Mentoring Female Undergraduates in Research-Centered Outreach

Abstract

Math Girls $\mathbb{R}$ock! is a year-long, two-tiered mathematics mentoring program that prepares female undergraduate mentors to facilitate high school girls' engagement in challenging mathematics concepts through a dynamic after-school program. In this article, we describe the distinct educational component of this program in which female faculty members mentor female undergraduate mathematics and mathematics education students in researching and developing the content to be presented at the high schools. In addition, we discuss some of the feedback collected from program participants about various aspects of this program relating to the involvement of the undergraduate students. In conclusion, we share advice for those interested in starting and running similar outreach programs.

Keywords:

• K–12 math outreach
• mentoring
• female role models
• women in math
• undergraduate research
• near-peer

1. INTRODUCTION

Role models are important in inspiring, recruiting, and retaining students in mathematics and science [9,11,12]. However, many students lack female role models from STEM fields. In addition, undergraduate research has been recognized as a high-impact educational practice [13] that provides undergraduate students with opportunities for increasing professional development and gaining valuable research and job skills [3,14]. Moreover, participation in K–12 outreach programs increases the STEM interests of undergraduates who are assisting in implementing those programs, and at the same time, these programs attract high school students into STEM fields [17]. Furthermore, research shows that near-peer¹ mentoring has long-term benefits for participating STEM students [1].

In this paper, we describe the Math Girls $\mathbb{R}$ock!² program, which incorporates all of the above components: female role models, undergraduate research in mathematics, K–12 outreach, and near-peer mentoring. It was founded in 2011 and designed as a high school outreach program that

• provides two-tiered mentoring and female role models for the participating student groups (female undergraduates and high school students). The female undergraduate students experience both sides of the mentor-mentee relationship: they are mentored by female faculty members and they, in turn, serve as near-peer mentors for high school girls;

• engages female undergraduate students in basic but meaningful mathematics research through investigating interesting mathematical ideas outside the standard undergraduate curriculum; and

• provides the undergraduates a central role in creating accessible, engaging, hands-on projects which they present to female high school students using teaching models and practices that have been recommended by the Scholarship of Teaching and Learning (SoTL) scholars [8,15].

It is beneficial for undergraduate women to socialize with other women in STEM fields and to serve as role models for female high school students [9,11,12]. In addition, the authors believe that mathematics undergraduates need to be exposed to fun, exciting, and relevant applications of higher mathematics and be taught to communicate complex mathematical ideas using clear, precise, accessible language. The Math Girls $\mathbb{R}$ock! program offers all of the above to the participating female undergraduate students.

An extensive literature search produced references on near-peer mentoring programs (mostly in medical fields and STEM fields) that include an undergraduate research component [2], but did not produce any references for outreach programs combining undergraduate research in mathematics involving high-level mathematics outside the curriculum and multi-layer mentoring.

In this paper, we describe the structure of the Math Girls $\mathbb{R}$ock! program (Section 2) and then elaborate on the unique educational component of the program aimed toward female undergraduate students that combine near-peer mentoring and basic undergraduate research in mathematics (Section 3). Furthermore, in Section 4, we share some of the feedback we have collected since 2011 using student surveys. In conclusion (Section 5), we discuss some of the challenges encountered and offer some practical advice for the implementation of the ideas used in this outreach program. In addition, Appendix 1 contains a description of one of the projects used during this program and Appendix 2 contains a representative sample of the survey questions.

2. STRUCTURE OF THE MATH GIRLS $\mathbb{R}$OCK! PROGRAM

Math Girls $\mathbb{R}$ock! is a year-long, two-tiered mathematics mentoring and engagement program. It prepares female undergraduate students to facilitate high school girls' engagement in challenging mathematics concepts during dynamic after-school meetings. Each year, the authors' home institution collaborates with high schools to run this program. It is worth noting that since the program started in 2011, three high schools, 17 undergraduate students, and 250–300 high school students have been involved in this program.

The Math Girls $\mathbb{R}$ock! program is designed with two main objectives in mind: to engage female university students in ways that will enrich their knowledge and learning in mathematics, and to convey to high school girls the value of a college education, in particular an education in mathematics. As was noted in Section 1, the focus of this paper is the benefit of the program for the female undergraduate students. The rich experience of the high school students participating in this program is the subject of a separate paper.³

Each year, female full-time mathematics professors (program directors) recruit up to three motivated and personable female mathematics or mathematics education majors (program assistants) to participate in creating and delivering two mathematics-intensive, hands-on projects per semester (for a total of four projects during the academic year). Some of the project topics were drawn from Graph Theory, Knot Theory, Topology, Cryptography, Probability, Advanced Geometry, Fractals, Tessellations, etc. Program goals are achieved through two-tiered mentoring during two kinds of meetings throughout the academic year:

• Weekly group meetings on the campus of the authors' home institution: during the five weeks leading up to the presentation of each project, program directors meet with the program assistants once each week for approximately 2 h to mentor them in researching and designing the hands-on mathematics projects and to prepare them for working effectively with the high school students while delivering those projects. This educational component of the program is discussed in more detail in Section 3.

