*By Jess Ellis Hagman, **Contributing Editor**, Colorado State University*

I’ve recently finished my third year as an assistant professor in the mathematics department at Colorado State University. Since my research area is mathematics education, I am often asked what it is like to be a math-ed researcher in a math department. Such curiosity points to a cultural difference between mathematicians and mathematics-education researchers, and alludes to a specific culture where it may be difficult to be an education researcher in a mathematics department. To me, this question sometimes feels akin to being asked what it is like to work at Hogwarts as a Muggle, surrounded by real witches and wizards. Certainly, this comparison carries with it some information about how I perceive the question: that mathematicians are the real researchers, and that as a mathematics-education researcher I am lurking in their world. While this may be how I hear the question, it is very far from my experience in my math department with my colleagues. There are about 30 faculty in my department and three of us are active mathematics-education researchers. I have had overwhelmingly positive interactions in my department and feel valued as a teacher and as a researcher. When asked how I have had such a positive experience in my department (i.e. how I have gained acceptance at Hogwarts by the wizards and witches), my answer is both that my colleagues are just great people and that we have good relationships because we have gotten to know each other and each other’s work through conversations rooted in curiosity. I think it’s been valuable that we respect each other both as people and as researchers. In this blog post, I want to share some of the substance of what I have shared with them about mathematics education research.

**Overview of mathematics education research**

Mathematics education research is the systematic study of the teaching and learning of mathematics. This means that the types of questions we ask are about how people (students of all ages, non-students of all ages) think about and do mathematics, and about how people teach, why they teach that way, and what students might learn from that instruction. We use quantitative (typically survey based) and qualitative (typically interviews and observations) research methods to answer these questions. By its nature, math-ed research is extremely broad. All sorts of people think about and do mathematics – including school children, college students, graduate students, nurses, research mathematicians, mathematics teachers, food card vendors, etc. So when we ask questions about how people learn mathematics, we can attend to different conceptions of learning (I’ll say more about this below), different populations of people, different types of mathematics, where the people are thinking about or doing the mathematics, how their experience with mathematics relates to other things, and more. When we ask questions about the teaching of mathematics, we can also attend to the components above, as well as different ways that we can teach effectively and how teaching is impacted by internal and external factors.

**Flavors of Math-Ed Research**

There are a number of ways of categorizing different types of math-ed research. I’ll go over a few, specifically: topic of focus, pure versus applied, and two different strands emphasizing the post-secondary level.

*Topic of Focus*

The most obvious and common ways to split up different areas within math-ed research are by population of students/learners (elementary, secondary, post-secondary, future teachers, in-service teachers, graduate students, mathematicians, etc.) and by content area (geometry, algebra, calculus, etc.) Thus, it is common to describe a math-ed researcher as: “She studies proof and reasoning of students across age levels” or “He studies how teachers understand proportions and fractions.” While these are overly simplified versions of how we might actually describe these two specific math-ed researchers, it illustrates my point.

*Pure and Applied*

Just like mathematics researchers differentiate between research conducted without any practical end-use in mind and research conducted in order to solve a specific problem, math-ed research also has pure and applied flavors. While it may be easier to make someone’s pure math-ed research applicable because it is focused on education, there is certainly an abundance of math-ed research done without an intended concrete application. Alan Schoenfeld (2000) delineates pure and applied math-ed research by their goals: Pure math-ed research is done in order “to understand the nature of mathematical thinking, teaching, and learning” and applied math-ed research is done in order “to use such understandings to improve mathematics instruction” (p. 641). Often in math-ed research, pure investigations quickly become relevant and other researchers are able to directly leverage such work in concrete settings.

Pure math-ed research may look at topics such as the cognitive structures people hold and develop surrounding calculus (e.g. Pat Thompson’s work). Applied math-ed research, on the other hand, is more directly focused on how to improve mathematics instruction. I consider much of the MAA Calculus Project’s work to fall under this category – we have focused on investigating what makes a calculus program especially good for students and how to support other mathemaics departments to improve their programs. Often I find that pure math-ed research relies and extends theory (this will be explained more below) much more than applied work.

