Hello! So glad to have you here for our discussion of chapter 11 of Teaching Student-Centered Mathematics PreK-2 by Van de Walle, Lovin, Karp, and Bay-Williams. To read past posts, you can visit our book study archive.
Chapter 11 – Developing Whole-Number Place-Value Concepts
This year I will be teaching my 15th year of 2nd grade. When I began teaching, and the few years following, I did not like teaching math. First off, I was never a “math person” and felt it was one of my weak areas. Second, we were using Saxon Math, and I felt it did not dig deep enough with the concepts and the spiraling was too fast for my students. We worked on place value, but I did not teach that conceptual understanding students need to fully get it.
My views on place value have changed 360 degrees when we implemented Common Core. Now, teaching place value is my favorite unit, and the reason is….I know how to teach it so my students grasp the concepts and use it for problem solving! This skill is so conceptual and I feel that textbooks do not use the concrete materials and learning that goes along with place value.
Now….to the chapter….in a nutshell 🙂 …
Children’s Pre-Place-Value Understanding
Kindergartners and first graders know a lot about numbers with 2 digits, but this understanding comes from the count-by-ones approach. They are not yet able to to separate the quantity into place value groups. There are 3 levels of understanding about the concept of “tens” through which students progress:
Level 1: Initial Concept of Ten. Children understand ten not as a unit but only as ten ones. When solving addition or subtraction problems involving tens, they count only by ones.
Level 2: Intermediate Concept of Ten. Children see ten as a unit that consists of ten ones, but they must rely on physical or mental reconstructions of models to help the work with units of tens.
Level 3: Facile Concept of Ten. Children are able to easily work with units of ten without the use of physical or mental reconstructions of base-ten models.
*The Formative Assessment on p. 176 is a great way to see which level your students “fall into”. This could be done at the beginning of the school year and in small groups, or one-on-one.
Foundational Ideas in Place Value
Place value can be a difficult skill for students to comprehend because it requires an integration of new and sometimes difficult to construct concepts of grouping by tens with procedural knowledge of how groups are recorded in our place value system and how numbers are written and spoken.
Children can count sets using three distinct approaches:
- Counting by ones
- Counting by groups and singles
- Counting by tens and ones
Regardless of how they count, the teacher’s objective should be helping them integrate the grouping-by-tens concept with what they know about number from counting by ones. This is where, in 2nd grade, I have to move some of my students. Many rely so much on “finding the right answer”, that they fall back on counting by ones. Some struggle, but eventually they realize, especially when we get to greater numbers, that this is the most inefficient way to count.
Counting by tens and ones results in saying the number of groups and singles separately: “three tens and five” for 35. Saying the number of tens and singles separately in this fashion can be called base-ten language. We have talked in our grade level, how the English form of numbers can be very confusing to kids…especially those numbers in the teens. Young children have to learn that 17 means 1 ten and 7 ones. Other languages often use base-ten terminology so this can be a good cultural connection for children. For example, in Spanish, 17 is “diecisiete” which is “10 and 7”.
Base-Ten Models for Place Value
This is where I think all teachers really need to understand that concrete materials are essential, no necessary, no CRUCIAL when teaching place value. I like how the authors recommend, when first learning bout grouping, students use beans, Popsicle sticks, straws, or linking cubes so that they can see how objects can be grouped and taken apart. Using base-ten blocks right away doesn’t show students about that because it requires trading, they cannot physically take them apart.
This year, I was introduced to Digi Blocks, and I love using them because students can put them together to make 10s and 100s, and take them apart. This was my first year using them, and my students were highly engaged when using them, and they were grasping place value concepts. I was able to order two 1,000 packs for Digi Blocks through a grant I had written. When I received them, all 1,000 of the ones were in a plastic bag. I dumped the bag in the middle of our classroom carpet and asked my students how many were there. They gave estimates, or guesses. I then asked them how we could find out. Some suggested counting by ones, but others said it would take a long time to do this. Because it was the first time they had seen Digi Blocks, I showed them how to group 10 ones into a ten container. They loved this and realized they could then count by 10s. After many 10s had been made, I showed them how to put 10 tens in a hundred container. Then they counted by 100s. We discovered that we actually did not have enough ones. However, I contacted the company, who was wonderful, and they sent out the ones we needed right away. I know not everyone has Digi Blocks, I have done the exact same activity, but with straws.
