Eva 0:00
Hello and thank you for listening to the math teacher educator journal podcast. The math teacher educator journal is co sponsored by the Association of mathematics teacher educators and the National Council of Teachers of Mathematics. My name is Eva Anheuser and today I'm talking with Kevin Vogt, who co authored an article with Kristin Vieira, which is titled filling a void. The mathematical quality and planning protocol for mathematics teacher educators, which was published in the September 2022 issue of the mathematics teacher educator journal will begin by summarizing the main points of the article and discuss in more depth the lessons they share in the article their successes and challenges, and how these lessons relate to their work. Kevin, can you briefly introduce yourself?
Speaker 2 0:48
Hi, thank you for having me. My name is Kevin boat. I am an assistant professor of education at Grace College in northern Indiana. I've just finished my second year here. And I had previously been getting my PhD at Michigan State University when this work took place.
Eva 1:06
Thanks so much for coming on. Let's jump right in. Can you give us a brief summary of the article?
Speaker 2 1:13
Yeah, so this article is about a tool that we developed at Michigan State, Kristin and I were working on a project where prospective mathematics teachers taught in an undergraduate teaching experience. So they would teach in maybe a college algebra or PreCalculus course something similar to what they would, they would teach at the high school level. And I did a lot of video analysis of the quality of their instruction, I helped assist in the courses in which they they taught in these lab settings. And I noticed there just a lot of errors happening in their instruction. And as I started noticing these errors and imprecisions. In their instruction, I started to think about well, how can we prevent these errors from happening ahead of times, so that this these in the moment coaching experiences don't necessarily take away from their perspective the students have a dominant classroom has mathematical authority. And so through analyzing these errors, we develop this protocol to the mathematics mathematical quality and planning protocol to help assist in Lesson Plan feedback ahead of their teaching, to try to catch errors that might be hinting at in their lesson planning. This is an article where we we look at one novice mathematics teacher educators using the protocol to provide feedback on lesson plans for her prospective math students.
Eva 2:29
And when you say error, you are referring to mathematical errors,
Speaker 2 2:33
correct mathematical errors. So these are teachers who have a lot of mathematical content knowledge math majors. This, our particular group scored high on an assessment of their mathematical content knowledge, yet they continually made errors in their instruction, either imprecision in their vocabulary, or overt mathematical mistakes. Okay, so who should read this article, this article would be useful for any mathematics teacher educator who's providing feedback on lesson plans, I think it'd be particularly useful for those who might be graduate students or early in the field in doing this, they haven't done this before, because it helps orient their minds who are looking at mathematical content more deeply, I think we, we all think about, okay, they're teaching in a more high stakes setting where they're responsible for students, whether this be student teaching, or in a lab setting, like what we had for the university teaching experience. And we think about their sequencing and what tests they're doing. But we don't necessarily pay attention to the mathematical content, because we assume they know it. I also think it could be useful for actually for prospective teachers as well to read this and for their thinking while writing lesson plans about about their mathematical content.
Eva 3:51
Okay, so what is the important problem or issue that you are addressing in your article?
Speaker 2 4:01
Yeah, so the problem that we noticed was there, we call it a void, or maybe there's a gap between mathematical content knowledge. And again, our assessments were showing that these students were proficient and their mathematical content knowledge, they could solve tasks. Well, some of these assessments that were out there and and then would they actually enact or enact a teaching episode, they were making errors, or they were having issues explaining or using correct notation for the mathematical content they're charged to teach. So there's like this gap between knowing the math and being able to teach the math and we wanted to see is there a way early in their experience and there it was, was still prospective teachers before they're in their classroom to help bridge that gap between their content knowledge that they seem to exemplify and their ability to teach on that knowledge or explain that knowledge? Yes,
Eva 4:54
it also seems like that knowing math in like a clinical setting You're by yourself, when you can just concentrate on it is different than, you know, like being able to draw on that knowledge when you're paying attention to like 100 million other things. When I was reading your paper last night, I was thinking, it's a really nice way to just help you focus pre focus on the math before you go in, because, like we all know, right? Once you're surrounded by like, 30, to 50 statements, there's a lot of other things to pay attention to as well.
Speaker 2 5:32
Absolutely, I was actually assisting in the course. And I had provided mathematical feedback on lesson plans, I provided feedback and lesson plans on their teaching, there were these, oh, no moments where they're teaching something, and I'm like, That's not correct mathematics. How did I miss that? And there's so much to pay attention to in lesson planning. And I think it's so easy to think about, okay, these are the lines of questioning they're gonna use, okay. So they're sequencing the tasks that you might miss an overt error.