• After-school meetings at the participating high schools: the program assistants, under the direction of the program directors, conduct one after-school meeting per project at each participating high school. Each meeting lasts approximately 2–2.5 h and includes presenting a slideshow on a chosen discussion topic (15 min), completing the project packet (1–1.5 h), and playing fun math games (15–30 min). At the conclusion of the meeting, the high school students fill out a survey about the project. At these meetings, the program assistants serve as near-peer role models for the high school students as they deliver the prepared project, engage them in various hands-on activities, and discuss aspects of college life.

This program helps young women at both levels strengthen intellectual and practical skills such as logical and critical thinking, quantitative and qualitative reasoning, and problem-solving. Both student groups are exposed to mathematical ideas and concepts not usually covered in their respective curricula. They are also engaged in fun, mathematics-based, hands-on activities for exploration and discovery. Discussions about college, mathematics, female mathematicians, and careers that use mathematics (which began with the slideshow presentation) continue naturally throughout the meeting.

Since 2011 there have been 17 program assistants (10 of them mathematics education majors) participating in this program. It is worth noting that the majority of them presented their experiences at international, national, or regional conferences, gaining valuable research experience and enriching their resumes. In addition, some of the assistants pursued a graduate degree in mathematics or mathematics education.

3. THE COMPONENT OF MENTORING THE UNDERGRADUATES

In this section, we discuss and elaborate on some of the important distinct and unique features of Math Girls $\mathbb{R}$ock!. As we pointed out in Section 1, Math Girls $\mathbb{R}$ock! offers an intentional and distinguishing educational component especially targeting the female undergraduate program assistants. This component was designed with a few objectives in mind: to offer female role models for the undergraduate students, to give these undergraduates an opportunity to enrich their knowledge with new mathematics topics, to introduce them to basic mathematics research while collaborating on designing high school projects and to provide them with the opportunity to use these projects to mentor and teach high school girls. These objectives are accomplished through the distinguishing feature of two-tiered mentoring offered by this outreach program.

To assess the impact of the program on the participants as well as to obtain feedback on various aspects of the program, participants were invited to take part in a series of surveys. Each year, the high school participants were given one pre-survey (before the program started) as well as post-surveys immediately following each after-school meeting. The undergraduates were given a post-survey after each project and/or a survey at the conclusion of the year-long program. These surveys contained both qualitative and quantitative questions (Appendix 2). The quantitative data were analyzed using descriptive statistics. In addition, some of the undergraduate alumni were invited to participate in a retrospective online qualitative survey in Spring 2021.

The feedback from these surveys is presented in this section as well as in Section 4.

3.1. Learning and Researching New Mathematics Topics

Before each semester starts, the program directors choose the mathematics topics for the two projects to be completed during that semester. Each project is led and facilitated by one of the program directors. After the topic is chosen, the lead program director creates a possible outline for the project, gives various research assignments to the program assistants, and provides them with resources (articles, books, website links, etc.). Some research assignments are group assignments, while others are given to each assistant individually. These assignments provide the experience and benefits of beginning mathematics undergraduate research as assistants discover what is known about that topic and present their findings for discussion [19].

During one year, for example, the following assignments were given to the program assistants to be completed outside the meetings:

• Project 1: Research mathematics used in card tricks. In addition, read and learn about Gergonne's card trick with 27 cards, counting in base three, and radix sorting.

• Project 2: Research Platonic and Archimedean solids and their properties.

• Project 3: Research various coloring theorems and graph-vertex coloring.

• Project 4: Research tessellations, their properties, and various types of tessellations.

For Project 3, each of the three assistants was given a specific aspect of coloring to research: the two-color theorem and games associated with it; coloring theorems for various surfaces; and the four-color theorem and vertex coloring.

During the first on-campus meeting, assistants share the material they have researched and the whole group discusses which parts should be included in the project and what other possible ideas should be considered. The lead program director takes these recommendations and produces a new draft of the project for the next meeting. Program assistants research follow-up ideas (assigned by the lead director) for continued discussion during the next meeting. For example, for the card tricks project, program assistants continued learning about card tricks they discovered during the previous week and created short activities as an addition to the main project. For the Platonic and Archimedean solids project (Appendix 1), they followed up on research about the occurrences of these solids in real life and suggested including their findings in the main project. Making substantial contributions that bring the project to life gives the assistants a sense of pride of ownership of the projects.

Similar iterations of the project drafts occur for an additional two or three weeks during the weekly on-campus meetings.

