Activities

There are two primary components to Circles and Pairs: Math Circles and One-On-One Tutoring. In both components, Circles and Pairs puts students into direct contact with professors, graduate students, and undergraduates, all of whom serve as teachers and mentors.


Math Circles: Math Circles are group problem solving activities that emphasize creativity, collaboration, and discovery-based learning. The topics introduced are designed to generate excitement, expose students to mathematics they often do not see in the classroom, and encourage the students to study more mathematics in middle school, high school, and college. The lessons are designed to foster interaction through hands-on activities and discovery-based learning. Students frequently break into groups of 2 or 3 to work on problems and explore topics, and there are several professors, UCCS graduate students, and UCCS undergraduate who serve as facilitators to help with the activities. We also encourage students to explain their solutions to others and to develop effective communication skills when talking about technical subjects.

The mathematics problems we give in Math Circles are designed to provide challenging, enriching experiences for students, while simultaneously building critical thinking skills. Often the problems we give seem more like puzzles or games than traditional math problems.

Some sample problems: 

1. If 10 people at a party each shake hands with every other person, how many handshakes occur?
2. Is it possible to cover a chessboard with rectangles of size 3x1?
3. If a disease infects 1 in 10,000 people, and you test positive for the disease using a test that is 99% accurate, what is the probability you actually have the disease? (Hint: It is not 99%.)

Goals

We have three main Outcome Goals for the students in that participate in Circles and Pairs:

1. Graduate from High School.
2. Attend College
3. Consider majoring in a STEM field.

In addition to the three Outcome Goals, we have Educational Objectives and want student to develop qualities that will make them more successful in academic pursuits. This includes

1. Acquiring Grit
2. Cultivating a Growth Mindset
3. Developing Positive Attitudes Towards Mathematics
4. Practicing Habits of Mind that aid in Critical Thinking

Grit
Grit is defined as "perseverance and passion for long-term goals". Individuals with high levels of grit are able to remain determined and motivated over long periods of time regardless of obstacles, challenges, or adversity. Over the past decade, research by Dr. Angela Duckworth at the University of Pennsylvania has shown that grit is a much better predictor of success than any of the previously used predictors such as intelligence, family income, innate ability, or scores on standardized achievement tests. Duckworth also developed a short questionnaire to measure grit, and in a study involving thousands of juniors in Chicago public schools, it was found that students with high grit scores on this questionnaire were significantly more likely to graduate. In addition, other studies have established positive relationships between grit and educational attainment, GPA, and retention in college. Duckworth has said that it is still uncertain how to build grit in students, but one of the best ideas she has found involves developing a growth mindset; i.e., the belief that the ability to learn is not fixed.

Here are some resources for learning more about grit.

• The Wikipedia Article on Grit
• The Duckworth Lab
• Angela Duckworth's TED Talk
• Grit and the Secret of Success, an excellent article at Brain Pickings

Growth Mindset
Research has shown that a student's mindset dramatically affects learning and can be used to predict academic trajectories. In a series of studies conducted over 20 years, Dr. Carol Dweck at Stanford University divided people into two groups: those having a "fixed mindset", who believe character, intelligence, and creative ability are static qualities that we cannot change in any meaningful way, and those having a "growth mindset", who believe these qualities can be cultivated and expanded through effort and that people can improve and grow through practice and experience. Dweck's studies showed that people with a growth mindset accomplished more, pushed themselves harder when encountering difficulty, and overall were more successful and happier.

Interestingly, Dweck's work also showed that regardless of what the the facts are regarding the malleability of intelligence and ability, students with a growth mindset (i.e., students who believe that these qualities are malleable) actually perform better. A large component of developing a growth mindset involves informing students that they can change -- for example, telling students that their mental capabilities can grow through practice or experience, that mistakes and failure are opportunities for learning, and that their levels of accomplishment are not predetermined. Likewise, it is important to help students develop Self-Efficacy (i.e., a belief in one's own ability to complete tasks and reach goals). It has also been shown that cultivating a growth mindset is one of the best ways to build grit. This is because students with a growth mindset believe that ability to learn is not fixed and can change with effort. Consequently, they are much more likely to perservere when they encounter challenges because they do not view failure as a permanent condition. Studies have supported this, and shown that telling students that they can grow, improve, and change, or convincing students that success is within their capabilities, can have a dramatic affect on their levels of grit, how hard they work, and their overall performance.
 

Here are some resources for learning more about growth mindset.

