Ask any elementary school teacher, and she or he will tell you that comparison problems are much harder for most kids than operations with other actions. For example, fourth-grade-teacher Jennifer Clerkin Muhammad asked her students to draw a picture of this problem from Investigations:

*Darlene picked 7 apples. Juan picked 4 times as many apples. How many apples did Juan pick?*

Her students are pros at representations and skillful multipliers, but we saw a lot of this:

Kids who were used to thinking of multiplication as either groups, arrays, or repeated addition had some productive, furrowed-brow time ahead of them. How can you represent comparisons? How is this multiplication? How do you draw the 4?

Then they started to get somewhere.

With each representation they discussed, Jen asked the excellent question:

“Where do we see the *4 times as many* in this representation?”

Seriously. Jot that one down. It’s a keeper.

Jen and her class played with a series of problems involving multiplicative comparison, (height, quantity, distance, etc.). For each problem, she asked student to create a representation. And then she’d choose a few to discuss, asking, “Where is the ______ *times as ______ *in this representation?” Lightbulbs were going off everywhere as students broadened and deepened their understanding of multiplication.

Teachers often see the same struggle with comparison situations that can be thought of as subtraction or missing addend problems, like:

*Marta has 4 stuffed animals. Kenny has 9 stuffed animals. How many more stuffed animals does Kenny have than Marta?*

If kids have been taught “subtract means take away” or “minus means count backwards,” then this problem doesn’t fit the mold. And if kids have been taught keyword strategies, they’re likely to hear “more,” pluck those numbers, and add them together.

Knowing this as I do, I jump on opportunities to explore comparison problems with my own kids. I want them to have a fuller sense of the operations from the beginning, rather than developing too-narrow definitions that have to be broadened later. So, subtract doesn’t *mean* take away in our house. They’re not synonyms. I actually don’t worry about defining the operations at all: we just play around with lots of different types of problems, and I trust that my kids can make sense of them.

Yesterday, I was talking with Daphne about a neighbor visiting tomorrow. Our neighbor has a little girl, whom I’ll call Katrina. She’s about 2. We also have more awesome girls who live across the street. Let’s call them Lily, 15, and Gloria, 13. Gloria helps out once a week, and my girls idolize her. Here’s how the conversation kicked off:

Me: “If Katrina’s mother comes over to help me on Sunday, could you play with Katrina for a bit so her mom and I can work?”

Daphne: “Yes!!! We love playing with her. And she loves us. She cries when she has to leave us. We’re her best friends.”

Me: “I think Katrina feels the way about you and Maya that you and Maya feel about Lily and Gloria. It’s fun to hang out with older kids. You learn a little about what’s coming for you.”

Daphne: “Yeah. But for Katrina, Lily and Gloria would be TOO old. Like, she’d think they were grown-ups because they’re SO much older than she is. We’re older than Katrina, but we’re less older than Lily and Gloria are.”

So there’s my opportunity. She was comparing already. With a little nudge, I could encourage her to quantify and mathematize this situation. I thought I’d start with the simplest numbers so she could think about meaning first.

Me: “How much older are you than Katrina?”

Daphne: “I think she’s 2. So…”

There was a pretty good pause.

Daphne: “Are we counting on here?”

That’s paydirt.

Me: “What do you think?”

Daphne: “I think so. Because I want to know how many MORE. So, 2. 3, 4, 5, 6. I’m 4 years older than she is. Yeah. That makes sense.”

Me: “OK. So how many years older is Maya than Katrina?”

Daphne: “6.”

Me: “You didn’t count on that time. What did you do?”

Daphne: “Well, I knew that I was 4 years older than Katrina, and Maya is 2 years older than me, so 4, 5, 6. Because Maya is OLDER than me, I knew they had to be farther.”

Me: “What do you mean farther?”

Daphne: “Like, it had to be more. I was starting with 4, and then it had to be more farther apart.”

Me: “Oh! So you were trying to figure out whether to go 2 more or 2 fewer?”

Daphne: “Yeah. And it had to be 2 more, because Maya’s more older than Katrina than I am. I mean, I’m older than Katrina, and Maya’s older than me, so she’s more older than Katrina than I am. It had to be 2 MORE.”

Hot damn!

Daphne’s life might have been easier right here if she had some tools she doesn’t have yet. I’m thinking about number lines, symbols, and a shared mathematical vocabulary including words like distance or difference. Those tools might help her keep track and facilitate communicating her thinking to somebody else. But I love that we were able to have this conversation without those tools, while driving in the car where I couldn’t even see her gestures. Her thinking is there, and clear. The tools will come as a relief later on, when the problems are more complicated.

We kept going.

Me: “OK. So you’re 4 years older than Katrina and Maya’s 6 years older than Katrina. I wonder how that compares to Gloria? Is she about the same amount older than you two as you two are older than Katrina?”

Daphne: “Well, she’s 13. This is gonna be hard.”

