Where the Digits Get Cut Decides Whether the Sum Works
Last timeThe Space Before the Word
A long number is chopped into arbitrary groups of digits that do not line up between the two operands. Much of what is blamed on reasoning is this, and it is fixable.
How digits get cut
Nothing in the merge algorithm knows that digits are different from letters. A
run of digits is a run of bytes, pairs within it are counted, and the common
ones merge.
What is common? Years, because text is full of them, so 19 and 20 and 2024
became symbols early. Round numbers: 00, 000, 100, 500. Small numbers written
out in prices and counts. The result is a vocabulary with a scattering of
multi-digit symbols chosen entirely by how often those exact digit strings
appeared in prose, which has nothing whatever to do with arithmetic.
| step | the number | how it cuts | symbols | why | what happened |
|---|---|---|---|---|---|
| 1 | 2024 | 2024 | 1 | a year, enormously common in text, so it | The model handles this number well, as a unit, with no internal structure available to it. |
| 2 | 2025 | 202 and 5 | 2 | less common as a string at the time the | One digit different from the row above and cut completely differently. Nothing about the values is involved. |
| 3 | 1234567 | 123 and 45 and 67 | 3 | three common digit runs, found greedily | The grouping has no relation to thousands or any other place value. |
| 4 | 7654321 | 76 and 543 and 21 | 3 | different common runs, so a different gr | The same length as the row above and cut on different boundaries, which is the fact the next section turns on. |
Why the addition fails
Written addition works by columns. Units under units, tens under tens, a carry
moving leftwards. Every method anybody uses depends on knowing which digit of
one number sits above which digit of the other.
Now give the model 1234567 cut as three, two, two and 7654321 cut as two,
three, two. There is no symbol in the first that corresponds to any symbol in
the second. The model cannot align them, because the information needed to
align them, which digit is in which position, is inside symbols it cannot see
into. It has to reconstruct place value from the identity of opaque units, for
this particular pair of groupings, having possibly never seen this pair of
groupings before.
This also explains the shape of the failure people report. Short sums are fine,
because short numbers are often single symbols the model has effectively
memorised. Medium sums are erratic, because sometimes the groupings happen to
line up. Long sums fail almost completely, and they fail in a characteristic
way: the answer has about the right magnitude and the wrong digits, which is
exactly what you would expect from a system that knows roughly what the numbers
are but cannot align their columns.
The commonly offered explanation, that the model cannot reason, does not
survive the experiment in the figure. The same model, with digits presented
singly, does the same sums. Nothing about its reasoning changed.
The lesson stops here
4 more paragraphs to go
You have read the opening. The rest of the argument, the problems that check whether it landed, and the lines worth keeping at the end all come with a plan.
The first lesson of every course in the library reads the whole way through, free, so you can see exactly what the rest of them are.
See the planThe contentsThis is the reading half
Starting the course gives you your own copy of it. Every idea on every page has problems standing under it, marked with a reason rather than a tick, and any sentence you do not believe can be opened and argued with. None of that can happen on a page nobody owns.
The contents