Today on the podcast we’re speaking with author Sue O’Connell about some key ways to build math fluency. Sue says that many of us were taught math in ways that focused on memorization, and regurgitation. But that doesn’t necessarily lead to deep understanding. To foster true fluency, she says we should start by taking some tips from literature and find ways to contextualize math problems so kids can make sense of them.
We started our conversation by asking, what does good math instruction look like?
Below is a full transcript of this episode!
Lauren: What does good math instruction look like?
Sue: Well when I walk into a math classroom, what I'd really love to see is students engaged in mathematics. So, I love seeing materials out on the tables, and students actively involved in creating models, definitely math talk. I want to hear some talk going in the classroom. Not the quiet little one person raises their hand to answer the question, but the constant turning, and sharing, and kids questioning each other, and talking back and forth with the teacher to share ideas. I love to see classrooms where you can see light bulbs go off, where there's insights during that class, where there are investigations going on and kids modeling, and talking about what they're seeing. I also want to feel energy in a mathematics classroom. I get the opportunities to go in a lot of math classrooms and I say so often that there is nothing that can suck the energy out of a classroom as much as the sit and do a worksheet in mathematics. I want to see kids having fun with math and being excited about mathematics.
Lauren: So then what are some common unproductive beliefs about the teaching of math?
Sue: Oh, there are so many of those, and most of them originated from the way we as adults today learned mathematics. Things like we believe that the more you practice the better you get. So, then that leads a lot of us to think that if we give students 50 of something they'll be better than if we give them five of whatever it is that they're practicing. That's not a productive belief about mathematics. That the idea is it's the answer that we're looking for. It's about a right answer, and math isn't about a right answer. It's about learning the skills to investigate and to figure out how to get to that right answer. Certainly it's unproductive for kids to think there are math people and people that are not good at math. We know that's not true. We see so many wonderful teachers out there today that struggled and believed they weren't good at math when they were students in the math classroom, and have come to be very confident and competent mathematicians.
Another extremely unproductive belief is that speed is connected to mathematics. That the faster you do something, the better you are at mathematics. We know that many mathematicians are not particularly fast. They're slow, and methodical, and thoughtful. I think another unproductive belief is that mathematics is about computations, that that's the ultimate goal in mathematics is that we can do a computation, when in fact what we've realized is just doing a computation doesn't make you proficient in mathematics. You need to be able to apply that. You need to be able to solve problems with that. You need to be able to use those computations when they make sense to use them. We tended to grow up with the idea it was about isolated computations and the ability to get the right answer, when in fact that's only the beginning of really being a mathematician.
Lauren: What do you recommend for teachers who were never taught math in a way that lead to true understanding? So, what if as a teacher math kind of scares me?
Sue: Well, that's probably a large chunk of teachers because none of us, or very few of us, were really taught to understand mathematics when we learned mathematics. We were taught to memorize algorithms, to do them over and over until we could get right answers. Now we turn around and our standards have changed to really focus on understanding. So now we're challenged to teach those same skills that we were taught to memorize. We're challenged to turn around and teach our kids to understand them. Well, we have many fourth or fifth grade teachers that don't understand why you do what you do when you add, subtract, multiply, or divide fractions. Why would we? We were never taught that.
We have many first, or second, or third grade teachers that struggle with some of the deeper place value understandings because all their teacher ever asked was, "Which digit is in the tens column or in the tens place? Which digit is in the hundreds place?" Really what it takes is relearning mathematics. Just looking at those concepts, browsing through any materials we can get our hands on to help us understand the why we do things and when we do them, not just the how we do them.
Lauren: In your work you talk a lot about how questions and questioning support student learning in the math classroom. What are some different opportunities for different questions that arise, and what kinds of questions are most effective to ask?
Sue: Well, probably best to start with the most common question asked, and that's, "What's the answer?" And why we want to veer away from that. It's not that the answer is not important, but what that question does is it only gives you the answer. It doesn't yield any student thinking. You don't know how that student got the answer, or whether there are any errors in thinking going on in that student's mathematical thinking. It lends itself to one student saying the answer, and then no discussions happen from that. We remember back in our math classrooms it was someone raising their hand and saying, "16," and that was the end of the discussion. It's the follow up questions that are so critical in our math classroom. So, it's okay to ask, "What's the answer?" As long as we're also asking, "How did you get that answer?" That means students are challenged for process to come out, for us to hear what they were thinking to arrive at that so we know whether that thinking was healthy mathematical thinking, or where errors might have occurred.
