← Atlas of Difficulties
S-08
The child doesn't know number bonds: doesn't see that 7 is 3 + 4
Type: operational
What the research says
The operation of classification develops the child's ability to split a given quantity into different groups. Without mastering classification and seriation at the level of hands-on actions with objects, it is impossible to move on to decomposing a number.
Lalaeva R.I. Dyscalculia in Children, pp. 11–12 (2005)
Among the symptoms of dyscalculia are insufficient knowledge of number bonds and difficulty learning the rule by which numbers are formed. Predominantly concrete thinking gets in the way of moving on to generalized operations with number.
Lalaeva R.I. Dyscalculia in Children, p. 23 (2005)
The symptoms of dyscalculia include: insufficient knowledge of number bonds, difficulty learning the rules by which numbers are formed, and a poorly formed sense of the quantitative relationships between numbers. Arithmetic operations require the ability to keep the end goal in mind, analyze the numerical data at the same time, and put together a step-by-step plan of action.
Baryaeva L.B., Kondratyeva S.Yu. Dyscalculia in Children, p. 26 (2013)
“Part — part — whole” is a key concept in mathematical development: the child understands that the number 8 is at the same time made up of 5 and 3, of 6 and 2, of 7 and 1, and so on. This isn't rote memorization but understanding number within a system of relationships. Structured visual materials (a twenty frame, an abacus) help make these relationships automatic.
Wittmann, 2011; Häsel-Weide, Nührenbörger, Moser Opitz & Wittich, 2014, cited in: Austrian Federal Ministry of Education. A school-based approach to difficulties with arithmetic., p. 14 (2023)
A linear arrangement of dots is not very effective for understanding the “part — part — whole” principle. Structured arrangements (in groups of 2–3) make it easier to perceive the subsets of a number.
Austrian Federal Ministry of Education. A school-based approach to difficulties with arithmetic., p. 14 (2023)
Exercises at home — free
Split it another way
Builds: Number bonds, the operation of classification
You need: 8 identical objects: counters or coins
- Put all 8 objects in front of the child.
- Ask the child to divide them into two groups — any way they like.
- Write down the result: for example, “4 and 4.”
- Ask the child to divide them a different way. Write it down again.
- Keep going until you find all the options: 1+7, 2+6, 3+5, 4+4.
- Lay all the written results out side by side. Ask: “How many were there in total each time? Why is it always 8?”
Lalaeva R.I. Dyscalculia in Children, 2005, ch. 1
Sort into groups — the foundation of number bonds
Builds: Classification is a prerequisite for understanding that a number can be broken into parts
You need: any toys or household objects — 10–12 of them
- Tip all the objects out into a pile.
- Ask: “How could we sort these into groups?” — let the child suggest a way.
- Sort them together. Ask: “How else could we do it?”
- Find 2–3 different criteria: by color, by size, by material.
- Ask: “How many groups did we get? How many are in each?”
- Goal: the child suggests a new feature on their own, without a hint.
Lalaeva R.I. Dyscalculia in Children, 2005, ch. 4
Grouping by shape
Builds: Classifying by a feature is the same skill as breaking a number into parts
You need: household objects, 4 sheets of paper with shapes drawn on them: a circle, a square, a rectangle, a triangle
- Lay the 4 sheets out on the table — one shape on each.
- Pick up an object: a plate, a book, a coin, a piece of cheese.
- Ask: “Which shape does it look like? Put it next to that one.”
- Go through them together: the plate is a circle, the book is a rectangle.
- Harder: one object fits several shapes. “Is the box a square or a rectangle? Why?”
- Goal: the child finds a basis for grouping on their own.
Lalaeva R.I. Dyscalculia in Children, 2005, ch. 4
Sort by color
Builds: Splitting a number into parts is the foundation for understanding number bonds
You need: Cuisenaire rods or colored blocks (2 colors), Explain Math to Me number frames (linear)
- Give the child 5 objects in two colors (for example, 3 red and 2 blue).
- Ask the child to sort them by color into two number frames.
- Ask: “How many on the left? How many on the right? How many altogether?”
- Write it down together: 3 + 2 = 5.
- Change how many are in each group and repeat: 4+1, 1+4, 2+3.
Lalaeva R.I. Dyscalculia in Children, 2005, pp. 163–164
How many ways?
Builds: A number is made up of different pairs — the foundation of flexible calculation
You need: 5 objects of the same color (counters, buttons), Explain Math to Me number frames, paper for writing
- Give the child 5 objects of the same color. Ask the child to split them into two piles any way they like.
- Say it out loud: “5 is 4 and 1.” Write it down: 5 = 4+1.
- Put the objects back together and split them differently. Write down the next option.
- Find all the options: 4+1, 3+2, 2+3, 1+4. Record each one in a number frame.
- Compare what you wrote: “It's the same number 5, but you can split it in different ways!”
