Shelly Scheafer, Assessment in the Early Years
ROUNDING UP: SEASON 3 | EPISODE 13
How is the work of assessing young children different from assessing students in upper elementary grades or in grades 6–12? And what actions can we take to ensure we understand our youngest learners' thinking? Today we're talking with Shelly Scheafer, senior manager of content development with The Math Learning Center, about the ways educators can understand and advance the mathematical thinking of our youngest learners. What makes the work of assessing young children different from assessing students in upper elementary grades or beyond? During this episode, we’re talking with Shelly Schaeffer, Senior Manager of Content Development with The Math Learning Center, about the ways educators can understand and advance the mathematical thinking of young learners.
BIOGRAPHY
Shelly Scheafer is the senior management of content development at The Math Learning Center. Prior to her work with MLC, Shelly was a K–3 classroom teacher, interventionist, and math coach in the Bend-La Pine Schools in Bend, Oregon.
RESOURCES
Catalyzing Change in Early Childhood and Elementary Mathematics by The National Council of Teachers of Mathematics
Children’s Mathematics: Cognitively Guided Instruction by Thomas P. Carpenter, Elizabeth Fennema, Megan Loef Franke, Linda Levi, and Susan B. Empson
The Developing Number Concept Series by Kathy Richardson
Number Talks in the Primary Classroom by Kathy Richardson and Sue Dolphin
How Children Learn Number Concepts: A Guide to the Critical Learning Phrases by Kathy Richardson
LearningTrajectories.org by Julie Sarama and Doug Clements
TRANSCRIPT
Mike: How is the work of assessing young children different from assessing students in upper elementary grades or in grades 6–12? And what actions can we take to ensure we understand our youngest learners' thinking? Today we're talking with Shelly Scheafer, senior manager of content development with The Math Learning Center, about the ways educators can understand and advance the mathematical thinking of our youngest learners.
Welcome to the podcast, Shelly. Thank you so much for joining us today.
Shelly: Thank you, Mike, for having me.
Mike: So I'd like to start with this question. What makes the work of assessing younger children, particularly students in grades K–2, different from assessing students in upper elementary grades or even beyond?
Shelly: Wow, there's a lot to that question, Mike. [laughs] I think there's some obvious things. So effective assessment of our youngest learners is different because obviously our pre-K, our first, even our second grade students are developmentally different from fourth and fifth graders. So when we think about assessing these early primary students, we need to use appropriate assessment methods that match their stage of development. For example, when we think of typical paper-pencil assessments and how we often ask students to show their thinking with pictures, numbers, and words, our youngest learners are just starting to connect symbolic representations to mathematical ideas, let alone put letters together to make words. So we need to take into consideration that primary students are in the early stages of development with their language, their reading, and their writing skills. And this makes it challenging for them to fully articulate, write, sketch any of their mathematical thinking.
So we often find that with young children, interviews can be really helpful, but even then, there's some drawbacks. Some children find it challenging to show in the moment what they know. Others just aren't fully engaged or interested because you've called them over from something that they're busy doing, or maybe they're not yet comfortable with the setting or even the person doing the interview. So when we’re working with children, we need to recognize all of these little peculiarities that come with working with that age. We also need to understand that their mathematical development is fluid; it's continually evolving. And this is why they often, or some, may respond differently to the same prompt or question, especially if the setting of the context has changed. We may find that a kindergarten student who counts to 29 on Monday may count to 69 or even 100 later in the week, depending on what's going on in their mind at the time.
So this means that assessment with young children needs to be frequent, formative, and ongoing. So we're not necessarily waiting for the end of the unit to see, “Aha, did they get this? What did we do?” We're looking at their work all of the time. And fortunately, some of the best assessments on young children are the observations in their natural setting, like times when maybe they're playing a math game or working with a center activity or even during just your classroom routines. And it's these authentic situations that we can look at as assessments to help us capture a more accurate picture of their abilities because we not only get to hear what they say or see what they write on paper, we get to watch them in action. We get to see what they do when they're engaged in small-group activities or playing games with friends.
Mike: So I want to go back to something you said and the way that you said it. You were talking about watching or noticing what students can do and you really emphasize the word “do.” Talk a little bit about what you were trying to convey with that, Shelly.
