# Using Precise Mathematical Language: Place Value

If we want students to use precise mathematical language (and we do!), then we have to be sure we model what that sounds like. Let’s look at some shortcuts we often take in our own language and see how they undermine our goal of having students deeply understand place value.

## Eight-O-Seven is a Time

The essential understanding of place value is that digits take on different meanings based on their place value position. We teach students that an 8 in the ones place has a value of 8, while an 8 in the hundreds place has a value of 800. So how important is it, then, that we say the number 807 asย *eight hundred seven*, rather thanย *eight-o-seven*? Very! When we take shortcuts naming numbers, we often strip all the place value meaning out of the number. No wonder students get confused. When numbers are spoken correctly, we should hear the place value of every digit.

We often do the same with decimals, reading the number 3.52 asย *three point fifty-two.* I’ll be honest, if I am not in front of students, I’d say it the exact same way. But when I have my teacher hat on, I have to sayย *three and fifty-two hundredths* to emphasize the place value.

When you think about it, word form is really the same as spoken form, right? However a number is written in word form, that’s how we should consistently refer to it.

Do your students need practice with the different forms of numbers? You can grab free sets of these cards in 6-, 9-, and 12-digit versions by visiting this post.

## It’s Not Borrowing if you Never Give It Back

Another area often devoid of precise place value language is adding or subtracting with regrouping. Consider this explanation for solvingย 423 – 158:

*Boys and girls, when we subtract, where do we start?*ย (on the right)*Correct–we always start all the way on the right.I can’t subtract 8 from 3, so I’ll go next door and borrow.The 2 changes to a 1, and the 3 becomesย 13.*

That’s probably how you learned to subtract. I know it’s how I learned. But it does nothing to help students understand the process of regrouping, which is all about place value. Contrast the previous example with this one:

*Boys and girls, when we subtract using the standard algorithm, where should we start?ย *(with the ones)*Why?ย *(because it’s the smallest place value position)*What’s my problem? *(weย only have 3 ones so we can’t subtract 8 ones)*So what can I do?ย *(trade a ten)*When I trade a ten,ย how many tens do I have left?ย (1 ten)*

*And what did I get when I traded a ten?*ย (10 ones)

*(13, the 3 ones weย had plus the 10 extra weย got from trading a ten.)*

How many ones do I have now?ย

How many ones do I have now?ย

Note that in the first sentence, I purposefully included the words *standard algorithm*.ย The reason for that clarification is that the requirement ofย starting with the ones is specific to the standard algorithm. There are other strategies for subtracting, such as splitting by place value, that don’t require you to start with the ones. Here’s an example of what Greg Tang refers to as “using funny numbers.”

Also, I useย *regrouping* andย *tradingย *interchangeably when working with students. I find thatย *trading* helps them make sense of the action that takes place when weย *regroup*.

Finally, I hope you notice one other difference between the two scenarios. Look how much talking theย *students* are doing in the second one. That’s huge. It’s my way to determine how well they understand the process–my formative assessment. I wantย *them to describe*ย the process, not me.

## Take the Precision Pledge

If you’ve been taking shortcuts with mathematical language, I get that it’s hard to change deeply embedded habits. I remember when I first started teaching and I found out you aren’t supposed to sayย *andย *in a number unless you areย referring to the decimal point. What? No more *two hundredย *and* thirty-five? *That’s right–justย *two hundred thirty-five.ย *I had to unlearn aย life-long habit of how I spoke numbers. Not easy, but important. To this day, I still cringe just a little when I hear a news anchor throw anย *and* into a number.

So let’s all take theย pledge to be more precise in our mathematical language. After all, precision counts!

I am bowing down to you right now!!! I am a HUGE stickler on pronouncing mathematical values correctly. Thank you for mentioning trading rather than borrowing!! It’s like fingernails on a chalkboard when I hear ‘borrowing.’

I’m with you, Kendra, but all too often we teach how we were taught. That’s why I love connecting with other educators on Facebook and Twitter–sometimes we don’t know what we don’t know.

Thx Donna for writing this-I often cringe when I hear teachers use imprecise language as well. The problem is that when teachers say says that we re borrowing in this process, they imply that somehow they are changing the value of that number! We need to teach kids that there are any different names for the same number–I like to use “renaming” because that is what we really are doing–renaming the number so that we can subtract!

Exactly, Mary! After we regroup, I always ask the students if we still have the same number. About 90 percent of the time they say no, until we’ve worked together for a while, that is. Then the percentage comes way down! It’s just additional evidence that they are just following steps they’ve been taught and don’t really understand what they’re doing.

Thank you for providing me with the specific terminology that I can use with my own students! “Trade” is in, “borrow” is out! I can’t wait to apologize to the kids tomorrow and let them know that the teacher made a mistake and was using the wrong word! They’ll love it!

You are AMAZING, Amy! Not only will you be modeling precise mathematical language, but you will also be modeling the power of life-long learning! Rather than framing it as a “mistake”, you could also tell them that you learned a new way of describing the process that you like better because it is more mathematical.

Fantastic post! Thank you!

I Iove, love, love everything you are modeling! I wonder about having kids says you can’t take subtract 8 from 3 and the misconceptions that might create for when they get to negative numbers. What are your thoughts?

*say

Yes, Krista, I understand your concern. We often say things in elementary school that don’t hold true in later grades–like when we add the answer is always bigger. I love this article from NCTM called 13 Rules That Expire. I was thinking that since I labeled the numbers “ones”, I was okay. If I only have 3 ones, I can’t take away 8 ones. Kind of like if I only have 3 M & M’s, I can’t take away 8 M & M’s. What do you think?

