Communicating mathematical reasoning

Good mathematical communication makes the logic of a solution visible. A correct answer can lose marks if essential reasoning is hidden, while a concise solution can earn full marks when every important claim is justified.

The aim is not to write more. It is to write enough that another mathematician can reconstruct why each step follows.

What you should know first

You should be able to:

• rearrange equations and use standard algebraic notation;
• distinguish an expression, an equation and an inequality;
• substitute into formulae and functions;
• recognise common command words such as show, prove, deduce and interpret.

Review algebraic fluency and proof and reasoning if these foundations are uncertain. For selecting the mathematics itself, see how to choose a method.

The claim, reason, conclusion pattern

A complete argument usually contains three ingredients:

1. Claim: state the mathematical fact being used or found.
2. Reason: show the calculation, definition, theorem or assumption that supports it.
3. Conclusion: answer the precise question.

Not every line needs prose. Correct notation often supplies the reason efficiently.

Worked example 1: classify a stationary point

For

$$f(x)=x^3-3x^2-9x+5,$$

show that the stationary point with positive $x$ coordinate is a local minimum.

Stationary points satisfy $f'(x)=0$:

$$\begin{aligned} f'(x)&=3x^2-6x-9\\ &=3(x-3)(x+1). \end{aligned}$$

Therefore $x=3$ or $x=-1$, so the stationary point with positive $x$ coordinate occurs at $x=3$.

Now classify it:

$$f''(x)=6x-6,$$

and

$$f''(3)=12>0.$$

Hence the curve is locally convex at $x=3$, so the stationary point is a local minimum.

The final sentence matters. The inequality $f''(3)>0$ is evidence, but the requested conclusion is that the point is a local minimum. See stationary points and curve sketching.

Make each implication honest

The symbols $=$, $\Rightarrow$ and $\Leftrightarrow$ do different jobs.

Symbol	Meaning	Appropriate use
$=$	the expressions have equal value	$2(x+3)=2x+6$
$\Rightarrow$	the left statement implies the right	$x=2\Rightarrow x^2=4$
$\Leftrightarrow$	each statement implies the other	$x+3=7\Leftrightarrow x=4$
$\approx$	approximately equal	$\sqrt2\approx1.414$

Do not use an equals sign to mean “and then”.

Incorrect:

$$x^2=9=x=3.$$

Correct:

$$x^2=9\Rightarrow x=3\text{ or }x=-3.$$

Also distinguish a reversible step from a one way implication. If $x=2$, then $x^2=4$, but $x^2=4$ does not force $x=2$. Thus

$$x=2\Rightarrow x^2=4$$

is true, while

$$x=2\Leftrightarrow x^2=4$$

is false.

Self-check 1

Insert $=$, $\Rightarrow$, $\Leftrightarrow$ or $\approx$.

1. $3(x-2)\;\square\;3x-6$
2. $x>5\;\square\;x>0$
3. $2x+1=9\;\square\;x=4$
4. $\pi\;\square\;3.142$ to $4$ significant figures

Answers

1. $=$, because these are identical expressions.
2. $\Rightarrow$, because $x>5$ is sufficient for $x>0$, but not equivalent to it.
3. $\Leftrightarrow$, because solving and reversing the step are both valid.
4. $\approx$, because the decimal is rounded.

Show enough algebra to expose the method

Routine arithmetic need not be narrated, but a marker must be able to see the mathematical route. Include:

• the equation or model you are using;
• substitutions before simplification;
• important restrictions or assumptions;
• the intermediate value from which a required result follows;
• units and degree of accuracy where relevant.

Worked example 2: a clear “show that” solution

The curve

$$y=\frac{x+1}{x-1}, \qquad x\ne1,$$

has a tangent at $x=3$. Show that the tangent has equation

$$2x+4y=13.$$

Differentiate using the quotient rule:

$$\frac{dy}{dx} =\frac{(x-1)-(x+1)}{(x-1)^2} =\frac{-2}{(x-1)^2}.$$

At $x=3$,

$$y=\frac{3+1}{3-1}=2, \qquad \frac{dy}{dx}=\frac{-2}{(3-1)^2}=-\frac12.$$

The tangent through $(3,2)$ is therefore

$$\begin{aligned} y-2&=-\frac12(x-3)\\ 2y-4&=-x+3\\ 2x+4y&=13, \end{aligned}$$

as required.

This does not begin by assuming $2x+4y=13$. It independently finds the point and gradient, then reaches the stated equation. The phrase “as required” marks completion; it does not replace reasoning.

Match the argument to the command word

Command	What your reasoning must do
Show that	Derive the supplied result without assuming it
Prove	Establish the statement in every allowed case
Disprove	Give one valid counterexample and explain the contradiction
Deduce or hence	Use an earlier result explicitly
Verify	Check directly that a value or object has the claimed property
Explain	Connect mathematical facts in words, not calculation alone
Interpret	Translate the mathematics into the original context

Checking several examples is not a proof of a universal statement. However, one counterexample does disprove a universal statement.

Worked example 3: proof by deduction

Prove that the square of every odd integer is odd.

Let $n$ be any odd integer. Then, for some integer $k$,

$$n=2k+1.$$

Therefore

$$\begin{aligned} n^2&=(2k+1)^2\\ &=4k^2+4k+1\\ &=2(2k^2+2k)+1. \end{aligned}$$

Since $k\in\mathbb Z$, the number $2k^2+2k$ is an integer. Thus $n^2$ has the form $2m+1$ for an integer $m$, so $n^2$ is odd.

The last two sentences close a common logical gap. The algebra reaches a useful form, then the definition of an odd integer turns that form into the conclusion. Explore other proof structures in deduction and exhaustion, contradiction and counterexamples.

