In his motivational classic “Tough Times Never Last, But Tough People Do!”, Robert H. Schuller introduced a powerful concept he called an “alphabet for action”, an A-Z list of action verbs designed to inspire what he termed “possibility thinking.” Schuller’s insight was that having a comprehensive, memorable framework of action-oriented words could serve as a mental toolkit for overcoming challenges and achieving success.
While Schuller’s alphabet focused on general life resilience, the concept has since been adapted across various domains. Leadership consultants have created A-Z lists for management excellence, business strategists have developed alphabetical frameworks for organizational success, and motivational speakers have crafted their own versions for personal development. However, one crucial domain has been surprisingly overlooked: academic research.
Mathematical research, in particular, demands a unique combination of analytical rigor, creative thinking, persistent problem-solving, and intellectual courage. The journey from mathematical curiosity to breakthrough discovery requires not just technical knowledge, but also the right mindset and approach. This is where an “alphabet for action” specifically designed for mathematical research can prove invaluable.
The Mathematical Research Alphabet for Action
Here’s an A-Z framework of action verbs specifically crafted for success in mathematical research:
A – Abstract – Extract essential patterns and structures from specific cases
Look beyond the details of individual examples to identify the underlying principles that govern entire classes of problems. This skill allows you to recognize when seemingly different problems share the same mathematical DNA, leading to powerful generalizations and elegant solutions.
B – Build – Construct proofs, frameworks, and mathematical objects systematically
Approach mathematical construction like an architect, start with solid foundations and add each element deliberately. Breaking complex arguments into logical steps not only makes your work more convincing but also helps you identify exactly where difficulties arise.
C – Calculate – Perform computations and numerical analysis
Concrete calculations often reveal patterns that pure theory might miss and provide crucial intuition for abstract concepts. Even in theoretical work, computing specific examples can guide your conjectures and help you avoid false starts.
D – Dissect – Break down complex problems into manageable components
Every intimidating problem becomes approachable when you identify its constituent parts and tackle them individually. This decomposition strategy prevents overwhelm and often reveals that the “impossible” problem is actually a collection of familiar, solvable pieces.
E – Experiment – Test mathematical ideas through examples and counterexamples
Mathematics thrives on experimentation; try special cases, look for patterns in small examples, and actively search for counterexamples to proposed theorems. This exploratory approach builds intuition and often leads to unexpected discoveries or reveals why certain approaches won’t work.
F – Formalize – Express intuitive ideas in rigorous mathematical language
The discipline of translating vague hunches into precise mathematical statements forces you to understand your ideas more deeply. This process often reveals hidden assumptions and gaps in reasoning that need to be addressed.
G – Generalize – Extend specific results to broader classes of objects
Once you’ve solved a particular case, ask yourself: what made this solution work, and where else might the same principle apply? Successful generalization multiplies the impact of your work and often leads to more elegant, unified theories.
H – Hypothesize – Propose testable mathematical statements and theories
Bold, well-informed conjectures drive mathematical progress by giving the community concrete targets to pursue. Even when a conjecture proves false, the attempt to prove or disprove it often yields valuable insights and new techniques.
I – Iterate – Repeat processes to refine understanding or reach solutions
Mathematical understanding deepens through repetition, revisit problems with fresh perspectives, refine your arguments, and don’t be afraid to restart with better approaches. Each iteration builds on previous insights and often reveals new connections.
J – Justify – Provide logical reasoning and proof for mathematical claims
Rigorous justification isn’t just about convincing others, it’s about convincing yourself that your reasoning is sound. The process of constructing proofs often reveals subtle errors in intuition and leads to deeper understanding of why results are true.
K – Kindle – Ignite curiosity and passion for mathematical discovery
Genuine enthusiasm for mathematical beauty and mystery sustains you through difficult periods and inspires creative approaches. Cultivating wonder about mathematical phenomena makes the inevitable struggles feel worthwhile and keeps you motivated during long research projects.
L – Link – Connect disparate mathematical concepts and fields
The most profound mathematical advances often come from recognizing unexpected connections between seemingly unrelated areas. These bridges between fields create new research directions and often lead to powerful techniques that illuminate both domains.
M – Model – Create mathematical representations of real-world phenomena
Mathematical modeling not only makes your work relevant to other fields but also provides a rich source of meaningful problems. The constraints of real-world applications often suggest natural mathematical questions that are both challenging and practically important.
N – Navigate – Find pathways through complex mathematical landscapes
Develop strategies for finding your way through intricate mathematical territories, know when to push forward, when to backtrack, and when to try a completely different route. Successful navigation often requires maintaining both local focus and global perspective.
O – Optimize – Find the best solutions under given constraints
Optimization problems naturally arise throughout mathematics and provide concrete goals that guide research. The techniques developed for optimization often have broader applications and help develop intuition about what makes certain mathematical objects “better” than others.
P – Prove – Establish mathematical truths through rigorous demonstration
Proof is the gold standard of mathematical certainty, but the process of proving often matters more than the final result. Constructing proofs develops logical reasoning skills and often reveals why results are true, leading to generalizations and new insights.
