Does Multiplying Always Make Bigger?

The idea behind Bigger or Smaller?

Let's try a small experiment. Answer without touching a calculator: is the result of 12 × 3/4 bigger or smaller than 12? And what about 12 ÷ 1/2?

Many adults hesitate for a moment on this question. For children it is even harder. The first multiplication examples we learn at primary school always make things bigger: 3 × 4 = 12. With division, things always get smaller: 12 ÷ 3 = 4. Over the years, a silent rule settles in our minds: “Multiplying makes bigger, dividing makes smaller.” This rule works for whole numbers (except 1), but with fractions and decimals it misleads. 12 × 3/4 is 9, which is smaller than 12. And 12 ÷ 1/2 is 24: this time division has made the number bigger.

The power of hidden models

In the 1980s, Italian and Israeli researchers noticed something interesting. Students knew the rule for multiplying fractions quite well, but when deciding which operation to choose for a word problem, they trusted their intuition rather than the rule. According to the researchers, the first models learned at school (multiplication = repeated addition, division = sharing equally) take root so deeply that they keep quietly steering behaviour even after students have learned the formal rules (Fischbein et al., 1985). Similar findings were seen in fourth and fifth graders working with decimals (Graeber & Tirosh, 1990).

A more recent study asked middle school students, university students and pre-service teachers a simple question: “Without calculating, is the result of this operation bigger or smaller than the larger fraction?” Answers for addition and subtraction were fairly good. But for multiplication and division with fractions less than 1, middle school students often did no better than chance, or even worse. Even pre-service teachers struggled with these questions (Siegler & Lortie-Forgues, 2015). In other words, students can do the operation, but they don't know what the operation does.

Why is it so hard?

Researchers say that part of the difficulty of fraction arithmetic comes from the nature of the topic itself (Lortie-Forgues et al., 2015). Addition needs a common denominator; multiplication doesn't. A rule is right for one operation and wrong for another. Because the rules look alike, they get mixed up. Studies using computer models also show that most students' errors come from carrying the rule of one operation over to another (Braithwaite et al., 2017). When conceptual understanding is weak, all the student has left is memorised steps. And when those steps get mixed up, the student has no compass to check whether the result makes sense.

What does research recommend?

The recommendation is simple but powerful: ask for a prediction before calculating. The IES guide specifically stresses that students should estimate the result of fraction operations in advance and understand why the operations work the way they do (Siegler et al., 2010). Checking whether the multiplier is bigger or smaller than 1 is the most basic clue to the direction of the result. Multiplying by 3/4 means “taking 3 of 4 pieces”, so of course it makes things smaller. Dividing by 1/2 means asking “how many halves are there in it?”, and there are 24 halves in 12.

How does Bigger or Smaller? use this?

In the game, the student works in a number factory. A number arrives on the belt, for example 12. The machine's screen shows an operation: “× 3/4”. The student's job is not to calculate but to decide: when the number comes out of the machine, will it be bigger, smaller or the same? They press a single button.

The heart of the game is the see-through machine. The idea comes from classroom activities where equal-length paper strips are split into halves, thirds and quarters and the pieces are used to do operations, and from research on Steffe's fraction schemes: the student splits a whole into equal parts, then builds a new length by iterating one part (Steffe & Olive, 2010; McCloskey & Norton, 2009). In the game, the number is a box. The box goes into the glass machine and is cut into equal slices before our eyes. With × 3/4, the box is split into 4 slices and one slice is lifted out and taken away; the box gets shorter. With × 3/2, the box is split into 2 slices and a copy of one slice is added, glowing green; the box gets longer. A dashed line always shows the box's old size. The rule is not memorised; it is seen: if there are more pieces than you split it into, it gets bigger; if there are fewer, it gets smaller. Division works in the same machine too: “÷ 3/2” does the same job as “× 2/3”.

