0:00  
Hello and thank you for listening to the mathematics teacher educator journal podcast. The mathematics 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 I'm talking with Kimberly Corum, who is an assistant professor at Towson University in the mathematics department. We will be discussing the article engaging pre service secondary mathematics teachers in authentic mathematical modeling, deriving Amperes law, published with co author Joker of follow you from the University of Virginia in the September 2019, issue of the mathematics teacher educator journal, we will begin by summarizing the main points of the article and discuss in more depth the lessons they shared in their article, their successes and challenges, and how these lessons relate to their other work. Kim, thank you for joining us. Thank

0:58  
you so much for having me. So

0:59  
let's jump right in and just get started with the first question. Can you give us a brief summary of the article including the results? Yes, of course.

1:09  
So a brief summary of the article is looking at pre service mathematics teachers strategies for developing a multivariable mathematical model to describe a scientific phenomenon. So my co author and I developed a model eliciting activity that we titled deriving Amperes law. And the activity is based around exploring what impacts the magnetic field strength of a solenoid. So solenoid is a coil of conductive wire that generates a magnetic field, when electric current runs through that wire. And within that solenoid, we can look at three independent variables and how that impacts the magnetic field strength, the number of reps of wire, the length of the solenoid itself, and the amount of electric current. And that relationship is known as amperes. Law. So what we did was we asked our pre service math teachers to try and derive this relationship experimentally, by collecting and analyzing the magnetic field strengths of different solenoids that I built. In doing that analyzing these strategies, we noticed some things that are pre service teachers were doing, in particular, their use of a systematic approach to data collection, identifying the types of variation, um, as a way to help them build their model, and then returning back to their collected data to verify and then revise and whether or not they felt their model accurately represented the relationship.

2:36  
So when you started, you use the term model as eliciting activity. Can you tell us what you mean by that term?

2:44  
Yes, in developing the the activity, we're looking at a lot of the work around modeling and modeling perspectives. And when we think about a model eliciting activity, it's one where the intent of the activity, or one of the primary instructional goals is the creation of the model itself, and sort of the work through the modeling cycle that the students do. So rather than the intent of this being sort of mathematics content, we really wanted to look at sort of what modeling strategies, our pre service teachers used when asked to solve a novel problem. Okay,

3:22  
so many questions, but I'm gonna try to wait till a little bit later.

3:25  
Okay, let's,

3:27  
let's start with who should read this article.

3:30  
I think that this article could be valuable for both mathematics teacher educators, and science teachers, educators, because the driving Amperes law, activity itself is a really integrated stem activity. We're looking at developing a mathematical model to describe a scientific phenomenon. And so I think that that has some nice applications in both the mathematics and the science, Teacher Education worlds. And then I think it could also be valuable for pre service and in service teachers. Within the article in preparing the article, my co author and I were very mindful of trying to share the materials and the activity itself in a way that other people could implement it in their own settings. So within the mathematics teacher educator article, we include not only links to the 3d print files that we use to create our solenoids, but we also include sample data tables that our participants collected. So it would be accessible to those who might not have the physical materials but still wanted to try and replicate that multivariable mathematical modeling activity.

4:39  
Okay. So anybody who would like to implement this activity should read it and then maybe people who want to implement similar model eliciting activities?

4:51  
Yes, I think so.

4:53  
Okay. So let's dig into what is the problem or issue That you're addressing in your article,

5:02  
one of the the big motivations that led us to do this work is thinking about how we can better prepare our pre service teachers for integrating mathematical modeling into their own practice. If we think about mathematical modeling, just sort of as an area of study, our teachers when they go out into their classrooms, there's an expectation that they're doing this in their own instruction. If you look at the nctm standards, or the Common Core standards, mathematical modeling is something that that comes up. And so we wondered how can we best prepare teachers to be able to incorporate modeling into their own practice? And we thought that if we had a way to engage them in rich mathematical modeling tasks, as part of their teacher preparation program and their math methods course, that might be a way to get them thinking about it.

5:55  
Okay. So it's an activity for pre service teachers, so that they experience modeling, and they have kind of an idea of what it looks like when they go into their own teaching.

