Episode #198: Assessing Multiplication Facts – A Math Mentoring Moment
In this episode we speak with TL Eller around developing assessment tools and routines to capture student understanding of their multiplication facts.
TL is a 3rd / 4th grade educator from Grove City Pennsylvania who has been on a career-long journey to help her students develop fluency and flexibility around math facts. In this chat we help TL focus her next moves on developing some easy to implement ideas on how to capture evidence of her students’ multiplication facts.
This is another Math Mentoring Moment episode where we talk with a member of the Math Moment Maker Community who is working through problems of practice and together we brainstorm possible next steps and strategies to overcome them.
- How listening to students can help us to reflect on “why” and “what” we do in the classroom;
- Why many students may feel that they have opponents in math class;
- What should proficiency for multiplication facts look like?
- What multiplication fact assessments can look like to promote growth;
- How we can develop an assessment tool to capture multiplication facts.
- How we can help students build fluency and flexibility around math facts.
TL Eller: The observation piece I'm getting, but we keep these data binders for standards and our kids are self reflecting all the time about where they're at. And I don't feel like we have a consistent way for those kids to really know what they know and what they don't know about those facts, the way we do the other standards. I'm wanting them to be able to really own and think about the strategies that they're using and even become more efficient with those strategies.
Jon Orr: In this episode, we speak with TL Eller, around developing assessment tools and routines to capture student understanding of their multiplication facts.
Kyle Pearce: TL is a third and fourth grade educator from Grove City, Pennsylvania who's been on a career long journey to help her students develop fluency and flexibility around math facts. In this chat, we help TL focus her next moves on developing some easy to implement ideas on how to capture evidence of her students multiplication facts.
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 problems of practice and together we brainstorm possible next steps and strategies to overcome them.
Kyle Pearce: But before we dive in my friends, have you thought about the pebble that's kicking around in your shoe and have you let us know about it? All you need to do is just give us a couple of sentences. We're not asking for a paragraph or a book on the idea, but just a question, the wonder that's going on in your mind right now, or that struggle that you're grappling with.
You can be invited onto a math mentoring moment episode by sharing that struggle with us firstname.lastname@example.org/mentor. Once again, that's makemathmoments.com/mentor so that you can be a guest on an upcoming episode. All Right, let's get into the discussion with TL.
Welcome to the Make Math Moments That Matter podcast. I'm Kyle Pierce.
Jon Orr: And I'm John Orr, we are two math teachers from makemathmoments.com 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, fuel sense making and ignite your teacher moves. Welcome my friends to another math mentoring moment episode. Today, we're going to be chatting with TL and it's going to be about one of those topics that everybody's thinking about, right? Regardless of what your stance is, everybody is wondering how much should I be doing math facts? Or is it straight memorization? What is the difference between automaticity and just rote memorization? Is there a difference? Does it matter?
And today we start to uncover and peel back what that might look like and sound like in a classroom, but then also how does that actually connect to assessment and more specifically evaluation? Because that's always something that is on our minds. When it comes to report card times, parent teacher interviews, how do we best help serve students so that they gain the skills, the fluency, the flexibility that is going to help them, not only now in your class, but also in future math classrooms and also to build their confidence in mathematics, to see themselves as math people, math students.
So this is a great conversation you'll notice through TL and John, you and I try our best. It can be difficult sometimes, but we try our best to ask more questions than give let's say answers because the reality is that every classroom, every context is going to be different, but hopefully friends here and in particular, TL took away some good ideas and thoughts to bring back to her team so that they can continue along this journey.
Jon Orr: All right, let's hear this interview.
Kyle Pearce: Here we go.
Jon Orr: Hey there, Terry, thanks for joining us here on the Making Math Moments That Matter podcast. We are excited to have you on this week's episode. We think the pebble that's in your shoes going to inspire and to be helpful for many educators. So thanks for joining us and how's everything going over on your head?
TL Eller: Yeah. Things are going really well. We're having a nice relaxing summer where it feels like when we are getting together as a teaching staff. We've had a couple of professional development optional days that we've been able to get together and work. But this is the first summer in two that we haven't just been talking about COVID and how-
Jon Orr: Right.
TL Eller: ... to survive. So-
Kyle Pearce: Exactly.
TL Eller: ... it's been a really reviving summer in that way to be able to start this fall, hopefully, normal, whatever that looks like, so-
Kyle Pearce: Absolutely. It's almost like you're hearing more and more people from all over the world starting to bring the focus back to pushing their own learning forward and continuing maybe where we left off before all of this began-
TL Eller: Absolutely.
Kyle Pearce: ... instead of, as you said, that survival mode that's occupying all of our time, our effort, our energy. So that's awesome to hear that you're seeing that light and it's coming to fruition here this summer. Before we dig in, tell us a little bit about your background story, where are you coming to us from? And what's your role in education and how did you end up in education altogether?
