At one location, the temperature increased

Why it is hard

Points are lost in two places. The phrase “3.5 times greater than” is usually taken as 3.5 times as great, as the key takes it, but its “greater than” invites adding the product onto the original reading, so the meaning has to be settled before multiplying. Then the rise from −7° to 14° must include the climb up to zero, which students who drop the minus sign leave out; with no choices to check against, neither slip announces itself.

1. Error-prone arithmetic moderate Working it out

   The last step is 14 − (−7): subtracting a negative starting temperature means adding 7, and the seven degrees climbed below zero are the part most often lost. (Brown & Burton, 1978; Junker & Sijtsma, 2001)

2. Turning words into math moderate Understanding the question

   The phrase “3.5 times greater than,” with “greater” in bold, has to become 3.5 × 4° as the key reads it, although “greater than” can suggest adding 3.5 × 4° onto the 4° reading. (Hegarty, Mayer & Monk, 1995; Koedinger & Nathan, 2004)

3. Much to keep track of minor Working it out

   Three times of day carry three temperatures, −7°, 4° and 14°, and “total increase” compares the first with the last, not either of them with the middle one. (Sweller, 1988; Carpenter, Just & Shell, 1990)

Strategy: Take “3.5 times greater than” as the answer key does, meaning the 9:00 a.m. reading multiplied by 3.5; use that product for 11:00 a.m., then subtract the 5:00 a.m. reading, minding its negative sign.

Traps

• Negatives and direction: Arithmetic with negative numbers, the order of a subtraction, or direction on a number line. (Moving a term across the equals sign without changing its sign is a faulty procedure.)
• Follows a key word: Picks the operation from a single word rather than the situation: "removed" so subtract, "of" so multiply, "more" so add, when the situation needs something else.
• Stops one step early: A choice is a number from partway through the solution.

The wrong answers, classified

• 7 Negatives and direction (Brown & Burton, 1978)

  Reveals a slip Expected pull: strong

  Finds 14° at 11:00 a.m. but subtracts 7 instead of −7, so 14 − 7 = 7, leaving out the 7 degrees the temperature rose just to reach zero.

• 25 Follows a key word (Hegarty, Mayer & Monk, 1995)

  Reveals a surface strategy Expected pull: moderate

  Reads “greater than” as adding on, so 4 + 3.5 × 4 = 18° at 11:00 a.m. and the rise is 25; the wording allows it, but the key means 3.5 times as great.

• 14 Stops one step early (King et al., 2004; Gierl et al., 2017)

  Reveals partial knowledge Expected pull: moderate

  Multiplies 3.5 × 4 = 14 and grids that 11:00 a.m. temperature itself, never subtracting the 5:00 a.m. reading of −7° to get the increase.

How the questions are analyzed, and the research behind it