A Correct Answer Is Not Yet a Complete Argument

In one study, beginning undergraduates and professional mathematicians read purported proofs and judged whether each was valid, with their eye movements recorded^[1]. The mathematicians shifted their gaze back and forth between consecutive lines far more than the undergraduates, who spent proportionately more of their time on the surface features of the arguments^[1]. The researchers took that as the mathematicians filling in reasons the proofs left unsaid, an inference from eye movements rather than a measurement^[1].

Take that as a picture of reading a written argument. Between any two lines of your work sits a claim that the second follows from the first. If you do not write it, the reader rebuilds it, and the rebuild is guesswork.

Nor is there a fixed bar: in that study the mathematicians sometimes disagreed about whether even a short proof was valid^[1]. So state your reason rather than leave the rebuild to the reader.

Show that $x^3 - 2x - 1 = 0$ has a solution between $x = 1$ and $x = 2$. Here is an answer that stops too early.

Let $f(x) = x^3 - 2x - 1$. Then $f(1) = 1 - 2 - 1 = -2$ and $f(2) = 8 - 4 - 1 = 3$. The values have opposite signs, so $f$ has a zero between $1$ and $2$.

The conclusion is true: $x^3 - 2x - 1 = (x + 1)(x^2 - x - 1)$, and the root in that interval is $x = \frac{1 + \sqrt{5}}{2}$, about $1.618$.

Now run the identical argument on $g(x) = \frac{1}{x}$ between $-1$ and $1$: $g(-1) = -1$ and $g(1) = 1$, opposite signs, so $g$ has a zero in between. It does not: $\frac{1}{x}$ is never zero.

Same form, one of them false. What separates them is a sentence neither wrote: the Intermediate Value Theorem needs the function continuous across the whole closed interval. Polynomials are continuous everywhere, so $f$ qualifies; $g$ is not even defined at $x = 0$. In full:

$f$ is a polynomial, so it is continuous on $[1, 2]$, and $f(1) = -2 < 0$ while $f(2) = 3 > 0$. By the Intermediate Value Theorem there is a $c$ in $(1, 2)$ with $f(c) = 0$.

That clause is all that stands between this argument and the false one beside it. Every theorem hides a condition like it: differentiability for the Mean Value Theorem, a nonzero factor before you cancel. Some questions are nothing but the clause: show that the solution is unique, explain why a derivative does not exist, justify why a series diverges.

Across a dozen experiments, people believed they understood how things work in far more detail than they did^[2]. The gap was widest for knowing why something works and narrower for plain facts and procedures, and those experiments asked about everyday things, not mathematics^[2].

A second habit pushes the same way: a review of how people work out what others know found that we start from our own knowledge and assume others share it^[3]. The assumption is usually fair, but we make it without checking, and wrongly taking it for granted is a well-documented cause of being misunderstood^[3].

Together these stop the missing line looking like carelessness^[2][3]: the step feels obvious, and you place your reader where you stand. Neither feeling is a measurement, so the remedy is a procedure, not more care.

Researchers once listened in as students talked their way through worked-out physics examples^[4]. The ones who understood best said why each step was allowed and tied it to the rule it came from; the ones who learned least mostly re-read the examples^[4]. The two went together, but a study like this cannot say which caused which^[4].

Writing has a firmer record. A review that pooled 56 classroom experiments with students in grades 1 to 12 found that writing about the material being studied helped them learn it, about as much in mathematics as in science and social studies^[5]. That writing was part of the lesson, and none of it went past grade 12^[5]. Putting the reason into words appears to be part of how the material gets in^[4][5].

A review of studies of mathematics learning found that students prompted to explain their own steps did somewhat better on tests straight afterwards, a small to moderate gain, larger still when they had first been trained in what a good explanation contains or given a structure to fill in^[6]. "Add a short line" is not a structure. This is:

1. Name the rule, theorem or definition you are using.
2. State the condition that rule requires.
3. Show that the condition holds in this problem.

The full justification above is exactly those three clauses. They cover the other common gap, where the algebra hands you two roots and only one can be the answer: the rule is the constraint the problem carries, the condition is that a length cannot be negative, and showing it holds names the root you discard. Limit arguments carry a longer list of their own, worth a separate pass.

A review that pooled 40 studies of practice questions answered on a computer compared several kinds of feedback^[7]. Feedback explaining the answer helped much more than feedback saying only right or wrong, and mattered most on questions needing deeper thinking^[7]. It helped more in mathematics than in other subjects, and less for primary and high-school students than for older ones^[7]. All of it is automatic feedback on practice items, not human markers or graded work^[7].

A red box on an online homework system is the feedback that did least for learning there, so treat it as a prompt to find the explanation^[7]. Take a solution to a classmate, a teaching assistant or a Learn4Less tutor, hand over the reasons, and ask whether they hold.

Whether leaving a justification out costs you marks is not something this research speaks to: none of these studies looked at markers, marking schemes or graded scripts^{[4][6][5][1][2][3][7]}. Anything anyone tells you about what a marker is thinking is experience, not evidence.

The support is uneven. The mathematics evidence on prompted explanation is mostly from tests given right afterwards, not ordinary classrooms or weeks later^[6]; the writing review stopped at grade 12, so it says nothing directly about first-year calculus^[5]; and the think-aloud work compared the strongest learners with the weakest, so it cannot show that explaining caused the learning^[4].

Two sources are not about mathematics at all: one asked about everyday objects, the other reviewed conversation in general^[2][3]. They describe a habit that plausibly reaches your written solutions, but nobody has checked. And the eye-movement work reads where people looked, not what they thought^[1].

Pick one problem from this week's set and read its instruction word first: justify, explain and show why all ask for the reason, not the number. For every step that is not pure arithmetic, write the rule, its condition, and why it holds. If you cannot name the condition, you have found something to learn.