TL Eller: All right. Great. First of all, thank you for having me. I was so excited when you invited me on the podcast. There's a pretty large group of our math staff at the elementary school that I teach that listen to your podcasts and you come up a lot in professional-
Kyle Pearce: All right.
Jon Orr: All right.
TL Eller: ... development. So they're all excited that I'm going to get to do this too. So it's pretty cool, but my name is TL Eller it's Terry Lynn, but I actually go by my initials TL.
Jon Orr: Okay.
TL Eller: And I am a fourth grade math, science and social studies teacher in Grove City, Pennsylvania, which is about an hour North of Pittsburgh. And I actually just finished my 20th year teaching.
Kyle Pearce: Nice. Congratulations.
TL Eller: Yeah. Thank you. And I actually spent many of those years, not teaching math at all. I got my masters in special education and so about... I would say 12 of those years, I was teaching pull out direct instruction and reading. So I would work with students with varieties of many different types of learning difficulties, but students with learning disabilities, speech and hearing impairments, visual impairments, you name it, students with autism, ADHD.
So working with a variety of students that were for whatever reason needed direct instruction pulled out. And then about eight years ago, our district started realigning what they were doing. And I had the opportunity to push into a co-taught math class. And I was super excited about that because even though I had spent all of those years teaching reading, I actually was always a math person, my undergrad, we had to pick an area of emphasis and I had picked science and math.
So I was excited to get the opportunity to go back into a math classroom. So pushed into that math classroom. And it was actually a class. We were ability grouping at the time. So it was a class of, we'll say the 25 lowest achieving students in fifth grade. So they were my students with IEPs were in that class, but it was also students without IEPs.
So there was myself, a regular ed teacher, and the title I teacher all teaching in that class. And about the second day in, I realized that we had a classroom full of kids who were all terrified of the subject. Some days we would just look at each other, we didn't even know what to do. We would ask a question and they would just stare at us blankly. And it hit me because it wasn't just my learning support students that were afraid of the subject. It was all of them.
So it was great because there were three of us and we were able to do a lot of small group things. But you guys often say, you're saying it like louder and slower, but we weren't really doing anything different. That's what we were doing. It never even really occurred to us that we could teach math differently.
Jon Orr: Right.
TL Eller: So then this is the moment where I became a math teacher. I had a specific student who was receiving emotional support services and he also was legally blind and we were doing long division and we were teaching it the traditional way with the algorithm. And you can imagine how frustrated and angry this student became pretty quickly because we were saying it louder and slower. And in this student's case, it was in accommodation that we had to enlarge his textbook. So we were also using this larger than-
Jon Orr: Larger, Louder, and slow.
TL Eller: Yeah.
Kyle Pearce: Yeah.
TL Eller: And so we were doing... I'm embarrassed to say this out loud, but so does McDonald serve cheeseburgers acronym? And we're going through the checks,-
Jon Orr: We all did. We all did.
TL Eller: ... we're going over the problems over and over and over again. And he finally looked at me and was like, "Why in bleepity bleep, bleep? Would you have a blind kid do this?" That was his comment to me. And we shut the book right then and there. And we went for a walk because I didn't have an answer for that. Why am I having a blind student do this? So that's what jump started thinking about. There has to be a better way and started really looking.
We had a district math coach who was awesome, who I started having some conversations with, my sister is actually also a math educator. She teaches pre-service teachers. So she's a professor of math education. I started talking with her and she led me to some of Joe Boaler's work and her YouTube site. So we started looking there.
And so eventually we just started making small systematic changes, started having some success. And it just became the thing that I was really passionate about. So a year or two later, I got to move into the regular classroom and start teaching math, science, and social studies. So I could really focus on math. That's when I found your podcast. And it opened me up to this whole new world of just... There's so many great people out there in mathematics, pushing good, solid research-based practices so that's kind of how I am, where I am now.
Kyle Pearce: Love it.
Jon Orr: Awesome story there. And I think your passion is showing just in the way you've told that story about your journey and you're right, I feel like our math community is so supportive and sharing of resources and helping each other change instruction. And I think it's helped us. I know that the Kyle and I wouldn't be here without that math community as well, doing-
TL Eller: Smoothly.
Jon Orr: ... what we're doing here.
Kyle Pearce: I think you shared a math moment about your teaching career that made you pivot and rethink the way that you were teaching. And that was a very clear math moment for you. But I'm wondering if we want to stretch back to think about your math moment, yourself, how you viewed math, maybe younger, if you can go back to then what would be a math moment that when we say math class, this moment just sticks out with you?
