More information on HCSSiM for parents and educators
Goals
The Hampshire College Summer Studies in Mathematics (HCSSiM) is an intensive six-week residential program that trains high ability high-school students in the mathematical sciences and mentors undergraduate and graduate students in teaching, research, and career navigation. Our primary goal is to identify, invite, and prepare motivated high-school students to pursue mathematics and mathematical vocations. We seek:
- – to enlist the support of the secondary school mathematics community in the identification of talented and highly motivated students,
- – to engage participants actively in the processes of mathematical thought, emphasizing the methods of discovery, proof, and modeling (rather than merely accumulating results),
- – to create an atmosphere in which the enterprise of sharing of insights, knowledge, and mathematical enthusiasm overshadow distinctions between teachers and students and competitiveness among students,
- – to explore active, accessible, and applicable areas of mathematics---for example, graph theory, dynamical systems, knot theory, and cryptography,
- – to provide varied role models of mathematical exposition and both guidance and encouragement for participants to develop their oral and written expository skills,
- – to encourage students to experiment with the learning process, take an increasing role in shaping their own education, and to evaluate regularly their own goals, tastes, and growth,
- – to encourage undergraduate and graduate junior staff to experiment with the teaching process,
- – to increase participation in mathematical activities by underrepresented students, including women, ethnic minorities, and economically disadvantaged students,
- – to introduce students to the larger mathematical community and to help them to an understanding of the joys and options it offers, and
- – to provide continuing support for our students and their teachers.
Training Future Scientists and Mathematicians
HCSSiM students make conjectures,
test examples, and construct proofs in the same way that mathematicians do;
our environment is characteristic of the mathematical enterprise. In 2006,
an alumn reunion conference (Yellow Pig Math Days) was held during the Summer
Studies; most speakers, all with very successful careers in math and science,
commented that HCSSiM was where they learned what research
is, that they were able to do it, and
that it was exhilarating. HCSSiM is characterized by its strong sense
of community and by an inviting, challenging, and empowering pedagogical style.
That spirit is visible in the pleasure and fascination with which pretty mathematics
is presented, and shapes the way problem sets are written.
For four hours each morning (six days per week), and in three-hour problem solving sessions (five evenings per week), the small classes of the Summer Studies engage students in the practice of mathematics. Students discover patterns, make and test conjectures and definitions, work together on creating proofs and counterexamples, build models, refine their results and formulate further questions, tackle challenging problems from mathematics and the real world, write programs, share their insights, develop their oral and written expository skills, evaluate their growth, learn about the range of opportunities in mathematics, and come to see themselves as mathematicians. The lively intellectual atmosphere is maintained throughout the day; the faculty live in the program dormitory and join the students for meals and recreational activities. Students discover the joy of doing mathematics, and that there are others who share their passion.
Given the goal of getting students to see themselves as real mathematicians,
the specific content of HCSSiM courses is less important than the approaches
taken to the development of the material. We view mathematics as a process
rather than as a collection of results. Students are rarely presented with
the finished products of mathematics, and are instead invited to pose their
own questions and to develop strategies for solving them. We make a determined
effort to impress upon students that they are entering a living discipline.
Throughout the program, high-school students are presented with examples of
recently solved and unsolved problems in mathematics; doctoral students and
mathematicians active in research are on staff and many others visit the
Summer Studies.
It is very easy to underestimate the speed with which alert and motivated students assimilate (often leading the way through) new material. Thus, while the content of the courses varies from class to class and year to year, it is never superficial. So as not to deprive participants of what mathematical opportunities remain for them in high school, we studiously avoid using or teaching the calculus; the ease with which discrete mathematics and computers handle topics such as dynamical systems makes this a minor constraint. We commonly cover undergraduate material equivalent to most of an elementary number theory course, most of a discrete math course, half of a modern algebra course, and a third of one or two elective courses (such as complex analysis and topology) within the first three weeks. During the second three weeks, we cover the equivalent of a semester-long undergraduate elective plus two one-fourth-semester courses. The undergraduate and graduate junior staff learn substantial mathematics while assisting the faculty.
A common reaction to such a description of the Summer Studies to people who
have not experienced it is: "You can't possibly do all of that." Nonetheless,
three decades have taught us that we can expect everyone to sample a lot of
new mathematics, to be aware of even more, to leave with resources to explore
later, a desire to share math with teachers and peers back home, and an enthusiasm
for research. It is also true that the pace and intensity of the Summer Studies
is such that some of the deeper lessons are not fully appreciated until after
the program is over. Faculty and students are fully exhausted at the end of
our six weeks.
All activities are designed to actively engage participants in the processes
of mathematical thought, to encourage them to see themselves as mathematicians,
and to increase their awareness of the world of mathematics. The long days,
the low student : staff ratio (usually better than 5:1), and the near-constant
interactions with faculty make the Summer Studies an intense, lively, diverse,
and exciting intellectual community.
Mentoring
The HCSSiM structure allows for vertical integration
of mathematical mentoring. Faculty, graduate students, and undergraduate students
mentor the high-school students in learning mathematics, in joining the mathematical
community, and in planning their futures. Faculty serve as mentors and models
to the junior staff members and high-school students; high-school students
learn about applying to college, college students are guided in the transition
to graduate school, and graduate students
receive advice for their professional careers. Some of the undergraduate and
graduate staff are alumns of HCSSiM, which allows for long-term mentoring;
other staff are new to the program, which produces
a broader impact in mentoring.
All HCSSiM instructors keep foremost in mind those aspects of presentation
that will pique students' curiosity and draw them further into mathematical
study; it is an intense and supportive teaching environment. The pedagogical
style is highly exploratory, encouraging generalization, collaboration, and
question-asking. Both graduate and undergraduate junior staff teach several
hours each week under the supervision of faculty. Undergraduate and graduate
junior staff members acquire rich experience in teaching, mentoring, and leadership
by assisting with collaborative learning and presenting material in classes
under the supervision of faculty; by the end of the program, they are designing
and teaching their own mini-courses. Each staff member is expected to deliver
a Prime Time sometime during the summer in order to practice presenting expository
mathematics or the staff member's own research results. The material must be
placed in context for it to be effectively communicated to high-school students.
By providing models, practice, and feedback, HCSSiM helps participants become
outstanding communicators. Everyone spends time at the board, shares questions
and insights in small groups, and regularly writes up
mathematics---and receives, almost immediately, praise, suggestions, and advice.
In this way, the junior staff members learn to use discovery-based active-learning
methods. Both students and staff take these skills, and a lot of mathematics,
with them when they depart Hampshire. Many junior staff members later become
teachers at the secondary, collegiate and graduate levels, and practice the engaging
and empowering pedagogy learned at HCSSiM.