Long pause.

Daphne: “She’s 7 years older than me.”

Me: “How’d you figure that out?”

Daphne: “I went 6. 7, 8, 9, 10, 11, 12, 13.” She held up her fingers to show me.

Me: “So how much older is Gloria than Maya?”

Daphne: “Umm…so Maya’s 8…umm…5. She’s 5 years older than Maya.”

Me: “How do you know?”

Daphne: “Because Maya’s 2 years older than me, so it’s 2 closer.”

This is the kind of problem where kids lose track of their thinking easily. Do we add or subtract that 2? In the last problem, she’d added. This time she subtracted. And she knew why!

When I glanced at Daphne in the rearview mirror, she had this awesome look on her face. She was looking out the window, but not seeing what was outside because she was so deep in thought. I wish I could have peeked inside her mind, and seen what she was seeing. Because, to use Jen’s question, where do we *see* the 2 years older? How did she see it? There was no time to ask, because she was onto pulling it all together.

Daphne: “So, Gloria’s a little bit more older than us than we are older than Katrina, because Gloria is 5 and 7 years older, and we’re 4 and 6 years older. But it’s pretty close.”

We pulled into the school parking lot on that one.

In our family, we count and manipulate objects all the time. Counting more abstract units like years, though, gives me a chance to open the kids’ thinking a little bit. In this conversation, Daphne was both counting and comparing years. Tough stuff, but oh so good.

I was thinking about just this the other day. Comparison is so natural for kids, especially when there are two of them. “She’s got more than me, it’s not fair” comes way before any ideas of addition or formal subtraction. I think more fundamental than counting on is pairing off. Multiplicative comparisons are a bit more deep, but many kids may be happy with “twice as many”, and probably happier with twice as long, or tall. I have written an outline of “fractions for adults” which gets at fractions by comparison of lengths, starting with “how many of these sticks, end to end, is as long as some number of those sticks”. Ratio, which is another name for multiplicative comparison, is so much closer to home than “John has 3 sweets and Mary has 5 sweets. How many do they have altogether”. So boring!!! The subtraction one is no better. “I had ten sweets, and then I gave Peter 4 of them. how many have i got left?”. Where is the realism here? If i am going to give Peter some of my sweets the first thing I do is to find out how many I have, then decide how many I want to keep, and only then decide how many to let Peter have.

The other problem is the mad crazy rush to symbols, and too quickly for the CCSSM, who write such garbage as “Add 3+4”. My immediate reaction to this was “Add 3+4 to what?”.

dang. that Daphne… she’s a keeper!

Deep thinking!!

Thanks!

I think Daphne actually does have a number line–she just doesn’t know that’s what it is. Her ability to describe the comparison as farther apart means she’s carrying the idea of a number line in her head. She has the ability to see numbers as a relationship.

I’m a math coach in an inner city elementary school and I struggle to get my kids to internalize number relationships in this way. I wonder what the preliminary work (as toddlers and preschoolers) is in order for them to develop this.

I agree with you. It’s not formalized, but she clearly has some visual in her mind that involves distance. I wish I could peek!

As for your school, that’s an interesting question. Obviously, I talk about math with my kids all the time, which puts them at an advantage for school. (That’s not why I do it, but that’s a reality.) I wonder what math your kids are doing in their lives that you could build on? For example, Daphne doesn’t have nearly as much familiarity with city contexts that are full of math, like numbered streets, city blocks, grids of streets and avenues, subway lines and stops, bus routes with distances, elevators for buildings with multiple floors (including basements), apartment numbers (often in odds and evens down the hall), and so on.

Whoops, premature reply. Anyway, my point is your kids might not think of all their knowledge as math, but you could certainly name it so, and build on it. I’m super interested in this idea, by the way, so I hope you’ll let me know as you mull it over. I’m thinking a lot about this quote from Danny Bernard Martin, from Martin, Danny Bernard. 2009. “Liberating the Production of Knowledge about African American Children and Mathematics.” In Martin, Danny Bernard, (Ed.), Mathematics Teaching, Learning, and Liberation in the Lives of Black Children. New York, NY: Routledge.

“Very little consideration is given to exploring patterns in the ways that low-income and African American children do engage in abstraction, representation, and elaboration….It is clear that school mathematics knowledge is privileged over children’s out-of-school knowledge and that low-income children’s out-of-school math knowledge is valued even less. Given such a restrictive view of mathematics knowledge, it is very likely that mathematical competencies linked to the cultural contexts and everyday life experiences of African American children are under-assessed and under-valued because these competencies do not fall within dominant views of what counts as mathematics knowledge” (Martin 2009, 16-17).

You’re making me think a lot about where they are finding math in their lives. I’m just outside Boston so not a city with a grid or numbered streets and most of my kids live in triple deckers with one apartment to a floor. I lived in a similar apartment when my son was little and we spend a lot of time counting the stairs up to our apartment on the third floor (28 in case you were wondering). That quote is really thought provoking–most of my kids are Latino/a so they have second language issues as well as those of poverty.

