The importance of watching students work…helping with place value misconceptions

expanded and standard form

We recently finished our unit on place value and I KNEW that not all students were solid.  As I usually do, I continued to give occasional entrance slips to try to keep my finger on the pulse of the class.  When I do this, I try to work in time to meet with strugglers–and I try to do it in groups of no more than 3 so I can WATCH them.

Here’s what I mean…I gave my students an entrance slip on Wednesday as class started.  As they turned them in, I did a quick check of them and sorted them into two piles–“no problem” and “better check it out”.

formative assessment
These assessments are from my “Formative Assessment Toolbox:  Place Value Edition” resource.  CLICK HERE to see it!

Looking at student work can tell you certain things…but I firmly believe we need to take things a step further and WATCH students to see HOW they are making their mistakes.  For example, I was looking at some addition with regrouping work a student had done–and time after time there was an error in the “regrouping”.  I could NOT for the life of me figure out what she was doing so I called her over and asked her to do a problem.  Within ten seconds I figured it out–she was starting on the left and regrouping to the right–showing a HUGE misunderstanding.  Turns out she had been pretty solid with partial sums (which can be done left to right or right to left) and was overgeneralizing that algorithm.  Without sitting right next to her and watching, I wouldn’t have figured it out and I wouldn’t have known to pull out the base ten blocks and model with her.

The same was true with a few of my kiddos who were making some mistakes with expanded and standard form.  I need to sit right with them and watch them work–and ask their thinking.
I worked with a few students at a time with my expanded form task cards (actually, to make it more fun, I put the cards in piles and the students took turns picking cards–it’s AMAZING how something as simple as letting students flip a card can keep them more engaged!)–and I watched them work on their white boards, asked them questions, and asked them to explain their thinking.

place value

As I was reassured that misconceptions were being fixed, I would send students away and fill their spot with new ones.  Other students were busy working on their math workshop options and I had a glorious 40 minutes to work up close and personal with these students.  The light bulbs kept going off-and I know the WATCHING was a huge part of it…putting myself in their shoes and trying to be a scientist figuring out where their thinking was going wrong.  It is SO validating for a child to hear, “I can TOTALLY see what you did it that way!  Let me show you something…” instead of only seeing problems marked wrong or sitting back in a large group discussion feeling confused and lost.

Watching students can help you see exactly what misconceptions students have–and for these place value concepts, several things come to the forefront and need coaching:

  • A lack of understanding of our base 10 system (especially once students get past 1,000 numbers get much harder to visualize)
  • A misunderstanding of “0” and how having “no” thousands gets represented
  • An inability to understand the organization of our number system…that we need three “places” before a comma, then three more and a comma, and so on–and that each of those “groups” or “periods” has a name and follows the pattern of one, ten, hundred.
  • The inability to recognize expanded form terms out of order (they might write 50 + 3 + 600 as 536)
  • Difficulties reading and writing big numbers and “hearing” the parts.  For example, if I say “four hundred thirty-two thousand”, students should hear that “432” and know it will come in front of the thousand comma.  If I say, “four hundred thirty-two thousand, seven”, they should know that they need to write a 0 in the hundreds and tens spots because there are no hundreds or tens.  Asking students to read and write big numbers is a great way to check for understanding.
So much of this is tricky whole class–you can present information this way, but when you see how many different ways place value understanding can break down, it becomes more and more clear how our interventions need to be much more personalized.

standard form expanded form

An added bonus of working next to small groups is the ability to “coach” them through some partner work.  After solving a card, I would ask students to compare work and try to reconcile any differences.  Learning how to check over their work, look for errors, and explain thinking is such a key part of the standards for mathematical practice.  As they got better at catching each other’s errors, my role became much more of an observer than a teacher–and the power got turned over to them.  Hearing things like, “Wait–you wrote 50,000 but that 5 is in the one thousands place” or “Remember that if there aren’t any tens you need to write a 0.” is enough to make THIS math teacher melt.  Seriously.

teaching place value and expanded form

Interested in checking out the task cards I used with my groups?  Just check them out below!  There are 36 cards with three different types of questions.  Each one has a “bonus” question too–so I used them with my entire class and then pulled my intervention kiddos and used them again.  See what you think!

expanded standard form task cards
I have a ton of place value activities in my store so check them out if you are looking for games, lessons, and more.  Just type “place value” into the search bar of my store!