A father is testing his daughter at the kitchen table. Eight sevens? Fifty six, at once, without a pause. Good. Seven boxes with eight pencils in each, how many pencils? Silence, then a guess.
She knows the table, but she cannot use it. That is why a course on the multiplication table has roughly forty tasks and not the eight or so a parent expects.
Why one task per row is not enough
A child who answers "fifty six" to "eight times seven" at once may have no answer at all to "fifty six is eight times what". And she will very likely get stuck when the same numbers come as pencils in boxes.
These are three different skills, and the school timetable gives them one name. The first is also the least reliable, because it often depends on order: the child recites the row in her head from the beginning until she reaches the right place. Knowledge like that fails exactly where it is needed, in a problem that has no row to recite.
So a course has to ask about the same cell in more than one way. Otherwise it confirms a trick, not a skill.
Five ways to ask about one row
The first is the direct question, "eight times seven". This is what everybody means by the table, and usually it is the only form a course asks.
The second is the same question reversed: "fifty six is eight times what". Reciting the row does not help here. The answer shows whether the child knows the cell itself or only the way to it.
The third asks nothing. It shows how to work the answer out: eight sevens are eight fives and two more eights. A child who has forgotten a cell needs a way to get back to it, not an instruction to remember harder.
The fourth is about neighbours: a choice between fifty six, forty eight and sixty four. Those are exactly the cells children mix up, so this question tests whether the child can tell them apart, not whether she knows the fact.
The fifth has no table in it at all. It is the pencils in the boxes. A child who can only answer the direct form stops here. It is much better to find that out at the kitchen table than in an exam hall.
Five ways for each of eight rows makes forty.
Eight rows, not ten
The rows for one and for ten have nothing to memorise. Each is a rule, and a rule is taught once. That leaves the rows from two to nine, and half the cells in them repeat each other. Eight times seven and seven times eight are one cell and get one task, not two. Otherwise the course would be twice as long, and the child would spend half of it on questions she answered ten minutes ago. Nothing teaches a nine year old faster that this app wastes her time.
Where the wrong answers come from
In the children's mode the child picks the answer from four options. The three wrong ones are built on purpose, not picked at random.
A random set tests nothing. Put fifty six next to three, nine and a hundred and twenty, and the child picks it without multiplying anything, because the others do not even look like answers to this question. The task is passed, nothing is learned, and the report shows a tick that a parent will trust.
So we build the wrong options from mistakes children really make. A neighbour in the same row: forty eight, sixty four. A cell from the next row: forty nine, which is seven sevens. Swapped digits in the answer itself: sixty five.
Every wrong option corresponds to a real kind of mistake. A child who chose forty nine has slipped into the next row. A child who chose sixty four has gone one cell too far. The explanation afterwards says exactly that, instead of "wrong, try again". The story of one family shows what an evening of this looks like.
The check takes out what is not needed
Forty is the maximum, not a target. The course starts with a short check, and the rows the child already knows are left out of it completely. A girl who is confident in everything except the eights and nines gets a course of eight or ten tasks, not forty.
What the girl sees on her screen is written in words: confident with the two times table, worth going through the eight times table together. There is no mark and no comparison with the boy at the next desk. We wrote a whole article on why a percent of a child says nothing.
One more thing about the maximum: a topic she made a mistake on does not disappear as soon as she gets it right. It comes back a few days later. One correct answer is a good sign, but it is not yet a skill.
The things we said no to
A timer on the answer. A timer measures what kind of morning the child had, but it looks like a measure of cleverness. A child who counts slowly and correctly is not worse than one who blurts out a memorised number, and we refuse to build a screen that tells her otherwise.
Forty tasks in a row. That is no longer a course. It is a drill, and a drill is built to different rules: a portion a day until the skill needs no thought. A course divides the table into skills, and a drill makes each one firm. Putting one inside the other breaks it.
Stars for a finished row. A reward for every cell turns the table into a collection, and then the child protects the collection. She goes back to the rows she already knows, because stars are easiest to earn there, and the eight times table stays untouched for as long as it is hard.
What still bothers us
We have one support for everybody: splitting the sum through five. Eight sevens are eight fives and two more eights.
For plenty of children the way through ten is shorter: eight sevens are eight tens minus eight threes. It works better on the high rows and worse on the low ones. We cannot yet choose between the two for each child, so the screen shows one way and not the other. That is a real limitation, not a matter of taste.
The second thing is the pencils. "Seven boxes of eight" is still a school question that only pretends to be about real life. In real life, recognising the table looks like a supermarket receipt, or a recipe doubled because guests are coming. We have not got there yet, and until we do, the fifth kind of question is the weakest of the five.