Another really critical question to ask is, "Why?" Or, "Can you prove that?" The justify kind of question, because that's the, "What were you thinking?" I'm thinking of things like, "Why did you choose that tool to do that investigation? Why did you choose that operation to solve that problem?" We get into the thinking pieces of their choices in mathematics, and again, those kind of questions lend themselves to deep discussions and ongoing discussions in the math classroom. Another huge one that I love today, and probably my favorite question today in math class, is, "What do you notice?" Because I love posing investigations, having students model ideas or gather data, and then instead of telling them what they're seeing, asking them what they're seeing. So, "What do you notice?" Becomes a critical question. "What do you notice about the patterns? What do you notice about our sums? What do you notice about your models that you've created?" Following up on that, "What do you notice?" I like questions that prompt kids to generalize.
So, "What's the rule? So, what's the big idea? So, what did we learn from this?" Again, that would have been something that even a few years ago our teachers would have jumped in and told us what we noticed and what it meant. Here's the rule. "Oh, you noticed that when you added five plus one and one plus five, you got the same answer. Oh, that's because the commutative property, the order of the addends doesn't affect the sum." I want kids to tell me, "Hey, we've done this over and over. It looks like it doesn't matter which order we add these in. The sum is going to be the same."
One other big question that I think we forget a lot is the summarize and closure kind of questions. Great near the end of class, great to get kids thinking about what they just heard and learned. "What did you learn in math class today? What's the big idea about whatever it is that you were talking about or studying today? What was easy about what we did in class today? What was hard about what we did in class today? What would you like to hear again tomorrow because you're a little confused about it and are not sure you quite get it yet?" Those kind of questions that really help us bring closure to ideas and to lessons, and really push kids to summarize their thinking.
Lauren: How is the teaching of math similar to and different from the teaching of reading and writing?
Sue: Well, that's interesting because we used to think they were completely different. Regardless of my math specialization right now, I actually started as a reading specialist. So, there are so many times when I see these connections between the teaching of reading and writing, and the teaching of mathematics. Growing up, I would have never told you there was any similarities between them at all. I guess one of the biggest similarities that I see is that we have recognized for so long in reading that reading is not about just what we'd consider those basic skills like phonics, and your word attack skills, sight words, that really the ultimate goal of reading is application.
The ultimate goal of reading is comprehension, and if students are great at phonics and sight words, and they can call out every word and read you a paragraph but they have no idea what they just read, we say they're not readers. They're just calling out words and yet in the past, we've said, "Oh, you're a mathematician if you know all your math facts, and you know the formulas, and you can get the right answers on this worksheet." We've realized how similar we should be thinking, and hopefully many of us are, to the way we teach reading and that those computational skills are important, but they're not the end all for being a mathematician. What we want for students is that they can apply the same application to comprehend mathematics, but in mathematics, we think of that application as problem solving. As knowing when and why to apply those skills to make sense of mathematics. So, I guess sense making is one huge connection between the two of those.
Now, I think in reading and writing there are many more repetitive skills over the years that you see a skill, and you're doing main idea and detail, or characterization, or cause and effect, over a number of years the complexity of the text changes and the precision to which your skills develop changes, but you have a lot of that repetition. Whereas I think a big difference in mathematics is the sheer quantity of content goals every year. The idea of in a given year, not only are they progressing in terms of their thinking skills like their ability to solve problems, or to model, or to build arguments with that gentle progression over years, but every year we have lots of content standards. Things they need to know about fractions, and decimals, and multiplication, and addition, and geometry, and measurement, and all of that to place in there. So, we need to really carefully balance our skills teaching with that vision of making sense, and applying, and making sure we're still thinking about the big overarching goal as being our ultimate goal. Not get tied down in all of the little skills that are introduced throughout a year.
Lauren: How can teachers integrate reading into math instruction?
