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Left-to-right addition algorithm

Left-to-right addition algorithm

Author: Christopher Danielson

To help the reader perform the steps of the left-to-right algorithm, and to suggest some critical questions to ask about understanding algorithms.

This packet consists of two videos demonstrating the left-to-right algorithm, including its relationship to the standard US addition algorithm and to the partial sums algorithm.

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There are many reasons to learn a new addition algorithm. Perhaps you are:

  • A teacher teaching an unfamiliar curriculum,
  • A parent of a child in an unfamiliar curriculum,
  • A student in one of my math courses,
  • A student who never quite understood why the standard algorithm works, or
  • A person who is simply interested in this sort of stuff.

This packet does not attempt to present the case that alternate algorithms are better than the standard algorithm. It is instead intended to demonstrate a particular algorithm-the left-to-right addition algorithm-in sufficient detail that the reader will be able to do it on his/her own, and to demonstrate important features of the algorithm and its relationship to the standard algorithm.

This packet is the second in a series on alternative algorithms. The reader might also be interested in the partial sums algorithm.

Unlike the partial sums algorithm, I am unaware of any school curriculum that teaches the left-to-right addition algorithm. Instead, I represent its purpose as being to help adults understand how abstract the standard algorithm is and how disconnected it is from the ways we think about numbers.

The algorithm

This video demonstrates the left-to-right addition algorithm and compares it to the standard algorithm.

Relating to the partial sums algorithm

This video relates the left-to-right algorithm to the partial sums algorithm. The left-to-right algorithm is a sort of shorthand for the partial sums algorithm.


So the left-to-right algorithm is similar to the standard algorithm in several important respects:

  1. We pay no attention to place value, thinking "eight" for an 8, regardless of what place it is in,
  2. When we get two-digit sums, we carry the second digit to the next place to the left (the difference is that in this algorithm, we carry to the sum; in the standard algorithm, we carry to the addends), and
  3. The algorithm is efficient; the amount of space and thinking required does not expand greatly with the number of digits.

It is the experience of most adults learning this algorithm that it feels confusing. We don't get why we are crossing out the number when we carry; we don't get why we have to start on the left, etc.

Reflecting on this reaction usually brings an understanding of the potential dangers of the more familiar standard algorithm, taught without meaning. Just as we feel uncertain and unclear about the left-to-right algorithm, so too do young children feel uncertain and unclear about the standard algorithm. Why do we write a little one above the next column when we carry? Why do we have to start on the left? etc. The same questions that children have about the standard algorithm are the ones we ask about the left-to-right algorithm.

The left-to-right algorithm is often described by adults as complex, involving too many rules to remember, and these rules seem arbitrary. Same for children learning the standard algorithm.

So if the left-to-right algorithm feels frustrating, it may have to do with not fully understanding the standard algorithm.

The second video in this packet suggested that one way to bring meaning to the left-to-right algorithm is to look for connections to the partial sums algorithm. Similarly, the partial sums algorithm is worth studying in more detail for understanding the standard algorithm better.