Featured Speakers:
- Angela Knotts, Research Associate and Professional Learning Specialist, Mathematics Education, WestEd
- Jill Neumayer DePiper, Senior Research Associate, Mathematics Education, WestEd
Host:
- Danny Torres, Associate Director of Events & Digital Media, WestEd
Danny Torres:
Hello, everyone, and welcome to the 26th session of our Leading Together series. In these 30-minute learning webinars, WestEd experts are sharing research and evidence-based practices that help bridge opportunity gaps, support positive outcomes for children and adults, and help build thriving communities. Today’s topic: What Teachers Need to Know About Mathematical Content Knowledge in Grades K-5. Our featured speakers today are Angela Knotts, research associate and professional learning specialist in our mathematics education team at WestEd; and Jill Neumayer DePiper, senior research associate for our mathematics team. Thank you all very much for joining us. My name is Danny Torres. I’m Associate Director of Events & Digital Media for WestEd. I’ll be your host.
Now, before we move into the contents of today’s webinar, I’d like to take a brief moment to introduce WestEd. As a non-partisan research, development, and service agency, WestEd works to promote excellence, improve learning, and increase opportunity for children, youth, and adults. Our staff partner with state, district, and school leaders providing a broad range of tailored services, including research and evaluation, professional learning, technical assistance, and policy guidance. We work to generate knowledge and apply evidence and expertise to improve policies, systems, and practices. Now I’d like to pass the mic over to Angela. Angela, take it away.
Angela Knotts:
All right. Thanks for joining us, everyone. Today we are gonna be talking a little bit about mathematical content knowledge in elementary. So grades K-5. My name is Angela Knotts, and I am a former math teacher, and at WestEd, I kind of do a mix of research related to teacher learning, math teacher learning, and I do a lot of work in schools with teachers and math leaders and other instructional leaders. And I will pass it to Jill to introduce herself.
Jill Neumayer DePiper:
Yes. Hi, I am Jill Neumayer DePiper. I’m also a researcher and professional development lead here at WestEd, a former middle school math teacher. And I also do research and evaluation and professional learning with teachers across the country.
Angela Knotts:
All right. So let’s look at a problem that you might see in third grade. And just think for a minute, if you were to show this problem to students and they seemed a little bit stuck or they weren’t sure what to do, what is something that you might say to them or a question that you might ask to kind of help them make progress? So just give, maybe like 20 seconds, and you can think about what you might ask or say and put it in the chat if you want. All right, what do you think you need to do first? Okay. Other thoughts? So feel free to, what is the problem asking you to do? What do you know about this equation? Okay, great. So I’ll put up here. These are just some things that we have seen. Oh, Danny, it’s not, oh, there it goes. Some things that we’ve seen teachers say, “Put the smaller number under the bigger number.” “Line up the ones place.” “Start with the ones place.” “Can you take 8 away from 7?” So all of these comments are things that could potentially lead to a mathematically sound strategy, but they’re really focused on using a specific set of steps, a specific algorithm to find an answer, rather than being focused on understanding the underlying math, inviting students to think about what they understand and do some reasoning.
So in order to figure out how are we gonna invite kids to do more of the second thing, rather than just focusing only on the first thing, we know the teachers need some special kinds of knowledge. So, of course, they need subject matter knowledge. Like any teacher, they need knowledge of their subject, and then they need pedagogical content knowledge. So that knowledge of how am I gonna teach that subject effectively, not just the knowledge. That said, the mathematical knowledge that elementary teachers need to teach math effectively is special. And it’s not limited to just knowing how to do the math in their grade, which is sometimes a misconception. So sometimes there can be a sense of, well, if you teach second grade, you really just need to know second grade math and maybe just a little bit more and that’s enough. But what we know is that teachers who are teaching elementary school, even though that is, they’re teaching foundational math, that does not mean that the mathematical knowledge that they need, it’s not that they need less knowledge, they actually need kind of a more and different, special kind of mathematical knowledge.
And so we really have two goals today. And the first one is to kind of dive into this specialized mathematics knowledge that K-5 teachers need to set kids up for success, not just in later elementary, but even going forward into secondary and into their adult lives. And then we’re gonna talk a little bit about how we might support teachers in developing that knowledge, what are some things that we can do? So we know the content knowledge matters. Obviously, teachers need to know the content. But in addition to this common content knowledge, just knowing and being able to do the math, being able to solve the problems that you’re teaching students, there’s also a kind of specialized content knowledge and these things go together, and just knowing the common content knowledge is not enough.