• The Wikipedia Article on Mindset
• Media on Dweck's Research on Growth Mindset
  • New York Times Article
  • Stanford News Release
  • Stanford Report
• Fixed vs. Growth: The Two Basic Mindsets that Shape our Lives, an excellent article at Brain Pickings

Positive Attitudes Towards Mathematics
Many of the students in Circles and Pairs have had difficulty with mathematics in the past or suffer from math anxiety. We give the students the opportunity to see aspects of mathematics that they typically do not get to experience in their school classroom, and we provide a supportive and welcoming environment for exploring these aspects. We want students to see a side of mathematics they have never encountered before, and to re-evaluate their attitudes towards mathematics.

• Here are the Attitudes Toward Mathematics that We want Students to Develop

Even if students do not attend college or major in a STEM field, we want them to understand that in our increasingly technical and scientific world, developing basic quantitative literacy will help them in many aspects of their lives and jobs.

Habits of Mind
We want Circles and Pairs students to become better learners and more effective problem solvers. We accomplish this by having the students practice Habits of Mind that are useful in learning and in thinking.

• Here are the Habits of Mind that We Want Students to Develop

When tutoring students we use best practices, informed by research on education and cognitive science, to make our efforts as effective as possible. The following handout (created by Nicky Case and dedicated to the public domain) is a nice summary of what cognitive science tells us what really works when it comes to learning:

• How Do We Learn?

Links to References cited in the handout:

• Piaget's Constructivism, Papert's Constructionism: What's the Difference? by Edith Ackerman
• Metaphors We Live By by George Lakoff and Mark Johnson (book)
• The Spacing Effect: A Case Study in the Failure to Apply the Results of Psychological Research by Frank Dempster (PDF)
• What Works, What Doesn't by John Dunlosky, Katherine Rawson, Elizabeth Marsh, Mitchell Nathan, Daniel Willingham, and Celia Johnson (PDF)
• Differential Effects of Incidental Tasks on the Organization of Recall of a List of Highly Associated Words by Thomas Hyde and James Jenkins

Other Components

In addition to the Outcome Goals and Educational Objectives listed above, there are several other components to Circles and Pairs, as well as additional purposes we hope the program will serve.


Mentoring Students
By having professors, graduate students, and undergraduates involved in Circles and Pairs as instructors, facilitators, and tutors, we provide the students in Circles and Pairs with a vertically integrated system of mentoring. The students interact with successful undergraduates in STEM, who are often just a few years older than they are, as well as with graduate students and professors who can provide more experienced advice, guidance, and instruction.

In addition, we provide a great deal of individual attention to the students. We typically have multiple professors, graduate students, and undergraduates at each lesson. In many of our meetings the ratio of students to instructors and facilitators is 2:1. We also make efforts to get to know the students personally, and provide a great deal of individual instruction, supervision, and feedback. We view mentoring as a critical aspect of the program, and we encourage the students to interact with the instructors and facilitators and engage in dialog on all topics.


Serving the Community and Broadening Access to Education
Part of the mission of the UCCS is to "broaden access to higher education for the benefit of southern Colorado, the state, nation and world." By promoting mathematics education, we are responding to one of the most immediate educational needs of our surrounding community.


Involving Underrepresented Groups in Mathematics
Certain groups are severely underrepresented in mathematics and STEM fields. These include ethnic minorities (e.g., African-Americans, Hispanics, Native Americans and Indigenous Peoples, certain Asians such as Vietnamese and Filipinos), women, and first-generation college students. We believe these populations provide a valuable untapped resource for mathematics and our nation, and that we would all benefit if more individuals from these groups were to become involved in and contribute to mathematics and mathematically-related fields.


Reciprocal Benefits for Volunteers
By engaging undergraduate and graduate students in the culture of outreach, we teach these volunteers the value and importance of community engagement.  Many students that participate in outreach programs such as Circles and Pairs continue to be involved in educational outreach and service throughout their lives -- and some even go on to create their own programs.  In addition, studies have shown that outreach program, such as Circles and Pairs, have numerous benefits for the volunteers themselves.  Besides the sense of fulfillment they often feel, volunteers have an opportunity to use their own education and put into practice what they are learning in their own courses.  This often results in undergraduate and graduate STEM students experiencing a renewed enthusiasm and uncreased dedication for their own studies, and the largest benefits in this regard are typically seen among volunteers who are themselves members of groups underrepresented in STEM (e.g., women and minorities).