As a teacher, curiosity around students’ mathematical thinking was the driving force behind the teaching and learning in my classroom. To better understand what they were thinking, I needed to not only have great, accessible problems but also create opportunities for students to openly share their ideas with others. It only makes sense that when I learned about routines that encouraged students to share the many ways they were thinking about math such as Number Talks, Notice and Wonder, and Which One Doesn’t Belong?, I was quick to go back to the classroom and try them with my students. It didn’t matter which unit we were in or lesson I had planned for that day, I plopped them in whenever and wherever I could because I was so curious to hear what students would say. Continue reading
Last week, I wrote a bit about place value after doing a Which One Doesn’t Belong activity with a 3rd grade class. Since then, I have been thinking A LOT about how complicated place value really is. I think you can get a feel for the various ways we handle place value with students in this Twitter thread.
I have been talking about this with my 3rd grade colleagues at school, so one of them did the same WODB activity and ended with the same discussion around the number 146. She asked how many tens were in that number and got a lot of 4’s and 14’s, but this time she also got 40, which I did not hear in the other class. She asked the students to defend their thinking in their journals.
The journal below is the one I anticipate the most, separating the places and naming the number of tens in the tens place. (Although, I am unsure what is going on with the 74, possibly was going to give another example and ran out of time?)
This journal shows a slightly different reasoning because now instead of saying there is 1 hundred, 4 tens, and 6 ones, the student is using the value (or quantity, again not sure what to call this here, its complicated) of the 4 in the tens place as 40 broken into the four 10’s so you can see them.
I want to pair the student above with the student below and have them chat. This student had the same train of thought in the beginning but broke the 100 into ten 10’s to arrive at 14 altogether.
The last one, that surprised us, was the 40 tens. He actually showed 40 ones that make up the 4 tens with his dash marks in the last speech bubble. I may want to pair this student with the second example in this post to have them chat about that 4 tens vs 40 tens.
All of this still leaves me wondering a lot. I know there are times it is helpful to think about the tens only in the tens place while there are times we want to be thinking about how many tens are in the whole number, but….
- When are those times?
- How do we best structure activities to explore these ideas with students?
- What assumptions do we make about student understanding of place value as we teach comparison and computation strategies?
Since the 3rd grade team begins the year with an addition and subtraction unit in Investigations the teachers and I were having a conversation about how students understand place value. While I don’t see teachers using the HTO (hundreds/tens/ones) chart in their classrooms, students still seem to talk about numbers in that sense. For example, when given a 3-digit number such as 148, students are quick to say the number has 4 tens instead of thinking about the tens that are in the 100. I think a lot of this is because of how we as teachers say these things in our classrooms. I know I am guilty of quickly saying something like, “Oh, you looked at the 4 tens and subtracted…” when doing computation number talks, which could lead students to solely see the value of a number by what digit is sitting in a particular place.
We thought it would be interesting to get a vibe of how this new group of 2nd graders talked about numbers since their first unit deals with place in terms of stickers. A sheet of stickers is 100, a strip of stickers is 10 and then there are the single stickers equal to 1.
I designed a Which One Doesn’t Belong? activity with four numbers: 45, 148, 76, 40
I posted the numbers, asked students to share which number they thought didn’t belong, and asked them to work in groups to come up with a reason that each could not belong. Below is the final recording of their ideas:
I loved the random equation for 148 that emerged and the unsureness of what numbers they would hit if they counted by 3’s or 4’s. One student was sure she would say 45 when she counted by 3’s and was sure she would not say 76 or 40, but unsure about the 148. I wrote those at the bottom for them to check out later.
Since the teacher said she was good on time, I kept going. I pulled the 148 and asked how many tens were in that number. I was not surprised to see the majority say 4, but I did have 3 or 4 students say 14. As you can see below a student did mention the HTO chart, with tallies, interesting.
As students shared, I thought about something Marilyn Burns tweeted a week or so ago…
So, I asked the students to do their first math journal of the school year (YEAH!):
“For the students who answered 14, what question did you answer?
“For the students who answered 4, what question did you answer?
After the students shared, I revisited the Hundreds, Tens, Ones chart. I put a 14 in the tens column, 8 in the ones column, and asked if that was right. The light bulbs and confusion was great! It was as if I had broken all rules of the HTO chart! Then I put a 1 in the hundreds, 3 in the tens, and they worked out the 18. I look forward to seeing them play around with this some more and wonder if when they go to subtract something 148-92, they can think 14 tens -9 tens is 5 tens.
I had to run out because I was running out of time, but snagged three open journals as I left! (I especially love the “I Heart Math” on the second one!