So glad you stopped to join us for our book study of Teaching Student-Centered Mathematics PreK-2 by Van de Walle, Lovin, Karp, and Bay-Williams. To read previous posts, you can visit our book study archive!
Today we move into the portion of the book that provides teachers with numerous strategies and activities for helping students develop important mathematical concepts. The authors also provide valuable research and rationale. If you haven’t already, we encourage you to get yourself a copy of the book so you can benefit fully from its contents. As we move on we will be highlighting portions of the chapters–there’s A LOT that lies within the pages of this book. We begin with early number concepts and number sense.
Chapter 8: Developing Early Number Concepts and Number Sense
This is a phenomenal chapter—for everyone! Understanding how children develop early number concepts and number sense is essential for all that follows in mathematics!
“Although students may come to school with many ideas about number, it takes time and lots of experience for them to develop a full understanding that will enhance all of the number-related concepts they will encounter in future years.”
This chapter helps readers understand the stages in which children develop early number concepts and stress the importance of understanding the stages, where a student “falls” within the stages, and the importance of not forcing or skipping any stages if a child is to develop a strong foundation in number.
A learning trajectory for counting is presented on p.104. As the authors state, “Teachers who are aware of this learning trajectory for counting are better able to design instructional tasks that are purposefully targeted at moving children from one level to the next.” If we don’t understand a student’s needs in this area, the likelihood of moving him/her on too soon or overlooking an important stage in development increases. This in turn can impact future learning.
I love the suggestion the authors give for helping students understand the structure of numbers in the teens by using a vertical number line (Figure 8.1) What a great way to help students recognize patterns and how the structure of numbers change as they begin to write two-digit numbers (Standard for Mathematical Practice 7: Look for and make use of structure).
There are many games and activities for helping students understand counting. An outstanding feature of this chapter is the inclusion of formative assessment notes that provide teachers with specific things to look for when observing students playing games and doing activities. This helps determine a student’s understanding of number relationships.
Fill the Towers is a game shared by the authors that requires students to count objects and make comparisons. Click here to learn about this game in a previous post about using beans for counting!
Once students have developed through the stages of counting, it is crucial that they begin exploring the relationships between numbers.
Early Number Relationships: More, Less, and Same
To help students understand these concepts, the authors suggest giving students the opportunity to create sets and make comparisons between the sets. They also stress how we need to spend ample time helping students understand the concept of less, which proves to be more difficult than more. Some students may need to match up counters/objects to understand if there are more or less but eventually need to be able to select groups of objects that appear to have more, less, or the same. A dot card activity, Finding the Same Amount (Activity 8.8) is included.
Number Sense: Building Number Relationships
We hear A LOT about number sense! Over the years, I don’t know how many times I have heard teachers say that kids don’t know their basic facts and this is often wht is solely equated as number sense. Why is this so wide-spread?? I think the authors hit the nail on the head—“…teachers too often move directly from the beginning ideas of counting to addition and subtraction, leaving children with a very limited collection of ideas about number to bring to these topics. The result is often that children continue to count by ones to solve simple story problems, and have difficulty mastering basic facts.” So what do we do about it? The authors present four different relationships that we MUST help students develop for numbers 1 to 10 (and 10 – 20 as described later in the chapter).
Students who have numbers sense understand:
- Spacial patterns – can recognize how many without counting (subitizing)
- One and two more, one and two less – know which numbers are one and two less or more than a given number
- Anchoring number to 5 and 10 – know how numbers relate to 5 and 10 (often referred to landmarks or benchmarks to 5 and 10)
- Part-Part-Whole – see numbers as being made up of two or more parts
As a second grade teacher, I need to make sure each of my students has developed the above relationships before moving him/heron to addition and subtraction. The authors provide me with MUCH support in this area.
Spacial Patterns: Dot Plate Flash
This is a subitizing activity that I learned several years ago and was also shared in the chapter. It’s simple to prep, takes little time, and is powerful! You can use paper plates and dot stickers found with the garage sale tags in any office or discount store. The number of dots you place on a plate, and whether you use two colors of dots, depends on the needs of your student/s. The authors suggest holding up a plate for 1 to 3 seconds and asking how many dots are seen and what the pattern looked like. Want more information about why subitizing is important? Here’s a great article I read about 5 years ago that I found to be quite useful in answering this question—Subitizing: What is it? Why teach it?
