How can parents help their children understand abstract mathematical concepts?

Mathematics is, at its core, an abstraction of the physical world. When a child writes 5 + 3 = 8 without asking "five what?", they've already abstracted. But that abstraction begins with concrete experiences — five fingers, five pencils, five people — before they internalize the idea that no matter the object, 5 + 3 = 8.

The same process applies when learning new concepts, such as fractions. Take the example 1/2 + 1/3 = 5/6.

Make It Concrete

Props are powerful. My favorite is a set of go pieces: plentiful, uniform, and easy to group by color.

• Fraction addition example: Pick 6 same-colored pieces and call that one set. Ask: "What's half of it? What's a third of it? Then what's half plus a third?"
• Why 6 pieces? Because 6 is the smallest number divisible by both 2 and 3. That's the physical embodiment of finding a common denominator — long before your child hears the term. If you'd started with 4 pieces, thirds wouldn't divide evenly.

Push the Idea Further

• Vary the set size. Try 1/3 + 1/4 with 12 pieces. Let your child discover that 6 pieces won't work here, and that they need a number divisible by both 3 and 4. That's them uncovering the concept of "least common multiple" through exploration.
• Try subtraction. For 5/6 − 1/3, build 5 out of 6 pieces, then physically remove 2 (a third of 6). Count what remains.

Transition to Abstraction

Once your child is comfortable getting answers using the props, take it away. Ask: "If I didn't have any pieces, how would you figure out 1/4 + 1/6?" That's when they begin narrating the abstraction themselves — "I'd need a number both 4 and 6 divide into… 12… then 3 twelfths plus 2 twelfths…"

It may take several iterations for your child to build intuition around a new abstract concept, but with practice they will get there, just as they once mastered 5 + 3.