This post is part of the Little Brains, Big Concepts series. Check out the previous posts: Introduction, Number, Counting Strategies, Part-Whole Relationships, Understanding 10.
Take a moment and add the following numbers in your head: 266 + 146. Don’t worry about getting the right answer, think about how you do it.
Did you stack the numbers and use the standard algorithm to add the columns from right to left, carrying some ones along the way?
If so, you may have noticed that keeping track of the ones and the numbers in each column is a lot to hold in your brain. Interestingly, many second graders are intuitively able to do this computation using a process like this one:
This approach may appear more complicated than stacking the numbers, but it actually uses the same amount of steps. Not to mention there are less opportunities for mistakes. How do second graders perform such magic?
They harness the power of place value.
The Big Concept
In a written number, each position, or place, has value – this is where we get the term place value. We use a base ten system, which means that the value of each place is centered around ten: as you move to the left, the value of each position is ten times greater than the value of the position to its right.
Place value enables us to break numbers apart. (Yep, another time to shine for part-whole relationships!) For example, we can use place value to break 185 into parts: 100, 80, and 5.
Another important player in our number system is zero. Zero acts as a placeholder that enables us to differentiate between 32 and 302.
Experiences that Build Understanding
Grasping all of the above characteristics is a lot to ask a little brain, or a big brain for that matter! If your child struggles to understand these concepts, don’t worry! It takes a lot of time and practice for children to understand place value.
Beware the Place Value Chart
There is a 100% chance that you will see a place value chart on your child’s homework. It is important for your child to know the order of each position – but beware. Memorizing the place value chart does not mean your child has mastered place value.
For example, a child might be able to successfully complete the chart for 34 by putting a 3 in the tens column and a 4 in the ones column. However, identifying that the 3 is in the tens’ place is not the same as understanding that there are 3 tens and that the value of the three 30.
Use the following experiences to ensure that your child goes beyond memorizing place value.
Use Tools to Model
Using manipulatives, such as linking cubes or base-ten blocks, is a great way to help your child physically work with our number system. They’re also great for helping you see your child’s thinking.
Let’s look at an example. You ask your child to represent 34 with linking cubes. What does each representation reveal about your child’s understanding of place value and ten?
The left representation shows a deeper understanding of place value because it breaks 34 into tens and ones. The right representation means that your child hasn’t quite grasped the “ten as a whole” concept – and that’s ok! They just need some more practice physically building ten.
Multiple Representations
Children need a lot of experience using place value to make numbers in different ways. We saw in the above example that 34 can be represented in two ways: 3 tens, 4 ones and 34 ones. However, those are not the only ways to make 34:
Representing the same number in multiple ways will help your child see math as flexible. Not only is this important for number sense, it also proves that math is more than just getting the one, right answer.
Expanded Notation
If you have a second grader, you’ve probably seen expanded notation on your child’s homework. Expanded notation is a way to write numbers that emphasizes their place value. For example, 453 would look like this:
This a great way to break apart numbers, but again, beware! Children are clever, and once they find the pattern, they can easily complete the task without developing any actual understanding of place value.
If you see expanded notation on your child’s homework, ask questions that encourage them to think beyond memorizing the place value chart: How many tens are in 453? What is the value of the 4 in 453?
Take Action
Play How Many Ways with your child!
- Write down a two-digit number and give your child some linking cubes – enough so that they can represent the number in at least two ways at the same time. You can also use base-ten blocks if you have them, but I prefer linking cubes because children can literally make and break 10. (If your child is in first grade or younger, use numbers from 11-20.)
- Ask, How many ways can you make 45?
- Ask, How many tens and ones are in each representation?
- Extension: Can you write an addition sentence for this representation?
References
Feikes, David., Schwingendorf, Keith. and Gregg, Jeff. (2018) Children’s Mathematical Learning. Retrieved from this website.
Dougherty, et al. (2010). Developing Essential Understanding of Number and Numeration: Pre-K-Grade 2. National Council of Teachers of Mathematics.