Watch the video or read the transcript below:

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• Mike’s Beyond Answers Mini-Course is now only available inside the Build Math Minds PD site
• Van de Walle book
• Beyond Answers book

So, any time that I can get him to do some trainings for us, you guys, it’s not just for you, it’s for me because I love learning from Mike.

So, here he is.

Hi. My name is Mike Flynn. I am the Director of Math Leadership programs at Mount Holyoke College and I’m the author of Beyond Answers Exploring Math Practices with Young Children. And, I’m going to open up here with a true confession and tell you that I never liked math until I was 29 years old. And I’m talking anytime, when I was a student and when I was a teacher. I didn’t like to learn it and I didn’t like to teach it. And it usually surprises people to hear that because this is all I do now in my career is I work in the math education field. And yet for most of my life, I didn’t like math.

What Has Learning Mathematics Looked Like Before

So imagine, if you were at my house, I live about two hours west of Boston, and you were visiting. And I gave you a destination in Boston. And I had you memorize my directions to get there. But, you can’t take GPS and you can’t use a map. And, the only thing you can do is hold on in your memory all of my steps to get somewhere in Boston. How would you feel if I sent you on your way?

Most likely, you’d be anxious the whole way because even though you may be able to remember the directions, and you remember the steps you are supposed to take, every single time that you’re making a turn, you don’t have an idea of where you are in the city. So, if you’re going straight on Kingston, great, you know Kingston.

You’re supposed to take a right on Bedford. Well, you know you’ve been on Kingston, you know you’re looking for Bedford, but you don’t know where you are in the city. And there’s something that’s unsettling about that. And it’s not unlike how I felt in math.

When someone said, line the numbers up, and then do this in this one column, and then do another thing in this other column, every time I’m going through those steps, I’m not thinking of the size of the number. I’m just looking at a very narrow part of the math. And that’s how math felt. I never really understood where I was.

And, in either case, whether you’re following someone’s directions to solve a problem, or you’re following someone’s directions to get somewhere in a city, if you make a mistake, then you’re pretty much lost at that point. You have two choices. You can back up and retrace your steps, or solve the problem again, or you can pull over and ask for help, or raise your hand in the classroom. Either way, it doesn’t feel good to have a vague understanding of what you’re supposed to be doing.

If we want someone to understand the area and if we want someone to know how to navigate somewhere, we need to give them more than just directions. We want them to actually learn the map. We want them to learn about the neighborhood. We want them to be able to explore it. To muck around it. To see the structure from that. And, so they can learn that there’s more than one way to arrive at a destination. And, in fact, sometimes, depending on the time of day, one way’s a lot more efficient than another.

And so we want people to have flexibility. We want people to be able to navigate a city really well. And the same is true for math. We want our students to have the same knowledge and understanding of mathematics as “townies” do of Boston. In other words, we want them to be “wicked smaht”.

So, what happens, is I think in math, when I grew up, and I’m sure when many of you grew up, math was all about answer-getting. And, I don’t think that’s what we should be focusing on. It really comes down to this idea of answer-getting versus problem-solving in mathematics. What does it mean to do math? What are we actually doing as math teachers?

To me, I grew up with answer-getting. Teachers showed me the steps to get to answers and we just practice the steps and then follow them over and over again until we’ve memorized them. Versus problem-solving which requires students to actually make sense of the situation and use tools and representations that make sense to them.

Answer-getting is reliant on tricks and memory devices to help kids find shortcuts and easy ways to get to an answer, versus problem-solving which pushes kids to develop an understanding of the operations and how they relate to one another. So, there’s a big difference here.

Shifting From Answer Getting to Problem-Solving

Classrooms that focus on answer-getting are really focusing on routine expertise. And this is defined as knowledge that’s memorized by rote. It’s the knowledge that can be used in isolation and understood in isolation, but it’s not understood in another context, when things are even different slightly. And, it’s really a challenge because when we focus on routine expertise in the classroom, we create what is called an illusion of understanding. And this is the appearance that kids know something, when in fact they don’t. And we’ve all seen this. And I can just share a quick story when this happened to me.

It was when I was tutoring some third graders for the state exam, and their teachers had told me that these kids had memorized multiplication. They’re all set with it. They don’t need practice with that. So, I thought, I’m going to start with a multiplication challenge for them. Just a quick problem. Just to give them a quick warm up and some success, since that was something that was going to be easy for them.

So, I gave them this challenge. And immediately all of the third graders put their hands up, started doing something with their fingers that I’d never seen before.

