Teaching Student Centered Mathematics Book Study: Chapter 11

Early ideas about place value... it's a big concept with so much nuance that it can be very easy to be a bit to narrow minded and miss an opportunity for a connection with your students. Think about the number 52 for a minute. What is it, how do you know? It could be 52 ones, a set of 5 tens and 2 ones. And in that case what is a ten? "One ten is ten ones". And if we wrote the digits 5 and 2 but switched the order it makes 25 which is an entirely different amount. Makes perfect sense... right?

WRONG! For our youngest students, forming ideas about place value means reconciling what they know about counting individual objects and applying that principal to counting sets of 10 objects to form a new unit. This topic is SO big in fact, that I am going to focus my discussion of this chapter on early place value concepts in terms of developing a concept of ten and tools/models for representing these concepts.

The text states that there are three stages a student goes through when understanding the idea of ten.

  1. Ten meaning ten ones. 
  2. "Ten" as the name of a set of ten ones with a model. 
  3. "Ten" as the name of a set of ten ones without the need for a model.  
I saw this transition so clearly in my students last year. A group of my tier 3 first graders did not have a concept of teen numbers come winter. In fact, they could only count up to about 12 before errors would occur. We played games daily counting around a circle, playing dice games on a number line, etc. and their rote counting to 20 improved dramatically. One game we played included a set of two ten frames and dinosaur eggs. Each egg had a number between ten and twenty. My kids would choose an egg and build that number on the ten frames with individual counters, they would then put the number into a bucket and build the next number. The first few days, I had kids who would clear the whole board in between numbers. We talked about how it took a long time to get out enough counters for the next numbers and they then transitioned to thinking about how to change the counters to make the next number. This activity was great for number relationships but I still had students who would count EVERY. SINGLE. COUNTER. to be sure they had the right number. I continued to prompt and ask the students how many counters were in the first ten frame "How many are there up here?" and without fail they knew that there were ten. From there, they began to use their knowledge of ten to count on when building a number or even to think about the relationship between teen numbers. For example, if I have the number 15 and I need to make 16, I know I can add just one more and that will make 16- I don't need to recount. A week and a half to 2 weeks in to this center, a few students started to notice the pattern in tens and ones. I started to hear comments like "14 is just a ten and a 4". I questioned about how they knew and how they could prove it counting by ones and counting on from 10. We "tested" the theory out with each number and found that their rule worked! At this point, we were still in stage 1 with my students recognizing a ten as ten ones and were dabbling in the idea of counting the first ten as "ten" but it was really only as a construct of counting on. 

The next time I made them math centers we moved from dinosaurs to fish and the novel new idea was that instead of counting out ALL of the counters to make the teen number, they could choose between individual counters and a connected ten frame that says the word "ten" on it. At this point, my students were becoming much more comfortable with the idea of calling something "a ten" as opposed to "ten" meaning ten ones. 

Even typing that sentence was confusing.... no wonder this is such a big concept for our little friends!! I'll fast forward to the end of the year quickly to tell you that my students did get to that third stage. I know because they were able to add and subtract ten from any number 10 to 99 in their heads without a visual representation and without the problem written out. They had a strong place value understanding that allowed them to understand two digit numbers and manipulate them! 

Now, rest assured that we did not spend 3 weeks just building dinosaur and fish numbers :) These were just a few activities that we did that represented their overall understanding of teen numbers. What I did not want to do was rush to the second stage of calling ten ones a "ten" before they were ready or really understood why we would want to use tens because they would have been able to rotely build these numbers for me but would not have understand how or why the math we were doing worked. 

So what else did we do during those three weeks? We used many, many, many representations of tens and ones and experimented with these tools. The text goes on to talk about models that represent place value concepts. These include: 
  1. Groupable base ten models
  2. Pregrouped base ten models
  3. Nonproportional models
Groupable models include items like straws, snapping cubes, beans, linking chains or any item that can literally be grouped together. A student can take a set of ten, connect or group them and that item is now called a "ten". This step is SO important for students. Without this step, kids will rotely use a tool called a "ten" but may have no idea of how it is made and what it means. Once students are comfortable with a groupable model, they can move to a pregrouped model. This is a tool such as strips and squares, ten frame cards or base ten blocks. They are great for convenience sake once a student has moved past the need to see ones grouped into tens and are less cumbersome to work with so a student can spend more time focusing on the MATH and less time focusing on sticking cubes together :) Again though, this tool is only useful once students understand what it is and how it was made. Used to early, these models can be confusing or students may use them rotely but lack understanding of the meaning behind the tool. When students fully understand the concept of how a ten is made and have an understanding of magnitude, you can move on to non proportional models such as coins, dollars or place value disks.

As I have said in most previous posts, this book is SO dense with information I really have only barely begun discussing this chapter. Really. Truly. So... be sure to check out the other blogs on the link-up below hosted by Adventures in Guided Math and/or grab a copy of the book for yourself!

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