Anchoring Numbers to 5 and 10: Number Medley
As a second grade teacher, this is a great introductory activity to do at the start of the year that helps me gain some valuable information about my students early on. Each student needs a five frame printable or ten frame printable and some counters. Call out a number from 1-10 and ask the students to represent the number on their ten frames. Call out another number and ask the students to change their ten frame to show that number. Continue in this way by calling out a series of numbers and asking students to change the amount on their ten frames. As the authors suggest, this is a perfect time to observe students and how they change their ten frames. Does a student take off all of the counters and start again or add counters on or take counters off? Does the student count or use the ten frame to anchor 5 and add on/take away? They also offer a variation of this activity where the teacher calls out a number and the students call out how many more need to be added (“plus”) or how many need to be taken off (“minus”).
Part-Whole Relationships: Number Sandwiches
I already have my dot cards printed out for this one! Dot cards are provided in the online toolkit for those who have a copy of the book, but they can also be found for free numerous places online. A set of dot card is used for each pair of students. Call out a number from 5 to 12. Each pair of students needs to find two dot cards totaling the number that was called, put the cards back-to-back with the dots facing out, and lay the pair on the work area. When they have found 10 pairs, students take turns trying to name the part of each whole (the amount on each card turned over on the table). I am excited to try this activity at the start of this year because I can, again, observe and take anecdotal records to formatively assess where my students are in understanding developing part-whole relationships as well as subitizing using the dot cards.
I think two of the best features of this chapter are the tips the authors give us for using story problems to help students develop number relationships and the quick, yet meaningful, formative assessments that are suggested for understanding our students’ number concepts.
The authors also go on to address pre-place value relationships (with 10) with activities such as Build the Number. If you don’t already have place value cards as used for this activity, feel free to download my place value arrows here. My students benefit greatly from activities and games using this tool (I have used them with fifth graders in past years as well, yet the ones I am sharing only go into the thousands.). Introducing numbers to 100 is also addressed with the use of various hundred chart activities. Finally, the authors wrap up by stressing the importance of using real-life contexts with calendars, estimation and measurement, and graphs to strengthen students’ understanding of number.
Ultimately, when we help students explore the relationships between numbers, they naturally begin to use these relationships to help them reason with more difficult problems. This will lead to flexibility when working with basic facts and beyond. This is something my second graders are able to do, yet it needs to be instilled in them through much exploration. An example of flexibility would be a student who cannot yet recall the sum of 8 + 9 but can use his/her current knowledge of number to create further relationships. He/she might choose to:
- use doubles, “I know 8 + 8. That’s 16. 9 is just one more than 8, so 8 + 9 is 17.”
- use tens, “Well, one more than 9 is ten, so I can move one from 8 to make a ten. Then 10 + 7 is 17.”
- create landmark/benchmark or “friendly” addends, “I can think 8 + 10, and that’s 18. 18 – 1 = 17.”
We will be exploring basic facts more in chapter 10, but I think the authors make clear how important developing number relationships FIRST is to future learning.
If you are interested in learning more about helping students build number sense relationships to 20, you might want to check out this past book review of Fluency Through Flexibility: How to Build Number Sense (Numbers 0-20) by Christina Tondevold. Christina’s book focuses on the four number sense relationships shared in chapter 8 as cited above. It’s a must have!
AND since Christina’s book is a MUST have, you have the chance to win a copy! You also have a chance to win my Missing Addend Fold-Its (0-10 and 11-20) and my ten frame clipart! For a chance to win, enter the giveaways below!
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Whoo–ee! That was a lot, but what an important topic for us all! Again, we recommend you get yourself a copy of Teaching Student-Centered Mathematics K-2, if you haven’t already.
We will see you back on Wednesday for Chapter 9 – Developing Meaning for the Operations.
Have a wonderful week—