Competition mathematics · AMC 10 · AIME
Competition mathematics coaching case studies
Three examples of the same coaching process at different starting points. Each begins with the student's actual work, reviews where a student tends to lose points (missing knowledge, problem recognition, method choice, execution, reading, or pacing), then targets the highest-priority pattern and retests it on unfamiliar problems.
Identifying details have been removed or generalized to protect student privacy, and each student is described only by the profile relevant to the coaching decision. These are selected coaching examples and should not be interpreted as representative or typical outcomes. They document how the coaching process was applied; where an outcome has not been independently verified, that limitation is stated in the case itself.
Case one
An AMC 10 student converting general ability into contest performance
This student entered AMC 10 preparation with strong general mathematical ability, limited familiarity with the competition format, and disappointment from an earlier contest result. Early lessons focused on converting that ability into reliable performance on unfamiliar, calculator-free problems rather than on broad review.
Diagnostic decisions
| Observed issue | Diagnosis | Coaching decision |
|---|---|---|
| Correct ideas with occasional avoidable losses | Some errors came from execution or stopping at an intermediate quantity. | Require a clean written path and a final check against the question. |
| Fast calculation before the structure was clear | Method choice was sometimes weaker than the underlying knowledge. | Use a two-path drill and require a reason for the chosen approach. |
| Difficulty with exact wording in counting | Reading and model selection were part of the error. | Restate the event before counting and verify that all cases are covered. |
| Limited experience under contest conditions | Untimed topic knowledge did not yet prove contest readiness. | Use calculator-free mixed sets, timing checkpoints, and full simulations. |
Training progression
Each lesson used the previous work to set the next priority. Strong areas moved quickly; recurring errors received direct instruction and a later retest.
| Stage | Work completed | Resulting decision |
|---|---|---|
| Initial diagnostic | Mixed algebra, geometry, number theory, and counting under calculator-free conditions. | Focus shifted to recognition, method choice, and execution rather than broad remediation. |
| Geometry and algebra | Medium-band problems exposed a part-to-whole ratio slip and an imprecise special-triangle setup. | Both errors became specific teaching targets instead of a general call for more practice. |
| Counting and probability | Order, selection, complements, casework, and exact wording. | The student began treating the wording as part of the mathematical model. |
| Number theory | Divisibility, modular cycles, divisors, and Diophantine reasoning through the medium and hard bands. | Number theory proved to be the strongest area, so lesson time moved to larger gaps. |
| Hard algebra | Sequences, functions, logarithms, and inequalities, with each shortcut derived before use. | Training addressed the largest remaining content opportunity. |
| Mixed consolidation | Previously taught material returned without topic labels and under time. | Practice moved from learning methods to recognizing and using them independently. |
Early progress checkpoint (Session 4)
Session-4 percentages are tutor-scored practice estimates based on session work. They are not official AMC or AIME scores and have not been validated as predictors of contest performance. They set the priorities for the remaining sessions: hard algebra, mixed recognition, error reduction, and timed practice.
| Measure | Initial coaching question | Session 4 position |
|---|---|---|
| Easy-band technique coverage | Could the first 60 points be protected? | Estimated at about 90 percent. |
| Medium-band technique coverage | How much of Questions 11 to 18 was accessible? | Estimated at about 70 percent. |
| Number theory coverage | Was number theory a gap or a strength? | Estimated near 85 percent overall, with reach into the hard band. |
| Content-reachable score | What score was supported by the techniques already covered? | Estimated near 95 to 100 before execution losses. |
| Remaining priority | Where would the next block of instruction add the most value? | Hard algebra first, followed by hard geometry and counting. |
The two-path drill
Before a long calculation, the student identified a direct method and looked for a shorter structural one. If the clean path appeared quickly, they used it and stated the key idea. If it did not, they chose a valid direct method, managed the time risk, and later recorded the structure they had missed. A parallel problem then tested whether the recognition transferred.