It is really important that children have had lots of “hands-on” experiences using real life objects so that the subtraction process makes sense, and is not an abstract concept. They need to have made numbers with groups of 10 and ones, and then re-named these. We like to play a game called make 100 with MAB blocks. Children roll a dice and collect what blocks they need. This involves lots of re-naming and re-building of numbers. When they have 100, they can play”Wreck 100″, which is simply the game in reverse. These games can be played over short periods of time, many days in a row. If students are ready they can move to Build 1000.

I teach in Australia(the state of Victoria) so our curriculum may be a bit more flexible than in the USA. At our school we have the flexibility to fit the curriculum to the needs of the child, not make the child fit into the curriculum.

Donna, I think I see an error in your blog. It’s this statement: Whatโs my problem? (we only have 8 ones so we canโt subtract 3 ones)

It should be reversed; yo have 3 ones so we can’t subtract 8 ones.

BTW, I am a principal and I often share your blogs with our teachers. I will definitely share this one with them!

Thank you!

Thank you for the head’s up! I have corrected it now. Thank you so much for sharing my blog with your teachers. ๐

Yes, yes and YES! Allow me to extend by noting that correct and precise mathematical language is in reach of all levels of learners, pre-k included. (In fact, that’s the BEST time if not in kindergarten.) Empower students with the gift of high expectations by discussing and noting digits when learning about number words and numerals. I promise, they CAN do it!

Another aspect of place value precision is when students are adding an extra “and” into spoken numbers rather than just at the decimal point. Many people say “8 hundred and 7” for 807. I write the number the way students say it (800.7) and they see the mistake. When we hit decimals, this becomes problematic as the decimal signifies the “and” in word form and should do so in spoken form as well!

Yes, exactly! I had to relearn that myself, as I had the bad habit of inserting an “and”.

Donna, I truly appreciate this post. I’ve recently commenced the Post Baccalaureate Degree program in Elementary Education and after a 20 year career in a different field, my cohort is already thrust into the classrooms to start our practicums.

I know I’m horribly guilty of these habits and after reading your post, it all makes sense to me. Over the past few weeks, I’ve experienced a huge shift in what I experienced in school and what is occurring now. Your post has already provided me with a new insight on how to deliver material effectively. The trade versus borrow language is spot on, thank you!

So glad to hear that you connected with this post, Darrell. Good luck with your studies and new career!

Speaking of language precision. What are your thoughts on using the word reduce as opposed to simplify. I personal believe reduce is such an incorrect word. The word reduce means to make smaller. But, you aren’t making the fraction lesser than it is. I believe it causes confusion and students then fall away from understanding that a simplified fraction is an equivalent fraction. Thoughts?

Christine, I have heard buzz around this topic. I think the common thinking leans your way–it’s not really accurate to say that we “reduce” a fraction. This is one that is deeply embedded in our mathematical language, though, so we probably won’t see it go away anytime soon!

I am an instructional coach who has been giving PD on the Standards for Mathematical Practice to my staff this fall. Yesterday we focused on Standard 6 (precision), and this was a huge topic of discussion (in a good way!). At the end of our PDs, I ask my teachers to leave sticky note feedback, and one of the questions yesterday was, “Where’s the line between kid-friendly language, yet precise?” I have my own thoughts, but was curious as to yours…thanks!

Great question! My response is that we should be using precise language from the get-go. I once heard someone say that if a 5-year-old can name every dinosaur that ever roamed the Earth, they can learn the word “vertex”. Why teach “corner” and then change it to “vertex” later? Why not call the answer to an addition problem the “sum” from the time addition is first introduced? So now I’m curious as to your thoughts…

YES!! Those were exactly my thoughts, so you just confirmed what I was thinking–thank you! (sometimes I tend to question myself and/or overanalyze! :-)). Throughout this PD series, the number one concern from the K/1 teachers has always been how to implement these standards at their grade level. And while I can appreciate where they are coming from in terms of their littles’ development, sometimes if we raise the bar of our expectations, they’ll often surprise us (which has been the direction I’ve been trying to steer them). So…thank you for validating my thoughts!

I’m so happy I found this blog post. I have been preaching for years about being precise about our mathematical language when we are in front of our students. I also cringe when I hear people use “and” in a number that doesn’t have a decimal and when teachers use the term “borrow” when teaching subtraction. I will be sharing this post with my teachers. Thank you for writing this post.

It is an important reminder that our use of precise math language must begin in the early grades and remain consistent from grade to grade to avoid creating more math confusion in the upper grades. I agree that it is hard to break away from the way we learned and it requires intentional rethinking when we are instructing our students. However, it is vitally important for our students’ long term success!

I love this! We just published an article with NCTM discussing some of the strategies that can be used with students virtually and in the classroom. I really appreciate your insights Donna!

https://pubs.nctm.org/view/journals/mtlt/114/9/article-p696.xml

Just finished reading your article! What a great example and specific strategies you guys provided!

I cannot find the download for the place value cards. Can you help me please?

There is a link in this post to another post. The link for the place value cards is at the bottom of the second post.

โTradingโ. Verses โ borrowingโ .. I also explain this as when we borrow somethingโฆ (

like in our financial management unit)when return itโฆ when we regroup and tradeโฆ we arenโt returning that 10โฆ ties in vocab and meaning

I was a high school math teacher (retired). Most math teachers in the upper grades are visual learners. Show us an algorithm 3 times and we got it. Makes it harder to know how to use language to explain things. Math language should be taught to all future math teachers.