Episode #129: What is Efficiency In Math Class? – A Math Mentoring Moment
Today we speak with Tom Marker, a 14 year veteran sixth grade teacher from Ohio! Tom reached out to us to tackle the big struggle of trying to convince his students to avoid rushing to standard algorithms and instead value strategies that can be more meaningful outside the classroom.
This is another Math Mentoring Moment episode where we talk with a member of the Math Moment Maker Community who is working through struggles and together we brainstorm possible next steps and strategies to overcome them.
- The right balance is between conceptual and procedural strategies.
- How can we manage differing student strategies;
- How to avoid rushing to the algorithm without devaluing this approach from a student; and,
- How calculators can hinder your quest to avoid the rush to the algorithm.
Tom Marker: So what it is, is in a training environment, there's skill acquisition, and then there's skill application. So when you're trying to acquire a new skill, you're more so like a zoo tiger that's taken care of, that's ... You feed the zoo tiger. They're safe. There's not a lot of struggle, versus a jungle tiger who now the environment has changed and they have to utilize their skills for survival. So it's a lot more productive struggle, but it's not as safe of a space, but also, if they're trained correctly, they're able to...
Kyle Pearce: Hey Math Moment Makers. Today we speak with Tom Marker, a 14 year veteran, a sixth grade teacher from Ohio. Tom reached out to us to tackle the big struggle of trying to convince his students to avoid rushing to the standard algorithm and instead value strategies that can be more meaningful for use outside of the math classroom.
Jon Orr: This is another Math Mentoring Moment episode where we talk with a member of the Math Moment Maker community, a person just like you who is working through struggles and together we brainstorm possible next steps and strategies to overcome them. Kyle, are you ready for this one?
Kyle Pearce: Let's do it. Welcome to the Making Math Moments that Matter podcast. I'm Kyle Pierce.
Jon Orr: And I'm Jon Orr. We're from makemathmoments.com and we are two math teachers who together-
Kyle Pearce: With you the community of Math Moment Makers worldwide who want to build and deliver problem-based math lessons that spark curiosity-
Jon Orr: Fuel sense making-
Kyle Pearce: And ignite your teacher moves. Welcome, everyone to another Math Mentoring Moment episode. These are some of our favorites and tonight, we get to talk to a veteran teacher who's got a challenge that I think we can all relate to.
Jon Orr: Before we get into this episode, we chat with Tom, and specifically talk about The Concept Holding Your Students Back, which is a course we have on our learning platform. Registration for that course opens in September, of 2021 but we wanted to just mention it here with you right now, because module one is completely wide open for you to dive into. This course, we spent a lot of time, almost a full year building this course because The Concept Holding Your Students Back is proportional reasoning, right Kyle?
Kyle Pearce: Absolutely. It's sort of like a passion of ours, because it's something that really holds most students back. A lot of students actually fall off the math wagon, when they get stuck in this additive thinking stage and they can't seem to break into that multiplicative stage and trust me, we've been there. We've had students stuck but the problem was, we didn't know what was really wrong. We just thought it was math facts, like if they just knew their math facts, everything would be okay.
In reality, there's so much more to the story and we unpack it in this nine module workshop and course online. You can go check it out at makemathmoments.com/proportions. As Jon said, the first module is wide open for you to explore and you can also add yourself to the waiting list. If you're interested. If you're from a district, and you're making district level decisions, and you're trying to beef up the content knowledge of your educators and help them reach more students and ensure access and equity for every learner, then this might be an opportunity for you to dive into as a school or a district. So reach out to us on the makemathmoments.com/proportion site, and we'd be happy to get on a call with you. All right, Jon, let's dive into our conversation with Tom.
Jon Orr: Hey there, Tom. Good morning, and thanks for joining us here on the Making Math Moments that Matter podcast. How are you doing today? It's actually pretty early today. How are things in your world?
Tom Marker: Doing well, doing well. I think that earliness for you guys is a little bit different for me. So seven o'clock is about three or four hours into my day normally.
Jon Orr: Wow. You got to be on East Coast time if that's true, too, because that is early stuff, Tom. You know what, tell us a little bit about yourself. Fill us in on some details. Where are you coming from? How long you've been teaching, and maybe what's your current role in teaching?
Tom Marker: So I'm about 14 years into teaching. I started as a in-school suspension teacher instructor before picking up one period of math later that year at the middle school level. Then I continued there for four years doing math and Language Arts in a seventh grade inner city school, and then transitioned over to one of the top districts in the state of Ohio, where I again did seventh grade language arts and math, and then moved on to a high school where I taught business courses.
I taught personal finance, introduction of business, entrepreneurship and a few classes like that before picking up and sliding over to algebra at the high school level for three years, and now I'm finally in my current role as a sixth grade math teacher at Olentangy Orange High School, which is about 15 minutes north of Columbus, Ohio and that's where I'm currently at. I'm a sixth grade math teacher and a high school baseball coach.
Jon Orr: Love it, love it and close to Columbus, that is not too far from us. What? Four hours down I-75, I think from Detroit. We're just across the border. So awesome stuff and listening to your 14 year vet experience, going from in-school suspension to some math, and now you've got the middle school math, you were in the high school. I've got to assume that all of those experiences are really helpful, maybe difficult at the time, people always say, especially newer teachers, as they get flopped around to different roles. It's going to make you so much stronger as you go.