In surveys given at the end of each project, the program assistants indicated that they felt they were engaged in meaningful research of new mathematics topics by learning new material, ideas, and connections that they had not seen before. It is worth noting that they take these assignments very seriously, work collaboratively on them, and enjoy the given responsibilities. In addition, some of them went a step beyond what they were asked to do, showing initiative and a deeper understanding of the topics as this survey comment shows: “Yes, I even extend[ed] this project into creating a Fractals WebQuest I intend to use when I start teaching.”

Ideas from program assistants for possible mathematics project topics are encouraged, welcomed, collected, and many times implemented for future projects. Usually the topic ideas the assistants suggest are topics that they have heard about but have not seen in their traditional curriculum. The program directors noticed that the assistants gained confidence in their mathematical abilities and had a sense of accomplishment at being heard and recognized when their topics were chosen for implementation.

3.2. Designing the High School Projects

Another aspect of the Math Girls $\mathbb{R}$ock! weekly on-campus meetings is designing, polishing, and producing the final project materials. During these meetings, program directors mentor the program assistants on creating the guided questions to be included in the project that will maintain a good flow of the material covered using active learning methods and practices. Each project is designed as a hands-on exploration activity with inquiry and discovery throughout, which creates a strong positive impact on student learning [10]. It was interesting to see the assistants' growth as they became accustomed to using these active learning strategies to design those types of guided discovery questions and contributed with such passion and dedication to the projects. As near-peers, the assistants' views and comments were extremely valuable for modifying the questions and the content to be appropriate for a high school audience. For example, their input on which card tricks should be included was essential. In the Platonic and Archimedean solids project, assistants were having trouble with definitions of polygons and their properties. They suggested that some review material on polygons be added at the beginning of the project so that the high school girls would not be lost during subsequent discussions of these solids.

In addition to being very hands-on, many of the projects have an added feature: they allow high school girls to use and apply mathematics in constructing a physical object that they can show to their family and friends. Some examples from different projects over the years include a three-colored origami dodecahedron, various paper models of a torus (using modular origami, sticky notes, etc.), each illustrating a different torus property, pop-up Christmas and Halloween cards using fractals, Valentine hearts formed by intersecting Möbius bands, Christmas ornaments using a hyperbolic paraboloid shape (hypar) [7], etc. It is worth noting that the assistants were again an essential voice in deciding what to include or not include in the construction part of the packet. For example, during the five-week preparation period for the Platonic and Archimedean solids project, the assistants made truncated tetrahedron and truncated octahedron jewelry. However, they strongly recommended to the directors that this activity should not be used with high school girls. Instead, they suggested making a simpler truncated tetrahedron Christmas ornament using bigger beads. Nevertheless, they were very proud of the more complicated jewelry they had made during the on-campus meetings and used their examples for demonstration when talking about the truncated tetrahedron and truncated octahedron properties during the high school meetings. They encouraged the girls to make similar jewelry on their own at home using what they learned in that project. Similarly, during the coloring project, assistants made a beaded seven-colored torus bracelet as well as Möbius band earrings that they used for demonstration during the projects but recommended to not include construction of those items in the project with the girls.

In the overall survey feedback about each activity, assistants used the following words and phrases to describe their impressions: loved, fun, worthwhile, interesting and doable, interactive, engaging, and informative. In addition, they described the mathematics topics as easy-to-understand and relevant, engaging and fun, interconnected, and entertaining. Anonymous alumni feedback on a new perspective gained through the program included: “We got to do higher/upper division mathematics such as Algebra, Geometry, and Graph Theory with the high school girls and finding ways to really teach it at a much simplified way that we both could understand, teach, and relate to.”

3.3. Preparing the Undergraduates to Teach and Mentor High School Students Through Active Learning

In general, during the five-week preparation period, the assistants learn and research more material than can be covered in the final high school project packet. However, in their survey comments, they expressed true appreciation for having been taught more and indicated they felt prepared for any questions that could come from the high school participants.

During the last two weekly on-campus meetings, each of the assistants has a chance to practice leading and facilitating the prepared project through active learning [4] while the rest of the group participates by pretending to be the high school audience. This practice allows the assistants to be well prepared for the upcoming after-school meetings and confident in answering anticipated questions. In addition, they are given strategies by the program directors for guiding the high school girls to do the discovery and exploration on their own. They become confident in leading and delivering active learning strategies through discovery and guided questions. Comments from program assistants regarding this aspect of the program included: “I have learned so much about helping to make mathematics exciting to young girls” and “It has been an incredible learning experience and has helped me become an amazing mentor.”