Sue: One huge place to take our reading skills and really use them in math is in the teaching of math problem solving. We are trying so hard to pull away from previous practices, things like keywords, where there were quick tricks that we told kids. "Just find this word and it will tell you whether to add, subtract, multiply, and divide." What we've realized is that kids then weren't really even reading that problem for understanding. They were reading it to look for a word, and they weren't learning about operations and what operations look like in real situations. They were just looking for a word. So, it really wasn't a very productive approach to teaching problem solving. I think the more we step back and think about word problems as reading comprehension, it really helps teachers see ways that they can support kids to better understand those word problems because that's a real issue across the country, is that we have a lot of students who cannot comprehend word problems. So, I think about some of the strategies that I talk to students about and to teachers about really comes straight from reading.
One of the first things we do when we're reading is I'll have them retell what was going on in the story. Don't look at the words, don't read that problem to me. Just in your own words, tell me what that problem is saying. That reading skill of summarizing. If your children in reading read a paragraph and you wanted to know if they understood it, teachers will say, "I would never ask them to read it out loud to me. That doesn't tell me they understand it. I'd ask them to summarize." We’ll do that in math. Don't say, "Reread the problem." Say, "Retell me the problem." Another thing we do in reading is we have kids visualize. We say, "Oh, you don't quite understand that? Think about what's going on in that paragraph you're reading. You'll get a picture in your head of what's going on in this story." That's another huge technique in mathematics. We have them visualize not just in their head, but grabbing materials, getting paper and pencil, drawing pictures because we know that visualization supports that comprehension.
We ask specific questions that get them back into that story problem looking for specific information. "Well, what information would we need to solve that? Can you find that information and justify for me why we need it? Why do we need that to solve this problem?" Again, just probing to get kids to comprehend. To me, the teaching of word problems is all about the teaching of reading comprehension, and if we take our best techniques from teaching reading comprehension and use those while we're working on story problems, we're going to get more understanding from our students.
Lauren: So then how does putting math in context help students to engage with the concept and make sense of it?
Sue: I believe putting math in context is critical. For so many people that did not understand mathematics, they were working with abstract numbers, and symbols, and equations. They would get wrong answers but those numbers didn't mean anything to them, and what we see is when students see a context for those problems, those computations, it makes more sense to them. So we used to think about, like the worksheets that we'd see in class. They would be 30 computations and at the bottom of the worksheet, there would be a word problem or two. Again, it definitely said the priority was computations and the problems come afterward, but what you notice in a lot of current materials, is those problems come first. They're at the top of the page. They're at the beginning of a unit. They're at the beginning of a situation that we're doing in math class, or at the start of a math lesson because they're going to give a context to everything else students are doing that day in class.
Rather than students then just manipulating numbers, they're thinking about whatever that story was, and those numbers make sense. We think about things like fractions and how difficult it is for students to solve word problems related to fractions, yet they can do the computations because they've memorized what you do when you add fractions with like denominators. What you do when you multiply fractions. We multiply the numerators, then we multiply the denominators, but they get a word problem and they can't figure out what to do. It's a huge advantage to say, "Let's start with this word problem. Somebody had 4 and 1/3 pounds of oranges, and he used 3/4 of them for a recipe. What would you do?" Then we start with, "Oh, it's 3/4 of 4 and 1/3. It's 3/4 times 4 and 1/3. Where do we see multiplication here? Let's do a visual of it. Let's talk about it." From that point on in class, as students are doing those computations, they may have that orange scenario in their head the whole class, but it's helping them make sense of why the product looks the way it does, of just understanding the numbers and symbols.
So, word problems are a great way to put math in context, and an easy way to put math in context. Just start the lesson with a contextual problem rather than an equation in which the numbers and symbols aren't meaning anything to students. Another really exciting way to do that is through stories, through situations. We've seen how excited students get in a math classroom when a piece of literature comes out and the teacher reads a story, and then mathematics investigations follow as a result of the context of that story. We know how kids love stories, we know the energy that stories bring, and certainly that context helps them see the mathematics in a situation. So, you might have little guys just learning what it means to add, and have a story where there are three little bears, and another bear joins them. Then another bear joins them. Then another bear joins them and all of a sudden, we're writing equations with three bears plus one. Now they're seeing that abstract equation, and it means something because of the story about the bears.
