds to the challenge of teaching. It may be obvious to me that the study of Calculus benefits one’s overall capacity to engage with the world, but it’s not always evident to my students.

When Vermont first adopted the Common Core State Standards for Mathematics, a veteran teacher-trainer told me, “I worry that because the Content Standards take up so many more pages, people will focus on them, and then this won’t work. The Practice Standards are key.” Since then, I’ve heard two of the lead writers of the CCSSM say that they worry about too much focus on the Practice Standards, and that they’re meaningless without the Content Standards. Conclusion: both are important, and they work in concert.

Analogously, our understandings of the liberal arts are about content and practice. Both are important, and they work in concert. We can’t develop our students’ intellectual capacities without carefully considering content, and we can’t rely on content alone to prepare them for the challenges they’ll face confronting questions we can’t even imagine right now. (I don’t remember Fortran, but the value of learning to write computer programs endures.)

What does this say about the “Liberal Arts Mathematics” course? Steve Strogatz has been tweeting about the one he’s been teaching, using materials from the Discovering the Art of Mathematics project. Jessica Lahey wrote about his course for *The Atlantic. * What I’ve seen suggests that it is indeed possible to engage students, even those who start the course with apprehension or even fear, in “authentic mathematical experiences,” as the project intends. For this audience, figuring out how to produce a scalene triangle with one cut of the scissors is an exercise in plane geometry, a foundational topic. It’s also an exercise in making mistakes and persevering, something that may not have been encouraged in their earlier mathematics courses. The authenticity is evident in both the geometric content and the exploratory approach.

What does it mean for our mathematics majors if we insist that mathematics is still one of the liberal arts? For one thing, it is one way I remind myself, and my students, that too much emphasis on coverage risks losing the longer-lasting lessons of careful, detailed analysis. Those lessons go beyond the specifics of any particular course. The other day I heard a student in our common room scoff at the notion that math is just about numbers; “it’s about logical thinking,” he said. From now on I’m going to ask explicitly for that sort of metacognition in my upper-level classes. What have your mathematics classes had in common? When have you noticed advances in your capacity to think abstractly? How have your upper-level math courses changed your thinking about the earlier ones? Asking students to look for coherence and connection will, I believe, make it more likely that their mathematical studies will have a lasting impact on them as members of society, not just as workers.

We mathematicians understand that our discipline involves creativity, beauty, and abstraction as well as precision and utility. An education worthy of a free person should include active, meaningful experience with all of those elements. I will continue to speak up about the historical importance of the quadrivium, the “mathematical arts,” in the liberal arts. At the same time, I will affirm the value of mathematical practice to intellectual development for all present-day students, not just those majoring in mathematics and the sciences. Finally, and most important, I want every course I teach to reflect the centrality and the value of mathematics, content and practice, to a modern liberal education.

[1] Grant, Hardy. Mathematics and the Liberal Arts. *The College Mathematics Journal*, **30**, No. 2 (Mar., 1999), 96-105.

[2] Grant, Hardy. Mathematics and the Liberal Arts. *The College Mathematics Journal*, **30**, No. 3 (May, 1999), 197-203.

Many thanks to Luisa Burnham, Ben Braun, and Julian Fleron for their assistance.

]]>When I teach classes for pre-service teachers, I typically have the students read and discuss a math education article about the teaching or learning of content they may eventually teach. This may include research articles (in journals such as *Journal for Research in Mathematics Education*, which typically report on research studies), or practitioner articles (in journals such as *Mathematics Teacher*, which offer practical insights without necessarily being rooted in rigorously conducted research).

Recently, however, I have also started to have students in more traditional postsecondary mathematics classes (not just those designed for pre-service teachers) read math education articles. Last term, for instance, after discussing counting problems in an advanced mathematics course, I had my students read an article by Batanero, Navarro-Pelayo, and Godino (1997) about effects of implicit combinatorial models on students’ solving of counting problems. Through such readings, my students can be exposed to research on students’ thinking about the very postsecondary content they are learning. I am always pleasantly surprised by the rich discussion such readings stimulate, and this made me reflect on the value of having students read such articles, even in their “pure” mathematics classes.

Both research and practitioner papers about math education can elicit valuable ideas and points of discussion from which math students can benefit. In this post, I make a case for three potential benefits of having students occasionally read math education articles in their math courses.

**Math education papers can help students learn more about a particular concept**

There are many math education researchers who focus on student thinking about specific mathematical concepts. This research tends to be qualitative in nature, allowing for the investigation of subtle mathematical details. There are a number of research methodologies that reflect this kind of work, including developing conceptual analyses (e.g., Thompson, 2008) conducting teaching experiments (e.g., Steffe & Thompson, 2000), or creating hypothetical learning trajectories (e.g., Simon & Tzur, 2004). While I will not outline the specifics of each methodology here, the point is that significant work is being undertaken to better understand students’ understanding of a variety of mathematical topics. I believe that having students read such articles at appropriate times could help them learn more about the mathematics they are studying.

As an example of the mathematical insight that can be gained from reading such papers, consider Swinyard and Larsen’s (2011) study in which they had students reinvent the formal definition of limit via a series of carefully chosen tasks. Ultimately, these researchers shared findings about how a pair of students thought about the formal definition, and they identified two central challenges that arose for students: “(a) students relied on an *x*-first perspective and were reluctant to employ a *y*-first perspective; and (b) students struggled to operationalize [that is, to clearly articulate] what it means to be infinitely close at a point” (p. 490). They then investigated ways in which students might handle these challenging ideas, providing in-depth discussion about details of the formal definition.

