Times As Many
Three times as many is not three more. Here is the difference.
For grown-ups
- How long: 10 to 15 minutes is plenty. Short and frequent beats long and occasional — the early-numeracy trials with the largest effects ran eight weeks or less.
- When they get one wrong: the page shows the working. Read it together rather than moving straight on. Feedback that explains is worth several times more than feedback that only marks right or wrong, and that gap is wider in maths than in any other subject.
- Not timed: thinking time is the point of a book. Let silences run.
- Print it too: doing the same problems on paper a few days later is worth more than doing twice as many today.
What this practises
Multiplicative comparison — reading "four times as many" as multiplication rather than addition.
Its own strand in Illustrative Mathematics grade 4, and a well-documented sticking point: "three times as many" gets read as "three more". Every page pairs the two comparisons so the difference is the thing being noticed, which is the variation-theory move — change one feature, hold the rest.
- The underlying idea
- Additive comparison asks how many more; multiplicative comparison asks how many times as many. Children routinely read the second as the first.
- Where this sits in Illustrative Mathematics
- 4th grade Unit 5: Multiplicative Comparison and Measurement
- Sources
- Institute of Education Sciences / What Works Clearinghouse (2021) · Illustrative Mathematics / Kendall Hunt (2024) all references →
Related measured skills: ALPACA · Elementary operations, ARF · Multiplication facts. What this means