TL Eller: Yeah. So my math moment, very different. I would say my math philosophy as a student was so different than as a teacher. So I was trying to think because math was always my favorite subject. I always really connected with my math teachers. And so, like I said, "I'd listened to the podcast knowing that you were going to ask this, I was thinking, why was that the case? Why did I love math so much?" And I traced it back to third grade.
It was the year that we were learning our multiplication facts. And I was really struggling to learn those facts. I was struggling to memorize them and do that and just struggling in math. And I guess I fully understand this. You need to know that my dad was a basketball coach. He was an old school. He was the varsity basketball coach.
And in our house, hard work was a thing that was very valued. And so when we were struggling with something, we were expected to work even harder. So in third grade I was coming home and we were playing around the world like everybody does, right? And at that time to practice and I was expressing my frustration that I wasn't doing well.
And so what we did was we dove into learning those facts. The only way they knew, which was straight memorization, right? So we would get up at 5:30 in the morning and practice before school started. And we would do all these things. But eventually I worked hard enough that I was the queen of around the world.
Jon Orr: Yeah.
TL Eller: Also, being a coach's kid competition was a thing I really liked that. So what I think happened for me is math became the subject that I could outwork my opponent. I could outwork. And because that just related so well with me and the values that I was brought up. With my dad for years had a copy of the textbook and I think secretly, he wasn't the best math student when he was younger. So he was learning it alongside of me. And it was more for him sometimes than me-
Kyle Pearce: Yeah, sure.
TL Eller: ... but we would get up in the morning and we would study together. And I remember some of my even junior high math teachers saying, "You can practice until you get this." And so I think that's why early on, even with that fifth grade math class, that's why I thought if we could just get them as passionate about working hard and feeling that success.
But it never occurred to me that there might actually be an easier way than what I was doing. And watching my own daughter who's in third grade now, well, she'll be in fourth grade next year, but this past year, she didn't have to study those facts the way we did because she learned strategies.
And so I think that as successful as I was, it was more in memorizing procedures and doing what I was told, but I found joy in that. I found joy in working hard and trying to be the best at that.
Kyle Pearce: Yeah. This is so interesting. It's so connected. So your memory is very connected to what we're going to be discussing today. So I want to almost segue right into it here. And it sounds like... And I'm going to make a couple connections here that I'm making, because hearing your story about long division, for example, and of realizing at the time that, well, really the student helping you to realize that huh, maybe what we're doing right now might not to say it's not valuable.
But maybe there's a different way or a better way. And you came to that realization and then reflecting back on how hard you worked in order to learn those facts yourself. I think something we also have to remember is that working hard is something we still want to value in math class-
TL Eller: Absolutely.
Kyle Pearce: ... but I wonder where's that spot before, especially for those students who might disengage. So you were a student who grew up in a household where that work ethic that no, that was something, that was instilled in you. So that worked for you-
TL Eller: Right.
Kyle Pearce: ... but then we start to zoom out and look at the other students that may not have had that same experience. So I wonder how do we help bring that to them, but in a way that isn't going to necessarily push them away. So I'm wondering when you think about your students in your classroom, you can probably imagine that if you come at it the way maybe your parents did after years of conditioning you to respond in a way, if you came out the gate like that, and a student's never had that, that might come off as a bad thing.
So I wonder if there's an in between there and I want to pause and give you an opportunity to maybe compare and contrast those, the world that you grew up in, and then maybe the world that your daughter's growing up in being a little bit different.
TL Eller: Right. I think that the biggest difference that I see at least is the way that we're approaching teaching multiplication now is a way that the students are encouraged to have some understanding and not just memorize those facts. That's so cute. So for me, it was frustrating for me to study over and over and over again, but eventually I was rewarded for that hard work and I had encouragement at home.
But I think for some of my students, they don't have maybe that encouragement at home. Maybe their parents are just as frustrated with math as they are. They don't have someone at home to practice with them. So I think that by providing them a deeper understanding of the facts and the strategies and what it actually is... And in our curriculum, we build tons of arrays and we call it hidden arrays where they're looking for hidden arrays within those facts.
And so it gives them that meaning where they're no longer afraid to take that leap. And then they start having small successes. The biggest difference for me is now when I say to a student what's six times four and they don't know, it's not just defeat because I can say to them, "Well, do you know, six times two? And how can that help you find six times four?" Where before if I said to a child, "What's six times four?" And they said, "I don't know." I'd say, "Well, darn we need to memorize that."
Kyle Pearce: Yeah. Here's a different one. That's completely unrelated, right?
TL Eller: Right. Exactly. So I think that might be the biggest thing is, it's just allowing students to have those little successes along the way, and it's not as defeating.
Kyle Pearce: Yeah.