I contend that having students in an Advanced Calculus course take the time to read, unpack, and understand this paper would help them develop a more solid understanding of the formal definition of limit. They needn’t focus on the methodological details of a given study, but rather they can engage with the results and reflect on what those results might mean for their own mathematical understanding. There are countless similar examples in other domains, such as linear algebra (e.g., Wawro, 2014), abstract algebra (e.g., Cook, 2014; Larsen, 2009), calculus (Dorko & Weber, 2014; Oehrtman, 2009), discrete mathematics (e.g., Annin & Lai, 2010; Lockwood, 2013), proof (e.g., Weber & Alcock, 2004), and many, many more. The point is that the work that researchers have done to unpack deep conceptual issues may help students better understand subtleties about a concept they are learning.

**Math education papers can help students think more about others’ thinking and learning processes, facilitating reflection on their own thinking and learning**

Another benefit to reading carefully chosen math education articles is that students can think more about how they, and others, think. For many students, doing mathematics can be an isolating activity, and they might not naturally reflect on how others think about or approach a problem. There can be two extreme aspects of this phenomenon. For students for whom math comes intuitively, it may be easy for them not to think about others’ thinking at all. They may (even without realizing it) assume that theirs is the only and best way to approach a problem, and that everyone else probably thinks of the problem in the same way. Less confident students, on the other hand, may assume that their thinking about a concept (including any confusion they have) must be unique to them, and that everyone else understands the concept perfectly (this belief is particularly prevalent among female students – see David Bressoud’s November 1, 2014 Launching’s column).

As we know, though, there are many different ways of thinking about a given problem, and chances are good that many students in a class will have the same conceptions or misconceptions about a particular problem or idea. Many math education articles contain data and results that could make students more aware of others’ thinking. For example, strong students may be surprised and intrigued to see that, in fact, people genuinely struggle with concepts that they find trivial. Hopefully (through some well-facilitated discussion) they could become more empathetic with fellow students and recognize that there may be alternative approaches to a problem. For students who feel less confident, it may be encouraging and empowering to realize that other students also struggle or have the same questions that they have. It might be less isolating for them to identify potentially confusing issues, and to be able to face them head on.

As an example, I recently had students read Yopp, Burroughs, and Lindaman’s (2011) paper about one teacher’s understanding of the decimal equality .999…=1, and a number of them said that the teacher in the article reflected many of their own (incorrect) ways of thinking about decimal representations. I feel that this empowered some of them to realize that they were not alone in their ideas, and it also encouraged them to be more reflective about the unproductive notions that they had held.

**Math education papers can help create a dynamic class environment**

Finally, there are benefits to having students think explicitly about ideas related to the teaching and learning of mathematics and then to discuss them with others. I have had students say that they loved papers they have read, and I have also had students strongly disagree with papers they have read. In either case, I view their responses and their critiques as something positive that contributes to their overall mathematical development. I want students to be able to think critically about ideas, and by reading math education articles, they are invited to think hard enough about an idea to evaluate and critique it. Even an unpopular article can spur thinking and discussion that stands to benefit the students.

Even more, such passionate responses can help create a dynamic learning environment in which students feel free to share their ideas and opinions. By facilitating reflection on an article, we can introduce an extra dimension to the mathematics under consideration. The emphasis can shift away from questions of “What do I understand or not understand?” to questions of “What is hard about these concepts?” or “Why do we all struggle to learn this?” Such discussion can provide a counterweight to some of the more solitary and isolating aspects of doing mathematics.

**Concluding thoughts
**

I am not suggesting that any math education research paper is appropriate for a given situation. Indeed, having students read math education articles requires some skilled facilitation of discussion, and it takes a certain level of mathematical maturity and buy-in for reading articles to be beneficial for students. For instance, I would not recommend that Calculus I students read Swinyard and Larsen’s paper about limit – they simply are not ready to consider the mathematical intricacies presented in the paper. However, that paper could be extremely useful for Advanced Calculus (or Introductory Analysis) students who have seen the formal definition and are in the process of thinking more deeply about it. (For an example of a paper that could be read by Calculus I students, see Trigueros & Jacobs (2008)).

I also acknowledge that engaging with math education research articles should be done with care, and we must not recklessly draw conclusions about mathematical or pedagogical ideas based on cursory readings of a few papers. The ideas discussed here are not meant to suggest an overhaul of existing classes but rather are meant to serve as a supplemental activity. An instructor who wants to explore this idea could experiment with incorporating one paper in a course at first, or maybe two.

To summarize, there is value in exposing math students to papers in mathematics education. Having students read carefully chosen papers in a math class has the potential to effectively enhance students’ mathematical knowledge, improve students’ understanding of others’ thought processes, and contribute to a more dynamic classroom environment.

**References**

Annin, S. A., & Lai, K. S. (2010). Common errors in counting problems. *Mathematics Teacher*, *103*(6), 402-409.

Batanero, C., Navarro-Pelayo, V., & Godino, J. (1997). Effect of the implicit combinatorial model on combinatorial reasoning in secondary school pupils. *Educational Studies in Mathematics, 32*, 181-199.

Cook, J.P. (2014). The emergence of algebraic structure: Students come to understand units and zero-divisors. *International Journal of Mathematical Education in Science and Technology, 45*(3), 349-359.

Dorko, A., & Weber, E. (2014). Generalising calculus ideas from two dimensions to three: How multivariable calculus students think about domain and range. *Research in Mathematics Education, 16*(3), 269-287.

Larsen, S. (2009). Reinventing the concepts of group and isomorphism: The case of Jessica and Sandra. *Journal of Mathematical Behavior, 28,* 119-137.