Shelly: Young children are doers. When they work on a math task, they show their thinking and their actions with finger formations and objects. And we can see if a student has one-to-one correspondence when they're counting if they group their objects, how they line them up. Do they tag them? Do they move them as they count them?
They may not always have the verbal skills to articulate their thinking, but we can also attend to things like head nodding, finger counting, and even how they cluster or match objects. I'm going to give you an example. Let's say that I'm watching some early first graders, and they're solving the expression 6 plus 7. And the first student picks up a number rack. And if you're not familiar with a number rack, it's a tool with two rows of beads, and on the first row there are five red beads and five white beads, and on the second row there's five red beads and five white beads. And the student solving 6 plus 7 begins by pushing over five red beads in one push and then one more bead on the top row. And then they do the same thing for the 7: They push over five red beads and two white beads. And they haven't said a word to me; I'm just watching their actions, and I'm already able to tell, “Hmm, that student could subitize a group of 5 because I saw them push over all five beads in one push.” And that they know that 6 is composed of 5 and 1, and 7 is composed of 5 and 2. And they haven't said a word; I'm just watching what they're doing. And then I might watch the student and I see them pause. Nothing's being said, but I start to notice this slight little head nodding. And then they say “13!” And they give me the answer, and they're really pleased. I didn't get a lot of language from them, but boy did I get a lot from watching how they solved that problem.
And I want to contrast that observation with a student who might be solving the same expression, 6 plus 7, and they might go, “Hmm, 6.” And then they start popping up one finger at a time while counting “7, 8, 9, 10, 11, 12, 13.” And when they get seven fingers held up, they say “13” again. They've approached that problem quite differently, but again, I get that information that they understood the equation, they were able to count on starting with 6, and they kept track of their count with their fingers, and they knew to stop when seven fingers were raised.
And I might even have a different student that might start talking to me and they say, “Well, 6 plus 6 is 12, and 7 is one more than 6, so the answer is 13.” And if this were being done [using] a paper-pencil as an assessment item or they were answering on some kind of a device, all I would know about my student is that they were able to get the correct answer. I wouldn't really know a lot about how they got the answer. What skills do they have? What was their thinking? And there's not a lot that I can work with to plan my instruction. Does that kind of make sense?
Mike: Absolutely. I think the way that you described this, attending to behaviors, to gestures, to the way that kids are interacting with manipulatives, the self-talk that's happening, it makes a ton of sense. And I think for me, when I think back to my own practice, I wish I could wind the clock back because I think I was attending a lot to what kids were saying and sometimes their written communication, and there was a lot that I could have also taken in if I was attending to those things in a little bit more depth.
It also strikes me that this might feel a bit overwhelming for an educator. How could an educator know what they're looking for?
Shelly: I do think it can feel overwhelming at first, but as teachers begin to make informal observations, really listening and watching students’ actions as part of just their daily practice, something that they're doing on a normal basis, they start to develop these kind of intuitive understandings of how children learn, what to expect them to do, what they might say next if they see a certain action. And after several years, let's say, of teaching kindergarten—if you've been a kindergarten teacher for 4, 5, 6, 20 plus years—you start to notice these patterns of behavior, things that five- and six-year-olds seem to say and think and do on a fairly consistent basis. And that kind of helps you know what you're looking at.
And fortunately, we have several researchers that have been, let's say, “kid watching” for 40 years, and they have identified stages through which most children pass as they develop their counting skills or maybe strategies for solving addition and subtraction problems. And these stages are laid out as progressions of thinking or actions that students exhibit as they develop understanding over periods of time. And listeners might know these as learning progressions or learning trajectories. And these are ways to convey an idea of concept in little bits of understanding. So when I was sharing the thinking and actions of three students solving 6 plus 7, listeners familiar with cognitively guided instruction, CGI, they might've recognized the sequence of strategies that children go through when they're solving addition and subtraction problems. So in—my first student, they didn't say anything but gave me an answer, was using direct modeling. We saw them push over five and one beads for 6 and then five and two beads for 7, and then kind of pause at their model, and I could tell with their head nodding that they were counting quietly in their head, counting all the beads to get the answer. And that's one of those first stages that we see and recognize with direct modeling. And that gives me information on what I might do with a student coming next time. I might work on the second strategy that I conveyed with my second student where they were able to count on, they started with that 6 and then they counted 7 more using their fingers to keep track of their count and got the answer. And then that third kind of level in that progression as we're moving of understanding was shown with my third student when they were able to use a derived fact strategy. The student said, “Oh, well, I know that 6 plus 6 is 12. I knew my double fact, and then I used that relationship of knowing that 7 is one more than 6. And so that's kind of how we move kids through. And so when I'm watching them, I can kind of pinpoint where they are and where they might go next, and I can also think about what I might do. And so it's this knowledge of development and progressions and how children learn number concepts that can help teachers recognize the skills as they emerge, as they begin to see them with their students, and they can use those to guide their instruction for that student or look at the class overall and plan their instruction or think about more open-ended kinds of questions that they can ask that recognize these different levels that students are working with.