So, I said, “What’s going on here?” And they all got the answer, three, really quickly, and they’re all excited. They were excited to tell me about this thing called the finger trick.

I said, “Well, how does it work?” And they said, “Well, if you want to do like five times nine, what you do is you drop your fifth finger, and then you count the fingers to the left of where you dropped your finger and that’s the first digit, and then you count your fingers on the right and that’s the second digit, so the answer’s 45.”

And I’ll be honest, I was floored. I’d never seen anything like this before. So I said, “Does that always work?” I really didn’t know. And they said, “Yeah.” And I said, “So for any multiplication?” They said, “Totally.” I said so, “Like for seven times six.” And they said, “Yeah.” And so I started to do on my fingers, and I thought, “Wait a second. That doesn’t work.” And a few kids said, “Well, I think it only works for nine.” But there was some debate. And I realized, something’s wrong. This is an illusion of understanding. So, their teachers were telling me they were fine with multiplication. They memorized their facts, which they had. They memorized their facts. But, did they really understand multiplication?

So, at the time, I was reading Van de Walle’s work, and he had been talking about this Conceptual Understanding Pentagon. This idea that if kids know something conceptually, they should be able to contextualize it, create models of it, express it in words or pictures or use equations. And, I thought, let me try some of this with the class.

So, I said, “You know what? Can any of you tell me a story for nine times three equals 27?” And Julian raised his hand and said, “Oh, you have nine pillows and you times them by three.” And I said, “Oh. What do you mean times it by three?” He’s like, “I don’t know, you times ’em by three.” And then another girl raised her hand and she says, “Nolan has nine books and he buys three more.”

And I thought, “uh-oh, we have a problem, here. So, they can’t contextualize multiplication. And that gave me a window that maybe there’s a piece of multiplication they don’t understand yet. But I wasn’t going to give up on them. I said, “Alright, well let’s forget that part of the pentagon. How about a model?” I said, “Sometimes we can create a model to show what the math means.” I said, “Can you make a model for nine times three equals 27.”

And all 10 kids I was tutoring, all of them got cubes out, and they got nine cubes and three cubes, and then they made an “X” with the other cubes. And that was a representation for nine times three equals 27.

And I laughed in my inside and then I thought, you know, I’m not going to give up on the kids yet. I said, “Alright, I see the nine. I see the three, but the model should show 27.” One kid actually went so far as to count the cubes he had. He saw he had 20, so he got seven more and made the “X” bigger. And that’s a true story. And it’s a funny story, but it’s also a sad story, because these are kids who presented this illusion that they understood multiplication, when in fact, all they’ve done is memorized it.

What I wanted to see was could they give me a story with nine bags and three apples in each bag, or show nine groups of three, or describe nine groups of three verbally or written writing. Or make a picture with nine groups with three things in each group. Or show with different equations what the math actually means behind it. And they couldn’t do any of these things.

Shifting Routine Expertise to Adaptive Expertise

And, so that’s the danger here with the illusion of understanding and with focusing on routine expertise. So, what do we do about it as teachers? We need to develop Adaptive Expertise. This is meaningful knowledge that can be applied to unfamiliar tasks as well as familiar ones. It’s the knowledge that is learned and stored in relation to other knowledge. It’s constructiveness. It’s building on ideas. And, It’s rooted in conceptual understandings. And so it has deep meaning built into it, and students have ownership of these ideas. It’s part of it. They have autonomy in this class.

But to do this, requires a big shift in our practice. We have to move away from the way math class used to be taught. Think about it. When you were in school, there was this I do, we do, you do approach that happened where the teacher would lead the lesson and show the kids what they were supposed to do. And then we would practice that together, and she would check with us and see how we were doing. And then, we would have to demonstrate it independently in some kind of quiz or test. And we’ve got to flip this.

I Do, We Do, You Do Shifts to You Do, We Do, I Do

So, the first thing we’re going to do is we’re going to get rid of this notion as the teacher as the giver of knowledge. The lecturer at the front of the room. And then, we’ll flip it around and we begin by putting the students at the center of this. We begin with a problem. The students get a problem, they work to make sense of it for themselves. And then, they collectively collaborate on this. They work with partners and share their ideas and have math arguments and critique the reasoning of others. And they work from places of understanding. And they grow from that. And then the teacher at the end comes in. Not at the beginning, but at the end to synthesize these ideas and pull ideas together and help kids make connections between them.