We do use Base Ten blocks a lot in our classroom as well. However, when using pre-grouped models you need to make extra efforts to make sure that children understand that a ten piece really is the same as 10 ones.
I never thought of this, but the authors also talk about using ten frames. The little ten frames are less common but are very effective. If children have been using ten frames to think about numbers to 20, the value of the filled ten frame may be more meaningful than the ten rods and squares of base ten blocks. They do warn that a significant disadvantage of the pre-grouped physical models is the potential for children to use them without reflecting on the ten-to-one relationships or without really understanding what they are doing…this is especially true if children have not had adequate experience working with groupable materials. Here are some ten frame tools you can download for free–personal ten frame sizes are included among other ten frame items for use in the classroom.
Developing Base-Ten Concepts
This section focuses on base ten concepts or grouping by tens. Children need to experiment with showing amounts in groups of like size and eventually come to an agreement that ten is a very useful size to use. This is where my Digi Block, or straw, activity really came into play because my students had the chance to group objects and experiment with counting groups.
In 2nd grade, we work with numbers up to 1,000. Here, the students are seeing how a group of 100 can be understood as a group of 10 tens as well as 100 single ones. It is important for students to use a groupable model, as with the Digi Blocks, so that they can see how the 10 groups of ten are the same as 100 individual items. This connection is often too implicit in the display of a hundreds flat or a paper hundreds square in the pre-grouped base ten models.
Oral and Written Names for Numbers
In kindergarten and first grade, students need to connect the base-ten concept with the oral number names they use. This is where using that base-ten language is important. Teachers should say “4 tens and 7 ones” instead of “forty-seven”. Emphasize the teens as exceptions. Say that they are formed “backward” and do not fit the patterns. This is also very helpful with ELL students.
When working with three digit numbers in 2nd grade, have students write the base ten name and the standard name. The book gives great activities and lessons to do with students so that teachers can formally assess student understanding and then form guided math groups based on students’ needs.
Everyday in my classroom we do The Daily Math Buzz (I have a bee theme going on in my room so hence the name. 🙂 ) I got smart this year and put the sheet in a plastic sheet protector and the students used dry erase markers. In the past, I have made a booklet with many of these in them for the school year. I have included the sheet we use—you can get yours here!
Patterns and Relationships with Multidigit Numbers
Now that students understand individual numbers, it’s time to move onto seeing the numbers in patterns.
The authors suggest using a hundred chart for students to recognize patterns. In kindergarten and first grade, children can count and recognize two-digit numbers with the hundreds chart. In first and second grades, children can use the hundred chart to develop base ten understanding noticing that jumps up or down are jumps of ten, while jumps to the right or left are jumps of one. There are many activities listed to do using the hundred chart.
Benchmark numbers are one of the most valuable features when using the hundred chart, and moving onto using a number line. Multiplies of 10, 100, and on occasion other special numbers, such as multiplies of 25, are referred to as benchmark numbers. In my classroom, we use benchmark numbers ALL THE TIME! This will lead to connecting place value to addition and subtraction. One relatively recent shift has been to blend instruction on numeration and place value. The NCTM Principles and Standards for School Mathematics suggest, “It is not necessary to wait for students to fully develop problems with two-and three-digit numbers. When such problems arise in interesting contexts, students can often invent ways to solve them that incorporate and deepen their understanding of place value, especially when students have the opportunities to discuss and explain their invented strategies and approaches”. This leads back to Chapter 1 when teaching for understanding was discussed….allow students to problem solve on their own.
The other point I would like to make, is the one about regrouping. First off, students in 2nd grade are not to use the traditional algorithm when adding and subtracting two-and three-digit numbers. From experience, they just don’t get it at this age. Using place value concepts, and tools such as the hundred chart and number line, students can add and subtract large numbers without using the ole carrying and borrowing. The teaching task is to get children accustomed to looking for combinations that work together and then looking for these combinations in computational situations. In fact, as you will see in Chapter 12, virtually all invented strategies for computation as well as mental strategies involve no regrouping at all.
Please feel free to share your thoughts about the chapter and helping your students develop place value concepts. We will see you back here on Wednesday for chapter 12, Building Strategies for Whole-Number Computation.