You know what, Tom, I'm really curious with all of that experience, going even further back. One of the questions that we love asking our guests is a math moment that you remember from your educational journey. Do you mind sharing with the Math Moment Maker audience?
Tom Marker: Yeah, I'd say one that sticks out for me, I transitioned from ... I went to a private school up until fourth grade and when I was going into fifth grade year, I transitioned to a public school near my house and the content, the curriculum worked a little bit quicker, I felt like at the private school. So we were a little bit ahead.
So when I went into the fifth grade classroom at the public school, I felt like I had already seen a lot of the material. I don't think I fully understood the material, but I've seen it before. So my ability to quickly complete work, whether it was in class, or homework or things along those lines, it made it seem like I was really intelligent in a math class and I would ask for more work and I feel like that's what I received.
I felt like I received more work as opposed to enriching activities or extension problems or real world application projects. So I think, I felt like I was really strong in math, but I wasn't getting stronger and now looking back at it, I don't even think I was strong in math. I think I was able to regurgitate information or concepts that I'd seen in the previous year.
Kyle Pearce: Interesting, and that's so common in our schools today is that if a student is ahead, or the teacher thinks the student is head, or is kind of say more advanced the object is, let's just give them more to do. I think we're definitely doing a disservice there. It sounds like you had that moment that's kind of ingrained into you. I'm curious, Tom, how would you think that moment has then helped shape the way you've become a teacher? Do you use that when students are more gifted than others or quicker at certain ideas and concepts? Are you remembering that when you're in class with those kids? How does that shaping your decision making in the classroom?
Tom Marker: Right now, I teach four core classes. So I have four different sections and some of them are somewhat ability grouped. So we absolutely have some stronger students in certain sections. What I try to get out of those students is just their ability to share and attack and approach problems differently and I think through dialogue, and allowing them to share out things along those lines, and then having them explain errors that they see in other students works.
I think the dialogue and the conversations that happen in the math class now, as opposed to just working on problems, allows students to stretch their thinking to see things from different perspectives. That's kind of my approach to it. So when I find out a student's very, very strong, I try to get them to be able to explain it, to be able to connect it to whether it's previous lessons or future lessons.
I think by teaching algebra at the high school level, I kind of see where they're headed in their math ... I guess I'd say their math career and try to say and stretch them that way, but the conversations that happen in class, and getting students to be able to articulate their thinking and to be able to really share out why they go about a problem a certain way, I think that's huge for me, as opposed to just answering more problems.
Kyle Pearce: I love that and you really had me reflecting as well, because I know even along my journey teaching, when I've had students that were asking for more work, I love how you put that. That's exactly what you got was more work. I'm sure there's so many people listening who might take a pause right now and reflect on their own practice. I think we often are really focusing in on trying to help students who might not be at grade level, let's say or maybe they have some gaps or there's things that we want to do and we really focus on differentiating for those students, but I think sometimes those students who are ready for something a little bit more, a little bit deeper, oftentimes, they get put as the last thing to think about, and what do we do?
We just hand them additional problems and I even think about just how much more impactful ... I remember in one particular class helping a student by basically giving them some knowledge type questions from the next grade level, rather than now thinking about it, I'm like, "Why didn't I actually look at my current grade level and look for some really challenging problems?" Less is so much more.
If I could give one really solid problem to that student, and maybe it's a problem that I never use, because it just wasn't developmentally appropriate for the rest of the class. You've taught me something, and had me reflecting here today. So I love that. It's like you've got some success there with getting students talking in your class and getting them working together. I'm wondering, why don't we dive right in.
I'm wondering if you have any challenges or struggles or what we like to call a pebble in your shoe, which I like to quote from Jim Strachan, a good friend of mine and mentor of mine, that pebble in your shoe. What can we dig into here with you today?
Tom Marker: I would say the most difficult thing I find is ... So we've been big with Desmos and using Notice and Wonder and trying to get kids to see things from different perspectives. So in sixth grade math at our school, I would say fractions and decimals are always challenging and I think they're challenging because of the way that we present them initially.
So we may present it as, we'll say, 2,500 or 0.25. And we don't see it as a quarter or as money or a quarter of a game, or whatever the situation may be. My biggest thing has been lately, I've been trying to get away from the procedural algorithms and things, but then sometimes with the struggling students, they'll go home, and especially we're on hybrid now.
So they might go to school on Monday and be off Tuesday and Wednesday. So they're working, we'll say independently, or with a tutor or a parent at home, who says, "Well, the way you get this answer is, I learned keep, change, flip," whether it's with fractions, and I'm trying to explain to them the why behind it and trying to give them other ways to approach the problem.
I think because, I won't say all parents, but some parents feel like the success is in getting the correct answer and they say, "Well, you'll get the correct answer always. If you do it this way, it'll get you the right answer," but I keep trying to explain to them, I don't know if I've ever used keep change, flip or move these decimal points in order to do long division. Because if I need to do that to do well on an assessment, that I'm probably just going to end up using a calculator when I get older anyways.
So I'm trying to get them to see it's not about answer getting, it's about problem solving. That's been my biggest struggle is just trying to get them past the point of people who say, "Oh, it's that new math, it's new math." No, it's the way that when you sit in, in class and somebody goes, "How does that kid get the answer so quick every single time?" It's because their approach to the problem is different and probably a little bit more efficient, and it's not that they're way more intelligent. They're just more efficient problem solvers. So I guess trying to get them into the "new math", what people are saying is new math.