Eva 6:01
So before we talk about the actual tool, and how it helps you focus, let's talk a little bit about what other work that's out there you're building on, like, what particular theories or previous articles do you build your work on?
Speaker 2 6:17
There's a bit of a this is another part of the void that we're we're trying to fill, I felt like we really struggled to find articles that talked about feedback on lesson plans. Other than we talked, we referenced Casper and colleagues in their work on categorizing the types of feedback that that are being given on lesson plans or characterizing the feedback given on lesson plans. We struggled to find areas where lesson plans are being used as almost kind of a remediation for mathematical content and mathematical knowledge for teaching. Building.
Eva 6:51
So one of the things you're drawing on is the mathematical knowledge for teaching literature, literature, right, and a mathematical quality of instruction.
Speaker 2 7:00
Correct. We see in mathematical knowledge for teaching, that a lot of the focus is on maybe thinking about anticipating student strategies, and likely strategies that they might employ. You might think about Hill and colleagues for that one. But we don't see a lot about their ability to enhance their knowledge during that lesson planning phase. So there's, there's more of kind of a look at potential students strategies, and then mathematical knowledge for teaching, then there is the practicing teachers own thinking and strategies, as well.
Eva 7:33
Yeah, and I think one thing you pointed out in the paper is that it's really hard for beginning students pre service and beginning students to anticipate student strategies, right, because they haven't been in the classroom yet. Versus if you've been a classroom teacher for a while, it's a lot easier to anticipate. Well, it's not easy. It's just easy are to anticipate, yeah,
Speaker 2 7:59
for sure. I mean, we require them to anticipate multiple likely solutions, correct solutions, as well as three potential incorrect solutions and their lesson planning. They did struggle to come up with incorrect solutions. It's hard when you have the curse of knowledge, you know how to solve it, how to think about students solve even correctly.
Eva 8:19
All right, so let's talk about this tool that you invented. So why don't you walk us through what it does and how it's used.
Speaker 2 8:29
So the tool itself, the mathematical quality planning protocol is basically a table headlined by three major categories of mathematical precision, content, error correction, and knowledge of content, and students suggestion. And these three categories, the mathematical precision category is, is basically looking at their ability to be precise in their mathematical language and their use of symbols, notation, their explanations and their questioning. So when looking at lesson plans, this would be looking, did they use the correct symbol? And they talk about greater than or less than? Did they inappropriately use an equal sign where they had a string of solutions that they kept saying equal but weren't actually equal? Did they use metaphors and they talk about carrying and borrowing, versus maybe re bundling or regrouping, and then the content error correction was where they made an overt mathematical error. They had an incorrect solution, perhaps to to a task, or they tried to do a composition of functions, and they didn't correctly or they had a conceptual error. So they proposed a mathematical idea that is not, in fact possible, perhaps. And then there's the knowledge of content and students suggestions. So this is where they might not be paying attention to the course curriculum. And they're reverting back to how they learn the mathematics versus how the course or the curriculum that the students are enrolled in how they might be reading about the mathematics in their textbooks, for example, so it helps the team Sure educator just think about all of these things as they are, are going through the lesson plan feedback phase. So the intent, and the way our novice pre service or novice sorry, MTE. And article used it was they did a normal pass through the lesson plan, they looked at all the sequencing and timing and tasks, choosing and all of that. And then they did a second pass, where they were looking at this protocol, reading the categories, and then looking at all the mathematical content of the lesson plan. So kind of used as a second pass to really orient their thinking toward the content itself. And it just helps to remind the reviewer where errors and issues might arise.
Eva 10:43
Yeah, so while you're talking, I pulled up the article. And in appendix B, you have this table you talked about, that has headings, to three headings, and then explanations for what those could look like. And so as you just said, and we'll get into this later, the person who was trying this out, did their regular feedback first, and then did a second pass with this table. And you could imagine that if you did that for a while, it might become second nature, or you it might go into your first pass. But to start with, you might need like a separate pass, just to pay attention to
Speaker 2 11:24
this. Yeah, I provide lots of lesson plan feedback. And I can say at this point, I don't look at these categories anymore, they're fairly engraved in my mind, I can't, I almost can't miss them. Now, these sorts of errors or issues arise.
Eva 11:38
And I feel like that mathematics education in this country is really divided in like, you know, there's two, there's more than two, but there's two major ways people come to it. One is through math, and one is through pedagogy. Right? And so whatever way you come first is probably gonna be the thing you notice first, which my background is more on the math side. And so those things like stand out to me right away versus the, you know, other things might be, I need a tool to kind of look through that. And so it's, it's interesting to kind of see, you know, depending on when this happens, I'm assuming and methods classes, right. And so in a methods class to focus primarily is on methods, right, and then the content has to be layered in there. So I was just thinking about that, as I was reading it, too. It highlights another way of how this artificial separation that we're having between the content and the pedagogy in this country.
