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Saturday, October 25, 2014

Fly on the Math Teacher's Wall - Place Value

I am teaming up with a great group of math teachers to bring you a series of blog hops called Fly on the Math Teacher's Wall. We're going to try and help you squash the mathematical misconceptions of your students on a variety of math topics. This time let's talk place value. There are 14 of us in this hop and we will be tackling place value from ages 6 to 16. (Ok, I know technically high school students are older than sixteen, but 6 to 18 didn't have the same ring to it.)

Right now I am in the middle of a Numbers and Operations course for my math endorsement and the timing couldn't be better for this hop because we just talked about some research on children's understanding of place value. In 1990, Sharon Ross wrote an article titled, "Children's Acquisition of Place-Value Numeration Concepts." She sets forth 5 developmental stages for place value understanding:
  1. A student interprets a 2-digit numeral as a whole number but assigns no meaning to the individual digits.
  2. A student recognizes place value terms (ones, tens) but attaches no meaning to the digits in a numeral.
  3. A student interprets digits with an understanding that the 3 in 35 means 3 of something, but not necessarily 3 tens.
  4. A student recognizes that in the numeral 35 it means 3 tens and 5 ones, but the understanding is limited and his/her performance is unreliable.
  5. A student recognizes 35 as 3 tens and 5 ones. His/her understanding is complete and performance is reliable.
When I think about my students with these developmental stages in mind, it helps me move past the "he just doesn't get it" frame of mind and into considering the types of activities that will help a student move from one stage of development to another.

I'm teaching first grade right now and I would say most of my students are in between the first and third stages. Although this week I pointed to the four in the number 42 and asked, "What does this mean?" One young mathematician yelled out 40 and another yelled out 4 tens. After my back flip, it really hit me how far the divide is with understanding place value.  I have a few students that I need to question further to see if their understanding is reliable, but I also have many who had no idea what we were even talking about. The CCSS for place value understanding in first grade is that a student can "understand that the two digits of a two-digit number represent amounts of tens and ones" and whatever core your state uses, you'll find something similar in first and second grade. On this list of developmental stages, my students need to move to at least level 4 and preferably level 5. I want everyone to be complete and reliable in their understanding, I'm just trying to be practical because the only reliable thing in first grade, is that you never know what someone will do or say.

So how do I get my students to understand that there are 3 tens and 5 ones in the numeral 35?

I think you should start looking at two-digit numerals at the very beginning of the year. Don't wait for your place value unit. Calendar time is a great place to begin. Are you counting the days we've been in school? Then just add a little more talk about the numbers after ten. Do you have a 100 chart or even better a 120 chart in the classroom? Look for patterns in the numbers, there's a whole plethora of stuff to be discovered in that chart and a lot of deeper understanding can have its foundation by just talking about numbers. And when I say talking about numbers, I mean the student's talking about numbers.  In 2000, Steven Reinhart wrote a great article titled, "Never Say Anything a Kid Can Say." While this article is aimed at middle school teachers, it is so applicable to all of us.

Free place value project by The Research Based Classroom
Take a look at a previous post I wrote about giving students a concrete discovery experience with place value - Beans, Beans, More Beans and Place Value. I'm not going to lie, this project feels hard, maybe almost impossible during the first day. But it is oh so worth it in the end.
Using beans to build a stronger understanding of place value

This experience will push my students up into that fourth stage of development, but how about getting to the fifth stage - complete and reliable? This is all about solidifying their understanding. You have to find fun and engaging ways to let your students practice working with their new understanding. Here are a few of my favorite ways to practice without worksheets:

Skittle Math
You'll need:
  • 1 large bag of Skittles
  • 1 container of frosting
  • popsicle sticks
  • plastic knives and cups/bowls that can be shared between several students 
  • Small portion cups or Dixie cups (1 per student)
Beforehand prep:
  • Put a scoop of frosting in each bowl
  • Divide the bag of Skittles into the portion cups. I aim for everyone getting between 20 and 40 Skittles, but I don't count them out. Just scoop them into the cups.
Now what?
  • Have students bundle into tens by putting frosting on a popsicle stick and putting 10 Skittles on it. You have to really squeeze them on and even slightly overlap them to make them fit.
  • After students have made all the tens they can, have them set out their tens and ones so they are easy for others to count without touching. 

  • Each student should record their Skittle math on the worksheet by sketching base ten pieces and writing the numeral. 
  • Students can then scoot around to other desks to count and record the work of other students.
Skittle Math - Solidify understanding of place value with a tasty activity
Click on the image above to download the student worksheet.
Place Value Scoot
This practice is a little easier--no beans, no candy, no mess! You can grab it from my TpT store for free. It includes the papers for both 2-digit and 3-digit practice.

For more great ideas on place value that will help you squash those mathematical misconceptions, head on over to Adventures in Guided Math.

 You'll want to watch for more in The Fly on the Math Teacher's Wall series of blog hops as we tackle some of the tough issues in math. And if you have some topic ideas you would like to see us cover, leave a comment. Thanks for stopping by.


  1. The Skittles activity looks like a LOT of fun. Would be perfect to do this week with candy corn! My real take-away from this post is your place value scoot activity. The idea of getting two digits and noticing the smallest and largest number that can be created with those digits is a great way at getting at value vs. digits for our students who are just breaking into the world of numbers between 20 and 100. Thanks for the great post! The Math Spot

  2. I love seeing the 5 steps of place value understanding. I had never thought of it that way. I love how you always link your ideas to research!

    The Math Maniac

  3. Wonderful post! Love how you included levels of place value understanding and some great activities. Thanks so much for sharing and for organizing this wonderful HOP!


  4. It's mentioned in the opening paragraph that the team "will be tackling place value from ages 6 to 16." What does place value understanding look like in, say, grades 7-HS? Or what symptoms arise in later grades, again, say grades 7-HS, that can be tracked to a lack of understanding (or misunderstanding) of place value in the earlier grades? Thank you. Tom

    1. Good questions, Tom. I think that difficulty in understanding the meaning of each numeral (43 is 40 and 3) is something that can be traced to a misunderstanding of place value in the primary grades. I think it is continually seen in upper elementary with students who have difficulty adding and subtracting larger numbers. They learn an algorithm that they don't understand and it becomes procedural with no conceptual understanding. My expertise is all elementary, so I wouldn't even want to venture a middle school or high school answer, but some of my team of bloggers are secondary math teachers. I'm sure their thoughts would be more valuable as to the symptoms seen in grades 7-12. Thanks for stopping by.

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    The Dots of Teaching