Mike: As a K–1 teacher, I remember that I spent a lot of my time tracking students with things like checklists. So I’d note if students “had” or “didn't have” a skill. And as I hear you talk, that feels fairly oversimplified when we think about this idea of developmental progressions. How do you suggest that teachers approach capturing evidence of student learning, Shelly?
Shelly: Well, we have to really think about assessment and children's learning is something that is ongoing and evolving. And if we do, it becomes part of what we can do every day. We can look for opportunities to observe students' skills in authentic settings. Maybe it's something that we're having them write down on their whiteboard, or maybe it's something where they're showing the answer with finger formations or we're giving a thumbs-up or a thumbs-down to check in on their understanding. We might not be checking on every student, but we're capturing a few and we can take note because we're doing this on a daily basis of who we want to check in with. What do we want to see?
We can also do a little more formal planning when we draw from what we're going to do already in our lesson. Let's say, for example, that our lesson today includes a dot talk or a number talk, something that we're going to write down. We're going to record student thinking. And during the lesson, the teacher's going to be busy facilitating the discussion, recording the students’ thinking, and making all of those notes. But if we write the child's name, honor their thinking, and give it that caption on that public record, at the end of the lesson, we can capture a picture—just use an iPad, quickly take a picture of that student's thinking—and then we can record that where we're keeping track of our students. So we have, OK, another moment in time. And it's this collection of evidence that we keep growing. We can also, by capturing these public records, note whose voice and thinking we're elevating in the classroom. So it gives us, “How are they thinking and who are we listening to?,” and making sure that we're spreading that out and hearing everyone.
I think, like you mentioned, checklists that you use.
Mike: I did.
Shelly: Yeah. And even checklists can play a role in observation and assessments when they have a focus and a way to capture students' thinking.
One of the things we did in Bridges Third Edition is we designed additional tools for gathering and recording information during Work Places. That's a routine where students are playing games and—or engaged with partners doing some sort of a math activity. And we designed these based on what we might see students do at these different games and activities. And we didn't necessarily think about this as something you're going to do with every student or even in one day because these are spanned out over a period of four to six weeks where that they can go to these games, and we might see the students go to these activities multiple times.
And so let's say that kindergarten students are playing something like the game Beat You to 10, where they're spinning a spinner, they're counting cubes, and they're trying to race their partner to collect 10 cubes. And with an activity like that, I might want to focus on students who I still want to see, do they have one-to-one correspondence? Are they developing cardinality? Are they able to count out a set? And those might be kinds of skills that you might've had typically on a checklist, right, Mike, for kindergarten? But I could use this activity to gather that note and make any comments. So just for those kids I'm looking at, or maybe first graders are playing a game like Sort the Sum, where they're drawing two different dominoes and they're supposed to find how many they have in all. So with a game like that, I might focus on what are their strategies? Are they counting all the dots? Are they counting on from one set of the dots on one side and then counting on the other? Are they starting with the greater number or the most dots? Are they starting with the one always on the left? Or I might even see they might instantly recognize some of those.
So I might know the skills that I want to look for with those games and be making notes, which kind of feels checklist-like, but I can target that time to do it on students [from whom] I want that information by thinking ahead of time: What can I get by watching, observing these students at these games? I mean, as you know, young children love it—older children love it—when the teacher goes over and wants to watch them play or, even better, wants to engage in the gameplay with them. But I can still use that as an assessment.
