← Atlas of Difficulties
S-05
The child thinks the quantity changes when objects are rearranged
Type: operational
What the research says
When comparing quantities, a three-year-old relies on perceiving the size and arrangement of objects: the child thinks there are more fingers when they are spread apart and fewer when they are held together. The child can't look past perceptual features and pick out quantity as a common property of the group.
Lalaeva R.I. Dyscalculia in Children, p. 12 (2005)
A formed understanding of conservation of number is a necessary condition for forming the concept of number. Grasping conservation of number is a prerequisite for moving from visual-action thinking to visual-figurative and abstract thinking.
Baryaeva L.B., Kondratyeva S.Yu. Dyscalculia in Children, p. 32 (2013)
The place-value system of writing numbers contains four traps: (1) the same digit has different values in different positions; (2) the digit 0 is not nothing but a placeholder; (3) in two-digit numbers, German names the ones first and then the tens (inversion: “one-and-twenty”), whereas the Arabic notation is the other way around; (4) the words “eleven” and “twelve” contain no explicit reference to ten. In languages without inversion (English), “number reversals” occur much less often.
Austrian Federal Ministry of Education. A school-based approach to difficulties with arithmetic., pp. 9–10 (2023)
Exercises at home — free
The same pile
Builds: Conservation of number
You need: 5–7 identical objects: coins, buttons or counters
- Lay 5 coins out in a row. Ask: “How many coins are there?”
- Gather the coins into a pile without taking any away. Ask: “And how many now?”
- If the child says “fewer”, don't correct them straight away. Suggest counting again together.
- Repeat 3–4 times with different arrangements: in a line, in a circle, close together, far apart.
- The goal is the moment when the child says on their own: “It's still five — you didn't take any away.”
Lalaeva R.I. Dyscalculia in Children, 2005, ch. 1
More water or less?
Builds: The amount doesn't depend on the shape of the container
You need: two identical glasses, a narrow jar, a wide jar
- Pour equal amounts of water into two identical glasses. Check together: there's the same amount of water in each.
- Pour one glass into the narrow jar and the other into the wide one.
- Ask: “Which jar has more water?”
- After the child answers, pour the water back into the glasses — show that nothing has changed.
- Repeat 2–3 times with different containers. Put it into words together: “There's the same amount of water, the shape is just different.”
Lalaeva R.I. Dyscalculia in Children, 2005, p. 163 (after M. Montessori)
The same — but different
Builds: Quantity doesn't depend on how the objects are arranged
You need: 10 identical objects in two colors (blocks, buttons, counters)
- Take 5 red and 5 green objects.
- Put the red ones close together in a pile and spread the green ones far apart.
- Ask: “Which are there more of — red or green?”
- Count them together. Check: they're equal.
- Gather the red ones together and ask again: “Has the number changed?” Say it out loud: “No — we didn't add or take away anything.”
Lalaeva R.I. Dyscalculia in Children, 2005, p. 163
Which are there more of?
Builds: Quantity isn't determined by the size of the objects
You need: large and small objects of two kinds (toys, cut-out figures)
- Set out two groups: a few large objects (for example, 3 big pears) and many small ones (for example, 5 small apples).
- Ask, without counting: “Which are there more of?”
- After the child answers, count each group together.
- Compare the result with the first answer. Discuss: why did it seem different?
- Repeat with different objects: big/small, far apart/close together — check by counting every time.
Lalaeva R.I. Dyscalculia in Children, 2005, p. 163
Worksheets — print and practice
Opposite features
↓ Download printable PDF (in Russian)Without help
- Can't move on to number as an abstraction — counts everything again every time
- Goes by the size and arrangement of objects instead of their number
- Doesn't understand that 5 apples and 5 blocks are the same number