I'm sitting here wide awake at 5:30 a.m. which is highly unusual for me. I am so not a morning person. However, I am in a hotel and have nothing else to do so I thought I would write a short post on place value. Why place value? I know that years ago, I had little understanding of how little understanding I was providing my students with the ways I "taught" place value. I know better know and have spoken about its importance within sessions. Yesterday, I saw a twitter post from one of my favorite math people James Tanton and it reminded me that I should take some time here to write about it.
Let's start with a question. How many tens are there in the number 234?
Some text to distance you from my answer. You can skip the rest of this paragraph and move on to the next one once you have answered the question above. Nothing important to read here. Honestly. Why are you still reading this? You don't follow directions very well do you? Neither do I so I'm not surprised. Ok. Enough chit chat. Let's get to the "answer".
Did you say that there are three tens? Twelve years ago, that's what my students would have said too. Why? Because when I asked that question in class, I was really asking the question "what digit is in the tens place?". At the time, I didn't really understand that the question I was actually asking has a very different answer. Now, I am much more careful about how I talk about place value before asking that question.
Let's explore one of the activities I use to build a better understanding of place value. You have base ten blocks in front of you and we have named them tens and ones. We aren't using the hundreds right now. Let's focus on building the understanding with smaller numbers first. Once you have that, you will see how it transfers to larger numbers.
Here is the process that I generally use when introducing place value now. Please keep in mind that this isn't meant to imply that this is the way you should be doing it but it is what has worked for me with the students and teachers I have worked with. Stop at the end of step 3 to see if you can answer the question I pose to students at that point.
Step 1: Make sure that students have had prior experience using base ten blocks. They must understand the relationships between the unit block, the tens and the hundreds. I am not spending time here discussing those relationships however.
Build me the number 23 using the base ten blocks. Let me clarify that. I don't want you to place the blocks on the table so they look like the digits 23. (Yes, I say this...it has happened more than once). I want you to represent the number 23 as a value. When I count your blocks, I will know that they have a total value of 23.
Step 2: Give students time to work. Walk around to formatively assess students. Which ones are struggling to build 23 in any way? Which ones are showing it just one way? Which have moved on to find a second way? Have any explored the idea that there is more than one way? Watch for a third representation (You are not going to call on that student. You just want to be aware that they already have that concept) Let's say that Johnny has represented them using all ones and Patty has represented them using two tens and three ones. I will make sure to let them I will ask them to share their representation with the class.
Johnny, how did you represent 23? Johnny will say "with 23 ones." Class, does 23 ones represent 23? Yes. Who else represented 23 using 23 ones? Write 23 ones on the board.
Patty, how did you represent 23? Patty will say "with 2 tens and 3 ones." Class, does 2 tens and 3 ones represent 23? Yes. Who else represented 23 using 2 tens and 3 ones? Write 2 tens and 3 ones on the board.
Class, who is correct? They both are because they both represent 23.
Step 3: I tell this story if I haven't told it to them before.
Class, I love math. I think about it all the time. I start walking around the room, looking dreamily up in the sky. Ok, you don't have to look dreamily...you can look up thinking hard or with a pondering look on your face. Your call.
Because I think about math all the time, I'm a little clutzy. Right now, I have 23 ones in my hand. Have a bunch of ones in your hand and wander around looking up...however you wish.
I'm wandering around, thinking about math...and I trip and fall. Don't actually fall...Just stumble a little bit. :) and a I drop my 23 ones. (Don't actually drop them. You don't want to be cleaning up that mess!)
Now, I have 2 tens and 3 ones in my hand. Grab those...doesn't have to be exact. No one is counting.
I'm wandering around, thinking about math...and I trip and fall. Which one's worse? Dropping the 23 ones because it's a lot to pick up and you probably won't find them all.
Because I'm clutzy, I will usually...BUT NOT ALWAYS...represent my numbers using the fewest blocks possible but I know that I can represent 23 either way. It will depend on what I want to do with that number. So, typically, if I ask you to represent a number, please use the fewest blocks possible but know that you can change that to suit whatever you are doing with the number as long as you represent it accurately.
Now, We have represented the number 23 using the fewest blocks possible: 2 tens and 3 ones and we have represented the number 23 using the most blocks possible: 23 ones. I wonder if we can represent the number 23 a different way. I'm going to give you 3 minutes to work on this puzzler. You might not figure it out in that amount of time and that's ok as long as you keep working on it. Who can you work with? Seat partner.
Note to reader: Remember to stop here and solve this question yourself!
Step 4: Wander around to formatively assess students as they work to find the answer. Choose a student who figured it out to share with the class and then ask who else found that representation.
Lise, how did you represent 23? Using one ten and thirteen ones. Who else represented 23 this way? Write one ten and thirteen ones on the board.
Reinforce that all three representations are accurate and acceptable and that it will be up to them to decide which one makes the most sense when working.
Question for the reader: When does representing the number 23 using 1 ten and 13 ones most often show up?
When you are working with the "traditional" algorithm for subtraction. On a side note, in Alberta, students are required to explore the standard/traditional algorithm for subtraction beginning in grade 2. Some cautions:
- This should NEVER be introduced until they have a solid understanding of place value.
- This should NEVER be introduced until they have had an opportunity to work concretely with the concept of subtraction. Spend lots of time building concrete understanding before representing any concept abstractly. If they truly understand the concept, you can tie that understanding to the abstract representation and it will then have meaning for them.
- You should NEVER use the term borrowing. What does borrowing imply? That you will pay it back. Do you ever pay back that ten you borrowed? No. I'm ok with the terms exchange or trade. I would love to hear any other terms you use.
- NEVER say that you can't take a bigger number away from a smaller number (this is usually said to help stop kids from subtracting upwards). You CAN take a bigger number away from a smaller number: 3 - 8 = -5. We just haven't learned how to do that yet. If they are subtracting upwards, that tells you they don't understand the concept. They are just trying to use the algorithm without understanding it.
After we have dissected the number 23, we move onto a number like 32. How many different ways can you represent this number? Since this will require a lot more blocks, they typically don't have enough to leave all representations on their desk so I have them build one representation and write down the "words" on their whiteboard (ex. 3 tens, 2 ones), or take a photo of it, etc.
Lots of time needs to be spent at this level with small numbers. Once they have this, move on to bigger numbers which is harder to represent using base ten blocks as you need too many. Can they find all the ways and write the "words" on their whiteboards without the blocks?
Afterwards, I might give them some questions to play with on a sheet or on the board. I let them choose an "easy one, a medium one and a hard one" to try.
When students are feeling confident and need a challenge, or if I am working with a grade that needs to move on to place value in hundreds, we look at three digit numbers, representing small quantities first (like 123) using base ten blocks (if needed as some students might not need them) and "words" (1 hundred, 2 tens, 3 ones). Here are some questions I let them choose an "easy one, a medium one and a hard one" to try.
Question for the reader: Now, go back to the original question I asked at the beginning: "How many tens are there in the number 234?" What would you say now? There are actually 23 tens.
Earlier, I had mentioned that James Tanton's tweet reminded me that I wanted to talk about this. He reminded me that we do this naturally for certain numbers such as twelve hundred, or six hundred fifty-seven thousand.
Reflection for the reader: How do you ensure that your students understand the difference between the questions "what digit is in the tens place" and "how many tens are there?"