Jon Orr: I think this is common when we're dealing with students who are going home and talking to parents and parents are understanding the way they've done math and they're showing them techniques and strategies, without thinking them through. It's a constant, I think battle that we all have. I think I understand your frustration, your challenge with trying to get students to see the underlying concepts of what you're working with in your classroom.
I wonder if you could fill us in on some details to give us a better picture of what you've tried so far. So I'm hearing you're struggling with getting them to understand that but what are you doing to combat that now, and then what is it about that, that you're doing that you just don't think you're having success?
Tom Marker: I'm okay with state assessments and everybody says, standardized tests and things along those. We don't teach to a test or anything along those lines, but I'm okay with any challenge that's thrown at the students whether it's a standardized test or not, because I think what we've done, at least what I've tried to do, and what I've grown to doing in my class, is to develop problem solvers.
So, for instance, will present a problem or a situation where I know that there are several ways to go about a problem, but some are more efficient than others. So we'll kind of talk through it. So just a situation may be a basic, your traditional textbook order of operations problem. Think about like a situation where it's a long expression with a fraction bar divided by, we'll say 2,500 or 0.25 and just present that problem and try to get students to talk about, "Okay, where would this problem ever come into play? Where would I ever see or have to attempt to solve something like this," and then we talk about different strategies.
So I think traditionally, if you take that problem home, and somebody sees who's using, again, I want to say, "old school math," they would say, "Okay, well, it's dividing by 0.25. So we're going to move this decimal two spots, we're going to move the decimal up top two spots, and then we're going to divide."
So then we come back and talk in a whole group situation. We'll talk about, okay, how did you approach this problem and what were your strategies? I think as students start to talk, maybe some of your students who are a little bit more, I'll say, they're able to work with numbers a little bit more fluently, they may say, "Okay, well, you're dividing by a fourth, or you could think about how many fourths are inside of 20," or something along those lines when they get there and a kid will say, "Wow, that's really good."
So I think what I try to do is get them in situations where there are several ways to solve a problem, and that when they come back, the conversation starts to go and kids can start to ask questions, but again, you got to create that space in your classroom where kids feel comfortable saying, "Well, why would you do it that way?" Or, "I disagree, I tried this way" and kind of get that conversation going.
So I think that's how I've tried to do is just to present problems that have multiple ways to solve it. What I would say is potentially more efficient ways, not better ways, but just more efficient ways and try to get the conversation started in the class.
Kyle Pearce: You're saying a lot of great things. Like even early in the conversation, you reference the idea of getting students talking and there was opportunity there for students to share their thinking. I'm hearing this mathematical discourse idea coming up again, where you're having students come in and reason and prove. These are all such important parts of math class and actually too, I also jotted down.
I thought it was great you mentioned, like, you look at a standardized test as just another challenge, and I tend to be on the same side of the fence. I don't think we should be completely worried about them, but at the same time, I look at it as it's a great opportunity when you have somebody else writing an assessment for students to work through, because then there's no bias there.
When I write my own assessments, or create my own assessments, there's some unconscious bias there. I'm teaching to my own test. The way I teach is the way I'm going to test and you can probably have a big debate about that, about some of the pros and cons there, but a lot of what you're saying really sounds fantastic. I hear this word efficiency coming up quite a bit and there's a bit of a worry about the rush to the algorithm.
For those who have been a part of our webinars or have listened to the podcast, you know that we talk about that a lot. We are concerned about those things as well. So I feel like we're all in one big happy family here talking about some of our concerns about rushing, and I also heard this idea of how parents sometimes can even influence that rush. "Hey, just do it this way. This is easier. There's that new math stuff again."
So there's all of these challenges, but I want to dive a little bit deeper here. I'm wondering, because I'm sensing a bit of a challenge between the idea of trying to help students get to this understanding of what is efficiency in math class, and what does it mean to you? Because, again, there are so many different varieties and flavors and perspectives on what does it mean to be efficient in math class.
Then there's this algorithm piece where some people look at the algorithm is the efficient way and clearly, I'm hearing that you don't believe that's necessarily where we want to rush students to. So I'm wondering, can you unpack that for us a little bit? What is efficient to you and how do you help try to notice and name it when it's happening in your classroom? Now, these are big questions, and it's early in the morning. So I apologize for that in advance, but I think it's worth us diving in here a little bit just so we can get a little more clear on what that means to you and how we can help shape that.
Tom Marker: I have three kids and my daughter is currently in third grade, and she brought home a math assignment the other day, and I was so excited because the approach that the teacher was taking in terms of teaching subtraction. I share a lot of these stories with my sixth graders, because I think they were taught borrowing and always borrowing. So I talked to them. I said ... Actually one of my co teachers, who's in the room with me during the class said, "You got to be careful, because I often hear you saying this is not a good way."
I try to apologize for that to my students at times because if they need to borrow and cross out and things along those lines, I don't mean to say it's a bad approach, but I want them to think of different opportunities. So for instance, she came home with a problem and I'm making this problem up. We'll just say 67 minus 49. I asked her, I said, "How do you get there? What are you going to do?"
