Experienced parents know that what children say and what children mean aren’t always quite the same.

Child: “Yes, Mom, I wiped the table.”

Translation: “Yes, I took a dishcloth and flicked the biggest crumbs onto the floor.”

Child: “No, I didn’t hit my brother.”

Translation: “Well, I was mad that he knocked over my block castle….and I may have pushed him a little…but I’m sure he fell over on his own.”

Child: “Yes, I understood my math assignment.”

Translation: “Please let me go play Legos now.”

When it comes to math, kids don’t always want to do the hard work of understanding the concepts. After all, it’s much easier to just follow the examples in the book without thinking too much about them.

For example, most first-graders faced with a place-value worksheet can easily figure out the pattern: “I just write the first digit in the first blank and the second digit in the second blank!”

87 equals ____ tens and ____ ones

87 equals _8_ tens and _7_ones

But a child who’s just matching a pattern doesn’t understand place value—even if he gets all the problems correct. He won’t know what 2-digit numbers mean, and he’s going to have a lot of trouble with more complex topics, like adding and subtracting 2-digit numbers.

It’s fine for kids just to go off and do math worksheets on their own when they’re practicing a topic they already understand well. We do not need to sit at our kid’s elbow for every multiplication problem! But when your child is learning a new concept, you’ll be glad in the long run when you spend the time teaching the concept thoroughly right from the start.

As Easy as (Stage) 1, 2, 3

To help your child understand math well, all it takes is to think about teaching new concepts in three stages.

Stage 1: Hands-on objects
Stage 2: Pictures
Stage 3: Written symbols

(You may have seen this approach called the concrete, pictorial, and abstract progression. I prefer to phrase it this way because it makes the purpose of each stage clearer. But no matter what you call, it will help your children understand math better!)

You probably followed this sequence naturally when you introduced your preschooler to the world of numbers. I’m sure you didn’t start by sitting down your bright-eyed little one and showing her a written number 4.

Instead, you began by counting real objects like stuffed animals or crackers.

Then, you showed your children how to count pictures of objects.

Finally, once your child understood the idea of numbers and counting, you showed her written numbers and helped her understand how the written numbers matched the real objects and pictures.

We teach this way naturally with very young kids because we know how concrete their little brains are. But older children benefit from learning math this way, too. Here’s how to use these three stages to make your math teaching more powerful with kids of any age.

Stage 1: Hands-on Objects

First, use hands-on, real materials to introduce the topic. Give the child lots of opportunities to move and touch real objects as she solves oral problems. These materials can be math manipulatives that came with your curriculum, but they can also be simple items from around the house.

For ideas on which manipulatives to use with which topics, check out my step-by-step guide to creating your free minimalist math manipulative kit. You’ll also likely find lots of good ideas in your math curriculum teacher’s manual.

This hands-on stage can be as short as 5 minutes or as long as five days (or even more!) depending on the complexity of the math topic. Be patient, and be willing to spend a good bit of time in this first stage. Once you’ve laid a solid groundwork with real objects, working with pictures and written symbols will be much easier (and go much faster).

Stage 2: Pictures

Next, work with your child to draw pictures that represent the hands-on materials, and use the pictures to solve problems. We’re not talking Van Gogh here—very simple sketches are just excellent! Ask lots of questions to help your child see the connections between the hands-on materials and the pictures.

If your textbook uses visuals, look at them and discuss how they illustrate the concept. Pictures are a great stepping stone between hands-on manipulatives and written symbols because they start to put the math on the page.

Stage 3: Written Symbols

Finally, transition to written numbers and symbols. Show your child how the manipulatives and pictures match the written numbers. As your child works through written problems, have him tell how the mathematical symbols match the manipulatives and pictures that he’s been working with. Any time he gets stuck, have him draw a quick picture or go back to the hands-on materials to help him reason out what he’s doing.

Three-Stage Math Lessons in Action

Addition Facts

It can be tempting to try to get your children to “just memorize” the addition facts, but they’ll master the facts much faster when they learn the addition facts with hands-on materials. Use small blocks, Legos, or counters to help your children learn the addition facts in 3 stages.

Area of a Rectangle

It’s much easier for kids to learn and apply area formulas when they understand where the formulas come from. You can use sticky notes or small tiles to give your child hands-on experience with filling in the areas of rectangles. Then, have her draw pictures of rectangles. Last, show her how the length x width = area formula matches the rows and columns in his drawing.

Long Division

Older children benefit from three-step lessons, too, especially when it comes to complicated procedures like long division. Play money from games like Monopoly or Life works perfectly for helping your child both understand why long division works and how to do it.

Long division is too long to illustrate with just one picture. But never fear, I’ve made a long division video that will show you step by step how to teach long division with play money.

Teaching 3-step math lessons do take a little more planning and energy than just handing your child a worksheet. But you’ll save yourself a lot of time and frustration in the long run. As your kids understand math better and feel more confident in math, you’ll spend less time correcting mistakes and reteaching. Plus, your kids will have more of those beautiful lightbulb moments that make it all worth it!

Your kids may still ~~lie~~ give overly-optimistic answers when you ask whether they’ve done their chores. But when they say that they understand their math assignment, you’ll know that they mean what they say.

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Kate

Kate blogs at Kate's Homeschool Math Help, where she's on a mission to help parents raise kids who are capable and confident at math (and have some fun in the process, too!) A homeschooling mom of two, she is the author of Preschool Math at Home, Addition Facts That Stick, and Subtraction Facts that Stick. She also teaches courses for parents at The Well-Trained Mind Academy.