Mike: That's helpful, Shelly, for a couple of reasons. One of the things that you said was really powerful, is thinking about not just the assessment tools that might be within your curriculum, but looking at the task itself that you're going to have students engage with—be it a game or a project or some kind of activity—and really thinking, “What can I get from this as a person who's trying to make sense of students' thinking?” And I think my checklist suddenly feels really different when I've got a clear vision of, “What can I get from this task or this game that students are playing?” and looking for evidence of that versus feeling like I was pulling kids over one-on-one—which I think I would still do because there's some depth that I might want to capture—but it changes the way that I think about what I might do and also what I might get out of a task that really resonates for me.
The other thing that you made me think about is the extent to which I remember thinking, “I need to make sure if a student has got it or not got it.” What you're making me think can really come out of this experience of observing students when they're working on a task or with a partner is that I can gather more evidence about the application of that idea. I can see the extent to which students are doing something like counting on in the context of a game or a task, and that adds to the evidence that I gather in a one-on-one interview with them. But it gives me a chance to see, “Is this way of thinking something that students are applying in different contexts, or did it just happen at that one particular moment in time when I was with them?” So that really helps me think about how those two different ways of assessing students, be it one-on-one or observing them and seeing what's happening, support one another.
Shelly: And I think you also made me think, it really hit on this idea that students—like I said, their learning is evolving over time, and it might change with the context so that they show us that they know something in one context with these numbers or this scenario, but they don't necessarily always see that it applies across the board. They don't make generalizations. That's something that we really have to work with students to develop. And they're also young children. Think about how quickly a 3-year-old and a 4-year-old change the same 5- to 6-[year-olds], 6 to 7. I mean, they're evolving all the time. And so we want to get this information for them on a regular basis. A unit of instruction may be a month or more long, and a lot can happen in that time. So we want to make sure that we continue to check in with them and help them to develop if needed or that we advance them, we nudge them along, we challenge them with maybe a question: “Will that apply to every number?” So a student discovers, “When we add 1 to every number, it's like saying the next number, so 6 and one more is 7, and 8 and one more is 9.” And you can challenge them. “Ooh, does that always work? What if the number was 22? What if it was 132? Would it always work?” So when you're checking in with kids, you have those opportunities to keep them thinking, to help them grow.
Mike: I want to pick up on something that we haven't necessarily said aloud, but I'd like to explore it. Looking at young students' work from an asset-based perspective, particularly with younger students, I've had points in time where there felt like so much that I needed to teach them, and sometimes I felt myself focusing on what they couldn't do. Looking back, I wish I had thought about my work as noticing the assets, the strategies, the ways of thinking that they were accumulating. Are there practices you think support an asset-based approach to assessment with young learners?
Shelly: I think probably the biggest thing we can do is broaden our thinking about assessment. The National Council of Teachers of Mathematics wrote in Catalyzing Change in Early Childhood and Elementary Mathematics that the primary purpose of assessment is to gather evidence of children's thinking, understanding, and reasoning to inform both instructional decisions and student and teaching learning.
If we consider assessments and observations as tools to inform our instruction, we need to pay attention to the details of the child's thinking. And when we're paying attention to the details, what the child is bringing to the table, what they can do, that's where our focus goes. So the question becomes, “What is the student understanding? What assets do they bring to the task?” It's no longer, “Can they do it or can they not do it?” And when we know, when we’re focusing on just what that student can do and we have some understanding of learning progressions, how students learn, then we can place what they're doing kind of on that trajectory in that progression, and that becomes knowledge. And with that knowledge then we can help students move along the progression to more developed understanding. For example—again, if I go back to my 6 plus 7 [example] and we notice that a student is direct modeling, they're counting out each of the sets and counting all, we can start to nudge them toward counting on. We might cover—maybe they're using that number rack—we might cover the first row and say, “Ooh, you just really showed me a good physical representation of 6 plus 7, and I noticed that you were counting the beads to see how many were there. I'm wondering, if I cover this first row, how many beads am I covering?” Six. “I wonder, could you start your counting at 6?” We can work with what they know, and I can do that because I've focused on where they are in that progression and where that development is going. And I have a goal of where I want students to go to further their thinking, not that being [in] any one place is right or wrong, or “Yes, they can do it,” “No, they can't.” It's my understanding of what assets they bring that I can build on. Is that kind of what you were after?