I showed it to my sixth graders and I would say, the overwhelming majority, borrowed. Crossed out and brought it over and those things. So I talked to my daughter, and she said, "We're only allowed to count by ones and 10s, and we've got to get there." I like that approach. So I think a lot of what I try to do with the students is say, "Okay, how do you do this and why do you do it?" Then we start to go from there and talk about, "Okay, well, what if you thought about it this way? What about counting up?"
Because I think when you borrow or you do long division, to me, if it's things that you can think about differently, without writing every single thing out. I know, a lot of people say, 'Show your work, show your work," but I like to say, 'Share your thoughts, share your thoughts," as opposed to show your work, show your work.
So I've been working really hard on trying to get students to see different ways to do it, and to be able to explain those ways, but at the higher levels, at seventh grade and eighth grade, I think sometimes the teachers come down me and they say, "Your student struggles to show their work on this particular problem and I can't give them full credit, because I can't see what they're doing." I think that's where the feedback and that conversations have to happen.
So to get back to your question, how I'm approaching it is just, I think at the younger levels, the teachers are doing a really good job and I just want to make sure that our sixth graders are seeing that the borrowing and the long division, I keep saying the keep, change, flip, all those methods, they don't need to be totally thrown out the window, the algorithms don't need to be thrown out the window.
I almost want kids to write their own algorithms, write their own formulas. I think that's the coolest. Remove the question. I don't know if it was Dan Meyer or who it was that talked about, "In a textbook, give the context, the story context, but remove the question completely and let the kids discover what the question's asking or what questions they think would go with that particular problem."
Jon Orr: Just discussing this is starting to emerge some things that you can see to address this struggling. When we first started this conversation, you wanted kids to see different strategies, but you wanted them to not rush to that algorithm. But then I heard you just say you want the kids to build towards the algorithm, or an algorithm that they've created. I think we would totally agree with that.
We want to put kids in situations. I like you described that you're going to give kids questions that have natural, different strategies to get there. The idea here, is that we want kids to build this fluency, but also create efficient strategies to get there and we want them to catch those strategies. Christina Tondevold talks about catching strategies, and also Cathy Fosnot and Pam Harris, we've learned a lot from them. They're like, "We want to emerge strategies."
I think it is about building towards an efficiency, but I think that efficiency comes from where that student is at that time. It sounds like you're on that path and you're working towards doing that. I think that's great. So I guess I'm wondering ... Because I think we got some tips here. I think you're on the right path but I'm wondering what do you think is not working with what you're doing because you're saying lots of great stuff here, Tom.
Tom Marker: The thing I worry about it is, am I going too much to the mental math side of things? Am I chasing too much of the ... I think every student can and absolutely can think about math differently. I guess I would relate it to and I talked about being a baseball coach before. We often talk about a zoo tiger and a jungle tiger. Are you familiar with that metaphor, I guess you'd say?
Jon Orr: Remind us and all of our listeners about what you're referring to.
Tom Marker: So what it is, is in a training environment, there's skill acquisition, and then there's skill application. So when you're trying to acquire a new skill, you're more so like a zoo tiger that's taken care of, that's ... You feed the zoo tiger. They're safe. There's not a lot of struggle, versus a jungle tiger, who now the environment has changed and they have to utilize their skills for survival. So it's a lot more productive struggle, but it's not as safe of a space but the struggle is amped up but also, if they're trained correctly, they're able to grow through that struggle.
So it's two different training environments and I can almost think of a classroom as being a training environment. So when you're first acquiring a skill, you're helped through that a little bit but at some point, you have to release the experience to the student and tell them, "Just let it rip," and you're going to have mistakes and you're going to struggle through things but through conversation and feedback and continually talking through things.
So I think sometimes I push kids into that jungle tiger mode quickly, but I think if you've created your classroom the right way, and there's trust and they understand that listen, a grade's never permanent. A grade is just where you are. It's a snapshot of where you currently are and that we can grow through this but you got to be okay with the struggle piece of it. I think I have to do a better job of gauging the temperature of the room and making sure that every student is ready for that jungle before just sending them on their way because I think some get worried and I definitely don't want shutdown mode.
So especially in the times that we are now, I always talk about active participation, and it doesn't always mean speaking in class, but it's the eye contact, the body language and the facial expressions. Well, the facial expressions are out the window with the mask. So I've lost that piece of your formative assessment of just trying to read the room.
That's my biggest struggle, I think right now is when you have a group of students, and just, I'm trying to push, push, push to thinking differently and if there's not a true love for learning or true love for math, I've got to find a way to get it there. Because you can't be exposed to the jungle if you don't have an interest or a curiosity, I guess.
Kyle Pearce: I think we're very similar in that thinking that we like to create sort of that jungle environment, we'll call it. I've never used that analogy, but I think I'm going to hang on to it. So thanks for that there, Tom, but something that we've talked a lot about as well is this idea of trying to build that culture. The word that comes to mind throughout this conversation is trying to and I always have to remind myself of this. So this is not a judgment by any means, but I'm the type of person as well.
I'm a kind of go bigger go home type person, that's my personality. I like to be all in or all out. The part that's challenging about that is that sometimes I lose sight, and I lose track of the balance piece, and it can be really hard because I want all students to achieve at high levels in my math class. I plan, I'll call it plan hard. I plan for the jungle.