3.4. Completing Assignments for Discussion Topics and Mathematics Games

During the five weeks of preparation time the program assistants are charged with other tasks. For example, they suggest various games that can be used to engage the high school girls in fun networking and competitive activities (e.g., mathematics Jeopardy, SET, SET cubed, Instant Insanity game, map- and graph-coloring games, etc.).

In addition, assistants work on choosing a discussion topic and preparing a corresponding slideshow presentation for use at the upcoming after-school meetings. The topics are chosen to be relevant to the high school participants and connected with mathematics and science. Some of the topics included college life, female celebrities that studied mathematics or science, mathematics in the movies, applications of mathematics to science, careers requiring mathematical skills, mathematics in art, etc. Choosing those topics and preparing the presentations is solely the assistants' responsibility. They take pride in these assignments and enjoy working on them collaboratively outside of the weekly on-campus meetings.

One participant reflected on this aspect of the program in the alumni survey: “Because we researched women in mathematics as well as the topics we were covering, I learned a lot about women in STEM that I hadn't heard of before. Having learned about them opened the door in my mind of what was possible for me and, when I started teaching, my own students.”

3.5. Mentoring the High School Girls During the After-School Meetings

Each after-school meeting is structured in the following way: for the first 15 min, the program assistants use the slideshow presentation they created to lead interactive discussions on the chosen topic and to network with the female high school students. Engaging in these discussions breaks the ice and forms a bond with the high school girls. Then the high school students are divided into small groups (5–10 students per group) with one program assistant assigned to each group. The program assistants lead the girls through the mathematics project for about 1–1.5 h, guide them through the discovery and exploration questions, and help them, when needed, with additional guided questions. At the end of the after-school meetings, program assistants engage their respective groups in the prepared friendly and fun mathematics competitions until all other groups have finished. Finally, surveys are completed by the high school girls at the end of the meeting. After presenting at both high schools, surveys are given to the program assistants.

Throughout these activities, the program assistants actively serve as near-peer role models for the high school girls and mentor them in several ways. Survey feedback indicated that both groups enjoyed these mentoring interactions (Section 4.2).

3.6. Networking and Socializing with Female Faculty Members

The weekly on-campus meetings are used as networking events where female faculty members share information about career opportunities, graduation and retention rates of female students, and prominent women in the mathematics and science community. During these networking activities, the program directors socialize with the assistants, but they also provide time for the assistants to socialize on their own and discuss topics that are relevant to their everyday college lives.

The program assistants' survey feedback indicated that they appreciated these opportunities for networking and socializing and that these activities had a positive impact on them (Section 4.3).

3.7. Mentoring to Encourage Conference Presentations

Here we address another aspect that is distinct to the program and relates to undergraduate research. Namely, as part of their growth as mathematicians and educators, the program assistants were encouraged to present their experience working on the Math Girls $\mathbb{R}$ock! program at various conferences. As a result, many of them presented at international (The Fourth International Women of the Mountain Conference), national (Nebraska Conference for Undergraduate Women in Mathematics (NCUWM)), and regional (Engagement Week-SoTL; Student Leadership and Mentoring) conferences. None of the program assistants had previously attended or presented at a conference or written a grant proposal. Under the guidance of the program directors, program assistants wrote and submitted conference abstracts for the first time. Program directors also mentored the assistants in writing and submitting grant proposals for travel funding.

The conference presentations by the program assistants addressed various aspects of the assistants' involvement with this program: mentoring the high school students, encouraging them to study mathematics, using technology to enhance girls' education, gaining leadership skills, etc. It is worth noting that many of the assistants later took the initiative in presenting at conferences they found on their own. Survey feedback contained many comments addressing this aspect of their involvement. The following is one representative comment: “I had the opportunity to go to the Nebraska Conference for Undergraduate Women in Mathematics when I was in Math Girls Rock! as well which was a great way to socialize with other women in mathematics.”

4. STUDENT SURVEY FEEDBACK

High school participants and the undergraduate program assistants were asked to provide feedback on the mentoring and networking components of the program. In this section, we present some of the provided survey feedback.

4.1. Feedback Received from Quantitative Survey Questions

4.2. Feedback on Various Aspects of Mentoring

In Section 3, we discussed various aspects of mentoring that were used during this program. In this section, we share some qualitative survey feedback on this component of the program.

Table 1 (Section 4.1) shows that high school students ranked the provided mentoring opportunities very highly. In addition, the high school girls' surveys mentioned that the interactions with the program assistants helped change their perspectives about mathematics in general. They saw the assistants as positive role models and were appreciative of their perspective on college and careers in mathematics. Moreover, they “liked having women in mathematical careers working with [them].” Furthermore, some of the high school students indicated that working with the program assistants was the most enjoyable or most interesting part of the program.

Table 1. Feedback provided by the high school participants on mentoring.