Mike: It is. And I think you also addressed something that again has gone unsaid, but I think you unpacked it there, which is: Assessment is really designed to inform my instruction. And I think the example you offered is a really lovely one, where we have a student who's direct modeling and they're making sense of number in a certain way, and their strategy reflects that. And that helps us think about the kinds of nudges we can offer that might shift that thinking or press them to make sense of numbers in a different way. That really the assessment is—it is a moment in time, but it also informs the way that you think about what you're going to do next to keep nudging that student's thinking.
Shelly: Exactly. And we have to know that if we have 20 students, they all have 20 little plans, that they're on 20 little pathways of their learning. And so we need to think about everybody. So we're going to ask questions that help them do them, and we're going to honor their thinking.
So again, I'm going to go back to doing that dot talk with those students. And so I'm honoring all these different ways that students are finding the total number of dots, and then I'm asking them to look for what's the same within their thinking so that other students also can serve to nudge kids, have them—let them try and explore a different idea or, “Ooh, can we try that Mike's way and see if we can do that?” Or, “Hmm, what do you notice about how Mike solves the problem and how Shelly solves the problem? Where is their thinking the same? Where is it different?” And so we're honoring everybody's place of where they're at, but they're still learning from each other.
Mike: You have made multiple mentions to this idea of progressions or trajectories, and I'm wondering, what are some of the resources that helped you build an understanding of children's developmental progression, Shelly?
Shelly: Honestly, I can say that I learned a lot from the students I taught in my classroom. My roots run deep in early childhood [education]. With that said, I think I stand on the back of giants, teacher practitioner researchers for early childhood who have spent decades observing children and recording their thinking. I mentioned cognitively guided instruction, which features the research of Thomas Carpenter and his team and their book. Children’s Mathematics: Cognitively Guided Instruction is a great guide for K-5 teachers. Another teacher researcher is Kathy Richardson, and some listeners may know her from her books, the Developing Number Concept series or Number Talks in the Primary Classroom. She also wrote a book called How Children Learn Number Concepts: A Guide to the Critical Learning Phases, which targets pre-kindergarten through grade 4. And then I think also the work of Julie Sarama and Doug Clements. They have a website that looks at learning progressions starting at birth all the way through grade 3. And this website is LearningTrajectories.org. And it's one of those that is always evolving, so it not only explains learning trajectories for all early childhood math concepts, but there are literally thousands of videos and lessons for teaching math, and new content is always being added.
So any of those would really give teachers some good ideas on how children learn the progressions that they go for and really help them notice and put a reference to what they're seeing kids do.
Mike: You mentioned giants, and those are some gigantic folks in the world of mathematics education. I had a really similar experience with both CGI and Kathy Richardson in that a lot of what they're describing are the things that I was seeing in classrooms. What it really helped me do is understand how to place that behavior and what the meaning of it was in terms of students' understanding of mathematics. And it also helped me think about that as an asset that then I could build on.
Are there resources you would invite our listeners to engage with if they want to continue learning?
Shelly: I think if listeners are interested in learning more about developmental progressions in math, the resources I mentioned—Children's Mathematics: Cognitively Guided Instruction by Thomas Carpenter et al., the [book] How Children Learn Number Concepts by Kathy Richardson, or Sarama and Clements’ LearningDirectories.org are good places to start.
But honestly, Mike, it's about teachers making purposeful observations, understanding what they're seeing and hearing, and then knowing what to do with that information. The Latin root of the word assessment means “to sit beside.” And I would like to invite our listeners to sit beside their students. Listen, watch, question, take note. Because developing the capacity to observe children in action, listening to their thinking and then acting upon what they see and hear takes practice, takes effort. And once teachers become fascinated with children sitting in front of us, we can become students of our students, as Allyn Fisher would say. That's when teachers really see the benefits. They'll recognize that all their students have math abilities, and that these math abilities are specific and actionable. And when we nurture our students with what they know and what they need to know, they will grow.
Mike: I think that's a great place to stop. Shelly Schaefer, thank you so much for joining us.
Shelly: Thank you so much, Mike, for having me. It's been a pleasure talking with you.
Mike: This podcast is brought to you by The Math Learning Center and the Maier Math Foundation, dedicated to inspiring and enabling all individuals to discover and develop their mathematical confidence and ability.
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