So if we think about, if we wanna dive into what is that specialized math content knowledge, let’s look at these two errors. So we’ve got some errors here to the subtraction problem that we showed a minute ago. So you can think about this and look at this and ask yourself, what did the student do in each case? The answer is obviously not correct, but can we tell what error the student made? So if it comes to you and you think you notice something, feel free to put that in the chat. But these are both errors that are going to stem from difficulties with the algorithm, right? So in the first one, the student considered the difference between the two digits, right? So Jill got it in the chat right there, just looking at the difference between the numbers, right? So that’s the first thing. And then in the second one, they’ve tried to do a little borrowing, like they realize that’s something they need to do, but they’re doing it from the wrong place, right? So of course, being an effective elementary third grade math teacher obviously requires recognizing that the answers are wrong, right? But just seeing that they’re wrong is not enough. In order to really be effective about what are we gonna do next, how are we gonna support the students to keep building their math knowledge, we need to have some understanding of what might have produced these answers.
So if I’m a third grade teacher and I’m looking at these two wrong answers and I’m mystified about how the student got those answers, it’s gonna be harder for me to choose some effective next steps and decide what I’m gonna do. So here’s another example of specialized content knowledge. Sometimes students do things that we don’t expect or that it’s not what the book said to do. And so here’s a strategy that is maybe not the traditional algorithm. And so of course, again, as a teacher, we do need to be able to look at it and say, yes, that’s the correct answer. But more important than is it the correct answer is what did this kid do? How are they thinking about the numbers? Is it a valid mathematical strategy? And is it something that’s gonna generalize? Does it just work in this one case or is this always gonna work? ‘Cause that’s something that I’m gonna need to know when I’m thinking about next steps. So what am I gonna do with this in my classroom?
And then another part of specialized math knowledge is the ability to look at a math topic or a problem and not see it just kind of in the moment, but to see it as situated within a larger mathematical storyline, right? So being able to subtract multi-digit numbers is certainly a skill that we want students to be able to do for its own sake, but it’s also important for other reasons. It’s important also because it’s building towards bigger mathematical ideas. And so, you know, as a teacher, when I’m thinking about how am I gonna respond to students’ struggles, how am I gonna set up learning experiences, it’s not just about how am I helping them get the right answer in this moment. I also, when I have this specialized content knowledge, I’m able to think down the road and think about the things that are coming up next. Where is this knowledge going? What’s the long game? That kind of thing.
And then the other thing I need to think about too is what language I’m gonna use, right? So I’m subtracting multi-digit numbers in third grade, but I know that pretty soon, students are gonna be subtracting decimals, right? And so if I know down the road that kids are gonna need to do things like 8 minus 4.53, then I might not want to use language like, “Put the smaller number under the bigger number,” right? So here’s two examples where we could imagine a kid doing either of these and they’re like, “Well, I did it, put the smaller number under the bigger number,” right, for different interpretations of what that means. And the issue here is not just that it’s not precise enough, that’s not the key issue here. The key issue is that, you know, if I say something like that, I’m really again focused on getting to the answer. Can I do the steps? Can I arrive at the answer? Rather than being focused on what’s the underlying math? Does the student actually understand what’s going on mathematically? And how am I kind of drawing on their knowledge?
Similarly here I could say, “Start with the ones place.” Great. I found the ones place, I subtracted 4 from 8. Now what? So it’s not that it’s not true, it just doesn’t super generalize well as that mathematical storyline is being developed over multiple years. And sometimes we try to solve this by saying, “Well, this is different.” So now we’re doing decimals, so now we need different rules. So I have a new process for you to memorize and be able to follow and reproduce. And, you know, maybe that works for a little while, but there’s still the issue here of we’re focusing on kind of superficial aspects of the problem rather than on the underlying math and thinking about the reasoning. So, you know, an experience that a lot of students have is every year they kind of see math as new year, new set of types of problems to solve, new things to remember, and they’re kind of experiencing math as a new type of problem and a new type of problem and a new type of problem and a new type of problem when really what we want them to see is the gradual expanding of a few central ideas and kind of understanding how it’s not only that problems’ types build on each other, but they’re the same problem type. There this idea of expansion and coherence that when we kind of just focus on what are the steps, can I get to the answer? Then we’re not doing that as much.
So, for example, let’s go back to third grade. And if I’m really focused on the reasoning and the underlying math, then I might ask different questions when I have that specialized deep content knowledge, instead of saying, let’s start with the ones, can you subtract 8 from 7, what if I said instead, “Is there another way you could represent this problem?” Or “What math tools could you use to help make sense of it?” Or, “What does 267 minus 38 mean?” And I think somebody actually put that question in the chat earlier, so good job. So if I ask that first question, then it might encourage me to think about, well, what tools do I have? And then maybe I can go back to base 10 blocks, and maybe I can think about how I can represent those and say, “Well, you know, I wanna take away 8 from 7, but I don’t have enough, so maybe I can decompose one of these tens.” And then I’m engaging with meaning and not just following an algorithm.
