# Which product of single digits do children usually get wrong?

(I was inspired by the comments in this answer to ask this question.)

I have some multiplication table cards from Kumon that have a list of commonly mistaken multiplications: $$7\times 8, 4\times 8, 11\times 12, 7\times 9, 6\times 7, 12\times 8, 4\times 7, 6\times 8, 9\times 12, 8\times 9, 11\times 11$$, and $$6\times 9$$ (in this order).

I assume that this is based on data they obtained from the numerous children who have answered their worksheets.

Is there some other source (a study, perhaps) that lists the products of single digits that children usually get wrong?

• Anecdotally, 7x8 was the only I struggled with when I was first learning my tables. Right before it was taught to us, my teacher said out loud "and 7x8 is the only one I struggle with sometimes....sometimes I say it's 54 but it's really 56." To this day I have to stop and take a moment to think "is it 54 or 56???" – ruferd Mar 3 at 14:51
• My adult students would agree with that. 7*8. – Sue VanHattum Mar 3 at 15:34
• For $7 \times 8$, try doubling $7 \times 4$ as a way of remembering it. Or thanks to @Adam try $8 \times 8 - 8$. – J W Mar 3 at 16:14
• @ruferd Just remember it like this: $56 = 7 \times 8$ (i.e. 5,6,7,8) – Jann Poppinga Mar 4 at 13:14
• And if comparing $7 \times 8$ and $9 \times 6$, just remember that (for a fixed sum) the product will become larger if the numbers are closer together. That's often more useful to remember than a specific product, anyway. – Misha Lavrov Mar 4 at 20:15

https://www.theguardian.com/news/datablog/2013/may/31/times-tables-hardest-easiest-children

There are links to a dataset in the article. As far as I can tell, this isn't a formal study:

But some new data generated by pupils at Caddington Village School in Bedford sheds light on which multiplications are actually the hardest – and how kids do overall.

The data is generated by an app produced by an app developed by education tech firm Flurrish, and in total the 232 children who participated produced more than 60,000 answers. Here's how they did

So the data is of unknown quality, but the graph is both pretty and pretty believable. (Except that I'd probably label the graph below inaccuracy rather than accuracy.)

It's notable that the data is slightly asymmetric but I'm unsure if that is statistically significant. i.e. Do kids use commutativity?

• Also the row/column of 10 that drops accuracy for 10*11 and 10*12. They've memorized a rule, but seem slightly uncertain if the rule applies to numbers higher than 10. – ruferd Mar 3 at 18:26
• Is it common to teach multiplication tables up to 12 in english-speaking countries? Or is this maybe specific to the UK (due to the historical 12 pence in a shilling) and the US (due to the still current 12 inches to a foot)? In the French system we only learn them up to 10. Or at least we used to, though I doubt this has changed. – jcaron Mar 4 at 11:24
• What surprises me about this chart is the asymmetry. 7 × 6 is wrong considerably more often than 6 × 7. Likewise for 12 × 11 vs. 11 × 12, 12 × 7 vs 7 × 12, etc. Not sure if this is just noise in the data or kids confused about commutativity? – Darrel Hoffman Mar 4 at 14:00
• @DarrelHoffman: Looks mostly like noise to me. I played with the raw data a bit, and none of the results seem to be more than 2 standard deviations away from what would be expected under the null hypothesis of commutativity. Furthermore, the biggest deviations are for "easy" pairs like 6×3 vs. 3×6 (74% vs. 82% correct, 1.96 σ), 12×1 vs. 1×12 (90% vs. 95% correct, 1.66 σ) and 10×3 vs. 3×10 (88% vs. 92% correct, 1.56 σ). For 7×6 vs. 6×7 the deviation is less than 0.6 σ, easily explained by chance. – Ilmari Karonen Mar 4 at 17:55
• The asymmetry is significant and expected. If you had presented a symmetric plot I would have downvoted because I know it's asymmetric. – Joshua Mar 4 at 19:11