Lockwood, E. (2013). A model of students’ combinatorial thinking. *Journal of Mathematical Behavior, 32,* 251-265.

Oehrtman, M. (2009). Collapsing dimensions, physical limitation, and other student metaphors for limit concepts. *Journal for Research in Mathematics Education, 40*(4), 396-426.

Simon, M. A. & Tzur, R. Explicating the role of mathematical tasks in conceptual learning: An elaboration of the hypothetical learning trajectory. *Mathematical Thinking and Learning, 6*(2), 91-104.

Steffe, L. P., & Thompson, P. W. (2000). Teaching experiment methodology: Underlying principles and essential elements. In R. Lesh & A. E. Kelly (Eds.), *Research design in mathematics and science education *(pp. 267-307). Hillsdale, NJ: Erlbaum.

Swinyard, C. & Larsen, S. (2012). Coming to understand the formal definition of limit: Insights gained from engaging students in reinvention. *Journal for Research in Mathematics Education, 43*(4), 465-493.

Thompson, P. W. (2008). Conceptual analysis of mathematical ideas: Some spadework at the foundations of mathematics education. In O. Figueras, J. L. Cortina, S. Alatorre, T. Rojano, & A. Sepulveda (Eds.), Proceedings of the Annual Meetings of the International Group for the Psychology of Mathematics Education, (Vol 1, pp. 45-64). Morelia, Mexico: PME.

Trigueros, M. & Jacobs, S. (2008). On developing a rich concept of variable. In M. Carlson & C. Rasmussen (Eds.), *Making the Connection, *(pp. 3-14). Washington, DC: MAA.

Wawro, M. (2014). Student reasoning about the invertible matrix theorem in linear algebra. *ZDM The International Journal on Mathematics Education, 46*(3), 389-406.

Weber, K. & Alcock, L. (2004). Semantic and syntactic proof productions. *Educational Studies in Mathematics, 56,* 209-234.

Yopp, D., A., Burroughs, E. A., & Lindaman, B. J. (2011). Why is it important for in-service elementary mathematics teachers to understand the equality .999 = 1? *Journal of Mathematical Behavior, 30,* 304-318.

*Editor’s note: The editorial board believes that in our discussion of teaching and learning, it is important to include the authentic voices of undergraduate students reflecting on their experiences with mathematics. We thank Ms. Andrews, Mr. Crum, and Ms. Laird for contributing their essay. More information regarding inquiry based learning can be found at **http://www.inquirybasedlearning.org/**.*

Inquiry based learning (IBL) classes inspired each of us to believe that we could go into mathematics. That we belonged. We may be able to prove something important or make an impact in the lives of other budding mathematicians. IBL classes have given us this confidence to believe in ourselves, and to have fun trying to discover for ourselves what math is and where it will lead us. It was not only this sense of being able to discover, however, it was also learning how to collaborate with others. Mathematics is not an isolated endeavor, but rather a concentrated attempt by groups of people working toward their common goal. In normal lecture-based classes, we would talk to our friends, and if we got stuck, we might ask one another what to do next. In the IBL classes, we would talk to each person in the class. Students would ask each other questions willingly. We would make new friends, and ask more questions, until each of us decided we were satisfied — we understood the material now.

In a typical math classroom, we go in, listen to a lecture about the infinite number of primes, or the procedure for finding a subgroup, and frantically copy the deluge of mathematics, all while trying to comprehend the complex concepts being presented. Then we go home, eat dinner, and attempt to apply our concepts to similar, but often new, ideas, without the guiding hand of a trained mathematician. In IBL classes, we receive materials, go home, and actively think about the new definitions and theorems. How do we apply these, how can we prove those, where can these be used to prove other statements? Then we marinate in these concepts, think about the math behind them for the rest of the day, and once we show up in class again, we are ready to discuss. We are ready to ask questions to further our understanding of the concepts, and attempt formally proving theorems using these concepts. This format motivated us to do more, to actively participate in the mathematics happening around us, and to think critically about theorems, rather than accepting them at face value.

The critical thinking that IBL instilled in us led to a deeper understanding of the mathematical process. There was creativity required; failure does happen, but you get up and try again. It was no longer black and white, it was a science — one that had experiments you could fail, but after having learned a lesson from the failure, you persevere and find a solution. From this, we learned proving mathematical results is a science experiment, at least when students are allowed to become active participants. We wanted to beat the problems, especially if they were challenging. We would sit there and think about why is the center of a group a subgroup of the overall group? Then, instead of giving up, we put various ideas on paper. Wrote down assumptions and useful theorems. How can these be applied? We got stuck. Instead of giving up though, or getting frustrated, we kept going. We emailed professors, got hints, and pushed forward. Then come class day, we could present a valid proof.

Looking back, this happened all the time. We would be at the library, banging our heads against the white boards in a study room writing down assumptions and wondering where to go. Normally, this is where we would stop. Where we would get frustrated. Where most students would give up, wait to go to class, and ask the instructor how to finish. However, in each IBL class, this is not the point where we gave up. This is the point that students strove for, to prove that they could get past that portion of the hardest problem on the set.

Becoming active participants opened the door to the mathematical community. We talked to professors and created a sense of community within our IBL classes that led to a deeper involvement within our department. We forged necessary relationships with our professors, and they instilled a drive to know the why behind the math we discussed. Instead of sitting in class watching the professors present a theorem and blindly accepting it at face value, we would question it. Another student was at the board trying to convince us that induction was a valid method of proof. Instead of just using the method and deciding that, yes, it worked, we could decide for ourselves if we believed what was written. We could critically think about the math being put on the board instead of just assuming that because the person is a professor, there is no need for them to convince us. After IBL classes, each of us have gone on to do some sort of undergraduate research or independent study. Taryn Laird specifically went on and did research with one of her IBL instructors. We became experienced independent learners who were driven to learn more and ask questions, and the relationships that we had forged were key in this process.