Or if I’m asking, what does it mean mathematically to take away 38 from 267, then maybe that encourages me to think about it as the distance on a number line, for example, and kind of draw it out and say, what does it mean? And then think about that base 10 structure and say, well, from 38 to go to 40, that’s 2, and then 60 more gets me to 100, and then I can add 100 more, and I can add 67, and then I’m to my 267. And then I can go back and say, well, how far was it? And I can add all those up. And so not only do I have the right answer, but I have engaged with meaning. And this is a way of thinking about numbers and place value and operation that’s gonna generalize into the future as this idea is developed. And then what that’s also gonna do is strengthen the student’s understanding of number and place value and the meaning of subtraction and also strengthening their ability to use all those tools and strategies that do generalize. So as their world of numbers expands, it’s not just, I have a new way and a new way and a new way, it’s that I’m taking the same tools, the same strategies and kind of expanding them and making ’em a little more sophisticated, a little more nuanced, but it’s all kind of the same mathematical ideas that I’m exploring here.
And so I can think about that with, if we go back to our decimal problem, thinking about this subtraction problem as distance, right, we can use the same strategy, going up to 4.6, then going up to 5 by adding 4/10 and then adding 3 more and saying, what was the distance and all? So kind of generalizing that same strategy. And then also, we saw earlier this piece of it where I wanna see what I’m teaching, this piece of mathematical content as situated within a larger storyline. So I’m thinking about what’s coming down the road and I’m thinking about how students are going to experience and explore those topics that are coming up, but another piece of that specialized knowledge is knowing what has come before. And not just the content that has come before, but what kind of experiences did students have in those past years, in those earlier grades that were helping them make sense of those ideas.
And if I know kind of the models and the language and the strategies that students have had in the past, then I can continue to build on those. So when I get to subtracting multi-digit numbers, it’s not totally new, it’s just expanding on things we’ve already seen in the past. So, for example, if I think about kindergarten, and I know that students are thinking about things like 8 plus something equals 10, and they’ve had to do some reasoning about how to find that missing number and how to kind of develop that idea of adding on. So starting an 8 and then thinking about how am I gonna get to 10? So if I know that in the past they’ve had experiences thinking about number bonds in that way and kind of, like, building up these fact families in their mind. And I know they’ve had experiences with things like 10 frames, so showing 8 and then adding 2 to get to 10 here, if I know they’ve had experiences with number lines.
And then I know that when they went to first grade, they kind of built on that, and then they’re thinking about decomposing and they’re thinking about representing maybe with number lines still, maybe that base 10 or that 10 frame kind of develops into base 10 blocks. And then that builds into this idea of thinking about bundles of tens and counting tens and unitizing by tens, right? If I know that those same representations and those same tools and strategies have been developed in this way, so we can see them here kind of using similar strategies, then when I get to problems like our subtraction problem in third grade, maybe it’s more natural for me to ask those probing questions and to encourage students to think back to those previous experiences and tools and think about, “You remember last year in second grade when we did the base 10 blocks?” and encourage them to go back to what they know.
And then if I’m asking about what does it mean mathematically, then they might think back to, “Oh, yeah, you know what we did distance, and I remember we were talking about subtraction is how far apart things are, and I can see it on a number line.” Then we have some coherence, and it’s not just a new topic and a new topic and a new topic. And so, you know, if we think about this in terms of grade bands, right, so we think about kindergarten, first grade, grade two, and we’re thinking about maybe in first grade, I’m thinking about, you know, two digits of a number represent tens and ones, and a bundle of a 10, a bundle of 10 ones is called a ten. And then the tens through 90 are telling me how many tens I have. That’s a really crucial piece of first grade mathematics. But, you know, part of that specialized mathematical knowledge is realizing that when I’m teaching first grade, it’s not just about teaching first grade math.
It’s also thinking about what experiences did students have before in kindergarten, what kind of language was used, what kind of models were used, what kind of representations, and thinking about how am I gonna build on that in a really structured way, a really intentional way, and then set them up not only for later in first grade, but also thinking about second grade when they’re gonna expand that idea of unitizing by tens and bundling ones together to make tens, and now I’m bundling tens to make hundreds. So kind of expanding on that same idea, and then that’s gonna continue down the line on into upper elementary school and even into secondary school. And so making sense of all that and having a good understanding of these mathematical storylines and the trajectories, the learning trajectories that students are experiencing, and which pieces kind of come together to generate which new pieces, is a big part of that specialized mathematical knowledge that’s gonna help teachers set students up for success in the future. And now I’m gonna pass it over to Jill who’s gonna talk a little bit about what are some things we can do to help teachers build that specialized content knowledge.
