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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.

Fall Favorites Blog Hop

Warm days, but cool nights, green grass that requires no water, mowing with only 2 bags of grass instead of six to compost, fresh apples off the tree and a backdrop of yellow, orange and red on the mountains behind me. What's not to like about fall? Here are a few of my favorite fall things in the classroom.

Art Projects
There are just so many fun fall art projects. Directed drawings are one of my favorites.
Directed drawing of Frankenstein that focuses on shapes.

Draw with pencil, trace lines with Sharpie markers and watercolor paint.
Maybe this is too much Frankenstein, but we changed the words from Witches Stew to Monsters Stew and then we just had to make monsters.
Frankenstein's Head

You can grab the cutting directions for making this project at Classroom Freebies Too.
Thumbprint creatures are always fun too. These little Halloween counting books are so cute. All the creatures are done with thumbprints or fingerprints. We came up with ways to draw a variety of Halloween creatures after starting with a thumbprint (skeleton, witch, spider, pumpkin, jack-o-lantern, vampire, bat, ghost, Frankenstein). This makes a fun Work on Writing center for this time of year. Warning: Have everyone use a baby wipe in between colors.

Nine vampires drinking

Six scary spiders

Creepy, Crawly Spiders
Spiders are a great way to start off with a little research. I love using a R.A.N. (Reading and Analyzing Nonfiction) chart to help us keep track of our spider knowledge.
Then I like to integrate our learning about spiders into our interactive writing and writing workshop. These ideas and more are included in my Spiders: A Unit of Reading, Writing and Research.

Draw the spider's body then add labels and even captions.

This is from my Cognitively Guided Instruction word problems with spiders.
And even a tasty spider! Use an Oreo, black licorice, mini M-n-M's and a little frosting as glue.

I wanted to put all my fall products on sale for this blog hop, but I just couldn't stop there. So all of my products are on sale 20% off through October 27. Plus, you can win all four of my spider products for grades 1 and 2. The winner will be announced on October 28.

a Rafflecopter giveaway

For some more fall favorites, you'll want to hop on over to Missing Tooth Grins. Happy Fall!

Saturday, October 11, 2014

13 Rules That Expire

Today I happened to find a must read article for all math teachers that really resonated with me called "13 Rules That Expire" by Karp, Bush and Dougherty in Teaching Children Mathematics.

There are just too many things that are overgeneralized in classrooms and the authors do a great job of not only pointing out some of the misconceptions that we allow students to use, but also how quickly they expire. Here's one that has been my little pet peeve for a while: the equal sign means find the answer or the answer is. This little misconception should be expiring in first grade, but I'm not sure older students understand the correct meaning of the equal sign. Change your language to "two plus two is the same as four" to help students understand that the equal sign represents a relationship not an answer.

More misconceptions:
  • when multiplying by ten, just add a zero
  • you can't subtract a bigger number from a smaller number
  • use keywords to solve story problems
  • addition and multiplication make numbers bigger
  • two negatives make a positive

If any of these are things you've taught, you'll want to read this article. There are more overgeneralized rules, plus suggestions for alternative language to help your students. Click on the link above to download the article from the NCTM website. Plus, the authors ask for more examples of rules to expire to keep the conversation going on the NCTM blog. So think about sending in some of your own examples.