Place value is the quiet giant of elementary math. When a child finally understands it, many apparently-separate topics collapse into one. Multi-digit addition, subtraction with regrouping, the multiplication algorithm, decimals, and even scientific notation are all the same idea wearing different clothes.
Place value is the notion that the *position* of a digit changes what it's worth. The 2 in 23 is worth twenty, not two, because it sits in the tens position. A child who truly understands this can tell you that 234 is "two hundreds plus three tens plus four ones" *and also* "two hundred plus thirty-four" *and also* "twenty-three tens plus four". Those three decompositions are the same number, and each one is useful for different problems.
The single most useful physical tool for teaching place value is base-ten blocks — ones cubes, tens rods, and hundreds flats. A child who has physically built 234 out of blocks and then physically *regrouped* ten ones into a single ten rod has seen the mathematical operation that 'carrying' in an addition algorithm represents. The algorithm is no longer a mysterious ritual; it is a written record of something they have done with their hands.
The Place Value Parade game in MathVenture uses a digital version of the same idea. The child sees tens rods and ones cubes on screen, and every regrouping event is animated — ten ones visibly merge into a single rod. Once a child has seen that happen ten times, the connection between the picture and the algorithm becomes obvious.
A warning sign that place value hasn't yet clicked is if the child treats multi-digit subtraction as a game of "borrow one". They can execute the borrowing, but they can't tell you what the 'one' is or where it went. If you see that, slow down. Go back to the blocks. The algorithm will wait, and it will be much stronger when you return to it.
A practical at-home check for whether place value has clicked is to ask the child to count out 27 objects from a pile, then to wrap them into bundles of ten. They should produce two bundles of ten and a loose group of seven. If they can do this without prompting, they have the concept. If they need to be told what to do, they need more time with the concept itself, not more two-digit arithmetic worksheets.
Once tens-and-ones is solid, hundreds usually follow within a few weeks. The pattern is the same: ten tens make a hundred, ten hundreds make a thousand, ten thousands make a ten-thousand. The number system is recursive in a way that almost every child finds satisfying once they see it. A common moment of delight is when a child realises that a million is just "one thousand thousands" and not some new mysterious quantity. The system has the same shape all the way up.
Place value also unlocks mental arithmetic that older children sometimes miss. A child who genuinely understands tens-and-ones can compute 47 + 26 in their head by adding the tens (40 + 20 = 60), adding the ones (7 + 6 = 13), and combining (60 + 13 = 73). A child who has only practised the column algorithm has to imagine writing the numbers down and carrying — slower, more error-prone, and eventually a wall. Mental arithmetic is the place-value pay-off; if it isn't there yet, the place-value foundation isn't either.
A useful follow-up game once tens-and-ones is solid: "how many ways can we make 47?". Bundles of ten and loose ones, broken open and re-bundled, give the child many concrete answers (4 tens and 7 ones, 3 tens and 17 ones, 2 tens and 27 ones, and so on). The play looks idle. The cognition is doing precisely what column subtraction with regrouping will require a year later. Time spent on this is among the highest-yield math investments any parent can make in the early grades.