## Monday, December 5, 2016

### Down the Rounding Rabbit Hole

Rounding has always been difficult for me to teach, and I suspect I'm not alone.  Here's a look at some of the many gimmicks, tricks, and tools teachers have used in an effort to get this concept across:

 Been there.

 Done that.
 Looks familiar.
 Monkeys?  Can't say I've ever tried this, but I'm skeptical.
 Not sure what this is.

 Seems like a lot to process, and I'll admit I've said similar things.

Our curriculum likes number lines:

 Grade 3 again.  Not a mountain or a roller coaster...it's a hill!

I've spent more time than usual this fall thinking about rounding; both the why and the when, but also simply the how.  Using a number line to round is all well and good, but it's not going to help you solve these problems...

...unless you can do the following:
• Identify place value locations and the value of the digit in a given place value.
• Identify the next multiple in that place value.
• Find the halfway point between the two multiples to use as a benchmark.
• Decide whether the number is more or less than halfway.
That's one big cognitive load.  No wonder many kids struggle!

Last year, I tried something different with some third graders who were having difficulty with rounding.  Here's a typical problem from their journal:

 An open number line won't help unless you know the multiplies of 10 that come before and after 68.

 Count forward by 1s until you get to a number you would say if you skip counted by 10s.
 Same, but this time count backwards.

 Find out if 68 is closer to 60 or 70.  This also reinforces complements of 10.

This was good as far as it went, but to round larger numbers, especially to units represented by a place in the middle of that number, students will need another strategy, hopefully one that doesn't rely on mountains, roller coasters, digits knocking on doors, or monkeys swinging on vines.
This fall I've been working with a fourth grader who, at the beginning of the school year, showed a complete inability to round.  She had a very limited notion of place value, and we spent several weeks shoring up some basics.

 We used this flip board...

 ...and these place value arrow cards.

I wanted her to be secure in her knowledge of place value locations and digit values before we ventured off into the world of rounding.  It took several weeks, but once she was able to complete tasks like this:

and this:

without the board or the arrow cards, I moved her on to rounding.  Here's the plan I had sketched out.  Let's take a grade 4 example.  Round 81,866 to the nearest thousand:

 Step 1: Write the number out in expanded notation.

 Step 2: Identify the place value you want to round to.

 Step 3: Identify the next multiple in that place value.  Here I might say something like, "If I was skip counting by thousands, what number would come after 1,000?"

 Step 4: Put the 80,000 back.  This can be combined with step 3.
 Step 5: Now we're ready for our number line.  We have our two possible responses for rounding 81,866 to the nearest thousand, so now we've got a fighting chance.

 Step 6: Where does 81,866 fall on this number line?  Can we benchmark halfway?
 Step 7:  This is now a matter of comparing 500 and 866.
We practiced on the big whiteboard in my room, and experimented with color:

 Rounding 697,654 to the nearest thousand.  After a while, she started to internalize some of the steps.

Here's an example of the student employing the strategy in her math notebook:

 Rounding 818,325 to the nearest thousand.  I would've liked to see her place the mark for 818,325 closer to 818,500, but that's for another day.
I'm happy to report that she's made lots of progress.  The mistakes she makes when rounding now come when she tries to do too much in her head.  Soon she will be able to visualize the number line in her mind's eye; the two boundaries, the halfway mark, the place where the number belongs.  Soon, but not yet.
So here's what I found out: 4.NBT.A.3 is a very interesting standard.

