Interpreting a Class VII integers test before moving on to algebra
Your Class VII Mathematics section has 38 students. The unit test on integers produced 31 scores below 40 per cent. Item analysis of the 31 low scorers shows: 27 answered −5 − (−3) = −8; 19 answered −2 × −3 = −6; 22 answered −7 + 4 = −11; but 26 of the same 31 correctly ordered −7, −2, 0, 3 on a number line, and 24 correctly answered 6 − 9 = −3 using a temperature context in the previous chapter's exercise. In class, several students recite "minus minus plus" fluently and still get the subtraction item wrong. The annual examination syllabus now requires factorisation and simple linear equations, and your head of department wants ten marks of drill on the sign rules added to the next test before you move on. You have two 35-minute periods before the next unit begins.
1. Diagnose what the students have and have not learned. Explain how the ordering and temperature successes change your reading of the sign-rule failures.
Read the worked answer
The profile is not "does not know integers". It is "has a procedure without a quantity model".
The successes are the diagnostic key. Twenty-six of 31 can order negatives correctly, and 24 can do 6 − 9 = −3 in a temperature context. Ordering on a number line requires a magnitude-and-direction understanding of negative numbers, and the temperature item requires interpreting subtraction as a move. So the underlying quantity concept is present in most of the low scorers.
The failures cluster where two operations interact: −5 − (−3) answered as −8 means the student added the magnitudes and kept the sign - the second minus was treated as a signal to combine, not to reverse direction. −7 + 4 = −11 shows the same additive-magnitude habit. −2 × −3 = −6 shows the rule "two numbers, multiply the values" with the sign rule not attached to any meaning.
The fluent recitation of "minus minus plus" alongside wrong answers is the decisive evidence: the rule is stored as a verbal formula, not as an operation on a number line, so it is unavailable when the item is embedded in symbols. The teaching task is to reconnect the rule to the move on the line, not to add more exposure to the rule.
2. Plan the two periods. Give the sequence, the representation you will use, and state what you will deliberately not do.
Read the worked answer
Period 1 - rebuild subtraction as a move (35 min). Five minutes: the temperature item they already got right, written on the board as the anchor. Ten minutes: number line on the floor or on paper, with a rule - the first number is where you stand, the second tells you which way to face and whether to walk forward or backward. Ten minutes: paired work on eight items of the form a − (−b) and a − b, every one drawn as a move before it is written as an answer. Ten minutes: board comparison of −5 − (−3) and −5 + 3 and −5 − 3, so the class sees that subtraction of a negative and addition are the same move.
Period 2 - multiplication as repeated direction (35 min). Start from 3 × (−2) as three walks of two steps backwards, then −3 × (−2) as three reversed walks, using the same line. Then ten minutes of mixed items where students must draw the first step before answering. Close with three items in words.
Deliberately not do: no new mnemonic, no "minus minus plus" chart on the wall, no ten marks of isolated sign drill. Drill on a formula that is already stored verbally and wrongly will strengthen the wrong association; the HOD's request is met instead by drilling the drawing-then-answering routine.
3. How will you answer the head of department's demand for ten marks of sign-rule drill, and how will you know within two weeks whether the reteach worked?
Read the worked answer
Answer with evidence and an alternative, not a refusal.
To the HOD: agree on the goal - accurate sign handling before algebra - and disagree on the instrument. Ten marks of bare sign items will be answered with the verbal rule and will not tell either of us whether the concept is fixed. Offer instead eight marks of items written in context (temperature, bank balance, lift floors) with one line of working required for each, plus two marks of a written explanation of why −2 × −3 is positive. Same marks, same topic, better evidence - and it protects the algebra unit, because the working line is what transfers.
Monitoring within two weeks: a five-item entry slip on three occasions, each containing one subtraction, one addition of unlike signs, one multiplication of two negatives, one ordering item and one word problem. Track the 31 low scorers by name on a simple grid. Success criteria agreed in advance: at least 24 of the 31 correct on both negative-multiplication items by the third slip, and no fall in the ordering items (which would suggest the new routine has confused a concept they already had). If the multiplication items do not move, the problem is the representation, not the practice time, and the next step is a different model - counters or the charged-particle model - rather than more drill.