Jon Orr: I think we have to agree that strategies is more important than facts, but TL, you reached out to us about some of the curriculum resources that you're using and the limitations that you're experiencing. And then you wanted to dive deeper into thinking about how to assess some mathematical facts. I'm wondering if you want to elaborate on your struggle or your pebble right now, that we can dig into here in this conversation. And so that we can figure out a pathway forward.
TL Eller: Yeah. That would be great. So our district, we moved to standards-based grading four years ago and also got a new curriculum, and also then had COVID all in the same year. So that was-
Jon Orr: Of course, of course.
TL Eller: ... yeah, those were some interesting challenges, but our third and fourth grade teams have been going back and forth with, first of all, is so we have proficiency scales and we have approaching at standard proficient is where we're looking at right now. And the first thing that we keep saying that we're going to come back to, and we're just talking in circles is what should a student who is proficient at their multiplication facts? What should that look like? What is the expectation? And then once we agree upon that, how do we assess it?
So I'm actually looking at our proficiency scale right now, and our curriculum encourages a bunch of different strategies. So right now it says approaching standard would be that students are familiar with those strategies and using them. And then the ultimate goal is that they're choosing the most efficient of those strategies, but I'm not sure that we all agree on what is the most efficient and is the most efficient for one student, the most efficient for another.
We've read a lot of Nicki Newton's work. And I love everything about what she has, but there are some things like they do talk about automaticity in there and it's what does automaticity look like? Does that mean that they have to be memorized or can they be using a strategy quickly and efficiently?
Kyle Pearce: Yeah.
TL Eller: There's just a lot of things that have come up in our conversations that we're just trying to iron out so we can be consistent with the way we're assessing these kiddos.
Kyle Pearce: Yeah. And you've brought up a lot of really interesting pieces here. So I just want to restate them because to just make sure everybody's clear in the way I'm hearing this pebble is obviously there's a value there. You value that students obviously, learn facts so that they don't have to maybe spend as much time and effort thinking them through. However, not saying that you just want them to know those facts. You also want them to leverage strategies.
And I loved your example of having one of the factors to help them maybe potentially use a doubling strategy there so that's a great facilitator move in order to help those students along. And what it sounds like is your real challenge here is, how do I assess? And then probably even more. So how do I evaluate students proficiency, fluency, automaticity you had brought up? But I always look at the word automaticity and I leverage a Cathy Fosnot's style definition of automaticity.
And the example I always use is memorization is something that, and you've maybe had to create some mnemonic to memorize because there's no logic, or reasoning to it. So for example, the code to my garage door, I memorize there's no rhyme or reason to it. It's just four digits or my phone number was assigned to me. So I make a mnemonic maybe to help me with some of the numbers there, but really there's no reasoning through that.
Whereas in math and in particular math facts, everything can be reasoned through. And that's where the entrance of strategies come in. And that's where I believe automaticity is where students have more known facts. So they build these known facts, but how they were built was based on connections, was based on reasoning.
So at the end of the day, I'm looking for evidence of students being able to reason through problems, but at some point this may not happen for all students, but the hope is that students will walk out of my classroom knowing a significant number of facts based on the grade level that I'm teaching, right?
TL Eller: Right.
Kyle Pearce: So whatever's realistic for that grade level and in standards, a lot of times here in Ontario, there are certain math facts that we focus in different grades. And we're using a lot of this similar language strategies, models in order to help them.
TL Eller: Right.
Kyle Pearce: Now, this puts you in a tough spot because I would argue from an assessment standpoint, right? I'm looking at this and I want to give students as many opportunities throughout the school year-
TL Eller: Absolutely.
Kyle Pearce: ... to continue moving along their own journey. But the evaluating part is where things get hard. And I think and this is my opinion, I'm wondering John, I'm going to tag you in here to get your thoughts on this as well. But my thought is when we evaluate that's more or less saying where students are after the learning has happened, right?
And I'm less interested in the whole evaluation part and really interested in the assessment part to help each student get further on their journey. Because the reality is that there are going to be some students who are not going to know all of their facts to 12 times 12, but if I can help all students arrive at those facts through strategies.
And when we talk about efficiency as well, efficiency is also based on the student themselves. What's efficient to them at their spot in the journey is going to be very different than what's efficient for maybe me or maybe you, who is at a very different place in our journey. John, what are your thoughts based on that before we flip it back?
Jon Orr: Yeah. I was actually wanting to flip it back because I'm curious TL to know what you've done so far in trying to think about this assessment journey that your students are on. Or I know you're trying to align assessment and what it looks like to be proficient at multiplication facts? But what have you done so far that has left you going? I don't think we're doing it right.
TL Eller: Well, I think we're doing pieces of it, right?
Jon Orr: Gotcha.