IBL classes gave each of us a glimpse into the real world of mathematics. It transformed our thinking, and gave us the extra confidence that we could do math. With this being said, much like the academic world of mathematics, our classes were predominantly male. Before IBL classes, this feeling of gender bias ranged from bothering us from the back of our minds, to being shoved down our throats. The IBL classes gave us confidence that, male or female, we could do math. When we went through these classes the instructors fostered an environment where we could ask all the same questions, come up with all the same answers, and prove all the same theorems. There are no gender reasons that should be able to stop us — each of us are just as capable as the others. Despite the obvious discrepancy in amount of males to females, we realized that anyone that told us math is not a woman’s field was wrong.

No lecture can compare to an IBL class. Lectures are useful, but IBL classes can transform students and create positive environments for people learning to question the world around them. They take students in that enjoy math, and allow them to fall in love with it completely. It helps people become less shy, gain necessary working and life skills, and truly learn to critically think. Inquiry based learning classes are about participating, learning, and growing as a mathematician.

]]>The notion of one quantity being proportional to another is certainly a very basic part of an understanding of mathematics and of its applications, from middle school through calculus and beyond. Unfortunately, the picture of proportionality that tends to emerge in school mathematics in this country is narrow and confused. Everyone learns the procedure of setting up and solving a proportion, but the connection of this to the idea of one quantity being proportional to another is tenuous.

In support of this statement, I summarize below the results of participant responses given in a workshop attended by teachers, mathematics educators, and mathematicians. The surprisingly shallow responses show a striking lack of a common, mathematically coherent understanding in this audience of the subject of proportionality.

**A. A simple problem**

In the workshop, participants first worked to solve this problem:

**Paper Stacks Problem:**

Suppose you want to know how many sheets are in a particular stack of paper, but don’t want to count the pages directly. You have the following information:

- The given stack has height 4.50 cm.
- A ream of 500 sheets has height 6.25 cm.

How many sheets of paper do you think are in the given stack?

All 18 participants found the expected result (360 sheets) by setting up and solving a proportion.

**B. What is proportional to what?**

Next, participants were asked this question:

Write down a sentence or two in response to this question:

* “In this paper stacking situation, is anything proportional to anything else?”*

The most natural response: “the number of sheets in a stack is proportional to the height of the stack” did in fact appear, but only in about a fifth of the responses. This response is in accord with a modern understanding of proportionality: a variable quantity *A* is proportional to a variable quantity *B* when there is an invariant *k* such that *A* = *kB*. In this situation the invariant is the number of sheets per centimeter.

Other responses suggested that “the height of the small stack is proportional to the height of the large stack.” But the ratio of these heights (about 0.72) is particular to these two stacks, and is not an invariant of the paper stacking situation. These two heights are not proportional in a modern sense of the term. What is getting in the way in these other responses, we feel, is a view commonly put forth in school materials: a ratio can be formed only between quantities of the *same kind*. The relationship between the number of sheets and the height of the stack cannot then be proportional, since the required “ratio” is between quantities of different kinds.

However, most disturbing is the number of responses that merely put together some scraps of remembered procedures, such as response number 4: “A proportion is the relationship of two ratios. The height of the two stacks is proportional since you are comparing one ratio to another; i.e. \(\frac{360}{4.5}=\frac{500}{6.25}\)”

**C. What does “proportional to” mean in general? **

Finally, participants were asked this question:

Write down a brief answer to this question:

* “What does it mean in general to say that one quantity is proportional to another quantity? Be as precise as you can.”*

The 18 responses are interesting enough that they are included in full:

- proportional relationship means that when one quantity in a relationship changes another will change according to some specific pattern (which won’t change in time / vary)
- “a” is prop. to “b: means that if b is altered by a factor (e.g., multiplied by t), then a is altered the same way.
- One quantity is proportional to another means the comparison is relating equal ratios.
- \[\frac{a}{b} = \frac{c}{d} \hspace{3em} ad = bc\]
- To be proportional means to have the same ratio in simplest form. The relationship between the two things is the same (in the real world like sugar:flour)
- As the numbers in the proportion change … there is a constant pattern of increase or decrease \[\frac{1\times 4}{2\times 4} \hspace{4em} \frac{4}{8} \hspace{4em} \frac{10\div 5}{5 \div 5} \hspace{4em} \frac{8 \div 2}{4\div 2} \hspace{4em} \frac{1}{2}\]

- It means that a fraction is equal to a fraction or that the two ratios are equal.
- As one part of the proportion changes the other part changes in the same relational way.
- If one quantity increases, the other quantity also increases. Or If one quantity decreases, the other quantity decreases
- It means that quantity “A” changes in a fixed or quantifiable manner as quantity “B” changes.
- The two ratios are equal. of, cross products are =
- As one quantity increases or decreases by a specified amount, the similar quantity also increases or decreases by the same amount.
- The rate of change between the two quantities is constant.
- quotients of 2 quantities are equal / constant if proportional
- The ratio of parts of each term is the same \(\frac{1\ \text{sheet}}{\text{ height}}\) is same for both

(each piece of the proportion is made up of like parts) - amount of an item will have a relation to another item
- As one quantity grows the other quantity also grows; it is a multiplicative relationship; ratio is constant; what about inversely proportional?
- When one thing is proportional to another, we can set up two fractions that are equivalent.