I want to make sure that every lesson is going to be a good experience and I want everyone to enjoy it as well. Not just get the math piece out of it, I want it to actually be something that people are enjoying along the way. Sometimes, if I lose sight of that balance piece that could be really tough and thinking about the culture of my classroom, and what am I saying and what am I doing in my classroom?
We talk about that a lot, as well is this idea of sometimes I say one thing and what I do suggests another. So am I articulating what I value through both what I say and what I do in my math class, and are students getting that? Are they getting that message? It sounds to me like you're giving a lot of open problems and these are a few assumptions. I'm wondering, what is the typical lesson look like? Can you give us, in this scenario, when you're saying this is my challenge that I'm bringing kids to the jungle.
What does it look like when you enter the jungle? When you get into class and let's say it's a newer concept. So if it is, you gave the example of grade three and borrowing and subtraction, that sort of thing. What would it look like in your grade six class, if you're entering into a new idea? You mentioned fractions earlier, maybe it's around fractions, operating with fractions or something along those lines. How does that look and sound for us just to make sure that we're getting a sense of what that math lesson looks like? Then we'll try to start hashing some ideas on maybe some thoughts and steps that we might take to move forward.
Tom Marker: So last year, I made the argument that in sixth grade math, based on our curriculum, the majority of problems can be looked at with ratios, proportions and tables. So I try to make a lot of connections to tables. So they have that skill. I know that they have that skill, that was one thing that I want to make sure they can see relationships between numbers within tables. So with that in mind, the majority of our units will start with what I'll call a hook or a story.
When I talked about getting up early, a lot of my preparation and things for class start through ... In all honesty, it's listening to this podcast and hearing some other ideas from other teachers and that starts in the morning. Then I just started to think through things and I'm reading, and some people say I'm addicted to Twitter, but I read on Twitter or I'll catch something. I'm like, "Okay, that connects to the lesson."
So a lot of my lessons start with a video, a picture, or a story. So I've had people in the past, say, "Can we get guided notes for student?" Whether it's an intervention specialist may ask for guided notes. I don't know what the notes look like until we start to go because they take over the class. So if you think about, I'll present a story or a situation or a video and I've gotten pretty good at Desmos.
So I'll usually put it in there with Notice and Wonder and just let them start to talk through that and just start to say what they notice about it, what kind of problems they think they can link to it, what kind of math concepts. That's how the day usually starts. If it's not a Desmos, and it's a situation with a SMART Board, where I'll just present a problem, we'll talk through a situation. I'll tell them ... It could be made up or fabricated in some way, shape, or form to kind of get us to the concept but I'll always start that way. So just to get them thinking about a situation, and that's every day.
So that's every day starts either like I said, a picture, a video or a story. Then we start to get into the concepts and then if we branch off into terms ... Sometimes we'll talk about terms and that's where my language arts background comes. We'll talk about root words and things along those lines just to tie it all together, but as far as the lesson ending, we don't have homework.
When I say traditional homework, it's not go home and do 10 to 12 problems on this. To me, the best homework is when a student leaves the class and continues to think about it. Like man, I wonder about this and that and they start to think about it in their own life. That's normally our homework and then they come back the next day and I say, "Did anybody make any connections with what we talked about yesterday?" The normal lesson, like I said, starts with a video, a picture or a story and that's how we jump into every single problem.
Kyle Pearce: I'm hearing lots of good stuff to encourage discussion in your classroom for sure. I think we articulated, Tom that most kids in my grade nine class, when I teach grade nine, when I think about this, I've got lots of kids who have come in with lots of different background knowledge, and especially algorithms or procedures, different ways that they've learned things through their parents or through their classrooms from different elementary schools, different classes.
What I notice my kids want to do is they want to get done the fastest. It's like, all of a sudden, before the culture has changed in my room, before ... Especially this is early on in the course. Kids will view math as a, let's get done kind of subject and they naturally will try to do the math the fastest they can get it done so that they can move on to get the next thing done.
It's like math, for some reason is a get done type subject instead of an exploration, discovery, naturally just absorb or learn. The quality of learning is not like ... Most kids don't view math as that subject. They just want to get it over with and especially if we're like, these are the questions to do then it's definitely like the faster I get them done, then the more free time I have.
I find that my students do that, and therefore they're going to try to naturally progress through to the fastest solution, like how do they get that done as quickly as they can. I've taken a lot of time to change mindsets, like change mindsets with how we view mathematics. It's a lot of discussions about not getting done subjects about what I'm valuing, like actually saying out loud that I value the strategies that you're coming up with. I want to see those strategies so that I can help you connect ideas together, and if you're using this standard algorithm in this particular way that I want to show you or help you see how another student's strategy has done differently is connecting to the strategy.
I guess this is a long way of saying, Thomas, and I don't think the algorithms themselves, especially when I get to grade nine, or maybe seven or eight is necessarily bad. I think they're developed with a purpose, but I think we need to make sure that kids understand the underlying concepts of them. If I feel comfortable in my classroom that a student understands or has demonstrated that they do understand how the algorithm is connected to say, a double number line or an area model, or other ways, ratio, tables.