Speaker 2 12:43
Yeah, well, I think about to even those who might be coming from more of a math perspective, if they're the teacher of the math course, they might have more more of a deep knowledge of the stuff, if the teacher of the math course or these pre service teachers are teaching, say to college algebra course, or perhaps their high school teachers and or elementary teachers, and these students are in their classrooms, and they're providing feedback, they might not really look at some aspects of this, because they're really familiar with their curriculum. But, you know, a lot of times they're more focused on, is this going to work in the timing? Is this going to work? You know, even if that math background, they might have their attention drawn elsewhere? And it just kind of helps bring it
Eva 13:24
back? Definitely. All right. So the innovation is this table that helps you kind of have categories for feedback. Now, let's talk about your research question, or the study that you did with this one participating teacher? Or empty? Sorry,
Speaker 2 13:42
yeah. So our guiding question was, we want to know, in what ways does this protocol help a novice mathematics teacher educator attempt to the content knowledge for teaching of prospective teachers, we were just trying to see from her perspective, and from what we observe in analyzing the lesson plan feedback. Did we notice any changes in her feedback? Do we notice any improvement and feedback on mathematical content? And, you know, it's a small sample size, but we were, we're fairly encouraged by the results. So let's share some of the results. Yeah, so we found that she seemed to attend to more categories of the protocol than she had before receiving it. She before receiving the protocol, she actually attended really well to the knowledge of content and students suggestion category where she was trying to advance their mathematical content knowledge and the feedback she was giving that stayed fairly consistent from before and after, but she, she added a few extra categories of feedback in the mathematical precision area. So she, she hadn't really attended to symbols notation or to mathematical explanations or mathematical questioning before, or while she did mathematical questioning, but she added those other categories in her feedback. She also gave slightly more feedback on mathematical content than she had before receiving the protocol, which was encouraging. But again, it was it was a fairly small sample size. And the other thing we noted was this particular group that she was providing feedback to, was, in the first semester have a stretched college algebra course. So they were in what might have been more of the familiar or mathematically easier content for them than then the second half of the course where they get an A logarithmic and exponential functions, where, in our past research, we noticed a lot of errors occurring. So they didn't have as many content Error Corrections, they were making mathematical errors like they had in previous studies that we had done.
Eva 15:44
All right, so let's wrap up our podcast with summarizing what is the contribution that this innovation and or your article makes to math, teacher education,
Speaker 2 15:58
I think it can be a little more widespread than we even suggested an article. But you know, we look at this as a means for helping pre service teachers are helping that nice teacher educators give pre service teachers feedback that prevents the kinds of mathematical errors we're often seeing our prospective teachers make in the classroom. And this wasn't just at our university, we had done video analysis across three universities that were part of this larger project and mathematical errors would happen. imprecisions would happen at all three locations, despite feedback on lesson plans being fairly extensive. And so we're thinking, how can we make that feedback on the mathematical side stronger, so that our in the moment coaching didn't have to be, you know, you made this error? Now, you have to correct it in front of your students, you know, how can we prevent that so that they're still seen as mathematical authority. In their very first teaching experiences, we see this as potentially beneficial as well to prospective elementary teachers. So next teacher educators who might be giving feedback to elementary teachers as well could use this to orient their thinking toward the types of errors and in decisions they're making. I know, in my current position, I provide quite a bit of feedback to elementary teachers on their lesson plans. And I think about these ideas more deeply, I think, than I did was secondary. That could also be because I have very few secondary students here. But yeah, I've noticed some of the same issues, misuse of the less than greater than sign misuse of the equal sign that it's pervasive across all grade levels, potentially. And so we see this as as helpful for mathematics teacher educators across all grade levels.
Eva 17:39
Well, thank you so much for coming on. Is there anything else that you would like to add?
Speaker 2 17:45
No, I'm just grateful that you reached out and I appreciate this opportunity to share more about this article. We really enjoyed seeing this article come to fruition after years of working with prospective teachers in this undergraduate teaching experience setting and so we're just appreciate her this opportunity to share our work.
Eva 18:02
Well, it was lovely to have you for further information on this topic. You can find the article on the mathematics teacher educator website. This has been your host Eva and Heiser, thank you for listening and goodbye.
Transcribed by https://otter.ai