TL Eller: The thing that I do that I like the most, I think gives me the most information is we do a lot of small group thinking task type things during. We have an intervention time. So when I'm working with those students and I'm seeing them either build a raise and think through those facts or as the year progresses. And we're working on multi-digit multiplication and they're using different strategies.
And then I can see how they're, a student who doesn't know seven times six, and then they figure it out. I can observe, were they drawing circles and doing loops in groups and they're still in the counting phase or did they say to me, "Well, I know seven times five and I can add another seven and get 42," the observation piece I'm getting, but we keep these data binders for standards.
And our kids are self-reflecting all the time, about where they're at. And I don't feel like we have a consistent way for those kids to really know what they know and what they don't know about those facts, the way we do the other standards, if that makes sense. I'm wanting them to be able to really own and think about the strategies that they're using, and even become more efficient with those strategies.
And I don't feel like I have the information or I think I know where they're at with their facts, but the only thing that our curriculum provided for us, are these timed assessments, which is so weird to me because it's the opposite of the way everything it's teaching. We're teaching all these great strategies and encouraging all these uses of strategies and different models with ratio tables and number lines.
But then we're assessing with these time facts. And it's funny because it's so not... We did one in nine weeks, just so I'd have some data. And it's so not part of the culture of our room. We're constantly saying speed isn't important. And then I give them this and that's just not even the way they think if that makes sense.
So I just feel frustrated that I don't have consistent information and this piece that's so important constantly from fifth grade, sixth grade, seventh grade, eighth grade teachers, they don't know their facts. I want to have a way that the kids can own it.
Jon Orr: Gotcha.
TL Eller: If that makes sense.
Jon Orr: Gotcha.
Kyle Pearce: When you referred to your other standards, not facts, you had this way for students to understand how they're doing. You're saying, you're doing a better job there, but not with math facts. What do you think is the difference between the facts and then the standards? Because certain standards are, can I do these things or do I know these things? But the multiplication math facts sound very similar in a sense of, can I do this or am I approaching this? What's different between what you're doing over here? Where you're feeling more success with students, understanding where they are on that journey and not over here.
TL Eller: I guess it's the assessments.
Kyle Pearce: Okay.
TL Eller: We feel like we have tasks that the students are doing for each of our different proficiency scales or we've combined. We don't always do unit assessments, but we do have some unit assessments, but we feel like their even moral task-based types of questions that the students are getting. And it feels like they complete that assessment and they have a really good picture of where they're at and then they can keep working.
And then we have other tasks where they have an opportunity to go back and revisit that standard and do that. And there's just so much with multiplication facts, I guess. So in fourth grade, you had mentioned certain facts for certain grades. And what's interesting is actually in Pennsylvania, our standard for third grade, it's zero through nine facts. And then in fourth grade it actually goes away.
We kept this proficiency scale in fourth grade because we know realistically not all of our students come proficient in their facts from third to fourth grade, but it actually disappears in Pennsylvania. It's not an expectation for us to continue to practice it, but we want to, because it's important to us, but it's a lot. I was thinking I wanted a way to know if they've become automatic with their twos, fives and tens.
Because those are foundational facts that help with all the strategies of the others. I was just trying to think of a way to organize it. So kids can self-evaluate and set goals the way we do with our other place value for example. There's comparing, rounding, and word forms and they work on that. And we have these tests and then they know I'm not proficient at this yet because I haven't mastered rounding yet.
And it's very intentional with what they know they're practicing, but with the facts, I can give them that too and say, "Well, you're approaching the standard because my gut is telling me that they're just not quite efficient enough yet," but I don't feel like I'm giving the students the feedback that they need to know where to go from there, if that makes sense.
Kyle Pearce: Right.
Jon Orr: Yeah.
Kyle Pearce: I wanted to ask one more question before, and then I've got an idea, not the answer, but something that might help you in terms of landing on maybe a tool or a process that you might be able to used to help you get closer to the school that you have. I'm curious about, would you say maybe in your classroom, maybe in your school, would you say that number talk routines? Is that something that is pretty common in your grade, across the grades? Maybe a bit of both. What do you think?
TL Eller: So Pamela Harris is three to five co-authors of the curriculum that we adopted. So problem strings in three to five are a part of our, at least weekly routine. There's a ton of problem strings that we do, which I know is a little different than a number talk, but same thing. But yeah, I would say that it's very much in culture of our school and the way the curriculum is written. Yes.
Kyle Pearce: Yeah. And I would be honest and say that the way Pam Harris runs her math talks, her problem strings very similar to Cathy Fosnot, very similar to math moment units for those who have never explored those math moment units that we have throughout our unit, we have problem strings related problems-
TL Eller: Yes.
Kyle Pearce: ... so that's great to hear.
TL Eller: Yes.