If I feel that that comfort level is there with their understanding, then I'm completely fine with them using, say an algorithm. I don't have to say, "Hey, go back and show me the table." Because I feel comfortable that if I did say that, then they could show me that. I think we want to make sure knowing our students where they are in their thinking is so important, and I think that can be helpful in saying, "Okay, I'm okay with the algorithm over here. This student might not be at the algorithm stage yet, because they're still moving into that progression," but also might be like, "Hey, I ran to the algorithm," but they didn't show these other techniques.
It's like, I think when I do that with students, or try to get them to see the underlying concepts, it's continually modeling. You as a teacher, I as a teacher continually model how that looks in a different way, like an area model or a number line. Then those strategies will get connected to the students because I think we want to move kids along so that, yes, they can process things a little bit faster, but also see those underlying concepts.
So I don't think the algorithms themselves are bad. I think it's just like, knowing where our kids are in their journey. I'm going to stop talking and see what you think about that and any comments you have about thinking that way.
Tom Marker: I think I've said it probably incorrectly or probably should have worded it differently in terms of the algorithms because I often feel that if you're using a standard algorithm, a lot of times that computer or a calculator does that for you now. So if you use a traditional and write out a traditional algorithm, are you going to ever do that outside of school? Where I feel like the skills that we're trying to teach are things that you can utilize outside of school, and that's my whole worries.
Yeah, the algorithms are great for passing an assessment, or getting an answer on a regular multiplication problem, but how can we think about it differently, because if we take the ... We're doing unit rates, and I talked to the students about, the algorithm would be inaudible the method or the procedure would be to move the decimal and then divide.
I try to talk to them about different strategies. If we were at a store, and you're going to pull out a piece of paper and move the decimals and divide, at that point, you're probably using your phone and a calculator. So not that the algorithms are wrong, but more so I want to give them something that they can utilize going forward. So where parents or tutors or people who say, "This is the new math," I would ask them, "How often do you use borrowing in your real life? In your real life, are you still using it or did it work really well to get you to pass that sixth grade assessment 10, 15,20 years ago?" Because I want to give you something that you're going to use day in and day out.
Kyle Pearce: Something to I'm hearing as well, Tom. I'm picturing how often people try, and I say ... This is mostly adults, I'm going to argue. When they're trying to do what we'll say is "mental math," it's like they're just doing algorithms in their mind. For many adults, some will just give up just like some students, where they're like, "I can't hold all of that information in my head." Oftentimes, it's because we call it, it's like, we strip the problem down.
We make it a naked problem. So you're in the real world, there's context, which is so helpful for us to use mental math strategies, and we've been taught or promoted throughout our own education to strip down the problem, grab these naked numbers, do some stuff to it, and then come bring it back and try to make it makes sense in the context again, and I'm hearing that from you and I completely agree.
I don't want to teach to algorithms, but one thing that I'm starting to realize myself is that one of the challenges we have is, there's two kind of groups of students in our classroom when it comes to procedures and algorithms and concepts. I'm going to argue that there's the group of students that are ready to have strategies and models and big ideas emerge. So we'll call them, they're like new to a concept, or they haven't memorized an algorithm yet.
It's like those are the students that are, to me, is so fun to work with, because it's like you can help these things emerge. Then you've got this huge group in our class were either at home or in a previous grade level, they've been shown these algorithms that they've been told, and they now trust these algorithms work, but they don't necessarily understand why they work. I kind of picture you standing in the middle of these two groups and the challenge is and this is only an interpretation based on this very, very short conversation, but the challenge is, is I feel like you're like I really want to emerge these ideas over here with this group, that they're ready to slowly let it emerge.
But then it's like the rest of the group over here has these strategies and it's almost like polluting the minds of these other students, saying, "Well, if you just move this over here, if you just change and flip, that's going to work." Then the other students are like, "Okay, I'm just going to blindly follow," and it's like, you're stuck in this sort of game. I feel like one of your, maybe, at least next steps could be something that I hear you're already doing, but I don't know if you're realizing that you're doing it.
I wonder if maybe just a little more intentionality around this idea might help, but you're talking about having students have conversations. There's math discourse, there's so many things going on here and what I'm wondering is, first of all, to us, calculators, let's push them away. I used to have calculators all the time. Kids were using calculators and calculators, oftentimes promote this idea of speed, as efficiency is speed.
What I realize now is like if I want these ideas, these strategies, big ideas and models to emerge in my math class. I want to be intentional about the quantities that I'm using, so that they lend themselves to students modeling it mathematically and using strategies without calculators. Then the second thing is, when students do come with the move, the decimal or the flip and change and all of these different, cross multiplying whatever it is that they're using, the one thing that I feel could help you get to this next place is, rather than us trying to demote the use of an algorithm, I love playing the play dumb role as the educator.
It doesn't matter which side of the room I'm dealing with. Students who are emerging strategies and models, or students who are already coming with a algorithm they trust, and they believe will work. I have the same response for both sides, which is, "Wow, that's really interesting. How does it work? I have no idea whether I should believe you or not." Sometimes I wonder if us just playing like we don't know this algorithm actually works, is a means to get students to ... Maybe they're going to keep using it.
It's going to be hard for them to break the habit. They've got this habit, that they've been using this thing, they trust that it works, they have no idea why it works. My goal for them is that conceptually, I don't know if you actually understand why this works and for me, mathematical proficiency, just like as they've outlined it in the adding it up document, comes down to those five strands of math proficiency and two of them, one of them is conceptual understanding. The other is procedural understanding.