*RUME and SoTL*

One subfield within applied math-ed research comes from college mathematics faculty who do scholarly work around their teaching, called Scholarship of Teaching and Learning (SoTL). This contrast the Research in Undergraduate Mathematics (RUME) community, which is the primary academic home for mathematics education researchers (pure and applied) who focus our work on undergraduate mathematics, undergraduate mathematics students, teachers of undergraduate mathematics, or undergraduate mathematics programs. SoTL is a community of academics of different disciplines who are interested in scholarly inquiry into their own teaching of their discipline. In mathematics, this community is primarily populated by mathematicians who engage in scholarship related to college level mathematics. Since both communities use scholarly principals to investigate the teaching and learning of undergraduate mathematics, there are many overlaps between questions of interest. However, there are also some key differences. Curtis Bennett and Jacqueline Dewer, prominent leader in the mathematics SoTL community, describe the differences between “teaching tips”, SoTL, and RUME as follows:

Teaching tips refers to a description of a teaching method or innovation that an instructor reports having tried “successfully” and that the students “liked.” If the instructor begins to systematically gather evidence from students about what, if any, cognitive or affective effects the method had on their learning, she is moving toward scholarship of teaching and learning. When this evidence is sufficient to draw conclusion, and those conclusions are situated in the literature, peer reviewed, and made public, the instructor has produced a piece of SoTL work…. Mathematics education research or RUME is more in line with Boyer’s “scholarship of discovery” wherein research methodologies, theoretical frameworks, empirical studies, and reproducible results would command greater importance. This naturally influences the questions asked or considered worth asking, the methods used to investigate them, and what the community accepts as valid. (Bennet & Dewer, 2012, pp.461).

To carry their progression of the teacher’s description of a good teaching innovation toward her production of SoTL work onto a RUME study, I will put this in the context of teaching proofs. A nice example of a teaching tip related to proofs is found on a blog post called “How to teach someone how to prove something,” where the author describes that when she teaches proof she has asked “each student to give a presentation to the class on some proof they particularly enjoyed, and I sat through a preview of their presentation and gave them extensive advice on board work and eye contact.” She says that though this took a lot of work on her end, it was beneficial for the students, claiming that it “really helped them prepare and also boosted their egos while at the same time increased their sympathy with each other and with me.” The author shares a teaching approach with an (unsubstantiated) claim about how this positively affected her students. Both SoTL scholars and RUME researchers would agree that this claim is unsubstantiated because she did not collect data (either from her classroom or others) to support it.

Suppose this same teacher wanted to provide some evidence for this claim that may convince others that her approach is beneficial. She may survey her students’ mathematical confidence before and after the class, and interview them to understand the role of the classroom presentations on their confidence. She could write a paper describing her approach and her findings, connect her work to other literature, and submit this work to a SoTL outlet (such as PRIMUS). The result may look similar to Robert Talbert’s 2015 PRIMUS publication describing the benefits of inverting the transition to proof class based the author’s personal reflections as the teacher of the course and responses to a questionnaire about the class from about 30 of the 100 students in class. One of the authors’ conclusions from this work was that a student-centered introduction to proof course shows promise for “helping students emerge as competent, confident, self-regulating learners”.

If this teacher then wanted to pursue this work in a way more aligned with RUME work, she would have to identify a specific research question. In RUME, the research question is a necessary component of the work to identify the scope of the question and ensure that the research methods are aligned with the research question, and that the results answer the question. One such question that would explore the role of proof on students’ beliefs could be: “What are undergraduate students’ beliefs about the nature of proof, about themselves as learners of proof, and about the teaching of proof?”, and explore the question on a scale larger than her own classroom. Such a research question partially guided the work of Despina A. Stylianou, Maria L. Blanton, and Ourania Rotou in their 2015 publication in the International Journal of Research in Undergraduate Mathematics Education. To answer their research questions, the authors surveyed 535 early undergraduate students from six universities and then conducted follow up written test and interviews with a subset of the students to better understand the survey results. One of the findings from this work was a strong positive relationship between students’ beliefs about the role of proof and with their views of themselves as learners.

The claims made by teaching tip, SoTL work, and RUME work all shed light on the positive relationship with engaging in mathematical proofs and students’ beliefs about themselves. The difference is the audience of the claims, the degree to which the argument may convince others of the claims’ validity, and the role of theory in the arguments.