It's stated so simply, and, compared to others, seems so uncomplicated.  But my dive into rounding this fall taught me that there's much more here than meets the eye.  Place value understanding is a complex understanding with many components, and all of them have to be secure and working in concert in order for students to be able to round multi-digit whole numbers to any place.
As I continued to work with the student, the nagging question of why are we doing this kept running through my mind.  Curious to learn more, I found this on page 12 of the Numbers and Operations in Base Ten, K-5 progressions document:

Rounding to the unit represented by the leftmost place is typically the sort of estimate that is easiest for students and is often sufficient for practical purposes.  Rounding to the unit represented by a place in the middle of a number may be more difficult for students (the surrounding digits are sometimes distracting).  Rounding two numbers before computing can take as long as just computing their sum or difference.  (emphasis mine)

Here's what I want to know:
• If rounding to the unit represented by the leftmost place is often sufficient for practical purposes, then why are we asking students to round, for example, 287,347 to the nearest hundred?  What headache do I have that rounding 287,347 to the nearest hundred will cure?
• Are tasks asking students to round numbers to units represented by places in the middle of a number simply exercises in place value?  Is there a way to make this task compelling?
• What distinctions do the progressions make between rounding and estimating?
• Should rounding be taught as a discrete skill, separate from any context?
• What scaffolds, tools, and place value models are helpful in promoting rounding understanding, and which subvert it?
And finally: What are your experiences teaching rounding?

1. It makes sense to me that you dislike rounding - I think it's purposeless preloading that we traditionally do so much of in math. It's not benchmarks, which are useful, it's not significant digits, it's not estimating. Do it b/c I told you to, with an arbitrary rule (5's round up) thrown in.

You instead turn to place value sense making, which pays dividends for the rest of their life. Kudos, Joe!

1. Thanks John! There was always something about it that didn't seem quite right. So this begs the question: Is it the standard itself that's faulty, or does it have to do with the way the standard is interpreted by teachers and curriculum writers?

2. Hey Joe,
I like your strategy you used with this student. The expanded form really makes them look at the value in the place. I see this with teachers I work with all the time, teaching rounding as a an abstract concept by itself without any context. Putting numbers on the board and asking for rounding to specific place values. Like you said we really only round for estimating, so I think it is so important to teach it in contexts that make sense to them. I like the term creating friendly numbers which also ties into many operations mental computation strategies. The number lines are also excellent models for this, that's why I think your strategy using the expanded form will work so well. Makes it easier to see the friendly number. Honestly though how often to we get asked to round to specific place values? Not often. LOL. I think possibly another missing piece that often gets looked over in place value is the key understanding of unitizing. I often see teachers teaching the place names, that the digit value changes based on the place but unitizing as it applies to place value is left mostly to calendar time when students bundle groups of ten as they track school days or something in that vein. Our board has addressed this as being a need in our primary classrooms. We also see a gap in students understanding of zero as a placeholder, which is also a key understanding of place value. I wonder if some students lack of unitizing also affects the area you are speaking of in your blog? Thanks for sharing again! Always look forward to your blog posts!

1. Thanks Mark. I would agree that unitizing is overlooked. This comes into play in a big way when students are asked to round, say, 289,962 to the nearest thousand. I think your board is smart to invest time there, especially in primary classrooms. Place value scaffolds, like the place value flip book I used with the student in the post, don't get close to this concept. And neither do the arrow cards, although I like them because they help with expanded notation. In don't think I've ever really gotten a secure foothold on how to best teach it. It's a work in progress for sure.

2. Joe I have added a link here for resource that Ontario created a few years ago called the Effective Guides to Instruction. This one is the K-3 Number Sense. There many more on all areas and up to grade 6. There is some really good info in here on the number sense big ideas with instructional strategies and what students should know in each grade. It may not line up perfectly with you curriculum but you may find it useful.

http://eworkshop.on.ca/edu/resources/guides/Guide_Math_K_3_NSN.pdf

3. Thanks Mark!

3. Great post Joe!
Thanks for tackling the rounding beast and sharing these strategies. It seems like were too quick to jump to "rounding" when many students don't have a foundational understanding of place value. Makes total sense why it's an issue.
Looks like this would be a great chapter for your book. That is... if you were to write one.

