In short: A good times tables game turns the maths itself into the move, instead of putting it in front of the game as a hurdle. Research on learning games shows that this kind of “built-in” maths can work, but not automatically. That’s why, in the game we are working on, a block 3 squares tall and 4 wide is the fact 3 × 4. And there is no clock on the calculation: the block only starts to move once the answer is in.

A made-up but familiar scene: your child sits at the table with a stack of flashcards. The front says 7 × 8, the back says 56. They read, think, flip. Correct. Next card. After ten minutes the stack is done, and 7 × 8 is ticked off for today.

Flashcards aren’t a bad tool. They make a child retrieve the answer before seeing it, and they give instant feedback. But a card never shows why 7 × 8 is 56. The back shows a number, not a picture. That’s where the idea for our game begins.

What makes a learning game a good one?

Many maths apps follow a simple pattern: first a sum, then you get to play. The maths is the entry ticket, the game is the reward. Early work on educational games already distinguished two ways of linking game and content: in an extrinsic link, the game world has little to do with the skill being practised. In an intrinsic link, the skill itself is what you do in the game (Malone, 1981).

How much this can matter is shown by a study of a maths game for 7- to 11-year-olds. In one version the maths was built into the game mechanics, in another it was asked separately, and there was a control version. 58 children played one of the three versions for two hours. The children with the built-in maths learned most. When 16 children could choose freely, they played the built-in version seven times longer (Habgood & Ainsworth, 2011).

Chart: two ways of combining maths and play. Extrinsic: first calculate, then play, the problem is a hurdle in front of the game. Intrinsic: the problem is the move, calculating is playing. Below: in one study, children spent seven times longer playing the intrinsic version in free time (Habgood & Ainsworth, 2011).

Do learning games always work?

No, and that matters to us. A meta-analysis of 77 comparisons found that serious games did slightly better on average than conventional instruction, but were not more motivating. They worked best when combined with other instruction and played over several sessions (Wouters et al., 2013). A second large meta-analysis found a similarly moderate effect and stressed that game design makes a big difference (Clark, Tanner-Smith & Killingsworth, 2016).

And built-in maths is no guarantee. In a study with 64 students in a vocational track, it was the players of the version with separate maths who learned more and found it more motivating (Vandercruysse et al., 2017). The researchers had expected the opposite. For us, this means it isn’t enough to hide maths somewhere in a game. It has to stay visible and carry the game.

Why are the block’s dimensions the multiplication fact?

The key idea didn’t come from game design but from multiplication arrays: a multiplication fact is a rectangle. 3 × 4 is 3 rows of 4 squares. If that’s true, a game piece 3 squares tall and 4 wide doesn’t just show the fact, it is the fact. Three decisions follow from that:

  1. The block is the fact. Height = first factor, width = second. The product is the number of squares the block covers on the board.
  2. Turning is the turnaround fact. Rotate the block and 3 × 4 becomes 4 × 3. It fits differently, but covers the same 12 squares.
  3. Big facts are big blocks. 9 × 8 is visibly bigger than 2 × 3. It’s well documented that facts with larger numbers are harder (see our post on core facts). So big blocks should fall more slowly.

We know from research that children don’t read arrays on their own (Battista et al., 1998). A game in which you constantly pick up, rotate and fit rectangles is our attempt to practise that reading without it feeling like practice.

Chart: a flashcard compared with a block. The flashcard shows 7 × 8 on the front and 56 on the back. The block is 7 squares tall and 8 wide, covers 56 squares, and turns into 8 × 7.

Why does the block only fall after the answer?

In our first idea, a child had to answer before the block reached the bottom. We moved away from that. What’s decided now: a new block waits until the multiplication fact is solved. Only then does it fall, and the child steers and rotates it into place. The time pressure sits in the fitting, not in the calculating.

Whether time pressure harms children’s calculating is less clear than often claimed. A review found only a few studies and no clear links between time pressure, maths anxiety and strategy choice (Caviola et al., 2017). Precisely because it’s an open question, we didn’t want to risk a child starting to guess under pressure instead of deriving. We’ve written elsewhere about maths anxiety in children and why we measure speed without rushing. Anyone who wants speed should choose it deliberately. And there should be a way to play where nothing falls at all.

And if the answer is wrong?

