Our base ten system is a huge milestone in the development of mathematical thinking. We are asking young ones to think about a group of ten as two things simultaneously: ten individual objects and a single group of ten. Ten and one at the same time! This big idea is called “unitizing”—thinking about (and counting) smaller, equal groups of items within a larger group.
Much like subitizing, unitizing and the overall structure of base ten is often “caught, not taught.” Our goal is for children to truly understand base ten ideas, rather than simply going through the motions. Young children will not understand how tens and ones work if we simply tell them; we need to show them. And this is where the learning trajectory comes back into play!
Early Number Systems
As you already know, base ten concepts are very complex and it can take a long time to build strong understanding. It took human civilizations thousands of years to arrive at the number system we have now! Take a look at the video below to learn more about the development of place value.
When they are ready to think more abstractly, children show their understanding of real life materials and scenarios with drawings. Much like using manipulatives, drawing helps children do and show their mathematical thinking.
Once students have a strong and stable understanding of how to organize quantities in our base ten system, they can represent numbers even more abstractly with symbols (numerals).
The video below focuses on the middle stage of the learning trajectory: young mathematicians’ pictorial representations of base ten concepts.
Looking at Student Work
In this video, you’ll get a little historical context to our base ten number system and look at the way that children’s thinking parallels this evolution of representation.
Questions for Reflection:
What do these drawings tell us about students’ developing understanding of base ten? (What ideas of base ten do students already understand that they can build on?)
What opportunities can you give your students to apply problem-solving and base ten concepts to real-world scenarios?
Foundations Are Essential
If your students are struggling with concepts in this section like unitizing and positional notation, return to Number Sense again and again to strengthen that foundation. As is the case with most early math concepts, strong number sense is a prerequisite.
Next, we’ll consider more closely the materials used in the concrete stage of the base ten learning trajectory. Continue on to Base Ten Manipulatives.
Additional Resources
The content of the videos above was adapted from an excellent book titled
“Young Mathematicians at Work: Constructing Number Sense, Addition, and Subtraction” by Fosnot & Dolk.