Survey statement	Average of the average rankings from each meeting
The opportunity to interact with [the] undergraduate students.	3.67 out of 4
	(with 4 being “very satisfied”)
Mentoring helps with math concepts	4.73 out of 5
	(with 5 being “strongly agree”)

It is worth mentioning that the high school girls frequently indicated that the assistants were good teachers and were good at explaining. The following is one representative quote: “I liked how they explained things and they cracked a couple of jokes unlike my teachers who are more monotone about teaching. Their teaching was not ‘book learning’ but learning.”

Furthermore, in the end-of-program survey, the high school students were asked to provide overall input about the opportunity to work with the program assistants throughout the year. Their answers contained the following words and phrases to describe the program assistants: have good attitudes, sweet, helpful, friendly, fun to be around, informative, patient, energetic, enthusiastic, happy, interesting, funny, engaging, and understanding.

In turn, the undergraduate program assistants benefited from the opportunity to mentor and interact with the high school girls, as indicated in the following quotes from their surveys:

1. “They were a lot of fun to work with. They were very receptive to the new information and got excited about the math, which was rewarding for me to see.”

2. “Interacting with the girls at [the high school] has increased my desire to develop fun and engaging math problems.”

In end-of-program surveys, undergraduates often referred to the positive experience they had while teaching: “I thought it was a great opportunity to teach others about ‘non-conventional’ math. I don't think the theory was too hard to grasp for [high school girls'] level.”

4.3. Feedback on Various Aspects of Networking

Table 2 (Section 4.1) shows that the program assistants ranked the provided networking opportunities very highly. In addition, alumni indicated in the retrospective online survey that their participation in the Math Girls $\mathbb{R}$ock! program increased their socialization with women in mathematics, broadened their mathematical knowledge, helped them “feel comfortable in presentation settings and around other people with interests in math,” and taught them how to “teach and simplify very complex math terms to people at all levels.” Some of them mentioned specifically that they appreciated the opportunity to work closely with the program directors on a weekly basis and to “get their insights professionally and personally.”

Table 2. Feedback provided by the program assistants on networking and mathematical content learning.

Survey statement	Average of the average rankings from each meeting
The opportunity to interact with [the] faculty – program directors	3.88 out of 4
	(with 4 being “very satisfied”)
The opportunity to interact with the other students – assistants	3.92 out of 4
	(with 4 being “very satisfied”)
The opportunity to discuss mathematics	3.88 out of 4
not typically learned in an undergraduate program of study	(with 4 being “very satisfied”)
The program increased my confidence in my ability to learn mathematics	4.18 out of 5
	(with 5 being “strongly agree”)
The program elevated my enthusiasm for learning mathematics	4.42 out of 5
	(with 5 being “strongly agree”)

4.4. Overall Feedback

The quote below from one end-of-program survey summarizes the program assistants' overall experience during the Math Girls $\mathbb{R}$ock! program.

1. “My overall experience has been great. It has been challenging at times when I have been exposed to new math and have been required to research deeper topics in math. I have needed to analyze the project w/ how the HS girls would see it and make suggestions for how to make the project understandable and fun. The project directors have been great mentors and teachers and have helped me gain a greater excitement about math and desire to learn more. I have gained a confidence in my abilities as both a teacher and a mathematician. I have been challenged, but it has been fun. …I also liked the experience of presenting at the NCUWM conference. It would be interesting to track these girls through HS and college to see if they do better in math and/or continue studying it in college.”

In addition, one of the alumni indicated that she hoped that the high school students were “able to leave feeling more confident in their math abilities and that any negative emotions toward math [had] been decreased if not eliminated.”

Table 2 (Section 4.1) also shows that the program assistants very much appreciated the various opportunities to learn mathematical content provided during the Math Girls $\mathbb{R}$ock! program.

The program assistants consistently indicated that they would strongly recommend this program to other college friends as well as to high school girls.

In conclusion, we share the following quote from one of the alumni:

1. “There were a lot of fun demos and teaching points in the program that really opened up my knowledge of the fun/puzzley side of mathematics that I don't think I would have really seen otherwise…I absolutely loved my time with Math Girls Rock! It was one of the best things that I did while I was in college.”

5. CONCLUSION

More and more employers (even outside STEM jobs) are seeking workers that possess skills that are gained through STEM classes: problem-solving, logical and critical thinking, innovation, creativity, strong work ethic, teamwork, and collaboration [6,16,20]. Hence, encouraging women to take more STEM classes builds their resumes with much needed job skills and makes them favorable and distinct job applicants. The Math Girls $\mathbb{R}$ock! program aims to encourage female high school students to take more mathematics and STEM classes. The distinct educational component addressed in this article offers the undergraduate students an opportunity to improve all of these skills. The program can be implemented in every college or university. One colleague who helped with this program for two years at the authors' very large, public, open-enrollment university later went on to implement a similar program at her new private, small college.