**D. What has gone wrong? **

The confusing jumble of responses here is disturbing. At the very least it points to a lack of a common understanding within the school mathematics community of this very basic and important subject. It would certainly be wrong to blame teachers. Rather, I believe the culprit is a general lack of mathematically sound grade-level appropriate presentations of proportionality that have been available to teachers. In addressing this lack, mathematicians must certainly play a major role.

**E. Comments**

The subject of proportionality in school has a long, complex, and fascinating history. Here, I will simply suggest the range of relevant issues.

**Euclid**

All school approaches to proportionality have their origins in Euclid’s treatment in Book V of *Elements*. This is where the brilliant treatment of ratio by Eudoxus appears. However, Euclid’s treatment of proportionality is essentially that of *discrete* quantities: four magnitudes that have the same ratio are called proportional. (See Definition 6.) Today, proportional relationships are understood as being between two *variable* quantities. In my view, the inadequate understanding of proportionality shown by many responses in the workshop is due to the failure of school mathematics materials to sufficiently stress the role of variable quantities in a modern understanding of proportionality. We elaborate on this idea in the next section.

**Proportions and missing the crucial invariant**

Finding the numerical solution to a problem such as the paper stacks problem by setting up and solving a proportion is fully reasonable, and we all do it. However, the mathematically interesting point in a situation such as this is that there is an ** invariant,** namely the number of sheets per centimeter (80 sheets per cm).

In an approach that focuses only on setting up and solving a proportion, this invariant never needs to be found. All that is found is an unknown (360 sheets) in one particular situation. This means that the crucial relationship between the variable quantities *n* = number of sheets and *h* = height of a stack is never seen: \[n = 80h.\] And in fact, seeing proportionality as involving a relationship between *variable* quantities was the key point missing from most responses in the workshop. Repeating this point from Part B above:

A variable quantity

Ais proportional to a variable quantityBwhen there exists an invariantksuch thatA=kB.

This statement includes two hard but very important mathematical ideas, the idea of an invariant, and the idea of a variable quantity. It is my feeling that work toward bringing out these ideas should begin as soon as the language of proportionality is introduced in middle school.

**Analogy: the Law of Sines**

To make an analogy, consider the Law of Sines for triangles: \(a/\sin\alpha = b/\sin\beta = c/\sin\gamma\). These three ratios are not only equal, but their common value is an important invariant of a triangle: the diameter of the circumcircle. Bringing out the invariant and its meaning is an essential part of a fully mathematical treatment of the Law of Sines. A focus on the invariant as the common value of a set of ratios should be an essential part of a mathematical treatment of proportionality as well.

**The Common Core State Standards in Mathematics**

The approach to proportionality suggested in the Common Core State Standards in Mathematics promises to be of real help, since the emphasis is directly on proportional relationships and the constant of proportionality. In fact, the approach is remarkable in that the term “ratio and proportion” does not appear at all, nor does the idea of “setting up and solving a proportion.” Instead, the central concept is proportional relationships themselves.

However, my observation is that old habits are hard to break. It will not be easy to overcome tradition in developing and implementing this far more reasonable approach.

**F. Conclusion**

I think we would all agree that a reasonable treatment of proportionality should lead to students being able to understand a statement such as this:

The gravitational force between two objects is proportional to the product of their masses and inversely proportional to the square of the distance separating them.

This requires a rather flexible understanding of the idea of “proportional to.” We have argued that a traditional approach to proportionality that focuses on setting up and solving a proportion is not adequate. Instead, what is needed is an approach that emphasizes the role of variable quantities and their invariant ratio. The responses in the workshop seen in Section C above would have been rather different if these ideas had been more prominent in school materials.

]]>The 2014 American Mathematical Society (AMS) Committee on Education (CoE) meeting took place on October 16-18 in Washington, D.C. I attended as a member of the AMS CoE. In addition to the committee members, there were many attendees from academic institutions, government, other professional societies, and the private sector. Like the recent CBMS forum that Diana White discussed in a blog post earlier this month, the focus of the CoE meeting this year was the first two years of postsecondary mathematics education. In this post, I will reflect on some of the key themes that stood out to me during the CoE meeting.

*The importance of collaboration*

The most prominent theme of the meeting was the critical role of collaboration and cooperation at many levels: among department members, at the institutional level between departments and administrative units, among professional societies with common missions, and at the national level to “scale up” successful models for effective teaching.

In talks about department-wide efforts to improve mathematics education in the first two years of college, Matthew Ando (Univ. of Illinois Urbana-Champaign), Stephen DeBacker (Univ. of Michigan), and Dennis DeTurck (Univ. of Pennsylvania) all emphasized the importance of collaborative support for department- and college-wide initiatives. This support includes faculty participation in specific programs, but was also more broadly framed through such lenses as thoughtful academic advising, working with members of other departments (such as engineering), and establishing clearly defined cooperative roles for departments and administrative units. I believe that this is particularly challenging at large universities, many of which are facing problems such as declining state support, increasing undergraduate enrollment, and severe constraints on instructional resources. For teaching environments such as these, where educational environments are increasingly large-scale, the message was clear that faculty need to work together in teams to create effective solutions to local challenges. However, these messages came with the caveat that collaboration is time-consuming, difficult, and requires sustained commitment from faculty.

It was interesting that, in addition to faculty and administrators from colleges and universities, participants at the CoE meeting included representatives from organizations involved in K-12 mathematics education, namely the National Council of Teachers of Mathematics and Achieve. This reminded me of the inherent connections between our challenges at the postsecondary level and the current national discussion regarding K-12 mathematics education (largely inspired by the widespread adoption by states of the Common Core State Standards). Institutions of higher education have increased their engagement in this discussion through the creation of advocacy organizations such as Higher Education for Higher Standards, demonstrating the type of collaborative efforts that are taking place at a national level. Another important aspect of post-secondary mathematics education that was pointed out during the meeting was the interaction between community colleges and institutions offering four-year degrees; the transition between these types of institutions is a rocky one for many students, and addressing this problem requires institutions to effectively work together.

