The rule that was the ceiling of arithmetic for a thousand years
There is a good mathematical case against teaching cross-multiplication. There is a better cognitive case for it, and the history is on the second side.
Abraham Lincoln described his own schooling in the short autobiography he wrote for Jesse W. Fell:
Of course when I came of age I did not know much. Still somehow, I could read, write, and cipher to the Rule of Three; but that was all.
Abraham Lincoln, 20 December 1859
For most people in most of recorded history, the Rule of Three was the end of mathematics education: the last thing you learned before the syllabus stopped.
The rule is this: given three quantities, find the fourth that stands in the same proportion. Given that 3 apples cost 2.40, what do 7 cost? Arrange the four numbers in a square, draw a cross through it, and the answer follows from where the numbers sit.
Set down in the seventh century
Brahmagupta set out the rule in the Brāhmasphuṭasiddhānta in 628 CE, and the striking thing about his statement is what is missing from it: there is no equation, and no unknown.
He names three quantities — pramāṇa, phala and icchā, conventionally the argument, the fruit and the requisition — and gives a rule for combining them, together with the observation that two of the three are the same kind of thing and the third is not. That last condition is doing real work. It is the constraint that stops you dividing apples by money, and in the grid form it is enforced by geometry: like sits under like, and if you have arranged the square correctly the units take care of themselves.
The formulation travelled. Jens Høyrup has traced how recognisably the same phrasing appears centuries later in Arabic sources and then in the Italian abbaco books, the commercial arithmetic taught in the reckoning schools that followed Fibonacci's Liber abaci of 1202. It reached England as the "Golden Rule". It reached Kentucky, and Lincoln.
The case against it
Cross-multiplication is a procedure, and procedures can be executed without understanding. A student who has learned the grid can produce correct answers to proportion problems while holding no model whatsoever of what proportionality is. Worse, the grid hides the multiplicative structure that the rest of mathematics is built on: the fact that a 20% rise is multiplication by 1.2, that repeated rises compound, that this is an exponential in disguise. A student who thinks in multipliers is ready for compound interest, radioactive decay and differential equations. A student who thinks in 2×2 grids is ready for more 2×2 grids.
And it breaks. It works only when one quantity is a fixed multiple of another. Inverse proportion needs a different arrangement, and the classical texts supplied one: Brahmagupta gives the vyasta-trairāśika, the inverse rule of three, immediately after the direct rule. So the honest complaint is not that the method fails there, but that nothing in the grid tells the learner which of the two rules they are in. Power relations are the real wall: there is no arrangement of four numbers in a square that will give you Kepler's T² ∝ a³.
The case for it anyway
Consider what fails, and how, when someone half-remembers each method.
The failure mode of a formula is misremembering it — and a misremembered formula fails silently. It returns a confident, plausible, wrong number, and there is nothing in the answer to indicate that anything went astray. The failure mode of a picture is not having one, which fails loudly and sends you back to first principles.
Almost every percentage error people make in practice is a base error: not knowing which quantity the percentage is a percentage of. Ask someone what a jacket cost before a 20% discount brought it to 60, and a great many will take 20% off 60 and answer 48. The multiplier form does not protect against this, because in new = old × (1 + r) the base is implicit: it hides inside whatever old refers to, which is exactly the thing being got wrong.
In the grid, the 100 is a cell. It is on the page. "Percent of what" becomes something you can look at rather than something you must remember, and the answer comes out at 75.
What makes a representation worth teaching
Four questions seem to separate representations that survive from ones that do not.
Does it collapse several procedures into one? Find the part, find the whole, find the percentage, find the new value after a change: four textbook procedures, four things to keep straight. In the grid they are one act: circle the number you want. Cognitive load tracks the number of distinct procedures more than the difficulty of any single one.
Does it put the invisible quantity on screen? See above: the base.
Does position carry meaning? In the grid the diagonal divides and the other two multiply, and that rule is the same whichever corner you circle. Spatial and verbal encoding together survive far better than verbal alone.
Is the step people skip made visible? A 20% rise is not "20" in the grid; it is 120 against 100. That intermediate step — turning a change into a ratio — is the one people omit, so a good representation shows it rather than presenting a finished 120.
The Cross-Multiplication Grid is built around those four tests, and it includes twelve worked cases so the versatility is visible rather than asserted.
Why the grid is legitimate
None of this would matter if the method were merely a trick that happened to work. Underneath it is a statement about similar triangles.
Stand a metre stick in the ground and measure its shadow. Measure the shadow of a tree beside it. The sun is far enough away that its rays arrive effectively parallel, so the stick and its shadow form a triangle with exactly the same angles as the tree and its shadow. Equal angles mean proportional sides — Euclid, Elements VI.4 — and a height nobody can reach becomes three lengths anybody can pace out.
That is the whole method behind measuring a pyramid by its shadow, and it is the same square you fill in on the grid. The proportion is not a convention about where to write numbers. It is a theorem about shape.
Teaching a method for cognitive reasons does not license teaching it as universal, and trading confusion for false confidence is not a win. But a representation you can rebuild on the back of a receipt beats a formula you have to recall correctly, and that is why people who learned it this way are still using it decades later.