Where to position the carried integers in long multiplication workings?

Question

When I was taught long multiplication many moons ago it would be to place the larger number on the top row and the smaller number on the bottom. Consider 25 x 531: $$ \;\;5 \; 3 \; 1 \\ \times \; \; 2 \; 5 \\ ---- $$ I am used to then working from right to left and doing 5 x 531 before doing 20 x 531 on a subsequent row. I have seen it done both ways and am apathetic towards a recommendation on this.

I would very much like an opinion on the following, however. Apparently the 'Swedish system' teaches that when you have a number to carry you place it to the right and then cross it out as it gets used.

$$ \;\;\;\;\;\;\;5 \; 3 \; 1 \\ \;\;\;\;\times \; \; 2 \; 5 \\ \;\;\;\;------ \;\;\; _{/1}\\ \;\;\;2\;6\;5\;5 \\ 1 \;0 \; 6 \; 2\; 0 $$

I have two objections to this:

• firstly the "1" in this carry associated with 5 x 3 = 15 means that the child ends up writing this right to left (1 and then 5) or writes a 5 first and then a 1 on the right subconciously thinking "51". This is confusing.
• secondly the carried 1 is not aligned with any associated column addition. When the multiplications are bigger, with more carried items, if a value is missed and not crossed out it can cause errors that cannot be easily be visually untangled.

On the other hand I am inclined to include the carried integer in a column. I prefer the bottom, but have seen it at the top as well.

Mine would typically look like: $$ \;\;\;\;\;\;\;5 \; 3 \; 1 \\ \;\;\;\;\times \; \; 2 \; 5 \\ \;\;\;\;------ \;\;\; \\ \;\;\;2\;6\;5\;5 \\ \; _1 \\ 1 \;0 \; 6 \; 2\; 0\\ ---------\\ 1 \; 3\; 2 \; 7\; 5 \\ _{1} \;\;\;\;\;\\ $$

If this was just one teacher I probably wouldn't ask the question, but, if it is a "considered system" where all students are taught the same in all schools and classes, my question is: Am I right and should this be taught differently, or does it not matter?

Answer 1

If I saw your lines with 2655, 1 and 10620, I would assume the sum to be 13375, since 6+6+1 = 13. This is probably just because I have learnt a slightly different way of writing down the algorithm than you have, and this is probably not very dangerous.

It might get a bit busy to write down the carried numbers in the sum, but otherwise I do not see any particular issue with it.

In any case, the current approach in didactic literature I am familiar with is that we should not start by teaching the standard algorithm. Teaching it early leads to pupils seeing it as a magical formula. Some might get it, many are likely to make all kinds of mistakes because they try to learn a complicated algorithm by heart, rather than after understanding the matter. Further, having an algorithm can reduce the will to understand; just use the algorithm to get the correct answer!

The recommended way is to have pupils calculate things and find their own ways of doing it and writing it down, with discussions where these are shared (and made sense of) to help everyone develop fruitful ways. After the pupils have a solid understanding of the basics, the teacher can introduce some standard algorithm.

To the extent that a standard algorithm is taught, it should be with focus on understanding what it does. Hence it might be productive to have several and see why they do the same thing. In a modern multicultural school these might even come from the pupils, spontaneously, if they went to school in different countries or their parents do mathematics with them. In any case, the value of standard algorithms is going down fast, since "everyone" has a calculator easily accessible "at all times". This is not to say there is no value, just that the value is smaller than it was back in ye olde days.

Some literature that supports this:

HINNA, Kristin; RINVOLD, Reinert A.; GUSTAVSEN, Trond Stølen. QED 5-10: matematikk for grunnskolelærerutdanningen. Bind 1. Cappelen Damm akademisk, 2016. Chapter 1.4.

VAN DE WALLE, John A.; KARP, Karen S.; BAY-WILLIAMS, Jennifer M. Elementary and middle school mathematics: Teaching developmentally. Pearson. One Lake Street, Upper Saddle River, New Jersey 07458, 2022. Chapter 11.

SOLEM, Ida Heiberg; ALSETH, Bjørnar; NORDBERG, Gunnar. Tall og tanke 1: matematikkundervisning på 1. til 4. trinn. Gyldendal, 2018. Chapter 4.

Those have references to the research literature. These are just normal books in teacher education. Other literature in didactics of mathematics is likely to have the same ideas in it.