This is where the findings from our recent posts come in. With neighbouring facts, children are often off by one rather than by a whole row, and they carry rules from addition over to multiplication (Squire, Davies & Bryant, 2004). So the wrong answer options shouldn’t be random numbers but typical mix-ups: for 4 × 6, say, 20 (that would be 4 × 5) or 10 (4 + 6 instead of 4 × 6). A child who picks one has made a confusion you can talk about.

A new, hard fact also shouldn’t arrive as a riddle, but together with its route from a core fact, such as 6 × 7 = 5 × 7 + 7. And after missed attempts, the game should show the answer with its route instead of leaving a child stuck. Why a mistake should be information, not a verdict, is the topic of why mistakes should not be red.

What you can do

  1. Flashcards with a picture. On the back, don’t just write 56. Draw the 7 × 8 rectangle on squared paper, with a line after the fifth row.
  2. A rectangle puzzle on paper. Draw a board 10 squares wide. Draw a fact card, cut out the matching rectangle and place it on the board. Turning is allowed. Fill a row completely and you may cross it out.
  3. Calculate first, then place. Play without a clock. The answer is said before placing, and afterwards your child counts the squares to check.
  4. Mistake detective. Offer four answers, for example for 6 × 7: 42, 36, 48, 13. “Which one is right? And how would someone get the wrong ones?”
  5. Check the games. For any learning app, ask: is the maths the move, or just the hurdle in front of it?

What we are working on

Under the working title Droptiply, we are building this game right now. The decisions described here are settled: the block is the fact, turning is the turnaround fact, there’s no clock on the calculation, wrong answers mirror real mix-ups, and new facts arrive with their route from a core fact. The game isn’t finished yet. When it is, we’ll show it here.

Frequently asked questions

Can a game really teach times tables? Games can help, but on average their effect is moderate (Wouters et al., 2013). They work best alongside teaching and conversations about the facts.

Isn’t this just Tetris with numbers? The movement is reminiscent of it. The difference: here the shape of the piece is the multiplication fact, and a piece only moves once the fact is solved.

Why no clock on the calculation? Because we want children to derive facts, not guess them. Speed comes with practice, and anyone who wants it should choose it themselves.

Sources

  • Battista, M. T., Clements, D. H., Arnoff, J., Battista, K. & Van Auken Borrow, C. (1998). Students’ spatial structuring of 2D arrays of squares. Journal for Research in Mathematics Education, 29(5), 503–532. https://doi.org/10.2307/749731
  • Caviola, S., Carey, E., Mammarella, I. C. & Szűcs, D. (2017). Stress, time pressure, strategy selection and math anxiety in mathematics: A review of the literature. Frontiers in Psychology, 8, 1488. https://doi.org/10.3389/fpsyg.2017.01488
  • Clark, D. B., Tanner-Smith, E. E. & Killingsworth, S. S. (2016). Digital games, design, and learning: A systematic review and meta-analysis. Review of Educational Research, 86(1), 79–122. https://doi.org/10.3102/0034654315582065
  • Habgood, M. P. J. & Ainsworth, S. E. (2011). Motivating children to learn effectively: Exploring the value of intrinsic integration in educational games. Journal of the Learning Sciences, 20(2), 169–206. https://doi.org/10.1080/10508406.2010.508029
  • Malone, T. W. (1981). Toward a theory of intrinsically motivating instruction. Cognitive Science, 5(4), 333–369. https://doi.org/10.1207/s15516709cog0504_2
  • Squire, S., Davies, C. & Bryant, P. (2004). Does the cue help? Children’s understanding of multiplicative concepts in different problem contexts. British Journal of Educational Psychology, 74(4), 515–532. https://doi.org/10.1348/0007099042376364
  • Vandercruysse, S., ter Vrugte, J., de Jong, T., Wouters, P., van Oostendorp, H., Verschaffel, L. & Elen, J. (2017). Content integration as a factor in math-game effectiveness. Educational Technology Research and Development, 65(5), 1345–1368. https://doi.org/10.1007/s11423-017-9530-5
  • Wouters, P., van Nimwegen, C., van Oostendorp, H. & van der Spek, E. D. (2013). A meta-analysis of the cognitive and motivational effects of serious games. Journal of Educational Psychology, 105(2), 249–265. https://doi.org/10.1037/a0031311

Lernwunder editorial team · Chaolan Lu

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