*The importance of student-focused teaching*

Multiple speakers emphasized the necessity of broad adoption of student-focused teaching methods. Talks by Michael Starbird (Univ. of Texas Austin) and Ryota Matsuura (St. Olaf College) provided interesting perspectives on alternatives to college algebra and on Hungarian problem-based pedagogy, respectively, with emphasis on creating engaging courses for students. In the talks by Ando, DeBacker, and DeTurck mentioned previously, the need for clear learning outcomes for students was also emphasized, with teaching methods selected directly in support of these outcomes. All of these speakers emphasized the key role that active learning environments play in student development; however, the implementation of active learning environments they described was varied. For some, this meant having calculus recitations be organized around a carefully-crafted worksheet, with teaching assistants serving as “coaches.” For others, this meant largely eliminating lectures from classes, capping class sizes at 30-35 students, and using extensive group work carefully guided by the course instructor. The main message on this theme that I took from the meeting was that while student-focused teaching methods are critical, there is no “one size fits all” method that works best.

Many of the discussions during the meeting, both formal and informal, centered on the core question of “what do we want our students to know and to be able to do?” Without a well-articulated answer to that question, it is challenging to decide which teaching methods faculty should adopt. A phrase that stood out to me, mentioned by Herb Clemens (Ohio State Univ.) during his introduction to Bernard Hodgson’s (Université Laval) talk about post-secondary mathematics education in Quebec, was that we need “systemic caring” for students to be embedded in our institutions. Regardless of their specific form or implementation, the articulation of student learning outcomes and the purposeful use of student-focused teaching methods are important components of systemic caring for students.

*The coherence of education initiatives in the mathematical sciences*

There is a remarkable coherence among current educational initiatives in the mathematical sciences, broadly defined. This was especially apparent during the talks by Mark Green (UCLA), Karen Saxe (Macalester College), and Nicholas Horton (Amherst College) about the Transforming Post-Secondary Mathematics Education group, the Common Vision for Undergraduate Mathematics in 2025 project, and the American Statistical Association Guidelines for Undergraduate Programs in Statistics, respectively. Other important initiatives and reports that are worth mentioning along with these are the 2015 Mathematical Association of America Committee on the Undergraduate Program in Mathematics Curriculum Guide, the National Council of Teachers of Mathematics report Principles to Actions: Ensuring Mathematical Success for All, and the National Research Council report The Mathematical Sciences in 2025.

These initiatives and reports share a strong focus on increasing the number of pathways for students into the study of the mathematical sciences, and on reducing the number of barriers for students to cross along the way. Speakers at the CoE meeting emphasized the important role that evidence-based teaching practices can play in this regard, and the need that faculty and departments have for professional societies to make such practices easily identified and accessed. I view the coherence of the recommendations arising from these non-coordinated efforts in the mathematical sciences as an extremely positive sign, as it provides encouragement for us to join efforts in pursuit of common goals; further, these reports provide a reasonably common language through which to do so.

*A final observation*

A recurring phrase used by participants through the meeting was “The Time Has Come,” hence the title of this article. I agree that the time has come for all of us involved in the mathematical sciences to work together to improve mathematics education at all levels.

]]>In early October, approximately 150 educators and policy makers gathered together in Reston, Virginia for the fifth Conference Board of the Mathematical Sciences (CBMS) Forum entitled *The First Two Years of College Mathematics: Building for Student Success*. Participants came from almost every state in the country and represented higher education institutions ranging from two-year colleges to top-ranked research universities. We spent two days reflecting, learning, and in some cases planning how to improve the last year of high school mathematics and the first two years of college mathematics.

As is often my reaction at these types of conferences, I found the two days both sobering and energizing — sobering because of the sometimes harsh realities and challenges we face, energizing because of the good work participants report on and the many people gathered together who care so passionately and who dedicate so much of their time and energy to moving us forward. For those who could not join us in Virginia, this blog post will present a few key highlights from the Forum, in an effort to open a broader conversation about the future of the first two years of collegiate mathematics instruction.

The Forum began by emphasizing many ways in which the mathematical sciences are thriving. The National Research Council report *Mathematical Sciences in 2025* notes that there have been many major research advances, both theoretical and in high-impact applications, with clear benefits to other STEM areas and to the nation. As the foundation of many recent STEM advances, the role of the mathematical sciences has expanded.

This creates a need to revisit many aspects of our mathematical training of students to ensure that we are meeting the needs of our diverse constituents. In addition, the overall student success rates in mathematics are concerning. The President’s Council of Advisors on Science and Technology (PCAST) report *Engage to Excel: Producing One Million Additional College Graduates with Degrees in STEM *notes that “Reducing or eliminating the mathematics preparation gap is one of the most urgent challenges — and promising opportunities — in preparing the workforce of the 21^{st} century.” PCAST considers the situation so dire that they suggest an experiment in which faculty from mathematics-intensive disciplines such as physics, engineering, and computer science design and teach college mathematics courses. The PCAST report has received a lot of attention from the mathematical community, see the blog post from David Bressoud for some historical context and the summary response from the Joint Policy Board for Mathematics. The full response is here.