However, before attempting to implement a similar program, interested faculty members should be aware of challenges pertaining to the recruitment and funding of program assistants. Clearly communicating expectations and deadlines to the undergraduates is another challenge.

Recruiting female undergraduate students is the most important step in running the program successfully. Choosing students with a good mathematics background who are approachable, easy-going, dependable, and pleasant to work with is difficult. But without each of these characteristics, a program assistant can potentially create difficulties for other members of the program. Most of the recruiting for this program was done by reaching out directly to female students enrolled in the program directors' upper-level classes or by receiving recommendations from colleagues. One of the program assistants enjoyed the program so much that she participated for two consecutive years. After the program was well-established, students who heard about the program from classmates sought out program directors and asked to join. For example, one of the Math Girls $\mathbb{R}$ock! high school alumni enrolled at the author's institution and volunteered as a program assistant. She pointed out that she wanted to give back to the community so others could benefit from the program the same way she had. Another female student wanted to write an honors thesis on the gender gap in STEM, so she asked if she could join the program staff and do research on this topic.

Ensuring funding to support the program assistants is another important challenge to consider. Most of the students at the authors' institution are full-time students who have families to support and at least one job. Without being paid for their time working on this program, many of them could not afford to participate. Over the years, the program directors have secured external funding from the MAA Tensor Foundation (2011–2014) and internal funding at the university, college, and department levels. Part of the funding was used to pay the program assistants for their work (hourly wage for working 2–3 h per week during each semester). The high school participants and teachers were not paid for their participation. Funding was also used for project materials, food during all meetings, and prizes for high school students. It is worth mentioning that the program directors volunteered their time and never received a stipend, credit in their teaching loads, or any other type of compensation for their time in this program.

Faculty members should be clear about the expectations for program assistants' participation and the time commitment involved. From time to time throughout the years, we have had a program assistant who was routinely unprepared for the weekly meetings. When that happened, the lead program director for that particular project regularly met with the student individually (outside of the weekly meetings) to reinforce research assignments and ensure that the program assistant would be adequately prepared to work with the high school students during the after-school meetings.

Another challenge is recruiting high school teachers. The program relies on highly-motivated high school teachers who are willing to volunteer their time and actively engage in recruiting some of their students to participate. The teachers must also clearly communicate expectations to their high school students (who are expected to participate the whole year as one cohort), advertise the program in a timely manner, and communicate effectively with the program directors.

In addition, faculty members with many years of classroom teaching experience should be aware of the pedagogical challenges the program assistants face due to lack of teaching experience. In the program surveys, some of the program assistants mentioned they were mostly challenged by teaching new mathematics concepts to the high school students, getting and keeping high school girls' attention, and working with each student in the group. However, they all expressed appreciation for the given teaching opportunity. Moreover, in the program surveys, the program assistants regularly suggested changes to logistics that the program directors tried to address adequately in future meetings (e.g., arriving earlier to prepare the rooms at the high schools, technology set up, etc.).

In conclusion, we offer the following advice to faculty members desiring to start a similar outreach program:

• Consider the culture and diversity of the population served and make the program appropriate for the level of the students participating.

• Involve undergraduates in the program in meaningful ways. Engage them in hands-on, collaborative activities and allow them to implement their ideas.

• Be the kind of role model we would like the undergraduates to be for others.

From our experience, the rewards of an outreach program (however small the program) are well worth the effort involved.

ACKNOWLEDGMENTS

The authors want to express their sincere gratitude to everyone who has influenced their work, helped them, and supported them in various ways while running the Math Girls $\mathbb{R}$ock! program: Florence Fasanelli, Meghan Dewitt, Kathy Andrist, Xiaoyi Ji, Jennifer Hooper, Ellen Backus, and the math teachers at the participating high schools. In particular, the authors are especially thankful to all undergraduate and high school students who have inspired and worked with them throughout this program. Special thanks to Stina Nyhus who helped with compiling and analyzing data and Kristen Hornberger for her continuous support of this project. The authors also want to express their appreciation to the Utah Valley University Department of Mathematics and College of Science for their constant support and encouragement to run this program. The first author wants to thank Yuliya Babenko for providing resources for building the truncated polyhedra jewelry. In addition, the authors thank the anonymous referees and associated editors for the helpful suggestions and comments that benefited the paper greatly. This study was approved by the Utah Valley University Institutional Review Board (IRB Protocol #739).

DISCLOSURE STATEMENT

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The authors are grateful for financial support from the MAA Tensor Foundation and the Utah Valley University Math Initiative.