Kyle Pearce: So again, helps to paint us a picture again about the importance of strategies, of reasoning, of models. All of those things are happening there if you're leveraging work like Pam or Cathy or the math moments resource. So awesome to hear, it sounds like to me, that what you're trying to figure out is, so you had mentioned about how you want students to know where they are. You obviously want to know where they are too, but you also want to give them an opportunity to know where they're at and where they want to move to next.
So you had mentioned this time testing thing. And time testing... I know there's lots of research that suggests why time testing isn't a great idea. There are some maybe reasons why you might do we'll call it hidden-timed testing, if that makes sense. Like we don't want it to necessarily be a race, but if that's a metric that's going to help you assess how students are building their automaticity.
It's all very fragile to talk about because personally, I'm not so concerned about a student getting X number of problems done after X number of time. I can tell pretty quickly having gone into schools, working and interviewing students from various schools, various areas of our district, 76 schools in total, I go in, I could sit down, I can ask a student a question.
And within about 10 seconds, I have a sense of whether it's automatic, whether they are reasoning through it or whether they're just totally guessing. So I guess my question to you might be, what do you hope that they have? So we talked about efficiency, right? Whether they have this automaticity idea, when you have automaticity, sometimes that's hard to tell if a student already has automaticity with a certain fact like they know, three times four is 12. It's very difficult for you to know whether it was built through strategies or whether they just memorize that, right?
TL Eller: Right.
Kyle Pearce: So that becomes problematic unless you're there for that entire journey. So I guess the question would be is like, "What do you want them to have? And how can I create or craft, I'm picturing, you could almost have your times table grid and you could almost have each box split up into little sections. It reminds me of something like what Nicki Newton would do with her running records. But here, if it's specifically for time tables, each students could have that.
And maybe each box is split up into three sections or four sections where you could allow them to self evaluate, right? Or assess, or maybe you could be assessing and giving it back and forth to them to try to help them individually improve in those areas. But ultimately for me, at the end of the day, in order to design this tool, you have to get clear on what it is you want, you personally want them to have.
Because for me personally, I build, you had mentioned how after grade three, the standard disappears throughout our entire math moments catalog. All the way into the high school grades. Everything we do is designed to be done without the use of a calculator to promote continually coming back. Because there's kids in grade nine that still don't know all their facts-
TL Eller: Right.
Kyle Pearce: ... and they don't have the strategies or the models yet in order to reason through them. So for me personally, and I know John believes this as well. That to me is, much more important than let's say them knowing the fact I want that, but I can't force that. What I can do though, is I can set them up so that in every situation that they have a strategy and a model, that's going to support them in getting where they want to get to.
And I'm going to obviously promote practice and I'm going to promote continually getting better in these areas. But me personally, I'm not so concerned on the evaluation side, but I do wonder if you were able to help students with... Let's say pick six times seven, because that's a fact that maybe students struggle with or six times eight.
I'm wondering about that fact, what do you want for students? What is it that you actually want to see when you ask students that question? And then what would you physically be able to quickly jot down so that you know what recorded what you want? And then you've also provided enough information for that students. I'll flip that back to you there-
TL Eller: Yeah.
Kyle Pearce: ... just to get your sense of, what do you hope in a perfect world? You ask that question and what would you like to see or hear?
TL Eller: Okay. So I think when I ask six times eight, what I'm trying to push them towards for fourth grade is that they have mental math strategies. So at the beginning of the year, they might model that in some way. But my goal is that they can do that eventually in their head. So six times four is 24, double it and get 48 and where my students usually come at the beginning of the year, some of them can do that. Some of them know it automatically.
But the ones who are struggling with the standard are the ones who they have to draw a picture to really get that out. Or they're drawing all of their sixes and combining them into twelves and they eventually get to 48. To me that's not quite efficient enough. I don't know that I feel like it needs to be automatic and that they can just tell me it's six times eight is 48, but I'd like them to be able to do it in their head. Does that make sense? Just have that strategy.
And maybe with a quick writing down six times four is 24 and then they know to double it is 48. But I think the goal is to get them to be able to do that mostly in their head. And then something else that you said that it just made me think. So as a math team, we've defined fluency as we want the students to be efficient and we want them to be accurate but we also want them to be flexible.
So I think for my students who really are nailing the standard, could they show me multiple ways? Could they show me more than just that doubling strategy? Can they also maybe use their fives fact and add another eight or something like that? So-
Kyle Pearce: Yeah.
TL Eller: ... those are the pieces that I'm looking for.
Jon Orr: Right. And when you say, because when we go back to the struggle that you had in your sense, how do I systematize this? But I think what you're elaborating right now is that you know what to look for and it's almost like, can you set up some progression? When you work with the student, you want to know where are they on that journey?
