GTLE Mathematics Specialism common mistakes
GTLE Mathematics Specialism · Understand the traps and how to avoid them.
When simplifying a ratio or rate problem, the candidate divides the two quantities in whatever order they appear in the sentence without checking which quantity should be the unit rate, then reports the reciprocal of the requested value.
Word problems bury the base quantity in a subordinate clause, and candidates pattern-match on the first two numbers they see. Because ratio language ('per', 'for every', 'to') is symmetric in everyday speech but not in mathematics, the inversion is invisible on re-reading.
What to do instead
Before computing, write an explicit unit statement such as 'answer is dollars per pound' and set up the division so the unit of the answer sits in the numerator. Then do a magnitude sanity check: if the answer implies a price of 0.02 dollars per pound for groceries, the ratio is inverted.
The candidate converts between units by multiplying or dividing by a single conversion factor even when the unit appears squared or cubed, so a 2 m by 3 m rectangle becomes 6 m instead of 6 m² and a volume conversion uses the length factor once rather than cubed.
Candidates memorize 'multiply when going to smaller units' as a one-step rule and do not track the exponent attached to the unit. Dimensional notation is often dropped mid-calculation, so the error surfaces only in the answer's unit.
What to do instead
Carry units through every line of work and treat them as algebraic objects that get squared and cubed along with the numbers. For area and volume conversions, write the factor with its exponent explicitly (1 m = 100 cm, so 1 m² = 10,000 cm²) before substituting any numbers.
The candidate solves an equation by squaring both sides, clearing a denominator, or multiplying by a variable expression and then accepts every algebraic solution without checking it in the original equation.
These operations are not reversible: squaring can introduce extraneous roots and multiplying by a variable expression can introduce values that make a denominator zero. Candidates treat algebra as a sequence of legal moves rather than as a chain of implications that must be preserved.
What to do instead
Adopt a standing habit: any time both sides are squared or multiplied by an expression containing the variable, substitute each candidate solution back into the original equation and discard failures. Also record domain restrictions (denominators nonzero, radicands nonnegative) at the start, before solving.
The candidate reads a normal-distribution or z-score problem and reports a probability for the wrong tail, or reports the area between the mean and z as if it were the area beyond z.
Standard tables give area from the mean to z, while the question usually asks for area beyond z or between two values. Candidates memorize 'look up the z-score' without memorizing what the table entry represents, and the symmetry of the curve makes a wrong-tail answer look plausible.
What to do instead
Sketch the curve on scratch paper every time, shade the region the question asks about, and label the table value as 'mean to z' before using it. Compute the requested area by subtraction from the sketch, not from memory of a formula.
The candidate computes a mean, median, or standard deviation for grouped data by averaging the class midpoints unweighted, or by using n as the number of classes rather than the number of observations.
Grouped frequency tables present two counts (classes and frequencies) and candidates default to the count they can see as rows. The distinction between 'number of categories' and 'number of data points' is not visually salient in a table.
What to do instead
Before any grouped-data calculation, write the total frequency Σf explicitly and use it as n. For the mean, compute Σ(f·x)/Σf with the midpoint x, and verify that the weighted mean lies between the smallest and largest midpoints and is pulled toward the class with the highest frequency.
The candidate answers a 'which statement must be true' or 'which could be a counterexample' item by testing one or two numerical examples and concluding from those, rather than reasoning about the quantifier.
Concrete examples are fast and feel rigorous, but a single confirming example cannot establish a universal claim, and a single example that satisfies the premises does not refute a 'must be true' statement unless it also violates the conclusion. Candidates conflate 'true in my example' with 'true in general'.
What to do instead
Classify the claim first: universal claims are refuted by one counterexample but never proved by examples; existential claims are proved by one example but never refuted by examples. Then deliberately test boundary cases (zero, negatives, fractions between 0 and 1, the maximum) where universal claims usually fail.
The candidate sets up a system of equations for a mixture, work-rate, or distance problem with the variables defined inconsistently, so one equation uses total quantity and another uses a rate, and the system has no meaningful solution.
These problems require two different kinds of equation (a quantity balance and a value balance), and candidates write both equations about the same aspect. Without writing definitions first, the mismatch is hidden until the algebra produces a nonsensical answer.
What to do instead
Write a two-line legend defining each variable with its unit before writing any equation, then label each equation as 'quantity' or 'value/rate'. If both equations reduce to the same relationship, one of them is the wrong kind and must be rebuilt.
The candidate writes a solution to a multi-step problem as a string of unsupported equations with no statement of what is being computed at each step, so partial credit is lost and the reasoning cannot be followed.
Candidates under time pressure treat explanation as decoration and assume the grader will infer intent. In constructed-response mathematics items, the scoring rubric awards points for stated strategy and justification, not only for the final value.
What to do instead
Use a three-part structure for each step: state what is being found, show the computation, and state the result with its unit. Begin with a one-sentence plan naming the strategy, and end with a sentence interpreting the answer in the context of the problem.
The candidate identifies a student's error in a work sample only as 'a computation mistake' or 'a careless error' rather than naming the specific misconception and the reasoning that produced it.
Candidates diagnose by comparing the student's answer to the correct answer and stop at the first divergence. Naming a misconception requires reconstructing the student's procedure, which is a different and slower task than checking the arithmetic.
What to do instead
Reproduce the student's method faithfully from their written work, then describe the rule they appear to be following in their own terms (for example, 'adds numerators and denominators'). Only after stating that rule should you classify it as a misconception and design a follow-up question that exposes it.
The candidate proposes a remediation for a struggling student that consists of more practice problems of the same type, without changing the representation or addressing the underlying conceptual gap.
Practice is the default response to error, and candidates assume volume fixes understanding. If the student's mental model is wrong, repeated practice rehearses the error and makes it more fluent.
What to do instead
Match the intervention to the diagnosed misconception: change the representation (concrete, pictorial, symbolic), use a contrasting case that the faulty rule would get wrong, and require the student to explain why the rule fails. Reserve drill for fluency after the concept is secure.