There's adaptive reasoning, strategic competence, and there's productive disposition. All five of those things are what go together. Those five strands are what make mathematical proficiency, and I am a true believer that if a student comes with an algorithm, and they can spit out answers, I'm like, "That's great. You have some procedural fluency." I would even argue they don't even have complete procedural fluency. Because they define procedural fluency as like a true fluency, like they understand how it works and they can get themselves out of tough spots using these procedures.
I would argue memorizing it, that doesn't actually mean that they've got a complete checkmark in that column. So when I'm picturing this, like, I've got this picture of Tom standing in the middle of the room, and you've got two different groups like these, and of course, they're all at different places along this continuum. You have students who are like fresh out, they're like eating up your open ended problems and trying to emerge strategies and models, but then this other side, they're like, "I already got the answer."
I guess my wonder to you is like, can I respond in the same way to both groups and say, "Wow, that's fascinating. I don't know why this works," or how do I trust that what you're saying is true, or any of those things to kind of nudge them along, so that they have to go back to the drawing board and go, "You know what? Mr. Marker, I don't think he's just going to accept 28 as the answer. He's not going to just say, that's good."
When students come to me with the algorithm, oftentimes, they'll say things like, "Is this right?" I'm sure there's teachers at home nodding their head, like, is this right? I used to say, "Yep," or I would say, "Oh, you're on the right track." They know right away. It's like, yep, I'm right. Okay, I'm done, or, oh, you're on the right, oh, I made a mistake and I have to go back and do it. Now, my response is always the same thing is, "I don't know, is it?"
If you just say, "I don't know, is it," to those algorithm friends, they have to go back and go, "Well, yeah, I borrowed right, or I changed and flipped. I did all of these things." But at the end of the day, if they come back to you, and you say, "Is it right?" They say, "Yes, it's right." I say, "Well, how come?" They say, "Well, I changed and I flipped." You say, "Well, I'm not very good with that type of thing."
I always say to kids, "I don't know, I'm not very strong with formulas and stuff. I need to see it. So help me see it." That could be maybe like a step in the right direction. It won't change it immediately, but I'm going to flip it back to you and get your perspective. Are you seeing yourself in that room with these two groups of students, or am I completely off base on that? I'm really curious to get your perspective on it.
Tom Marker: It's dead on and I love the approach that you take with that. I think the one thing I try to do, too as far as like, I don't know, is it is, can you put context to it? Because I think you guys talked about it. I don't remember the podcast number, but you talked about reasonableness of answers and that's, I think, one of your standards or something along those lines, where students have to be able to make sure that their answers are reasonable.
I think, like you said, "I don't know. Is it right?" I would moreso say, "Is it reasonable?" But they have to have context. They have to give the problem context. So I love the idea of explaining the algorithm or explaining the method. We have a method that we've been using for a couple of years that I got upset and frustrated with this year in terms of least common multiple and greatest common factor in that I don't know if you're familiar with like a birthday cake method, but it was really frustrating to me because the problem may say, "This bell rings every four seconds and this thing happens every 12 seconds. When will they occur again?"
Kids kept getting the answer and I couldn't explain why a birthday cake method worked for the answer. It was frustrating to me. So they would say, "Well, should I use birthday cake method here?" I'm like, "I have no idea what you should do here. Here's how I would approach it. How would you approach it?" They're like, "Well, I want to know whether I should use birthday cake method." I'm like, "Gosh, one, we're calling it a birthday cake method, which is embarrassing to start with, and secondly, I don't even know why this thing works perfectly." It does.
So that's my big struggle too. I know if they say, "Well, I'll just give it a shot and try birthday cake method and get it right," and I'm like, "Oh gosh, I don't know where to go with that, because the students are struggling, they got the right answer, but they have no idea. Should I correct them? Should we talk more through it?" There's some things that we've done in the past, right or wrong. I think you continue to evolve and develop and reflect and gosh, I don't know if I'll ever use birthday cake method again for, because it's so frustrating to me that one, the name. It just makes me cringe a little bit and then the idea of why it works.
Jon Orr: There's a perfect opportunity to say like, "What is that? Explain it to me. Why is that working?" Tom, this has been a great conversation. I'm wondering at this time, how are you feeling right now about the call? I'm thinking, what do you think a big takeaway from our conversation here has been today and also what is going to be your next step?
Tom Marker: I love the part about probing and just saying, never answers it right with yes or no. To me, I liked the idea of, well, explain to me why you ... Justify why you believe it's correct or why are you even asking the question? What makes you nervous about why it could be wrong?
I've listened to every podcast, every one that you guys have, and there's so many different takeaways, people's different perspectives, people's different approaches, right or wrong and I just think of it as a big professional development opportunity for me and listening to the podcast. So hopefully just talking with you guys and hearing your approaches to things that I've been nervous about. I just think I need to get better at ... I think you mentioned it just on the podcast that was released this week about asking the right questions and being able to make sure you're asking the right questions to get the students to not only think, but to reflect and to evolve. So I think that's some of the notes that I've taken is just the importance of how you ask a question and how you answer their questions as well.
Kyle Pearce: I love it. It is something as well, I know with Jon and I, especially doing this podcast and I'm in a role in my district where I get the opportunity to learn with educators. So we've done a lot of reading on coaching and oftentimes if you pick up a good coaching book and we always talk about The Coaching Habit as a book that has really influenced how we do things as math coaches, but I would argue it also has influenced how we interact with our students as well.
