Lesson 23 of 26Write a result another reader can reconstruct
Connect definitions, argument and evidence in clear prose.
Try
Read the sentence “the equilibrium is stable.” It omits the equation, parameter range, state space and meaning of stability. A reader cannot determine what was proved. Replace it with a statement about xₙ₊₁=rxₙ+b for real parameters and |r| below 1, then show the error formula that supports the conclusion.
Good mathematical writing is not decorative prose added after the mathematics. Defining objects and stating hypotheses are part of making the claim correct and checkable.
Predict
What changes at r=1? Predict why a statement that omits the parameter restriction can fail even when the derivation appears familiar. Then identify where division by 1−r occurs in the argument.
Experiment
Compare three versions of the same result.
| Version | What the reader can check |
|---|
| “The method converges.” | Neither method nor target is defined |
| “The recurrence converges when stable.” | Circular condition, still incomplete |
| “For real | r |
For r=1/3,b=2, the fixed point is 3 and xₙ−3=3^(−n)(x₀−3). Include this concrete example after the general statement to help the reader connect notation with behavior. Do not substitute the example for the general proof.
Understand
A results section should state the question and result early, then develop the evidence in logical order. Define symbols near first use, including units and domains. If Q=a/b represents an equilibrium stock, say that a is input per time, b is a positive removal coefficient per time and Q has stock units. Explain the balance producing the ratio rather than presenting it as an unexplained formula.
Separate a theorem from its illustration, a fitted estimate from its uncertainty and a numerical observation from a proof. Use phrases that identify the evidence precisely: “the finite search found,” “the bound proves,” or “under the stated noise model.” Avoid universal conclusions that exceed the investigated regime.
A figure should answer a question. Give axes, units, legend, observation versus prediction labels and an informative caption. If two models are compared, shared scales make differences interpretable. If an uncertainty band appears, state whether it is a parameter interval, predictive interval or scenario envelope and how it was constructed. A graph that cannot be interpreted independently of the author’s narration is incomplete evidence.
An abstract can follow four roles: question, method, supported result and limitation. For example, an investigation may derive a two-compartment invariant, estimate one identifiable rate combination and show that an additional observation is needed to separate two rates. That limitation is a substantive result, not a detail to hide.
Attribute borrowed ideas as well as direct quotations. Rewriting a proof in your own words creates original exposition, not automatically a new theorem or proof idea. State what was adapted and what your investigation contributes.
Apply
Write one complete result paragraph and one caption from your investigation. Ask a reader to identify the assumptions, output and evidence without additional explanation. Revise every undefined symbol or unsupported transition they find. The assessment concerns reconstructable reasoning, not a minimum word count.
Continue
Review and revision test whether the report survives another person’s questions. Preserve the original version so improvements and unresolved objections remain visible.
