The Three Mathematical Stages Your Child Needs in Order to Succeed in Math

If your older child is struggling with math, or your younger child just doesn't seem to understand a math concept, it may be because they skipped over a mathematical stage. There are three mathematical stages that follow the natural development of a child's thinking. If a stage is skipped over while learning a mathematical concept, your child will start to have problems.

The first mathematical stage is the manipulative stage, and as the name suggests, it involves having children use real objects to learn mathematical concepts. The manipulative stage is foundational to everything else a child does with math, so it is very important to teach your child math using real objects. If you skip this crucial step, your child may have problems with math down the road. Don't require memorization of math facts at this stage, just let your child acquire as much practice as possible with physical objects.

Within the manipulative stage, a child will observe and learn about:
o the number of things
o the size of things
o the shape of things
o the pattern of many things

Once your child has had enough practice and experience with real objects, he will be able to imagine objects mentally without seeing or touching them. This next mathematical stage is the mental image stage which is often used along with the manipulative stage until the age of twelve or so.

If you are teaching a math skill and your child says they don't understand it, the most likely problem is that he can't imagine it mentally. The way to solve this problem is by providing your child with lots of manipulative experience so that he can naturally make the switch from the manipulative stage to the mental image stage in a particular math skill.

The last stage, the abstract stage, is where children don't need a mental picture to understand a concept (i.e. they can understand the number eight without seeing eight things in their mind). This stage is more common in upper mathematics courses like algebra and geometry than in elementary mathematics.

Although children go from the manipulative stage to the mental stage to the abstract stage in that order, they can always go back to a previous stage. Where you can start seeing problems is when math workbooks combine several stages in one lesson (i.e. a child is shown pictures of objects (mental image stage) and then asked to use abstract symbols). You will achieve more and go faster if you stay within a child's stage strengths, especially for a child who struggles in math.

You can help your child transition between stages. For example have your child use manipulatives when they are learning to use abstract symbols like 5+5=10. Have him do this until it becomes easy to use the standard arithmetic notation.

If you work with your child's natural mathematical thinking stages, your child's frustration level will drop and math will seem much more doable. When learning a new mathematical concept, just remember to focus first on the manipulative stage, then the mental image stage, followed by the abstract stage. If you do this, your child will succeed at math!