6:08  
Yes, I think that's one way to think about it. I think that another way to think about the problem is just sort of how we can engage classroom students in mathematical modeling as well. So and I'm not sure if this is something that might come up later. I don't mean to jump around to your question,

6:27  
go ahead and jump,

6:28  
okay. The task was originally developed for middle school students. And we then implemented it with our pre service teachers initially as a way to test the feasibility of the activity, but then in doing so realized that the activity led to some really interesting moments with our pre service teachers of grappling with the content and grappling with mathematical modeling, that we felt that the pre service teacher audience was also a good audience for this task beyond just doing it with middle school students.

7:01  
Okay, so if I'm summarizing and tell me if I'm not summarizing, well, there's kind of like, two goals. One is for them to experience modeling, so they know kind of what it is. But one is also for them to learn about modeling by engaging into a modeling activity,

7:19  
correct? Yes. Okay.

7:21  
Sometimes listening is difficult. Alright, so how does your work build on existing work in the field? what particular theories or prior work? Are you grounding your work in

7:36  
for this one, we were looking primarily at sort of Richard lushes work related to modeling and model eliciting activities. And we use that a framework to help us think about how we would design the activity itself. And then we also looked at work related to obstacles that pre service teachers face when doing modeling tasks. in other contexts. My co author, Joe did some work around this looking at pre service teachers modeling, using trigonometric functions. And then we also referred to some of Rosemary's avex work about modeling. And that helped us inform our analysis of the modeling strategies that our pre service teachers used as they progress through the task in the modeling cycle.

8:25  
Okay, there's a question that's been in my brain. So I'm going to ask it, and then I'm going to fold it into the next question. So this task is situated in a physics context. Would you agree? Yes. So physics is fairly closely related to mathematics. but bear with me for a second. What if somebody said to you, while this is a physics problem, why are we doing this at a math class? What would you say to that?

8:53  
That's a great question. The cover story would be sort of physics and electricity and magnetism. But if we think about sort of the structures that we're looking at, and being able to describe with a model, the relationship between these different independent variables that impact the magnetic field strength of a solenoid, we're tying in a lot of mathematical content. So we're looking at types of variation, and sort of collecting data, analyzing data interpreting data. And I would argue that when we think about sort of the subjects that are related, if we're forcing ourselves to think about them as sort of independent fields of study, then we're losing a lot of opportunities for not only faculty and teachers to make connections across their disciplines, but also opportunities for our students to make connections between the different things that they're setting. And I think that connection making is really important. So if you're able to bring in physics or another science into a math classroom, why not what's the harm and That

10:00  
I think there's two things that I take from your response. One is, well, even though it's a physics context, it is mathematics that we're doing. And so it hasn't placed in math class. And the other one is, well, it's important to connect across different disciplines.

10:19  
The summary, were initially that even though that it's a physics context, there are a lot of underlying mathematics involved. And then their second summary statement that I heard was that there is value in sort of making connections across disciplines. And I agree with both of those summary statements.

10:38  
Is there anything good you might want to add to that? Or does that sum up how you think about this?

10:44  
I think that those two points summarized it very well

10:46  
tell us a little bit more about the innovation you created, and the motivation for creating it.

10:53  
Sure. So our motivation for creating this activity, actually went back to an existing NSF project that's happening at the University of Virginia. So Joe was working with his colleague Glenn bowl, and had partnered with the Smithsonian Institute, and then local schools and other colleges to develop these series of invention kits. That was intended as a way for students to learn sort of eighth grade physical science concepts by recreating historical inventions like the solenoid, the speaker, and the linear motor. And they envision these kits being used in engineering classes, middle school engineering classes in middle school science classes. So then Joe and I work together to see if within these existing invention kits, could we find ways to integrate these kits into a mathematics classroom. So we started with the solenoid kit, because that was the the first kit that they had finished. And then also because the solenoid itself is an ever present object in pretty much every piece of technology that we use that we don't often think about. So for example, our ability to record this conversation here is because of the role of solenoids, and speakers and things like that, Joe likes to think of the solenoid is the ultimate stem object. So we wanted to create a math task related to the solenoid kit, and found Amperes law which describes one relationship between the variables and a solenoid. And so we questioned whether or not that can be experimentally derived by measuring the magnetic field strength generated by different solenoids. And we found that it was possible to collect data and sort of recreate this relationship. But we weren't sure whether or not we would have success with providing this task to our intended audience of middle school students. So we piloted it with some students, and then also with our pre service teachers. And then we also implemented it with a group of rising eighth grade students as well. So

13:04  
that's cool. And I kind of want to know if you have math activities with the other kids as well. But let me just because it's so nicely smoothly going. Ask about what research questions you investigated

13:17  
for our work with the pre service math teachers. The two questions that we looked at were, what strategies do pre service secondary mathematics teachers use when experimentally deriving Amperes law? And our second question was, what our pre service secondary mathematics teachers thoughts about incorporating mathematical modeling tasks in classroom instruction, after participating in this model, eliciting activity?