Multiple speakers at the Forum called on the mathematical community to wake up and heed the call for change this proposal implies. They called for a broadening of localized efforts to respond, while taking note of illuminating data and promising programs that may make this possible. Speakers at the Forum emphasized that we do not lack demonstrably successful and promising programs to meet many of the challenges the first two years of college mathematics present. However, widespread adoption of these programs has lagged.

*STEM Careers: Building for Success in Calculus*

The *Characteristics of Successful Programs in College Calculus* study from the Mathematical Association of America provides a large-scale, evidence-based understanding of who takes calculus, why they take it, and what happens in the course.

Students come out of calculus courses with greatly decreased confidence and frustration at their lack of understanding. Further, many students who initially declare or express interest in Science, Technology, Engineering, and Mathematics (STEM) majors change their mind in calculus class. The lack of persistence in STEM trajectories is sufficiently dire that we can currently view a moderate decrease in overall student interest in pursuing a STEM career after taking calculus as a success.

However discouraging this may seem, it is also encouraging that we now have such extensive data, as it allows the mathematical community to transition from small scale studies and so-called autobiographical reasoning — based on personal experience at our own institutions or throughout our own careers — to evidence-based reasoning from a large scale study.

*Affecting Student Success in General Education Mathematics*

As noted by Tony Bryk, President of the Carnegie Foundation for the Advancement of Teaching, “developmental mathematics is where aspirations go to die”. Yet the vast majority of first year community college students require some form of developmental education. Lack of student success in these developmental mathematics courses often hampers degree completion, as less than half successfully go on to complete a transfer level mathematics course (defined as a course that meets the general education mathematics requirement at a four year institution). The likelihood of persisting to successful completion of a transfer level course decreases drastically as the number of developmental courses needed as a prerequisite to a transfer level course increase, dropping to approximately 10% for those needing three or more developmental courses.

The* New Mathways Project*, an initiative of the Dana Center and the Texas Association of Community Colleges, is thus far showing remarkable results in both decreasing the time required for students to be ready for a transfer level course, as well as the success rates in such courses. It consists of three primary pathways — the statistics pathway, the quantitative pathway, and the STEM pathway.

The statistics pathway, known as Statway, prepares students in disciplines such as nursing, social work, and criminal justice for the college level statistics course their disciplines require. Now in its fourth year of implementation, it has a success rate (defined as earning a C or better in a college level statistics course within two years) of approximately 50%, as opposed to approximately 15% for those in a comparison group.

Results in the other pathways are also promising, and scale-up efforts are in place. The New Mathways Project is now being implemented in additional states, including my own state of Colorado, with optimism based on solid data that it will positively affect degree completion.

*Moving Forward*

This blog post cannot possibly do justice to the scope of topics addressed throughout the two days. For example, I have not addressed the Common Core and secondary to post-secondary articulation, faculty instructional approaches, diversity and gender concerns, or the reality of the fiscal challenges associated with these efforts. The complete list of abstracts, many of which contain the corresponding slides, give a sense of the scope of the Forum.

There remain many challenges related to entry-level mathematics, and to put the importance in context, the mathematics community would be wise to keep in mind that the number of students taking these courses, including the calculus sequence, by far eclipses the number of mathematics majors. College mathematics instructors would benefit from increasing their awareness of the extensive developments related to curriculum and teaching, especially at this lower level. Several Forum speakers also noted that the reward structure for mathematicians in academia, especially those at research universities, is another obstacle that we need to address to further increase participation, especially amongst active research mathematicians.

The intensive two days ended with a final challenge. Noting that the Forum would be a failure if participants only used what we learned as information for ourselves, we were encouraged to talk broadly with colleagues about what we learned and more broadly about the first two years of college mathematics education. Toward this end, I presented a summary of this meeting to our undergraduate committee, and I am going to share a copy of *Mathematical Sciences in 2025* with both my department chair and dean. It doesn’t feel like much, but it’s a start. This blog entry is also a contribution to that discussion, and an attempt to reach out to colleagues beyond my institution.

What are your thoughts about the issues the Forum addressed? How can we improve the first few years of undergraduate mathematics instruction, either locally or nationally? What do you view as the barriers to wider implementation of programs with demonstrated success? What are you willing to do to contribute to this effort?

I encourage readers to contribute to this discussion in the comments section and/or under the announcement for it on the AMS Facebook page.

]]>*Comment from the Editorial Board: We believe that in our discussion of teaching and learning, it is important to include the authentic voices of undergraduate students reflecting on their experiences with mathematics. This article is our first such contribution. We feel it provides a window into many of the subtle challenges students face as they transition to advanced postsecondary mathematics courses, and that it mirrors many of the themes discussed in previous posts. We thank Ms. Mattingly for being the first student to contribute an essay to our blog.*

In previous math classes, I was the quiet worker who kept to herself and didn’t know when or how to ask questions. After improving my skills in a problem solving class, that has changed. The group work we did each day allowed me to be around other people who think significantly differently than I do. Being in this environment was difficult at first because I actually had to work through problems with other people, which was somewhat unfamiliar to me. My classmates and I were not just sitting down and reading information about specific math problems. We had to analyze and make sense of the best methods and strategies to use and present our ideas to each other. Confusion would set in when other students introduced different approaches. The only way I could understand their ways of thinking was to ask them to explain. Asking questions in math initially intimidated me, especially because my questions had to be directed to my peers. I did not want them to think that I could not keep up with the material or that I did not belong in the class. But I also did not want to misunderstand major mathematical concepts as a consequence of not asking questions. So I started asking my group members each week what strategies they used in their solutions. Although it may have seemed repetitive to them or obnoxious to have to explain their approaches, it helped me immensely. Through my question asking, I was able to talk and think about math in a unique way. I could compare my peers’ techniques to my own, which further stimulated my interest in the particular subjects that were covered in the class. This skill has been and will continue to be essential in my future relationship with mathematics.