Lauren: So then how can teachers come up with good contexts?
Sue: When I'm looking for stories, literature to use in a math classroom, I don't necessarily always go to the literature that was created to teach mathematics. There are many books out there that are being published that are intended to teach math concepts, and many of those are wonderful, but there's also opportunities just to think about the context and say, "When would you use that mathematics?" So, if I was thinking, "Well, I want to get into perimeter with my students." I might just stop and think, "When would I ever need to know perimeter? Ah, maybe if I were making a garden and putting fencing around that garden, or how about a playground, and we're doing some kind of fencing around that playground? Or, let me see what kind of stories I can find about gardens or playgrounds," and then we can jump from just a lovely piece of student literature into a context that makes sense in mathematics.
So, if I'm thinking about adding two digit numbers, I'm thinking I need more than just a couple of something. I need a story where I'll have lots of something. So, let me look at, ah, I know this great book about a pumpkin patch and the farmer is putting all these pumpkins in the back of his truck. I'm thinking, "Whoa, I could have some pumpkins in the truck and more go in. Some pumpkins could fall out of the truck while he's heading to the market where he's bringing these pumpkins." I just really try and think about the math skill first. What is the skill and when would I use that skill? Then identify some context in which I'd use that skill and look for stories that do that. They can be such fun with kids, and they don't have to have the math in the story. They just have to set the tone, be that springboard to jump off into that mathematics.
Lauren: Once you have that idea, how do you find the book that matches it?
Sue: When I'm looking for stories to match a context, sometimes a story just pops out to me. I just know that book and I know kids are excited about it, and that's great when that happens but sometimes that doesn't happen, and it's a struggle to find that context for multiplying multi digit numbers, or adding multi digit numbers, or things that we don't generally see a typical math book that addresses those. I just think about, again, what's the core of that mathematics? So if I'm multiplying multi digit numbers, then I need something that's going to have very large numbers in it. There's this great book, Pop's Bridge, and it's a story of the building of the Golden Gate Bridge. There's just so many great stats about the length of the bridge, and the height of the bridge, and all the celebration they had when the bridge was constructed, and how many people walked across the bridge. I found ways to weave those into lessons.
Sometimes it takes more of a search to find the right book. So, sometimes I find the math connections when I'm not looking for them. There can be a great book, like How Many Days to America by Eve Bunting. It's a story of an immigrant family and their voyage to America, and as they're going on their voyage, they don't have much food and they have to share it. This family of four has a few pieces of food and they're sharing it between the four of them. It wasn't an intent to use that, I wasn't looking for a math story on showing fractions as division. If I have three mangoes and we share them between four people, how much will each person get? In reading that, that just popped out at me. So sometimes reading the books, it's a nice surprise that you read it and this math idea pops out of it. That one I put aside saying, "Whoa, that would be perfect for showing fractions as division, because that's what they're doing. They're taking a certain number of whole objects and they're splitting them between a certain number of people, and that will work for this context."
At other times, I'm out there searching for the book. I have a math idea in mind and it's looking through my own library of many, many children's literature books, asking friends, sending it out there to media specialists who know books so well and saying, "Do you have anything? Here's what I'm looking for. Something that I could do two digit addition with, so that has 20, 30, 40 of something in it." Give them some ideas and see if they can. Media specialists are great for being able to come up with some of that. Or, I'll just go on Google or go on Amazon and start typing in things like, "Lots of," and all of a sudden it's lots of cats, and lots and lots of books come up with that. Then I'm looking through them and reading them, and seeing if they work for that context.
So, it's a real exploration. Sometimes it just happens and you're reading the book for a literature purpose, and that math just jumps out at you. Other times, it's a real search to just find a book that works. Talk to colleagues, talk to media specialists, get online and Google it. There are so many of them out there. There are so many possibilities for math topics.
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Susan O’Connell has decades of experience supporting teachers in making sense of mathematics and effectively shifting how they teach. She is a nationally known speaker and education consultant who directs Quality Teacher Development, an organization committed to providing outstanding math professional development for schools and districts across the country. She is the lead author of Math in Practice and also wrote the bestselling Now I Get It. Connect with Sue on Twitter @SueOConnellMath