Notes on contributors

Violeta Vasilevska

Violeta Vasilevska is Professor in the Department of Mathematics at Utah Valley University (UVU). She received her Ph.D. from the University of Tennessee and joined Utah Valley University in 2010. Her research interests are diverse, ranging from topics in pure mathematics (topology, algebra, and graph theory) to topics in math education/SoTL. She loves to engage her students in actively building their knowledge in class as well as outside of the classroom setting. Vasilevska co-founded and has been running the UVU outreach program Math Girls $\mathbb{R}$ock! for high school women since 2011. In her spare time, she loves doing Origami, reading, hiking, and traveling.

Carolyn Hamilton

Carolyn Hamilton is Associate Professor in the Department of Strategic Management and Operations at Utah Valley University. She received her M.S. in Mathematics from the University of California, Riverside and joined the UVU Mathematics Department in 1993, chairing the department from 2004 to 2009. Hamilton co-lead the Math Girls $\mathbb{R}$ock! program from 2011 to 2015. Her awards include the UVU Trustees Award of Excellence and the Deans Award for Excellence in Teaching. She currently coordinates the business calculus program at the Woodbury School of Business.

Notes

1 Near-peer mentoring is used to describe communication or activity between people or organizations that are very similar or nearly equal. A near-peer teaching model is one in which a more experienced student acts as the instructor for less experienced students. [5]

2 The stylized $\mathbb{R}$ is meant to represent the set of real numbers.

3 In progress at this time.

APPENDICES

APPENDIX 1. DESCRIPTION OF THE PLATONIC AND ARCHIMEDEAN SOLIDS PROJECT

The project “Platonic and Archimedean Solids” has been used twice in the program. The version of the project described here was delivered during the fall semester of 2013. The full project package can be found at [21]. In this appendix, we provide an overview of the project and describe the guided interactions between the program assistants and the high school girls.

Program assistants received mentoring during weekly meetings that included instruction for beginning each project by introducing everyone in the group. They were also coached to affirm each high school student's answer in positive and constructive ways.

For this particular project, the assistants were mentored to begin by prompting high school students to discuss the concept of a polygon, properties of polygons, types of polygons, etc. before starting with the project package. These discussions helped the program assistants ascertain the high school students' level of understanding of the material covered by the project. After these initial discussions, the assistants used the material on polygons contained in the project packet [21, Section 2.1] to review polygons: the definitions used, properties of the polygons, and different types of polygons. For each of the defined concepts, high school students were asked to demonstrate their knowledge and offer explanations of their answers (which encouraged profound discussions among the high school students). Most of the project questions were open-ended, for example “What kinds of polygons have you seen before?” After asking “How many different regular, convex polygons exist?” [21, Section 2.1], the assistants were mentored to ask follow-up questions to guide the high school students' answers (in case students struggled with it). The aim of the follow-up questions was to lead the high school students to discover that:

• Increasing the number of sides of a regular, convex polygon will approximate a circle, and

• Extending the concept of a regular, convex polygon to three dimensions will create a regular, convex polyhedron.

Discussing these questions allowed the high school students to brainstorm and make predictions about the nature of regular, convex polyhedra. Program assistants then introduced Platonic solids [21, Section 2.2] and confirmed some of the high school students' brainstorming ideas and guesses. The assistants asked the high school students to determine the effect of increasing the number of faces of a regular, convex polyhedron. They used plastic-interlocking shapes [18] (triangles, squares, and pentagons) as polygonal faces to try and form regular, convex polyhedra. The students discovered the maximum number of each of those faces that could be used to form a vertex of a polyhedron that is both regular and convex. Discussion followed about why the number of polygonal faces used must have a maximum in this three-dimensional setting.

In addition, a guided discovery was implemented about why regular n-gons (for $n\ge 6$) cannot be used as faces of a regular, convex polyhedron. Each of these questions generated a discussion concerning the sum of the angles at a vertex in a convex polyhedron. The program assistants had been mentored through the above-guided discussion method and use of manipulatives during the weekly on-campus meetings. In addition, the assistants had been coached to ask the high school students about any Platonic solids found in everyday life. They received many different answers, many of which were surprising and novel to the program assistants.

Next, various properties of the Platonic solids were discussed and the assistants led the high school students in discovery of Euler's polyhedron formula. Models of the five Platonic solids were used to help students count faces, vertices, and edges of the solids. Assistants guided the high school girls in a discovery discussion introducing the concept of a dual of a Platonic solid, including how a dual could be drawn. They used inquiry questions as follows:

• How could the property you just discovered be used to draw the dual of a given Platonic solid?

• What would you choose as the vertices of your dual of the given Platonic solid? Why?

• Can you guess what the dual of each of the Platonic solids would be based on the table provided?

The next section on Archimedean solids [21, Section 2.3] contained material that was completely new to all participating groups. The assistants first introduced Archimedean solids and then discussed some of their properties. After introducing uniform truncation, the assistants led a discussion on how to obtain a uniform truncation of a Platonic solid. Each assistant's group used the provided set of Platonic solids during their discussion for better visualization of the properties discussed. In addition, the assistants used truncated tetrahedron jewelry (Figure A1) built during the five-week mentoring period as visual aids when discussing truncation.

