PSLCE Mathematics common mistakes
PSLCE Mathematics · Understand the traps and how to avoid them.
When subtracting, the candidate subtracts the smaller digit from the larger digit in each column regardless of position (e.g. 503 − 178 written as 4 7 5, or 62 − 38 = 36), producing an answer that is never checked by adding back.
Borrowing/regrouping is taught as a procedure of crossing out and rewriting, so learners memorise the marks rather than the exchange of one ten for ten ones. When a column shows a smaller top digit they fall back on the intuitive but wrong rule 'take away the smaller from the bigger', which works often enough on easy items to survive unnoticed.
What to do instead
Have them do every subtraction twice: once with place-value counters or an abacus to show the exchange, and once in columns, then always verify by adding the answer to the subtrahend. Give a five-item set of pure 'zero in the middle' subtractions (400 − 236, 702 − 458) until each is checked correctly by addition.
The candidate adds or subtracts the numerators and the denominators of fractions directly (2/3 + 1/4 = 3/7) and multiplies across without simplifying (2/3 × 3/4 = 6/12 stated as final).
Whole-number operations were learned as 'do the same thing to the top and bottom numbers', and nothing in the symbolic form of a fraction signals that the denominator names the size of the parts. Learners also lack a firm sense of equivalent fractions, so they do not see that 6/12 and 1/2 are the same number.
What to do instead
Before any symbolic work, make them add fraction strips or shaded circles: 2/3 + 1/4 physically cannot become 3/7 because the pieces are different sizes. Then drill converting to a common denominator with a written step ('rename both fractions with denominator 12'). For multiplication, require the answer in lowest terms and check it against an estimate.
The candidate misaligns decimal points when adding or subtracting (3.6 + 12.45 written as 3.60 + 12.45 aligned at the left, giving 16.05 or 48.45), or compares 0.7 and 0.65 by saying 0.65 is larger because 65 is bigger than 7.
The learner treats the decimal as two whole numbers separated by a dot rather than as a position on the number line. Place-value language ('tenths', 'hundredths') has not been connected to the digits, so trailing zeros look like they change the value.
What to do instead
Make them write the numbers with the decimal points stacked vertically and pad with zeros to equal length before computing. For comparison, place both on a marked 0–1 number line or convert to hundredths (0.70 vs 0.65) and state the comparison in words: 'seventy hundredths is more than sixty-five hundredths'.
The candidate converts between units by multiplying when the conversion needs dividing, or uses the wrong power of ten (2.5 km written as 250 m, 350 cm written as 3.5 m, 3 hours 45 minutes written as 3.45 hours).
Conversion is remembered as 'move the decimal point' with no anchor to the relative size of the units, so the direction is guessed. Time is especially vulnerable because learners apply base-ten thinking to a base-sixty system.
What to do instead
Require a unit-size check before computing: 'a metre is smaller than a kilometre, so the number of metres must be bigger'. Build a two-column reference table (1 km = 1000 m, 1 m = 100 cm, 1 hour = 60 min) and have them write the conversion as a fraction with units cancelling. For time, always convert to minutes first, compute, then convert back.
The candidate computes the area of a rectangle by adding the sides, confuses area with perimeter, or uses the perimeter formula when the question asks for the distance around a field.
Both quantities are introduced together and both involve the two side lengths, so the labels 'area' and 'perimeter' are stored as words rather than as different measurements (square units versus linear units). The unit in the answer is usually ignored.
What to do instead
Make them state the unit of the answer before starting ('square metres' means area, 'metres' means perimeter) and draw the rectangle with squares inside for area versus a line traced around for perimeter. Give mixed sets where they must first write A or P next to each question, then solve.
The candidate calculates change from a purchase by subtracting the wrong way round or by ignoring the note value given (bill of 500 for goods costing 375 answered as 875, or change given as 125 when the customer paid with 500 and the goods cost 375 but a 200 note was also tendered).
Word problems about money are read for the numbers only, not for the relationship between them. Learners also do not habitually identify which quantity is the total and which is the part, so the operation is chosen by the order the numbers appear.
What to do instead
Train a fixed routine: underline what is asked, write 'amount paid − cost = change', estimate the answer first (about 100), then compute, then check that the change plus the cost equals the amount paid. Practise with realistic note-and-coin combinations so the tendered amount must be built from the notes described.
The candidate reads a timetable or clock incorrectly: treating a 24-hour time such as 14:45 as 4:45, computing a journey that starts at 09:50 and ends at 11:25 as 2 hours 35 minutes (subtracting 50 from 25 and borrowing 100 instead of 60), or misreading an analogue clock's minute hand.
Base-ten borrowing is over-generalised to minutes, and 12-hour clock habits conflict with 24-hour notation. Reading an analogue clock requires coordinating two hands and skip-counting in fives, a skill that is often only partly secure.
What to do instead
Convert all times to minutes after midnight before subtracting (11:25 = 685 min, 09:50 = 590 min, difference 95 min = 1 h 35 min) and show that borrowing in time gives 60, not 100. For 24-hour time, drill 'subtract 12 and add p.m.' with a table of paired times, and have them read analogue clocks daily and write both notations.
The candidate finds the mean by adding the values and dividing by the number of different values listed rather than the number of data points, or reports the range as the largest value instead of the difference between largest and smallest.
The three averages are taught as three formulas in quick succession, so the divisor in the mean is recalled as 'how many numbers I wrote down', and the range is confused with the maximum because both involve the biggest number.
What to do instead
Have them rewrite the data set as a list with a count written above it ('n = 8') before dividing, and always write the mean as total ÷ n with both numbers visible. For range, force the sentence 'largest minus smallest' to be written out before subtracting. Use small data sets where the mean can be checked by balancing.
The candidate continues a number pattern by a constant difference when the rule is multiplicative or two-step (2, 6, 18, 54 continued as 90), or substitutes a value into an algebraic expression incorrectly, evaluating 3n + 2 as 32 when n = 4.
Pattern work in early grades is almost always 'add the same number', so that rule is applied without checking. In algebra, the notation 3n is read as the two-digit number '32' because juxtaposition meaning multiplication has not been made explicit.
What to do instead
Require the learner to test the rule on at least two consecutive pairs and to state it in words ('multiply by 3') before extending. For substitution, insist on rewriting the expression with brackets: 3n + 2 becomes 3 × (4) + 2, and drill the convention that a number next to a letter means multiplication.
The candidate solves a multi-step word problem by using only the first two numbers encountered, or leaves the answer as the result of an intermediate step (finding the total cost but not the change, or the number of buses but not the number of seats left over).
Working memory is loaded by the reading, so the learner grabs the first operation that suggests itself. There is no habit of planning the steps or of re-reading the question to see what is actually demanded.
What to do instead
Teach a three-line plan before any calculation: 'What I know', 'What I must find', 'The steps in order'. Have them number the steps and tick each one off, and finish by re-reading the question and writing the answer with its unit. Use problems with a deliberate distractor number so that selecting relevant information is practised.