So if you think about some of that technique that we described with our students, it's like in that book, a key thing that I try and I try to use it in life just in general, is this idea of stay curious a little bit longer and hold off on advice giving just a little bit longer. I love how open that line is because even in this scenario, it's like, even though I want to say to a student, "Using the algorithm when you really don't understand what the heck you're doing is not going to be helpful for you in the long run."
That's what I want to say, but how that's interpreted or whether that actually has any valuable impact on the student is probably pretty low. The chances that that's going to work is pretty low. So it's like if I can ask a question, that's this curiosity part that he says is like, stay a little curious, just a little bit longer. It's like, if I can ask students those questions, maybe I can help nudge them along this journey without them necessarily realizing that I wasn't a huge fan of the algorithm.
So again, it seems like just small little tweaks that you can try just to see how they work with students and see where they go. Now, you said grade six, you talked earlier about how so many problems can be solved through ratios, proportions, and tables and we tend to agree. Middle school concepts, and then even here in Ontario, into grade nine, proportional relationships is such a big and such an important concept, but I would argue it's like one of the muddiest areas of mathematics.
There's so many people who know how to solve proportion problems, but the understanding of ratio, the understanding of what is a rate, how do we reveal rates? Where do they come from, and all of the journey leading up is really muddy. So for you taking the time, joining us here today on the podcast, we're going to be hooking you up with a seat in The Concept Holding Your Students Back, which is a full course on unpacking key concepts in proportional relationships. Now, we're not suggesting that you aren't doing a great job with that, but we did want to give you something that might help push your thinking as well and give you some tasks that we go through throughout the course.
It's a nine module course and usually done over 12 weeks, but we'll give you all the time you need there and we'll just give that to you, because I'm really curious to see whether there's anything in there that helps you along this method, because we'll take like full lessons and we'll unpack how we'll approach them and some of the questioning that we might have for students along the way, noting that we know that some students are going to try to resort to an algorithm and something I think is really important too, as being that Mr. Marker, standing in the middle of these students who have some students over here who are emerging strategies and models, because they're like fresh coming into this concept. Then you have these students over here who know the algorithms, and the hard part for me as the educator is I need to be able to notice and name where students are.
Jon mentioned this earlier, where they are in their journey and I have to help try to connect them because only when a student on one side of the spectrum and the other side of the spectrum, only once they see that what they're doing is actually the same stuff in disguise, that is something that I think is so helpful for us, especially for someone like yourself, who's in this scenario where you feel like it's almost like two completely competing priorities in a classroom based on where students are coming from. We would argue that if we can help connect those ideas in our consolidation, that will help both sides of the room value, oh yeah, that student's approaching it this way and I can see that idea in my solution. We do a lot of that inside of that course. So we're going to give that to you.
We'll email you all details after the fact, but my friend, we want to thank you for hanging out with us. In the show notes for those who are listening, we're going to have a ton of links. For example, I heard about multiples. So we're going to add Hour Glass Multiples, which is a great task on multiples. You talked about fractions earlier. We're going to add the Wooly Worm Race, which is a complete unit of study for folks to access.
So lots of good stuff that are going to be in the notes, but at this point, I'm wondering Tom, is it okay if we get in touch with you, let's say like nine to 12 months down the road to see how things are progressing on your end, to see how this journey you're on is coming along?
Tom Marker: Yeah, absolutely. I'd love that. I'd love that.
Jon Orr: Awesome stuff, Tom. We want to thank you for joining us here on the podcast and we're looking forward to seeing you in that course, The Concept Holding Your Students Back, and we'll talk soon.
Kyle Pearce: As always, both Jon and I learned so much from these Math Mentoring Moment episodes. It's really great to be able to not only hear what educators are struggling with, but it reminds us of those same struggles that we experience when we're in the classroom and it really gives us an opportunity to dive deeper and think and reflect on ways that we can overcome these challenges to ensure we reach every learner in the classroom.
Jon Orr: A great way to hold yourself accountable is to write it down or even better, share with someone or your partner or a colleague, what you have learned. Maybe head on over to the Make Math Moments community area, and tag us at Make Math Moments in our free private Facebook group or a hit us up on Twitter.
Kyle Pearce: Awesome, and Jon, we don't go and just randomly email people to join us for these episodes. We need to hear from you. What are you grappling with? A lot of times people think like what they're grappling with is sort of unique to them, but in reality, there are so many struggles out there that so many other people are thinking about and struggling with. So why not apply over at makemathmoments.com/mentor.
You just add in a couple sentences, basically on what it is that you're grappling with and we will get in touch with you to discuss a time and date that we can hop on for a Math Mentoring Moment episode.
Jon Orr: In order to ensure you don't miss out on new episodes as they come out each Monday morning, be sure to subscribe on your favorite podcast platform.
Kyle Pearce: Show notes, links to resources, and complete transcripts to read from the web or download and take with you, you can grab them over at makemathmoments.com/episode129. Again, that's makemathmoments.com/episode129. Well, Math Moment Maker friends until next time, I'm Kyle Pearce.
Jon Orr: And I'm Jon Orr.
Kyle Pearce: High fives for us.
Jon Orr: And high five for you.
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