And we can see this with a nice visual that I think about. It begins with a launch. That the whole class starts together and we can give them a problem. We lead with a problem.

And then there’s this nice release, where the class goes off and they work either alone or with partners. And they collaborate on ideas together, and they share their thinking. And this whole time the teacher’s walking around the room and observing and talking with kids, offering support and challenges, interventions. Anything that has to happen, but also beginning to orchestrate the conversation that’s going to happen later. So that, the ideas that are generated from the class, the teacher’s then beginning to find the connections and planning a really rich conversation at the end for this big Return.

So the Return is when everyone comes back together and the teacher starts to facilitate a conversation where kids share their ideas. But everything’s chosen with intent. So that ideas begin to build off one another and connect. And so here, we have an opportunity for teachers to facilitate learning. Not deliver the knowledge, but facilitate the learning among students. And this is what provides ownership of the ideas. It makes math engaging for kids. It makes it make sense for kids. It helps them understand that area.

And this often requires a big shift from teachers. Even if you’ve already started doing this with students, there’s a lot to the craft. There’s a lot of finesse that needs to happen, and there’s a lot of learning along the way to help us get better at that.

Go Beyond Answers

And this is why I chose this topic for my mini course that I’m doing with Build Math Minds. So, if you’re interested in exploring it, it’s called “Beyond Answers: Exploring Mathematical Practices and Developing Mathematical Reasoning in Elementary Classrooms“. And what we hope to do is dive into some of this work together. We’re going to do some math together where you’re learners and you have a chance to experience some struggle. And you get to analyze what it’s like to be a learner. So then, as a teacher, you can have a better sense of what your students need and how we can work to facilitate that learning versus teach it directly.

And so you’ll walk away from this class with:

• Concrete strategies, frameworks and routines that support our shift from being deliverers of knowledge to facilitators of learning.
• You will deepen your own mathematical knowledge so you can deepen your students’ mathematical knowledge because we know the more teachers understand the math, the better able they are to help their students understand that math.
• And finally, you’ll create, adapt and learn about rich tasks that support students’ mathematical reasoning.

So, I opened this video by saying that I didn’t like math until I was 29 years old. So, you’re probably wondering what happened. And, at 29, is when I had my first real meaningful professional learning experience in math. I had a chance to actually dig into some of these ideas as a learner and see math for the power that it is. I actually got to understand the math. And once that happened, it awoken something in me. It awoken a passion that math is worth exploring. That math is powerful, and that I actually can make sense of these ideas, and it’s fun to do.

And it changed how I taught. And it changed everything in my career. And, it’s been my mission ever since. After thinking that, “Why couldn’t I have learned this way when I was a kid?” It’s been my mission to now make sure that kids can learn that way. That they don’t have to wait until their 29 years old to find a love for math. And so if you’d like to explore how to make that happen in your own classroom or just for yourself, then consider joining the mini course with me, and we’ll dig into some of these ideas. And, we’ll learn and grow together. So I look forward to working with you, and we’ll see you soon. Take care.

Back to Christina

Thank you so much, Mike. Again, he does not disappoint. And, I hope that you guys took away from some of that journey of what Mike has gone through. It’s very similar to my own journey that I did not learn mathematics beyond just rules and procedures until I was an adult. And it wasn’t until I was getting professional development, getting my Master’s degree, that I learned that there’s a different way to think about mathematics.

And every time that I start sharing stuff like this, just like what Mike’s talked about, is that we wish we had learned this when we were kids. So the biggest piece for me, of why we should be changing our teaching, is so that we can help our students not have that same experience. We want them to get answers, yes, but we want them to go beyond answers. That’s why I love the title of his book because it’s not about disregarding answers. We still need kids to be able to get an answer, and get the correct answer in mathematics, but the learning of mathematics and the teaching of mathematics should be so much more than that. We need to go beyond just answer-getting into truly understanding the landscape of mathematics.

So, I hope that you’ve taken some bits away that you can go back to your classroom and start to tweak a little bit here. But if you want more help in how do we start shifting our own teaching, then I would love to have you join Mike in his upcoming mini-course.

I hope that through that course, you will get that support to help you go deeper than what we can in these short videos. My whole goal with every week when I do videos is to help you build your math mind a little bit. Get a little bit further in our understanding of the mathematics, but sometimes we just want to go a little deeper. So, if you want to go a little deeper with Mike, in exploring how we go beyond just answers in the learning of mathematics, come join him in his mini-course.

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