Q – Question – Challenge assumptions and probe deeper meanings
The most important mathematical advances often come from questioning what everyone takes for granted. Develop the habit of asking “why” and “what if” about standard assumptions, this skeptical mindset can reveal hidden depths in familiar concepts.
R – Refine – Polish arguments and improve mathematical understanding
Mathematical ideas improve through refinement, simplify proofs, clarify definitions, and eliminate unnecessary complications. This polishing process often reveals the essential core of an argument and makes it more likely to generalize to new situations.
S – Synthesize – Combine different mathematical ideas into unified theories
The ability to weave together insights from multiple sources creates powerful new mathematical frameworks. Synthesis requires seeing beyond the surface differences between approaches to identify common underlying principles.
T – Transform – Convert problems into more tractable forms
Learning to recognize when a change of perspective, coordinate system, or representation can make a difficult problem simple is a crucial skill. The art of transformation often involves finding the “right” way to look at a problem that makes the solution obvious.
U – Unify – Bring together seemingly separate mathematical concepts
Unification reveals the deep structure underlying mathematical phenomena and often leads to more powerful, general theories. When you can show that different approaches are really the same approach in disguise, you’ve achieved a profound level of mathematical understanding.
V – Visualize – Create mental and graphical representations of abstract concepts
Visual thinking makes abstract concepts concrete and often reveals patterns that algebraic manipulation might miss. Even when working with objects that can’t be literally drawn, developing visual intuition helps guide your reasoning and suggests new approaches.
W – Wonder – Maintain curiosity about mathematical mysteries
Cultivate a sense of awe about the beautiful patterns and unexpected connections that mathematics reveals. This wonder drives the motivation needed for long-term research projects and helps you appreciate the elegant solutions that make difficult work worthwhile.
X – eXamine – Scrutinize mathematical objects and relationships carefully
Careful examination of mathematical structures often reveals subtle properties that casual observation might miss. This close attention to detail prevents errors and often leads to insights about why certain phenomena occur.
Y – Yield – Allow mathematical insights to emerge through patient work
Some of the deepest mathematical insights can’t be forced, they emerge naturally when you’ve prepared your mind through careful study and patient exploration. Learning to work steadily while remaining open to unexpected discoveries is essential for long-term success.
Z – Zoom – Focus intensely on crucial details while maintaining the big picture
Successful mathematical research requires the ability to alternate between microscopic focus on technical details and macroscopic vision of how those details fit into larger patterns. This cognitive flexibility helps you avoid getting lost in technicalities while ensuring your work contributes to broader understanding.
Extensions and Applications
This mathematical research alphabet represents just the beginning of what could become a valuable framework for academic and intellectual development. Several natural extensions suggest themselves:
Discipline-Specific Adaptations: Other fields could benefit from their own specialized alphabets for action. Physics researchers might need verbs like “Approximate,” “Measure,” and “Simulate.” Computer scientists might emphasize “Debug,” “Optimize,” and “Scale.” Literature scholars could focus on “Interpret,” “Contextualize,” and “Analyze.”
Pedagogical Applications: This framework could be powerful in graduate education, helping new researchers develop a comprehensive mental toolkit for approaching mathematical problems. Advisors could use it diagnostically, identifying whether students need to strengthen their ability to Experiment with examples, Abstract general principles, or Question underlying assumptions.
Research Group Integration: Mathematical research groups could adopt this alphabet as a shared language for discussing research strategies and methodologies. Teams could regularly check in on which verbs they’re emphasizing and which they might be neglecting.
Personal Research Development: Individual researchers could use this alphabet for self-assessment and goal-setting. Are you strong at “Abstracting” but weak at “Experimenting”? Do you excel at “Proving” but struggle with “Questioning”? The alphabet provides a framework for identifying growth areas.
Interdisciplinary Collaboration: As mathematics increasingly intersects with other fields, this alphabet could serve as a bridge, helping researchers from different disciplines understand and adopt mathematical thinking patterns.
Overcoming Research Obstacles: For researchers experiencing blocks or frustration, systematically working through the alphabet could provide new approaches and renewed motivation. Sometimes the key to breakthrough is simply trying a different verb.
The power of Schuller’s original insight, that having a comprehensive, memorable framework of action-oriented thinking can transform our approach to challenges, proves just as relevant in the realm of mathematical research as it does in personal resilience. By providing mathematical researchers with their own “alphabet for action,” we offer not just a tool for success, but a reminder that mathematical discovery is fundamentally about taking thoughtful, purposeful action in the pursuit of truth and understanding.
Try This: Pick three letters that resonate with your current research challenges. For the next week, consciously try to embody those action verbs in your daily mathematical work. Notice what changes in your approach, your insights, or your motivation. You might be surprised by how deliberately practicing these actions transforms not just your research process, but your relationship with mathematical discovery itself.
Whether you’re a graduate student facing your first major research challenge, a seasoned mathematician exploring new territories, or an educator looking to inspire the next generation of mathematical thinkers, this alphabet for action provides a comprehensive framework for approaching the beautiful, challenging world of mathematical research with intention, strategy, and hope.