I’m thinking about asking my first grade teachers to begin the year asking the kids to tell everything they know about math–numbers, shapes, money, distance, even things they don’t think of as math but are math. They have experience with food shopping, (I’m remembering the little girl–about 5–who was reading her father the unit price and item price for all the different cheeses today at Trader Joe’s), they have experience with packaging, possibly cooking, taking steps, bus numbers and times.

My kids seem to struggle most with number lines which is why I found your lessons so interesting. I don’t think they have much exposure to things that show numbers in sequence (your cow weight tapes blew me away–I’m all of an hour an a half away from you but a cow weight tape is as far from my daily reality as giraffes in the wild). I saw a great thing at NCTM about making number lines with a pipe cleaner with a bead on it and marking 0 and 10 at the ends and asking kids to show where different numbers are. Just like in your lesson the idea of interval needed to be developed.

I’m sorry I missed you at NCTM but so happy I found your blog.

Jo! We moved to Portland from JP. I worked for BU, Tufts, and the Boston Teacher Residency, so was in schools all around Boston. What you’re describing is so familiar to me. I wonder if we met? Which school are you in? Feel free to make this convo private at tracyzager@gmail.com if that’s easier.

As for this quote, I LOVE it. It really pushes me away from thinking in terms of deficits. Hard work, but essential.

I love the way you’re thinking about what the kids do know and how to build on it!

Given that your kids are multilingual also, I’m wondering if you want to visit a few great bilingual math teachers in Boston? I was at the Hernandez for a bit, and the K1, K2, 1 team was stellar. Happy to put you in touch. Also, the 4th grade team at the Hurley is outstanding. Those two schools (both two way bilingual) have figured so much out. Again, happy to put you in touch. Just let me know. 🙂

Sorry we missed each other at NCTM too, but great to be in touch now. 🙂

Hi, I’m Liz Hofschneider, Math program Coordinator from Northern Marianas Islands. I enjoyed imagining I was with you and your kids in that car on your way to dropping them to school. What a way to engage the children to math without telling them it is. I have four children that were in my math class from when they were in jr high to high school. All four followed me even when I have to move to another more “getto” below poverty school just to have me as their math teacher. All four are grown up, the older two are both officers in the US Airforce as pilot and nuclear commander. The youngest almost perfected the SAT and received full scholarship in Georgetown U.

Parents are a big factor to children’s success if they are exposed to early literacy in mathematics. Simple conversations can spur math skills and young children exposed to these kinds of conversations develop active minds.

On another note, I attended the NCTM annual conference in Boston last April, 2015. I wish I could have attended your presentation.

A dilemma as a math coach that I struggle with is finding ways to help our elementary teachers make sense of mathematics in the classrooms and stay away from rote memorization. I need PD ideas or speakers to invite for these PDs.

Thank you for the ideas you post here.

Hi Liz! Great to e-meet you. Sorry we missed each other in Boston. You came a long way!!!

Have you read Children’s Mathematics? http://www.amazon.com/Childrens-Mathematics-Second-Cognitively-Instruction/dp/0325052875/ref=sr_1_1?ie=UTF8&qid=1437963640&sr=8-1&keywords=children%27s+mathematics+second+edition+cognitively+guided+instruction It is a fantastic book and a great introduction into children’s mathematical thinking.

Are you able to bring people to you for PD, or do you use technology to have virtual meetings? I have different ideas for both.

Here are a few elementary blogs that I highly recommend:

https://mathmindsblog.wordpress.com/

http://exit10a.blogspot.com/

https://bstockus.wordpress.com/

https://mathexchanges.wordpress.com/

Best,

Tracy

Hi, I am Kalyn and I was introduced to your blog through a professor. I am an elementary education major and in my math class we just talked about the different types of addition and subtraction problems, comparison being one of them. I loved reading on how you brought that math concept into a real life setting and ran with it. Do you ever feel that students get bogged down with terms and techniques that they struggle to see the actual situation at hand? Thanks for the great read.

Hi Kalyn, glad to meet you, and I’m glad you’re becoming a teacher! As for your question, it all depends on how you’re teaching math. If you teach it from a place of sense-making, then you can help kids focus on the actual situation. If you teach it procedurally, then yes, kids get bogged down in rules and terms and things. In this blog: https://tjzager.wordpress.com/2014/10/18/making-sense/, I cite some teaching techniques that can help you slow things down a bit so kids don’t just pull out the numbers and do something to them. Also, Dan Meyer’s blog is great, and he has developed lots of techniques so kids can enter a problem and make sense of it without rushing to calculate. Here’s one example: http://blog.mrmeyer.com/2011/the-three-acts-of-a-mathematical-story/ Good luck with everything and keep in touch. Tracy