Figure A1. Truncated tetrahedron beaded jewelry.

Finally, the assistants led their groups in making the truncated tetrahedron Christmas ornament (Figure A2) following the provided instructions [22]. Throughout this hands-on activity, the assistants used the truncated tetrahedron properties discussed in the project package as a framework for constructing the ornament. For instance, the high school students were instructed that the large beads should be placed on edges shared by two hexagonal faces while the small beads should be placed on edges shared by a hexagonal face and a triangular face.

Figure A2. Truncated tetrahedron Christmas ornament.

As was pointed out in Section 3.2 of this paper, each final project packet went through several iterations during the weekly on-campus meetings. For example, the first iteration of this project did not contain review material on polygons. The review material was added following suggestions given by the assistants. The second iteration originally asked high school students to find the number of edges, vertices, and faces of each Platonic solid on their own. Assistants suggested a filled-in table should be provided containing this information since the task was very easy but relatively time-consuming. One of the iterations contained additional properties of the Platonic solids, such as the three spheres associated with each Platonic solid (the circumscribed sphere, the midsphere, and the inscribed sphere), the connection between the Golden Rectangle and the Platonic solids; etc. In consideration of time constraints, the program members decided to cut out this material.

The program assistants were given the following tasks to research during the five weeks:

• Polygons and their three-dimensional generalization (polyhedra);

• Platonic solids and their corresponding duals;

• Archimedean solids;

• Truncated polyhedra, etc.

The material that the assistants researched and discovered was more than could be used in a timed project. However, the assistants' surveys indicated they appreciated the opportunity to explore these concepts more deeply and learn about topics that they had never seen before.

APPENDIX 2. SAMPLE PROGRAM ASSISTANT POST SURVEY

The survey questions below were given to program assistants after completion of the project on Platonic and Archimedean solids. Similar surveys were given after the completion of each project. Some of the questions pertaining to specific mathematical content changed slightly from project to project, but the same objectives were assessed.

• Has the program so far met/exceeded/failed your expectations? Explain!

• How well organized were the Math Girls $\mathbb{R}$ock! meetings at [the participating] high schools for Project 2? What would you change to make future high school meetings better?

• Describe your overall impression so far of the Math Girls $\mathbb{R}$ock! program directors (including organization, conduct, interaction and preparation of the Math Girls $\mathbb{R}$ock! program). How well organized were the weekly meetings? What would you change to make future weekly meetings better?

• Express your thoughts about Project 2: Platonic and Archimedean solids.

• During Project 2,

  1. Did you learn new math topics? Explain.

  2. Were you exposed to new math ideas that you might not have seen before? Explain!

  3. Were you involved in researching math questions?

• Describe your feelings about the Platonic and Archimedean solids portion of the program. (What did you like or not like about this portion? Explain.)

• Describe your feelings, thoughts, and impressions about the truncated tetrahedron ornament portion of the program.

• Describe your overall impression of the second Math Girls $\mathbb{R}$ock! project (including topic, activities, games).

• Describe your experience with and the opportunity to interact with high school participants and discuss math during each of the Math Girls $\mathbb{R}$ock! meetings at [the participating high schools].

• Please indicate how satisfied you were with the second seven weeks of the Math Girls $\mathbb{R}$ock! program.

         (1) Not at all satisfied  (2) Slightly satisfied

         (3) Moderately satisfied  (4) Very Satisfied

  1. The materials and information delivered deal with Platonic and Archimedean solids.

  2. The opportunity to interact with [the] program directors.

  3. The opportunity to interact with other undergraduate students [program assistants].

  4. The opportunity to discuss mathematics not typically learned in undergraduate program study.

• To what extent, if any, do you feel you experienced each of the following types of learning as a result of your participation in this program?

         (1) Not at all  (2) Small extent

         (3) Moderate extent  (4) Great extent

  1. I increased my knowledge of careers and women in mathematics.

  2. I gained a greater understanding of the usefulness of mathematics.

• To what extent do you agree or disagree with each of the following statements concerning the impact of this program on you personally (so far)?

         (1) Strongly disagree  (2) Disagree  (3) Neutral

         (4) Agree  (5) Strongly agree

  1. It increased my confidence in my ability to learn mathematics.

  2. It elevated my enthusiasm for learning mathematics.

• After participating in the second part of the Math Girls $\mathbb{R}$ock! program, what aspects of the program did you find to be

  1. most interesting?

  2. most challenging?

  3. most enjoyable?

• Do you have any other comments or suggestions about improving the Math Girls $\mathbb{R}$ock! program?