# A Family Math Session: Number Studies

Homeschooling Math has unique challenges like teaching several grades every day. What if you could teach math once a day as a family?

A unique characteristic Gattegno’s textbooks is that it is quite possible to do math as a family. Gattegno sets up his first couple of books as number studies. You pick a number and you study it.

Number studies let every child participate at their own level. Today, I will show you how I manage a family math session using number studies.

I follow a simple system of validate, challenge and extend with each child. I will show you what this looks like with each type of child.

## The Young Child

A young child has gone through the first three chapters of Gattegno much like their older siblings. They might not be comfortable with every operation and they still need the rods handy.

At this level in the number study, a young child uses the rods to build a number. The child can simply record their rods as simple addition.

### Validate the Young Child

Validate the child’s train by reading back the train to the child. The goal is just to connect with the child’s mathematical thinking. Ask the child if you read it correctly.

### Challenge the Young Child

Is the train equal to the number being study? How does the child know? Challenging the child to verify their answer is a crucial step in helping the child learn to communicate clearly and find confidence in their solutions.

A young child may gravitate to using mostly orange rods and one rod equal to the remaining amount to verify their answer. This is perfectly acceptable. The goal is getting the child to articulate their understanding.

### Extend the Young Child

If the child recorded their equation with simple addition, ask the child to substitute a number with subtraction, multiplication, fraction or division.

For example, let’s look at substituting the value of four using subtraction. Ask the student, “What two numbers have a difference of four?”

The child can use rods to find the answer. Once the child finds two numbers, replace the four in their equation with the new operation and numbers. For example, 5-1. This is very much like the substitution game which is another great family math activity.

Depending on the child, you may go through each operation that day or stick to just one operation. We seem to find it helpful to rotate through each operation throughout the week.

## The Early Abstract Child

The early abstract child is just learning to work numbers abstractly in their mind. They don’t always need the rods, and often reject the rods even when they need them.

### Validate the Early Abstract Child

The child will often write a few equations down for a number study. Read the child’s questions and ask them about what is similar and different about each equation. Just connect with their thinking at first.

### Challenge the Early Abstract Child

Do you see something not right about their equation? Simply say you aren’t sure about an answer. Ask them to explain their equation or to show you using the rods.

Connect with the child’s thinking before you tell them it’s wrong. Often, in trying to explain how they know their answer is right, they will often catch their own mistake. If they don’t catch their mistake, as they explain their answer, you will be better able to provide them insight into why it’s wrong or give them better language and notation to clarify their ideas.

If the child is obstinate about using the rods, you use the rods to build their equation. As you build, they will tell you what you are doing wrong, and this may help you see their thinking better. Often, I found that the child is using the wrong language and not the wrong thinking.

### Extend the Early Abstract Child

All children will tend to gravitate to one operation over the other. Strangely enough in our house, we seem to like fractions. Subtraction and division seem absent often from our initial equations. Just ask the child to play the substitution game with one of their equations. Constrain the child to use the operation they are avoiding.

## The Abstract Child with Small Numbers

The abstract child doesn’t really need to use rods at all. The child is very confident using all the operations and doesn’t really need to be encouraged to be creative.

### Validate the Abstract Child

Read their equations. Ask the child to tell you what is similar and different about each equation.

### Challenge the Abstract Child

There is no answer sheet. This may be the most challenging part to Number Studies. Your child will get very creative that sometimes they are more susceptible to mistakes. It is important to check their work.

You can have siblings trade worksheets to verify each other's work too. Encourage the younger children to use rods to help them solve the problems. You can work as a family to solve the older child's problems. The younger children feel quite accomplished by the end of it.

### Extend the Abstract Child

This child often deals with small numbers (numbers under 100) at first. Every number study can extend beyond 100, so you can extend the child with constraints that push them beyond their comfort zone.

For example, if the target number is 50, the child could think of two fractions of 120 that equal to 50. Or have them think of two numbers over 100 with a difference of 50. This abstract child should be encouraged to use rods if this is a difficult task.

## Beginning with Constraints

In the previous examples, we viewed each stage with a simple number study where the child begins with the freedom of creating any kind of equation equal to the target number. However, you could begin the number study with a specific constraint.

If I already noticed in the previous week that everyone is avoiding subtraction, I will begin the class by targeting that operation. For subtraction, I pick something simple like represent this number as a difference of two numbers.

This is a relatively simple constraint to begin things, and I found it is better to start simple and work from there. In the next constraint, I add in another operation like fractions or multiplication. So the child is now finding the difference between two fractions equal to the target number.

## Number Study Worksheets

I can’t always whip these constraints up on the whim. It is why I created the Number Study Worksheets. I only require eight problems to begin. I find it is best to do too little. We can always add when we see a child's enthusiasm, but it is very hard to take a way when a child enters the realm of overwhelmed.

The realm of overwhelmed equals quitting time in my book. It requires too much energy for the child at this point to continue. We want to avoid this realm as much as possible.

We go through a couple of these types of constraints in each family session. I don’t think every child needs to do the same constraint during the session. Sometimes I have one kid working with one constraint and another on a more difficult constraint. What we have in common is that we are all studying the same number.

Whether you use Gattegno or not, you might find number studies a fun addition to your math routine. It certainly shows children the more creative side of math. It is a great way for families to bond over math.

A parent also introduced to me a fun game using a simple deck of cards that is very similar to number studies. There are two versions of the game: Make 24 and Target Number and you can read the rules over HERE .

Next blog post on Family Math sessions, I will show you another way to setup family math sessions using Cuisenaire rod structures.