TL Eller: Yes.
Jon Orr: Can they be flexible? Have I seen them use multiple strategies to show this multiplication fact? Or have I only seen them use one strategy always? So there's that, "Hey, I'm checking off the box that they're flexible because I've seen this flexibility, but I'm not checking off the box of flexible if I haven't," but you could also say, "I'm checking the box off on that student if I've seen them memorize, but I haven't seen them strategize."
So I can see them do it with their heads, but I haven't been able to, with confidence say, they're doing all of these things and once those boxes have been checked off that I'm seeing, "Hey, I'm seeing flexibility. I'm seeing strategies and I'm seeing consistency along that line." That's when I can go, "Hey, they're at the standard or they've exceeded the standard." So what does that progression look like for you? Can I create that? And then use that to help me measure students and where they are in that journey?
Kyle Pearce: And one thing to that you've brought up here, John, that I think is important for us to at least be thinking about, because I think we're all with you. I bet you everybody who's listening is we want students to be flexible, and fluent, and we want them to not have... Let's say one, go to strategy because maybe that's not always going to work. Maybe it's not always going to be efficient.
It's the example of subtracting 1001 minus 999. If my only approach is the algorithm, then wow, that's a lot of work, that's showing a lack of flexibility, which brings me back. Well, you said something about the end goal being... We want to help students so that they can do more in their head, but do be cautious because we don't want students to believe. And I know you're not saying this, but this is more for people listening.
We don't want students to believe that not being able to do it in their head and having to use a visual model. We don't want them to look at that as a bad thing because something that's interesting is some students that know all their facts, can't model it at all on paper and that's not good either. So it's when we talk about that proficiency standpoint, we want students to be strategically competent, right?
We want them to have some adaptive reasoning skills where they could actually in one scenario, maybe do something in our subtraction scenario, it doesn't really make sense to use the takeaway standard algorithm model. And then the other piece is I wonder if we want to get to assessing the flexibility, maybe it's less about the day-to-day and more about during those problem strings doing more assessing in terms of their flexibility.
Because the reality is for those who haven't checked out Pam Harris's work, Cathy Fosnot's work, our work on the math moment site through all of our math talks. Every string of problems has a specific model and strategy and or big idea that we're trying to elicit. So through that, the questions we craft are trying to get students to come to this realization that, "Hey, I can get there a different way and we want to see if they're able to do that work."
So I guess what I'm saying is if I'm a person who really likes partial products and that's my go-to strategy, I also don't want to make them feel as though, I know you're not punishing them. We don't want them to feel as though it's like, "Well, I didn't use the strategy that Mr. Or wanted, right? Or I guess my strategy is not good enough, right? So a fine line. So I would say trying to draw out those additional strategies through those number talk routines, and you had mentioned earlier about, if not every day, maybe weekly, I would argue almost every day, anytime we're in between problem-based lessons, having a string of problems, those strings could be crafted by ourselves.
Because I want to answer a question of like, "Hey, do they know a doubling and having strategy or are they able to do that work?" If so let's do this string. And let me assess how they're doing because the reality is if I see six times seven, I might resort to my same go-to strategy, but it doesn't necessarily mean, I don't know other strategies if you know what I mean. So it's a really tough thing to try to assess well, unless the problems we're giving them is crafting that path or that story.
And that's where those related problems are so helpful because you essentially are working a pattern or a behavior of the mathematics out. And that behavior is what makes us as humans go, "Hey, I recognize this pattern and I'm going to maybe leverage that on the next problem in the string. Whereas if I just saw that one problem out in the wild, I might not leverage that same strategy." So there's all kinds of thoughts.
And as we mentioned, we didn't want give necessarily answers, but maybe things to think about. In particular, when you go back to your next PD session with your colleagues to discuss, and maybe as a team, try to get that information out on the board as to what is it that we really want to see and how do we want to track this with our students? So that we can feel good about the work that we're doing, because it sounds like you're doing amazing work-
TL Eller: Thank you.
Kyle Pearce: ... already. So I'm wondering before we wrap up here, what would you say is a big takeaway from today's conversation and what's your next steps in your mind on where you're going to go next with this idea?
TL Eller: So I think my big takeaway is that because just in hearing you talk about the problem strings and the different things, I do think that is just part of our math culture and we're doing that a lot. And I think I have a lot of information about my students up here that I see, but I think that I do need to develop a way of tracking.
And I think I was really caught up maybe on having some task that they could actually do. But I think that the daily tasks that we're doing, it might be enough information. It's just that I'm not recording that information. Even with problem strings. There are things that I could see that if I could develop a way that I'm just quickly writing down this student did this.