The game has two modes. In the See-through machine, the box is sliced in front of the student as soon as the question appears, and the student decides by seeing how many slices there are. In the Closed machine, the machine's lid is shut: the student sees only the operation and the incoming number, and the numbers are chosen very close to 1 (7/8, 9/8, 0.9, 1.1). The lid opens after the answer, and the same show plays as an explanation. In this way, we first show and then withdraw the support. The screen is deliberately kept simple: one machine in the middle and three buttons. Slice numbers and a recipe line such as “split into 4, take out 1 piece” are optional; they are turned on with “Details” in the menu.

The levels are ordered to untangle the misconception step by step: first multiplying a whole number by a fraction, then a fraction by a fraction, then division, and finally a shift that mixes in decimals too. After a wrong answer, the game doesn't just say “wrong”; it names the misconception: “Multiplying doesn't always make bigger!”

Tips for teachers and parents

• Before every fraction operation, pause for a second and ask: “Will the result get bigger or smaller?”
• Read “× 1/2” as “take half of it”, and “÷ 1/2” as “how many halves are there in it?”
• Use everyday examples: “If we make 3/4 of the recipe, do we need more flour or less?”
• Once the answer is calculated, compare it with the prediction. If they don't match, that's a great chance for a discussion.
• In class, have students cut paper strips of equal length: “Split the strip into 4 and lay 3 pieces side by side. Is it longer or shorter than the first strip?” Then have them split it into 3 and lay down 4 pieces.

References

• Braithwaite, D. W., Pyke, A. A., & Siegler, R. S. (2017). A computational model of fraction arithmetic. Psychological Review, 124(5), 603–625. https://doi.org/10.1037/rev0000072
• Fischbein, E., Deri, M., Nello, M. S., & Marino, M. S. (1985). The role of implicit models in solving verbal problems in multiplication and division. Journal for Research in Mathematics Education, 16(1), 3–17. https://doi.org/10.2307/748969
• Graeber, A. O., & Tirosh, D. (1990). Insights fourth and fifth graders bring to multiplication and division with decimals. Educational Studies in Mathematics, 21(6), 565–588. https://doi.org/10.1007/BF00315945
• McCloskey, A. V., & Norton, A. H. (2009). Using Steffe's advanced fraction schemes. Mathematics Teaching in the Middle School, 15(1), 44–50. https://doi.org/10.5951/MTMS.15.1.0044
• Lortie-Forgues, H., Tian, J., & Siegler, R. S. (2015). Why is learning fraction and decimal arithmetic so difficult? Developmental Review, 38, 201–221. https://doi.org/10.1016/j.dr.2015.07.008
• Ni, Y., & Zhou, Y.-D. (2005). Teaching and learning fraction and rational numbers: The origins and implications of whole number bias. Educational Psychologist, 40(1), 27–52. https://doi.org/10.1207/s15326985ep4001_3
• Siegler, R. S., Carpenter, T., Fennell, F., Geary, D., Lewis, J., Okamoto, Y., Thompson, L., & Wray, J. (2010). Developing effective fractions instruction for kindergarten through 8th grade: A practice guide (NCEE 2010-4039). Institute of Education Sciences, U.S. Department of Education.
• Siegler, R. S., & Lortie-Forgues, H. (2015). Conceptual knowledge of fraction arithmetic. Journal of Educational Psychology, 107(3), 909–918. https://doi.org/10.1037/edu0000025
• Steffe, L. P., & Olive, J. (2010). Children's fractional knowledge. Springer. https://doi.org/10.1007/978-1-4419-0591-8
• Van Dooren, W., Lehtinen, E., & Verschaffel, L. (2015). Unraveling the gap between natural and rational numbers. Learning and Instruction, 37, 1–4. https://doi.org/10.1016/j.learninstruc.2015.01.001
• Wilkins, J. L. M., & Norton, A. (2011). The splitting loope. Journal for Research in Mathematics Education, 42(4), 386–416. https://doi.org/10.5951/jresematheduc.42.4.0386