1. Thanks Graham. It's another example of piling a concept on top of a shaky foundation. And I wonder if students who can round successfully really understand what they're doing. Maybe they're just better at memorizing and following the "rounding rules."

2. Who knows if you'll see this or it's too long ago! First of all I don't even know you but love your name because for 38 of my years on this earth I was Natalie Schwartz. Now I'm married but anyway!

I have been struggling with ROUNDING for the last few weeks as well as last year! I'm in Missouri and our standards say that students should use rounding as a means of estimating. Our curriculum (mostly engage) uses number lines which I really love because of the number sense but after reading your blog I'm starting to understand what you meant by a cognitive load. A few things have happened to me recently and I think I'll blog about it today. Trying to get better at that. Can I put the address for your blog in my blog and respond with my thoughts?

4. This idea that rounding follows place-value understanding is very interesting, and rings true for me. I have to help university students in science classes who do not know how to round, and it is a very useful insight to relate it to their place-value understanding. It seems the students who have trouble with rounding often also have trouble with scientific notation, which is also a place-value thing.

1. Thanks for your observations. It's fascinating to hear how the concepts we deal with in elementary school play out for students further on in their mathematical educations. I'd be curious to know the kind of role rounding plays in university science classes. What kinds of rounding would they do? To what place value?

5. I’ve been working with fifth graders recently on estimating answers to multi-digit multiplication problems, and the issue of rounding has been on my mind. I remembered that I wrote a response to a teacher’s query about rounding and included it in About Teaching Mathematics. It addresses rounding and estimating together. (I've tweeted it -- can't figure out how to paste here.)

First and foremost, I think both estimating and rounding have a purpose, and I work to keep the reason for doing either evident. About the standard: 4.NBT.A3, you’re right about an enormous amount being crammed into that seemingly simple statement. Many of the standards fall into this category―seeming simple but really embodying an incredible about of math. I think that many CC standards are packed this way, but I keep in mind that the standards aren’t and do not claim to be a curriculum, so deciding how to teach what’s needed for students to be successful with any standard is still up to curriculum developers, teachers, etc.

There is so much wrong, in my view, with your curriculum materials and the sadly funny collection of best-intention charts and rhymes you included to point out the gimmicks and tricks teachers rely on. But teachers are trying their best to help students be able to answer the questions posed in the text you’re using.

1. Thanks Marilyn. I agree that teachers are doing their best to help students answer the kinds of questions that 4.NBT.A.3 generate. There's like a rounding cottage industry going on! To me it's indicative of something. Maybe something wrong with the standard itself? I think what's bothering me is its dissociation from any context. Estimating has an important purpose, and rounding helps us estimate. So rounding exists in service to estimating. Does rounding have a purpose on its own? Can lead a purposeful existence without estimation?

2. We see this consistently with our expectations from our Ontario Curriculum, there is a lot more math going on underneath the expectations and many teachers are lacking content knowledge for unpacking and it affects their ability to set up learning opportunities for the kids. We are doing a PLC at one of my schools with that right now. Having the teachers look at the expectation and then find what are the key understandings that make up that expectation. A good example is one of our grade 3 fraction expectations. "Divide whole objects and sets of objects into equal parts, and identify the parts using fractional names without using numbers in standard fractional notation."
One example of key understanding that is required for students to truly understand this expectation is that the parts have to be equal area but not necessarily the same shape, they also don't have to be adjacent.
When we did our PLC not one of our teachers knew that about fractions. So unpacking these expectations is critical in my opinion. Obviously from Marilyn's comment and yours Joe that many of the Common Core Standards are similar. Love these conversations!

6. Hello! This post was recommended for The Best of the Math Teacher Blogs 2016: a collection of people's favorite blog posts of the year. We would like to publish an edited volume of the posts at the end of the year and use the money raised toward a scholarship for TMC. Please let us know by responding via http://goo.gl/forms/LLURZ4GOsQ whether or not you grant us permission to include your post. Thank you, Tina and Lani.