Jill Neumayer DePiper:
Yes, thanks, Angela. I think, you know, I heard, you mentioned that idea of storylines, other people have noted coherence, and that’s really what we’re talking about here. What should we do to help teachers build this specialized math content knowledge? And we have three actions we’d say be good steps to take. So first, to build that understanding of those mathematical storylines and those specialized content knowledge or the rising content knowledge, teachers should have an opportunity to review and make sense of those detailed standards-based progression documents like the one that Angela was showing, and others are through, we have one we’re sharing here from Achieve the Core. It’s just an example of how another group has linked in group standards. Your districts may have done this as well. And these documents present a coherence map and can help teachers understand the ways in which standards from one group lead to another.
As teachers come together around these things, questions you can ask are, where did my grade level standards come from? What standard or standards is this standard building upon? And how does this standard matter for my students’ future learning and understanding? It’s really great to know, you know, where did this come from? We talk about that with students as well as teachers. The reason why this matters is because it’s leading to this later, just like Angela mentioned with that idea of a storyline. As teachers look at these documents, they can have an opportunity to explore these learning progressions and those storylines, and then they can understand the coherence that we’re trying to create for students. So the next step though is to consider how those skills and concepts are taught. That was another thing Angela really mentioned. It’s not just what those standards are, but what experience did students have? You know, what materials did they use? How did those standards look for them? What was the progression of those activities? What additional activities and experience do students now need?
So knowing what happened in a previous grade can help you understand what you should be doing now, as well as knowing that they’re gonna be doing number lines next helps you know appropriate language to use, to talk about those things. So we really think those experiences are helpful. So it’s not just those standards maps. Like really those are lovely, but we have to make those come alive for teachers in their context and with their tools and with those experiences. You know, how have students done something is really the detail around those standards that can help students, like Angela said, know, “Oh, I saw that before.” Because it’s both the standards and those activities. So the next activity is you need to bring teachers together to engage in vertical collaboration, co-planning, and community building. We have found this to be very powerful. When you bring teachers together in a grade band, they need this protected time to sit together, here they are, here they’re sitting together, during the school day and across the year for that vertical collaboration, really in grade bands. You know, a K-2, a 3-5 really brings second and third grade teachers together too, so they can really see what do these things look like.
So then teachers come together, they co-construct a shared understanding of those learning trajectories. So I would say, see, bring the standards documents that you have, and bring the pieces of the activities you did so you can really together say, hey, all of those things together. So they thought about those standards in their own grade level, and now they’re gonna discuss them with their grade level, their grade band colleagues and say, “How did you teach that standard? What does the standard look like for you?” When teachers come together, they can also reflect on what they noticed. “Wow, that isn’t what I thought.” Or, “This is new to me.” “I didn’t know we talked about it in that way.” As well as what activities else, what other activities do students need? “Oh, we’ve done these, but maybe in this earlier grade, I should do this because I see how you’re doing this next step.” We have found that teachers find this really very powerful and helps them carefully craft experiences that then bring these connections to life to students, just as Angela was talking about.
When you have these conversations with teachers and other people in grade bands, it’s really important to think about them related to certain content as well, right? So Angela’s examples here, you’d have a very different conversation if you’re talking about geometry and measurement standards. So this isn’t just a one-off activity, it should be done regularly throughout the year. These are some possible questions to guide. I’m sure you all could come up with additional ones too. So, thank you for joining us to talk through this. I’m now gonna pass it back to Danny to close out our session.
Danny Torres:
Well, thank you, Angela and Jill, for a great session today. And thank you to all our participants for joining us. We really appreciate you being here. Feel free to reach out to Angela and Jill via email if you have any questions about the work we discussed today. You can reach Angela at [email protected]. And you can reach Jill at [email protected]. And if you’d like to learn more about our mathematics education work at WestEd, visit us online at WestEd.org/math, or you can scan the QR code displayed on the screen here.
You can check out recordings of our past Leading Together webinars online. We’ve covered a range of topics, including literacy, assessment, special education, artificial intelligence, and other sessions on mathematics. To access our Leading Together webinar series recordings, visit us online at WestEd.org/leading-together. And if you’re interested in learning more about WestEd and staying connected with us, you can sign up for WestEd’s email newsletter to receive updates about free resources, research, services, and more. Subscribe online at WestEd.org/subscribe, or you can scan the QR code displayed on the screen here. You can also follow us on LinkedIn and Bluesky. With that, thank you all very much. We’ll see you next time.