With this question-asking skill came a respect for other students’ speed of thinking. As I was learning to think like my group members in certain situations, I also was learning that these students were thinking at different speeds than I was. In many instances, I would attempt to understand what the problem was asking for and in the meantime, my peers were already halfway to a solution. I knew that I wasn’t misunderstanding anything. I simply was not making connections as quickly as others in the class. It constantly surprises me how swift some students are in accurately assessing problems and understanding what is needed to get to a solution. After working with all different types of students, I have learned to respect and accept that others may be working at faster cognitive speeds than I am. In previous classes, I thought that being quick with my math skills was most important. I have now seen that understanding the material is essential to becoming a fast thinker in mathematics.

Through the homework problems and quizzes in the problem solving class, I realized that math problems require perseverance. The problems that are actually difficult, that actually require a student to think about how to apply his or her mathematical knowledge to the solution of a problem, are the problems that take time. I found myself working hours on different math problems, trying to get closer and closer to the right answer. Oftentimes I went through the trial and error process. Failing in math is not as scary as it once was, because of my experience with this problem-solving course. Sometimes I would successfully solve a problem, while other times I didn’t come close to the solution. Either way, I was able to see that simply working with math was enhancing my problem solving skills. I will always remember Paul Zeitz’s quote in his book, *The Art and Craft of Problem Solving *[1], that says, “*Time spent thinking about a problem is always time [well] spent. Even if you seem to make no progress at all,*” (pg. 27). As I encounter future math classes and harder math problems that seem unsolvable, I will keep this quote in mind. Any time spent toying with a problem is enriching my mathematical knowledge.

Understanding Zeitz’s important quote about mathematical thinking prompted me to see the open mind that math problems require. When I took a number theory class in my first year, I was under the impression that all solutions to problems were very obvious and the methods to solve them were evident. Going through a geometry class as a sophomore challenged this belief because I began to see that not everyone knows which method to use immediately to solve or prove a problem. With the problem solving class, this belief was completely put to rest. I have seen firsthand that it occasionally takes experimentation to figure out which method or tool to use in problem solving. Through discussion and group work with my classmates, I noticed that it is not always blatantly obvious that we should draw a picture or use induction or reformulate a hypothesis to find the crux move in a solution. After discovering this, I attempted to open my mind when reading assigned problems. Instead of honing in on one specific method or strategy, I have accepted the fact that one specific method or strategy might not be the only way to achieve my goal.

An open mind in problem solving has allowed me to experience mathematical thinking outside of the classroom. My high school math experiences bred the idea that students don’t necessarily need to think about math outside of the classroom. I brought this idea to college and had no problem passing my classes. But in this math problem solving class, where we were challenged to think in different ways and to explore math on our own, thinking about math strictly inside the classroom was insufficient. One incident that had a deep influence on my problem-solving experience occurred on a walk home from class. In small groups, my classmates and I were trying to determine the difference between permutations and combinations. After working for an entire class period, I did not fully understand the difference between these two basic combinatorics concepts. As I walked home, I reviewed the strategies and methods that I used and compared them with the explanations of my peers. I was still stuck and still frustrated. Upon emailing my professor about my misunderstanding, I realized that although I had encountered an obstacle with the content, I was gaining an invaluable way of thinking. I was extending my mathematical thought processes and interests to outside of the standard classroom environment. I am still constantly left wondering why something in math works. By pondering on my own, whether it is with pencil and paper or not, I am able to see the impact of this class. No class before had prompted me to take my own time to figure out why I did not understand the material. This was a huge win for me as a math student. I was and am still experiencing the effects of struggling with math in a way that will always benefit my problem solving skills.

One of the most interesting proofs that I saw was dealing with the number of subsets of a set with \(n\) elements. Earlier that semester, my probability professor had mentioned that the number of subsets of an \(n\)-element set is \(2^n\). I didn’t think I would be seeing this anymore, so instead of trying to understand why this was so, I just accepted the information. Later on, in my problem solving class, we had a question about the number of subsets in an \(n\)-element set. Obviously I knew it was \(2^n\), but I had no idea why. I eventually understood after I showed interest in the problem through question asking, when another interpretation of all of the subsets of an \(n\)-element set was written up on the board. I saw that another way to write a subset of an \(n\)-element set is by exchanging the actual numbers in the set with 0’s and 1’s. A “0” indicates the absence of a specific element in the subset, while a “1” indicates the presence of a specific element in the subset. For example, the set \(\{1, 2, 3, 4, 5, 6\}\) has the subset \(\{2, 4, 6\}\). This subset can be written as 0, 1, 0, 1, 0, 1, where the 1’s correspond to the numbers found in the subset. Since each space has either the option to be in the subset or not, then each space has two options. Since there are *\(n\)* spaces, then there are \(2^n\) subsets. By asking questions, expressing curiosity, and actually attempting to understand why the answer was \(2^n\), I was able to see a portion of my growth as a problem solver.

I am left with questions regarding my future mathematical experiences. Instead of simply thinking about mathematics outside of the classroom, I am now wondering how to discover and develop new insights about mathematical concepts on my own. Instead of learning about how to use certain strategies, I am wondering how to present these strategies to a group of peers in an orderly and effective manner. Instead of asking questions that do not necessarily prove to be productive, I am slowly learning how to ask the *right* questions.

**References**

[1] Zeitz, P. (2007). *The Art and Craft of Problem Solving*. Hoboken, NJ: John Wiley & Sons, Inc.