13:45  
Okay, so I'm going to fold the next two questions together, which are more or less, what evidence are you providing? And what are your results? So just tell us what you found? And how you know you found it? Sure.

13:58  
What we found was that in terms of the first research question, looking at the strategies that our pre service teachers used, when deriving Amperes law, we noticed a clear task progression that they went through that we summarized as collecting and analyzing data, structuring their model, recognizing the need for a constant, and then testing and revising their model to make it fit the data. So some of their data collection strategies included, being very systematic, for example, isolating each variable, and then identifying the types of variation for each individual or variable individually. Before looking at the data set holistically. We saw that our pre service teachers are very confident in what the model structure should be by identifying types of variation. But they realized that without a constant in their model, their model did not map back to their collected data. And it was in that identifying the constant that we noticed a lot of productive struggle. Aminatta grappling with the constant is how we described it, we noticed that our pre service teachers were recalling unrelated prior knowledge. So for example, thinking they had to convert units to metric pulling in known constants like pi, or E, but then we're always assessing the reasonableness of this. And what we found ultimately allowed them to be successful was that they were always going back to their collected data. And using that to see whether or not their model makes sense. So it was this sort of process of testing, revising, retesting, revising, in terms of sort of the value in the activity that our pre service teachers found, or the value in modeling activities in general, we found that our pre service teachers felt that this activity gave them a better sense of what it means to actually do mathematics. So thinking about sort of collecting data, and then just sort of what's the nature of mathematical formulas. They also thought about it from a teacher perspective and thinking about the benefits of these types of tests or their future students, while also recognizing that implementing tests like this, in classroom settings is very difficult. And so the data that we use to help us come to these conclusions, were sort of when this when our pre service teachers were engaged in actually working through the task, we were thinking about their work with modeling as students. So we collected sort of fine grained level of data, video recordings of their problem solving session, creating audio transcripts of that, so we could code and look at the work that they did, analyzing their written work. And then also using our field notes, to look at the way that the pre service teachers engaged with the task is future teachers. And we asked them to complete a post tasks survey and more they sort of debrief their experiences, and then thought about how the task might be incorporated into classroom settings.

16:57  
Okay, thank you so much. We talked a little bit about who should use this innovation already. So I would like to close out with a question that's been on my mind since I read this paper. So I don't know if you've read in the last issue, I think there was a paper that used the Flint Water task, which was also a modeling paper. And so that one used not a physics context, but different context. And I don't know if you've thought about it, and if you haven't, that's okay. But if you have, I'm kind of curious, because yours is so situated in math and science versus what makes this task different from that other task is that that one was situated in like a real world social political context. Do you see similarities, differences between those settings? I have not had a chance to read that article yet.

17:56  
But I think that what I appreciate about the article that you described, is, you know, situating, sort of mathematical modeling within the idea of sort of mathematics or social justice, I think brings in real world contexts that are immediately relevant to our students lived experiences. I think that the this task in particular, the Amperes law task is absolutely an authentic context. But it might not be one that students think about as being applicable to their daily lives, even though they use solenoids all the time, they might not be thinking about that. But I think any work, that sort of bringing mathematics into the real world, is important work. And I think that mathematical modeling, and the idea of using mathematics to solve real problems is a great way to do that. And a great way to sort of build those connections between what students see in their mathematics contexts and what they're experiencing outside of school.

18:59  
Yeah, you really motivated me, like reading your paper and talking to you is really motivated me to think more about the similarities and differences between these two types of modeling. And, you know, this is an open question that the listeners can think about as well. Alright, is there anything that you would like to add before we close this podcast?

19:24  
No, I don't think so. I really appreciate you taking the time to let me share this work with you. It's work that I'm very excited about, and that his servant had a big part in in sort of defining sort of the research that I've been doing so I was excited to be able to talk about it.

19:41  
Thank you, Kim, for joining us. And for further information on this topic. You can find the article on the mathematics teacher educator website. This has been your host, Eva Anheuser. Thanks for listening and good bye