Lalaeva R.I. Dyscalculia in Children, 2005, p. 164
Part circles
Builds: Moving from objects to a diagram: a number as a whole made up of parts
You need: a sheet of paper with circles divided into sectors (2–4 sections), a pencil
- Draw a large circle divided into 2–4 sectors.
- Ask the child to draw small circles in the sectors so that there are 5 in total.
- Below the circle, write the number sentence: for example, 3+2=5.
- Suggest three ways to split it: 3+2, 4+1, 2+2+1.
- Explain Math to Me number frames can be used as support: first fill in the frame, then transfer it to the circle.
Lalaeva R.I. Dyscalculia in Children, 2005, p. 164
Match the same amounts
Builds: Number as an invariant: 4 pears and 4 stars are the same number
You need: picture cards (different groups of 3–5 objects)
- Lay out 6–8 cards with pictures of different objects.
- Ask the child to draw lines connecting pairs with the same number of objects — first by eye, without counting.
- Then check by counting.
- Harder: add “traps” — cards with similar but different quantities.
Lalaeva R.I. Dyscalculia in Children, 2005, pp. 164–165
Digit and picture
Builds: The link between digit and quantity in different directions
You need: picture cards, digit cards 1–5
- Task A: put the right digit under the picture — find “4” for a group of 4 pears.
- Task B: find the picture for a digit — we see “5” and look for the matching card.
- Task C: connect with a line — digits and pictures in different columns.
- Task D: fix the mistake — digits are written under the pictures, and some of them are wrong.
- Work in all four directions — each one uses a different link.
Lalaeva R.I. Dyscalculia in Children, 2005, pp. 165–167
Write the number sentence under the picture
Builds: Moving from actions with objects to mathematical notation
You need: cards with two groups of objects, paper, a pencil, Explain Math to Me number frames for checking
- Show a picture with two groups of objects in one circle (for example, 4 fir trees and 1 fir tree).
- The child names the number in each part.
- Writes the number sentence: 4+1=5.
- Next step: the child picks a picture to match a ready-made number sentence.
- Final step: sees only the picture — writes the number sentence independently.
Lalaeva R.I. Dyscalculia in Children, 2005, p. 167
Number bonds: “Two paths”
Builds: Splitting a number into two addends
You need: dot cards from 1 to 10, digit cards, 2 sheets of paper as “paths”
- Put the card with the digit 5 on the table.
- Task: “Find two dot cards that make 5 together.”
- Show an example: a card with 2 dots and a card with 3 dots.
- Ask the child to find other options: 1+4, 5+0.
- Put each pair on the “paths” — two sheets side by side.
- Ask: “How many options did we find? Are there more?”
- Harder: the child picks a number, and the adult looks for pairs.
Baryaeva L.B., Kondratyeva S.Yu. Dyscalculia in Children, 2013, pp. 84–85
One more, one less
Builds: Understanding “number neighbors” — the link between order and quantity
You need: digit cards 1–10, 2 cardboard arrows — a big one and a small one
- Arrange the digits in a circle on the table.
- Point to a digit with the small arrow and give a clue: “My number is one more than 4.”
- The child points to the answer with the big arrow and explains.
- Next riddle: “My number is one less than 5.”
- Once the child has got the hang of it, switch roles.
- Harder: “two more,” “two less.”
Baryaeva L.B., Kondratyeva S.Yu. Dyscalculia in Children, 2013, p. 85
Ducks on the lake
Builds: Addition and subtraction with visual materials
You need: 10 identical small objects: buttons, pebbles, figurines, 2 sheets of paper — the “lake” and the “shore”
- Put 3 objects on the “lake”: “Three ducks are swimming on the lake.”
- Add one more: “Another one flew in. How many are there now?”
- The child counts and answers. Write it down together: 3+1=4.
- Move one to the “shore”: “One climbed out onto the shore. How many are left?”
- The child counts and answers. Write it down: 4−1=3.
- Repeat with other numbers: 5+2, 7−3.
- Harder: you tell the problem out loud — the child lays out the objects and solves it on their own.
Baryaeva L.B., Kondratyeva S.Yu. Dyscalculia in Children, 2013, p. 87
Number house
Builds: Number bonds, understanding structure
You need: a sheet of paper with a “house” drawn on it: the roof is the number, the two windows are the addends; a pencil
- Draw a house: the number 7 on the roof, 3 in one window, the other one empty.
- Explain: “The number 7 lives in this house. There are 3 in one room. How many are in the other?”
- The child thinks and writes in: 4.
- Check together: 3+4=7?
- Draw the next house with a different number.
- Try houses for 5, 6, 8, 9, 10.
- Harder: both windows are empty — the child comes up with an option on their own.
Baryaeva L.B., Kondratyeva S.Yu. Dyscalculia in Children, 2013, p. 90
Without help
- Can't add without counting one by one: doesn't know that 7+3=10 because 7=10−3
- The addition table doesn't stick — every problem is solved from scratch
- Bridging through ten is impossible without knowing number bonds
What to work with