State conditions when using a theorem or model

A named result is not a magic instruction. State or demonstrate the condition that makes it applicable.

Worked example 4: independence in probability

Events $A$ and $B$ satisfy

$$\operatorname P(A)=0.4,\qquad \operatorname P(B)=0.5,\qquad \operatorname P(A\cap B)=0.2.$$

Determine whether $A$ and $B$ are independent.

For independent events, the defining condition is

$$\operatorname P(A\cap B)=\operatorname P(A)\operatorname P(B).$$

Here,

$$\operatorname P(A)\operatorname P(B)=0.4\times0.5=0.2 =\operatorname P(A\cap B).$$

Therefore $A$ and $B$ are independent.

Writing only "$0.4\times0.5=0.2$" leaves the purpose of the calculation unstated. The definition and conclusion make the argument complete.

Communicate statistical conclusions precisely

A hypothesis test conclusion should include:

1. the comparison with the significance level or critical region;
2. the decision about $H_0$;
3. the strength of evidence for the contextual claim.

Worked example 5: conclusion from a p-value

A test of

$$H_0:p=0.30 \qquad\text{against}\qquad H_1:p>0.30$$

gives a p-value of $0.018$. Interpret the result at the $5\%$ significance level when $p$ is the proportion of customers choosing a new product.

Since

$$0.018<0.05,$$

the result is significant at the $5\%$ level, so reject $H_0$. There is sufficient evidence to suggest that more than $30\%$ of customers choose the new product.

Do not write ”$H_0$ is false” or ”$H_1$ is proved”. A hypothesis test measures evidence under uncertainty. Also keep the direction of $H_1$ in the conclusion. See hypothesis testing language.

Make mechanics reasoning directional

Mechanics equations are meaningful only after directions, forces and modelling assumptions are clear. A strong solution normally:

• defines a positive direction;
• identifies forces on the chosen body;
• resolves only those forces;
• applies $F=ma$ with signs consistent with the chosen direction;
• interprets a negative result rather than silently changing its sign.

Worked example 6: signs communicate direction

A particle of mass $4\text{ kg}$ is on a rough horizontal plane. A horizontal force of $10\text{ N}$ acts to the right, while resistance of magnitude $18\text{ N}$ acts to the left. Find its acceleration.

Take right as positive. Applying $F=ma$ horizontally gives

$$10-18=4a.$$

Hence

$$a=-2\text{ m s}^{-2}.$$

The negative sign means the acceleration is opposite to the chosen positive direction. Therefore the particle accelerates at

$$\boxed{2\text{ m s}^{-2}\text{ to the left}}.$$

The equation records the direction convention; the final sentence translates its sign. Review forces and free body diagrams and Newton’s laws.

Use notation that reduces ambiguity

Small notational choices prevent large misunderstandings.

• Define variables: “Let $t$ be the time in seconds after release.”
• Use brackets in substitutions: if $f(x)=x^2+1$, then $f(a+1)=(a+1)^2+1$.
• Write limits on definite integrals: $\int_0^2 f(x)\,dx$.
• Distinguish a vector from its magnitude: $\mathbf v$ and $|\mathbf v|$.
• Include constants of integration: $\int 2x\,dx=x^2+C$.
• Keep exact and approximate equality distinct.
• Attach units to contextual final answers, not to every algebraic intermediate line.

When introducing a symbol that is not given in the question, define it immediately. In a proof, specify its domain, such as $n\in\mathbb Z$ or $x>0$.

Avoid both extremes

Too little

For the equation $2^{x+1}=7$, writing only

$$x=1.807$$

hides the method and may not earn method marks.

Enough

$$\begin{aligned} 2^{x+1}&=7\\ (x+1)\ln2&=\ln7\\ x&=\frac{\ln7}{\ln2}-1\\ &=1.807\ldots \end{aligned}$$

Therefore $x=1.81$ to $3$ significant figures.

Too much

Long prose describing every arithmetic operation can obscure the mathematics. Prefer a displayed chain of valid equations, with prose at decision points: why a theorem applies, why a root is rejected, what a sign means, or what the final result says in context.

Self-check 2

Question 1

A student differentiates $y=x^2e^x$ and writes

$$\frac{dy}{dx}=2xe^x+x^2e^x=e^x(x^2+2x).$$

They were asked to find the gradient at $x=1$. What is missing, and what is the complete answer?

Question 2

Disprove the claim: “If $a^2>b^2$, then $a>b$ for all real $a$ and $b$.”

Question 3

A hypothesis test at the $1\%$ significance level gives a p-value of $0.032$. Should $H_0$ be rejected? State the reason.

Answers

1. The differentiation is correct, but the requested evaluation and conclusion are missing:

   $$\left.\frac{dy}{dx}\right|_{x=1} =e(1+2)=3e.$$

   Therefore the gradient at $x=1$ is $3e$.

2. Take $a=-3$ and $b=2$. Then $a^2=9>4=b^2$, but $a=-3<2=b$. This counterexample disproves the universal claim.

3. No. Since $0.032>0.01$, the result is not significant at the $1\%$ level, so there is insufficient evidence to reject $H_0$.

A final communication check

Before moving on, ask:

1. Have I answered the exact command and used any requested method?
2. Can a reader see the key equation, theorem or model?
3. Have I justified restrictions, rejected values and important assumptions?
4. Are $=$, $\Rightarrow$, $\Leftrightarrow$ and $\approx$ used correctly?
5. Does the conclusion state what the result means, with units and context if needed?
6. Is the precision appropriate, with exact values kept until the end?

Next, practise identifying avoidable weaknesses in common A-level Mathematics exam mistakes and learn systematic ways of checking answers.