How can I help my child use abstract thinking in math?

Question: My 8-year-old can manage any addition and subtraction problem when he can use physical manipulatives to help him work through the math. But he’s lost when we try to move from the physical/tangible into anything abstract. Also, the idea of money, say one quarter being worth more than three nickels, just stumps him. Any advice on how to help him make this connection?

These difficulties aren’t uncommon for kids who learn and think differently, and there are some good ways to address them. The important thing to know upfront, though, is that it takes time. I don’t know of any quick fixes. But here’s a three-step process that can help math students work better in the abstract.

Step 1: Use three-dimensional objects.

You mentioned manipulatives. That’s a great starting point. Lots of schools use buttons, beads, and other objects that are easy to pick up and move around. That’s because hands-on learning helps students master math concepts (rather than just memorizing math facts).

Using Unifix cubes or Stern blocks can be particularly helpful. Stern Structural Arithmetic is a hands-on instructional program that addresses issues like your child’s very well. The blocks in Stern’s counting board set can help students understand what’s called “relative value.” Kids your son’s age don’t need to use this phrase, but they need to learn the concept it describes.

Here’s how these kinds of blocks can help your child understand relative value: The smallest block in the counting set is one unit long and is worth “1.” The next block in the series is two units long, and it’s worth “2.” This goes on up to the longest block, which is 10 units long and is worth “10.” Kids can also piece together different combinations of these blocks to see how different numbers add up to 10.

The blocks can help your child focus on the numbers themselves and their relative value to each other. These blocks can also help your child start to understand that, for math, the essential details in the sentence, “What is six buttons plus eight buttons?” is 6 + 8, not the buttons.

Step 2: Focus on using two-dimensional objects.

Once your child shows that he understands relative value using the three-dimensional blocks, it will be easier for him to work with two-dimensional objects that he draws using a pencil on paper. For example, he can draw dots or other shapes to use in the same way he’s used the blocks.

This is harder because he won’t be able to pick up the dots or shapes and move them around. The goal is for him to start making the change in quantities in his head. If this is difficult, he can erase or cross out marks to “move” them. This can be a helpful step as he’s moving from working with three-dimensional objects to working with two-dimensional objects.

Step 3: Shift the emphasis onto visualizing the objects.

Once your child shows that he understands the math process using two-dimensional “manipulatives,” he can move on to visualizing objects in his head. This can require lots of practice. You can help by encouraging him to talk about what he’s visualizing and how he’s changing the quantities based on the math problem.

For example, asking him to describe what he “sees” in his mind and how he “mentally moves” things around will help you make suggestions and offer corrections. You may also want to encourage him to use the blocks or things he draws on paper to check his work. This can help him develop a more concrete sense of the relationship between what he does in his mind and what he does with physical objects or drawings.

Moving through each of these steps will take time and effort, but it will lay an excellent foundation for abstract thinking in math. This will serve your son well as the material he learns in school becomes more complex over time.

Try relating blocks to coins.

You mentioned that your son has trouble working with quarters, nickels, and other coins. To help with this, it’s a good idea to work with a set of counting blocks like the one I described above. Once your child can use the blocks, then you can start to relate the blocks to the coins. Talk about how a penny is worth “1” just like the one-block is worth “1.” A dime is worth “10” just like the ten-block is worth “10.”

This can be tricky. But explain to your child that a pile of 10 pennies is worth as much as one dime. Talk about how we assign a number value to each type of coin so that we don’t have to lug around a big bag full of pennies every time we want to buy something! You can even tape the coins to the appropriate blocks so that your child can see the differences in relative value.