**Role of Theory**

Since mathematics education researchers are concerned with the teaching and learning of mathematics, math-ed research draws on theories of learning, often from psychology. In pure math-ed research, this theory is often made very explicit, such as in Pat Thompson’s work where he draws very explicitly on Jean Piaget’s constructivist perspective. In applied math-ed research, such as the work through the MAA’s Calculus projects (that I am involved with), the theory may be more implicit in the work, meaning that it is not at the forefront of the work but that there is an underlying theory guiding the work. In SoTL work, there is often no implicitly or explicitly mentioned guiding theory of learning. This is mostly due to a combination of differences in expectations and goals of SoTL versus mathematics education research. To wrap up this post, I will give a (very!) brief overview this idea.

A theory of learning is an explanation of how people learn – observe that this is subtle, as other ways of phrasing this sentence carry with them different assumptions about learning.

- “…how people gain knowledge” assumes that knowledge is something to be acquired, and draws on an
**acquisition metaphor**of learning where our brains are vessels for carrying around our knowledge. - “…how people develop knowledge” draws on a
**constructivist perspective**, where each individual reconstructs mental images based on interactions in the world. - “…how people become more proficient in certain practices” is a more
**participation-oriented**phrasing, highlighting that knowledge is not something one owns but something one does by participating in the practices of a community to growing level of expertise.

Based on which explanation of how people learn a researcher subscribes to, they will ask different research questions and answer these questions using different approaches. For instance, suppose a researcher were interested in exploring student learning of derivatives. Taking the first approach to learning (i.e. drawing on an acquisition metaphor), a math-ed researcher may create a research study to investigate “How much do students learn about derivative in teaching approach A?” To answer this question, the researcher could develop a test with a number of derivative questions and administer the same test to students at the beginning and end of the class and compare the results. This approach assumes that what students have learned in the setting of the classroom is carried with them into a testing situation and how they do on the exam is indicative of what they know. If, instead, the researcher draws on the second approach to learning (a constructivist approach), then she may ask the question “What are different student conceptions of derivative?”. To answer this question, she may create a think-aloud interview where the students are filmed or recorded working on various problems about derivative, and asked to explain how they are thinking about the problems. This approach assumes that an interview setting can recreate a situation where students can access and share how they make sense of derivative. Lastly, if the researcher ascribes to the third perspective of learning (a participation-oriented approach), then the question asked may be “How do students use their understanding of derivative in mechanical engineering classes?”. This question could be answered by observing student interactions in the classroom as they work in groups on problems that rely on the derivative. This approach assumes that it does not make sense to decontextualize student thinking from the real learning environment.

**Conclusions**

This brief introduction to mathematics education research was written to shed some light on the aspects of math-ed research that are often of interest to mathematics researchers, from my perspective as a Muggle in a Math department. If you have more questions about what we do – ask us! For more information, check out the SIGMAA on RUME page which has more information on publication venues and conferences related to RUME. This book about SoTL has more information about the community, and much more information can be found online. Lastly, both RUME and SoTL sessions appear at the Joint Meetings, which are great ways to get a small taste of this work. In closing, please enjoy these musings about Muggles:

“Muggles have garden gnomes, too, you know,” Harry told Ron as they crossed the lawn… “Yeah, I’ve seen those things they think are gnomes,” said Ron, bent double with his head in a peony bush, “like fat little Santa Clauses with fishing rods…”

―J.K. Rowling,Harry Potter and the Chamber of Secrets“The wizards represent all that the true ‘muggle’ most fears: They are plainly outcasts and comfortable with being so. Nothing is more unnerving to the truly conventional than the unashamed misfit!”

―J.K. Rowling, Salon 1999

Hi Jess. I think that you’ve done a wonderful job of communicating about research in math education here. I want to also add that mathematics educators study how teachers of mathematics learn to teach in teacher preparation contexts (and what they learn) and how practicing teachers learn more about teaching in professional development contexts (and what they learn). Much of this type of scholarship is showcased at conferences like the annual conference of the Association of Mathematics Teacher Educators (AMTE) and in journals specifically focused on research in teacher education contexts. In mathematics education, those journals include the Journal of Mathematics Teacher Education (JMTE) and Mathematics Teacher Educator (MTE).