Build the Skills for Serious Research
A structured introduction to research thinking, where students learn to use mathematical ideas and analytical tools to explore meaningful questions across a wide range of fields.
From a Topic to a Researchable Question
Students learn how to turn a broad topic into a meaningful, mathematically grounded research plan—identifying the essential features of the problem, determining what should be preserved or simplified, establishing appropriate assumptions, and formulating questions that can lead to substantive investigation.
Frame the Problem
Identify the essential features of a topic and determine how they can be represented, modeled, and investigated mathematically.
Choose the Assumptions
Determine which features of the original problem must be preserved, and which can be simplified, while keeping the model meaningful and mathematically tractable.
Formulate the Questions
Use the resulting model to formulate precise, nontrivial questions and identify outcomes worth investigating.
Identify Paths Forward
Explore possible approaches and determine what mathematical tools, ideas, or further knowledge may be needed.
A Guided Process from Idea to Formulation
Students work one-on-one with their mentor in an online setting to develop an initial idea into a clear, mathematically grounded research formulation, refining the problem through discussion, analysis, and careful choices about assumptions and objectives.
Explore Topic
Begin with a topic, phenomenon, or question of genuine interest and examine the ideas needed to understand its essential features.
Develop Model
Translate those essential features into a mathematical framework by identifying variables, relationships, and appropriate assumptions.
Refine Problem
Test and revise the formulation, balancing mathematical tractability with enough complexity to preserve what makes the original problem meaningful.
Identify Directions
Formulate precise questions and identify mathematical tools or research directions that could support a deeper investigation.
For Students Ready to Explore
Foundations is designed for intellectually curious students interested in exploring questions from a wide range of fields. Projects need not be purely mathematical. Instead, students learn how mathematical ideas, models, and reasoning can be used to understand a problem, identify its essential features, and investigate meaningful questions.
Intellectual Curiosity
Interested in exploring questions beyond the standard school curriculum and understanding how ideas connect across different fields.
Analytical Readiness
Comfortable engaging seriously with quantitative ideas and learning new mathematical tools when needed.
Research Potential
Ready to work patiently through ambiguity, discussion, revision, and questions that do not have predetermined answers.
A Foundation for Deeper Research
Foundations can stand on its own as an introduction to serious research thinking, or it can prepare a student for more advanced work. Depending on the student’s interests and readiness, the experience may lead naturally into the Topical Review Track or the Research Fellowship Track.
Skills, Evidence, and a Clear Next Step
Foundations gives students tangible evidence of their work while helping them discover whether deeper research is the right next step.
Certificate of Completion
Receive a REACH@Lyceum certificate recognizing successful completion of the Foundations Track.
A First Written Report
Develop a structured written report and learn how mathematical ideas, assumptions, reasoning, and conclusions are communicated clearly.
Research Readiness
Gain a clearer understanding of personal research interests and readiness for a Topical Review or, in exceptional cases, a Research Fellowship.
The Foundations Track builds the habits required for deeper work: framing a real problem, choosing meaningful assumptions, and forming precise mathematical questions. It can lead naturally to the Topical Review Track—or, for exceptionally prepared students, directly to a Research Fellowship. Progression depends on the student’s interests, readiness, independence, and completed work; it is not automatic.
Start with a Conversation
Every student brings different interests, preparation, and goals. A consultation helps us determine whether the Foundations Track is the right starting point and how the experience can be shaped around the student.