Because I keep a lot of anecdotals in our small group and different things, but I haven't done that in problem strings. I've done it in other places, but maybe having a way to organize specifically for those fact, strategies and the types of things that I'm seeing, maybe I don't really need an assessment, so to speak, but just a better way of keeping track of what I'm seeing in my classroom every day.
Jon Orr: Got it. Yeah. I'm envisioning something that can be helpful is oftentimes we talk here on the podcast, is that you don't have to give the same assessment to the whole class all the time and it can be different things on different days you could be looking for, you know what? I'm seeing this from these students today. I got to make a point to make sure I see that from those students on another day.
So keeping some record so that you can make those check boxes and I can help you be quick, but also be accurate along that student's journey on whatever that progression that you're looking for. So I like that you're moving towards that. It makes sense that, "Hey, we've got all this information up here and sometimes we need to translate that down here."
So it feels more real, but we know that we've got so much up here that is worthwhile and also valid. Your memories are completely valid as an assessment, a piece of evidence for that student. So moving forward here, we're excited for your journey and what your next steps are, which means that we do want to check back with you.
And let's say six months, maybe 12 months. So next year we'll set up a time to check in so that see how things are going. Would you be open to that coming back on and sharing your journey so far?
TL Eller: Yeah, no, I'd love to do that. That would be awesome.
Jon Orr: Awesome stuff. Awesome stuff. So we'll make sure we reach out, but, "Hey TL, we want to thank you so much for joining us here on this episode and can't wait to have you back."
TL Eller: Awesome. Thank you so much.
Kyle Pearce: Take care my friend and enjoy the rest of your summer.
TL Eller: Thank you.
Kyle Pearce: Well, my friends there you have it. A great conversation with TL, talking about a pretty complex, pretty difficult idea, difficult pebble because we're talking about two pebbles, right? We recently did an episode that was all about assessment and valuation on its own. And here we're talking about how do we actually do assessment? And how do we evaluate students when it comes to something like math facts, especially when we're talking about the efficiency?
What is efficient for one student might be different than another? What math facts are really important for students to know how quickly should they come to their mind? And do we want to assess, evaluate? Do we want to do both? What is the answer to that question? I think for a lot of us and for your school teams have to have these conversations to really get a clear vision on what do you want for students?
And what are the implications on the choices that you do make? Especially when it does come to assessment and more importantly evaluation, right? When you have to actually stick a number or a letter on that work at the end of a term, or a semester, or school year.
Jon Orr: And I think it was particularly important in this discussion we had with TL that we came or she came to realize that we needed that progression. We needed that success criteria. What are we looking for to pinpoint meeting the standards on this particular learning goal and thinking about there are different ways to look at the evidence to help you get to that success criteria.
So I think it was important for her and us to look at that and go, "Hey, we actually are missing that piece." And I think that can help in that goal that she was having ongoing, how do we know they're at the right spot at the right time?
Kyle Pearce: I love it. I love it. So my friends, how are you going to reflect on this particular episode? We're on social media, Make Math Moments is all our social media accounts. We love hearing from you, our Facebook group, Math Moment Makers K through 12 is always a great active place, but right now, and lately we've been really encouraging everybody to pause and just take a moment to write down a line or two as a rating and review on Apple Podcasts.
Because that goes such a far way to not only let us know because we'll get a notification saying, "Hey, this particular person said this about what happened on the podcast or what was said on the podcast?" So that obviously fills our heart, but then also it tells Apple and other podcast platforms that, "Hey, you should be sharing this particular podcast with more people around the world." And that makes a stronger Math Moment Maker community and in turn helps more students love and see themselves as math doers and learners.
Jon Orr: Right. And we're also looking for conversations just like this one. We talked with TL because TL reached out to us and said, "Hey, I have a question. I have a pebble." And hey, we said, "Why don't we just chat about it? We'll hit the record button and we'll share it with the community because all of these chats we think can benefit that community."
So we would love to hear from you and your pebble so that we can chat about it. So head on over to makemathmoments.com/mentor, fill out a quick form and tell us what your pebble is. And we may reach out to you to discuss that pebble here with us. So again, makemathmoments.com/mentor.
Kyle Pearce: Awesome. My friends show notes, links to resources, and complete transcripts are available in the podcast section of the makemathmoments.com website. But remember there's all kinds of other goodies. We've got our three part framework that you can read on the web or download and take with you. Our how to start the school year off right guide is on the makemathmoments.com website.
And we have over 50 problem-based math units. These are units of study, not just single 50 lessons, 50 units, and that number is growing each and every day. So head on over to makemathmoments.com to find anything, and everything you could possibly be looking for as you prepare for the next lesson. All right, my friends until next time I'm Kyle Pierce.
Jon Orr: And I'm John Orr.
Kyle Pearce: High fives for us. And a big high five for you.
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