Case two
An AIME-track student turning scattered preparation into a plan
This student was preparing for the AMC 10 with the goal of qualifying for the AIME and had already invested in strong books, adaptive practice, contest videos, and a timed diagnostic. The effort was real, but the resources were not yet organized around a diagnosis. Coaching turned that work into a structured sequence with direct feedback and a controlled increase in difficulty.
Changes to the preparation
| Starting point | Change made | Practical effect |
|---|---|---|
| Several strong resources used independently | Organized the books, adaptive practice, videos, and diagnostic around ranked gaps. | Every assignment had a defined purpose. |
| A general sense of weak fundamentals | Used timed papers and live problem solving to identify the exact source of misses and slow work. | The next steps became a short, teachable list. |
| A request for gradual difficulty | Sequenced fundamentals, recurring contest structures, and timed mixed sets. | Challenge increased only after the prerequisite method was stable. |
| An AIME-qualification goal | Centered the plan on score conversion, problem selection, timing, and when to move on. | Training matched the contest objective. |
| Strong materials already owned | Built the plan around resources the student already had. | Coaching added diagnosis, sequencing, feedback, and accountability without replacing useful resources. |
Resulting direction
The student moved from choosing work by resource to choosing work by diagnosed need. The plan named what to study next, why it mattered, how difficult the work should be, and how to test it under time. No independently verified contest result is available for this student. This example therefore documents the coaching process and observed practice performance only.
Case three
A strong performer working toward more consistent results
This student had strong underlying ability and high accuracy, but a practice-score range showed that access and consistency still varied. A recent near-perfect diagnostic meant the next set had to be harder, especially in geometry, so that conceptual, strategic, and execution errors could be separated rather than grouped together.
How the process applied
| Evidence | Coaching response | Planned area of improvement |
|---|---|---|
| Strong number sense and high attempted-question accuracy | Protect the strength and spend lesson time on higher-value gaps. | More of the session becomes available for Questions 11 to 20. |
| Geometry accurate but slower | Raise the difficulty and time recognition, setup, and execution separately. | The program addresses the actual source of the slowdown. |
| One lesson problem required a hint | Retest the structure without help and classify any failure as concept, strategy, or execution. | The method becomes independent rather than prompt-dependent. |
| Tiling and parity produced the clear miss | Teach coloring, invariants, and lower-bound proofs, then use spaced transfer. | Convert a confirmed gap into a scoring opportunity. |
| Practice scores range from 13 to 20 | Track first-pass choices, stalls, returns, and Questions 11 to 15 conversion. | Work toward a more consistent practice floor. |
Objectives describe what the coaching plan was designed to improve. They are not predictions or guaranteed outcomes.
What the family receives
- A written diagnosis based on prior papers, written work, timed results, and live problem solving.
- Lessons chosen from the student's current error record rather than a fixed chapter sequence.
- Separate tracking for geometry concepts, strategy choices, execution, pace, and hint dependence.
- Clean-path scoring across geometry, algebra, number theory, counting, and probability.
- A brief recap after each lesson, a midpoint report, and a final readiness report.
- Official mock results and a specific test-day plan before the exam.
The program is designed to work toward more consistent practice performance and to raise the level of problems the student can solve independently within the time limit. These are the plan's objectives, not predicted or guaranteed contest results.
What these cases do and do not demonstrate
They demonstrate a diagnosis-first process: reviewing where a student tends to lose points, teaching the highest-priority gap, and retesting it under contest conditions. The Session-4 percentages are tutor-scored practice estimates, not official scores, and have not been validated as predictors of contest performance. One case has no independently verified contest result, and the planned areas of improvement describe what the coaching plan was designed to work on. These are selected examples and are not representative or typical outcomes. None of this guarantees an AMC score, AIME qualification, or any specific outcome for another student.
Start with a readiness check, not a guess.
A diagnostic reviews where a student tends to lose points (knowledge, recognition, method, execution, or pacing) and returns a prioritized plan for the next block of work.
No guaranteed scores, qualifications, or contest outcomes