Use the links to step through the entire series of posts:
Chapter 1, Understanding Counting
Chapter 2, Understanding Number Relationships
Chapter 3, Understanding Addition and Subtraction: Parts of Numbers
Chapter 4, Understanding Place Value: Tens and Ones
Chapter 5, Understanding Place Value: Numbers as Hundreds, Tens, and Ones
Chapter 6, Understanding Multiplication and Division
Which brings us to Chapter 4, Understanding Place Value: Tens and Ones. Richardson packs a lot of information into this chapter. So much so, in fact, that it’s divided into two sections: (1) Learning the Structure of Two-Digit Numbers and (2) Learning to Add and Subtract Two-Digit Numbers.
I absolutely love the way this chapter emphasizes over and over the connection between place value and the operations of addition and subtraction. This is such a common breakdown in mathematics instruction in many classrooms. Teachers teach multi-digit addition and subtraction with no reference to place value, and they don’t understand how important it is for students to understand place value prior to attempting multi-digit operations. After reading this chapter, it should be crystal clear how integral place value is to truly understanding the process behind multi-digit addition and subtraction.
As with other skills, Richardson reminds us that students can appear to know more than they do. Just because a student can identify the digit in the tens place of a number, it doesn’t mean they understand what that digit means. To demystify the concept of place value, students need lots of opportunities to group objects into groups of tens and count those groups. Translation? They need lots of concrete practice composing and decomposing numbers in different ways.
As I read the critical phases related to understanding the structure of tens and ones, I noticed how well aligned the 1st-grade common core standards are with the phases outlined by Richardson. It’s all in there: understanding that a ten is a bundle of ten ones, understanding the structure of the numbers from 11-19, counting tens, mentally finding ten more and ten less, and subtracting multiples of 10. This chapter is an excellent resource for teachers trying to understand and implement the CCSM.
We are reminded that knowing the parts of numbers to 10 supports an understanding of place value. Students who still don’t know parts of numbers should continue working on that skill as they move forward with place value.
Up next is adding and subtracting two-digit numbers. This section makes my heart sing! I love the use of ten-frames to show how numbers can be “reorganized.” Isn’t that a much friendlier term than “regrouping.”? The pictures and captions Richardson uses make the concept very clear. But what’s also clear is that teachers need to be working in small groups or individually with kiddos to truly assess their understanding of the phases. Take, for example, the activity on pages 98-99. Unless the teacher interacts with the student while doing that activity, there’s no way to determine how the child arrived at the answer of 13. And it’s how the child did it that indicates which phase they are in. Moving to larger numbers, the examples on pages 104-110 show how vital it is that children have the opportunity to discuss their strategies.
As the chapter wraps up, there’s a final reminder that models help students make sense of the math they are doing.