Welcome to Episode 161: Making Math More Human with Thomas Colclough. Dr. Tom Colclough is a Postdoctoral Scholar in the Center for Knowledge, Technology, and Society at the University of California, Irvine. He earned a bachelor’s of science degree in Mathematics and Philosophy with Specialism in Logic and Foundations from the University of Warwick, UK, and then a doctorate in Philosophy from the Department of Logic and Philosophy of Science at UCI. This background will be very important to our conversation as you’ll see in a few moments. In this conversation, Tom and I discuss the various trauma-informed practices such as safety, choice, empowerment, trustworthiness, and collaboration and how they can be applied in higher ed math courses. There are specific design choices that can be used to combat negative and restrictive mindsets that some students bring into math classes. And once these affective parts of the learning process are addressed logically and systematically, students can find greater success where they often previously found frustration. You’ll also see many connections with the UDL guidelines throughout this conversation with specific emphasis on multiple means of engagement.
Resources
Contact Tom Colclough via email at: tcolclou@uci.edu
Tom Colclough’s Introduction to Inductive Logic Syllabus
Math and Me: Constructing mathematics identities through trauma-informed pedagogy poster
Center for Knowledge, Technology, and Society
King and Kitchener’s Reflective Judgement Model
Ken Bain’s What the Best College Students Do
Transcript
1:06:52
SUMMARY KEYWORDS
Universal Design for Learning, trauma informed pedagogy, math anxiety, student engagement, reflective practice, intellectual tenacity, student barriers, collaborative learning, student empowerment, course design, student reflection, mathematical reasoning, student confidence, learning environment, educational psychology.
SPEAKERS
Thomas Colclough, Lillian Nave
Lillian Nave 00:00
Welcome to Think UDL, the Universal Design for Learning podcast where we hear from the people who are designing and implementing strategies with learner variability in mind. I’m your host, Lillian Nave, and I’m interested in not just what you’re teaching, learning, guiding and facilitating, but how you design and implement it, and why it even matters.
Welcome to Episode 161 of the think UDL podcast, Making Math More Human with Thomas Colclough. Dr Tom Colclough is a postdoctoral scholar in the Center for Knowledge, Technology and Society at the University of California Irvine. He earned a Bachelor’s of Science degree in mathematics and philosophy with specialism in logic and foundations from the University of Warwick in the United Kingdom, and then he earned his doctorate in philosophy from the Department of Logic and Philosophy of Science at the University of California, Irvine. This background will be very important to our conversation, as you’ll see in a few moments, and in our conversation, Tom and I discuss the various trauma informed practices such as safety, choice, empowerment, trustworthiness and collaboration and how they can be applied in Higher Ed math courses, there are specific design choices that can be used to combat negative and restrictive mindsets that some students bring into math classes, and once these effective parts of the learning process are addressed logically and systematically, students can find greater success where they often previously found frustration. You’ll also see many connections with the UDL guidelines throughout this conversation, with specific emphasis on multiple means of engagement with, of course, things like choice, flexibility and collaboration. We mentioned a few resources in this conversation, and you’ll find them in the resource section just before the transcript on think udl.org as always, thank you for listening to the think UDL podcast. I’d like to welcome to the think UDL podcast. Tom Colclough from University of California at Irvine. Welcome to the podcast, Tom.
Thomas Colclough 02:42
Thank you very much. It’s great to be here.
Lillian Nave 02:44
I’m really glad to talk to you about what I saw at a recent conference about math and a lot of the anxiety that goes around it. And you have a lot of very good ideas that I think are worth sharing. So I wanted to ask first of all about you, and my first question is, what makes you a different kind of learner?
Speaker 1 03:07
Yeah, thanks. Lillian, so a bit of my background. I did math and philosophy as an undergraduate over in the UK, and then I went on to do a PhD in logic and philosophy of science. So math and logic have been kind of a through line for my academic career. So over that whole stretch, I’ve found that I learn those subjects best when the pace is kind of quite steady and quite patient, rather than trying to get to the right answer quickly. And so often, the point where things kind of start to make sense to me is after something hasn’t worked. So you know, you might try an approach to a problem, and then you realize halfway through that it’s not going to plan, or it’s not going through, and then you have to sit with that and kind of work out exactly what’s gone wrong and where it’s broken down. So that process of kind of writing things out, spelling things out, following the reasoning a bit further than maybe you’re confident about in the moment, and then trying to diagnose the mistake has been, you know, pretty central to how I’ve learned over the over the years. So I think, you know, I’ve come to see getting something right on the first pass is almost like a bit of a missed opportunity sometimes. So you know, if everything lines up straight away, you don’t always get a clear view of why it works or what would happen if it didn’t. So you know, when something goes wrong, when you make a mistake, you’re you’re forced to kind of engage a bit more closely with the the structure of the problem that you’re working through. And that kind of, I’ve found as has tended to produce a deeper kind of understanding,
Lillian Nave 04:56
wow, that is like the most meticulous and. A wonderful way to say that you’ve learned from failure, that we we, which I have found is true, is we learn from our mistakes. So that’s our my biggest teacher is when something goes wrong and but you’ve actually, like logically laid it out for me, which I really appreciate that.
Thomas Colclough 05:19
Thank you. Thank you. Yeah, so it theme for for the rest of the chat as well. I thought I’d open with that,
Lillian Nave 05:26
well, well, how logically thought out that has been already. I love it. Okay, so I kind of met you in the context of this math, and me a poster you’re presenting and about ways that you have found you and your colleagues have found to really help students, especially students who are have some anxiety about math to succeed. And I will tell you that I often hear a criticism of Universal Design for Learning is that, Hey, that sounds great for the Humanities, but how does it work in things like STEM subjects? And so I was very much intrigued, and you seem to have really connected both of those in in your work about how important it is to be thinking about the student as a whole. So I wanted to ask you about that research and what you’re doing. So my first question is about, well, first, a definition. Can you explain? Or what is trauma informed pedagogy first, and why do you think it’s helpful in particular for a math class?
Thomas Colclough 06:36
Sure. So this is the you know angle that we’re coming at this class that I’ll be talking about through. So to kind of introduce the concept, I’m going to borrow a definition of it from two people, Janice carello and Lisa Butler. They’ve written kind of extensively about this. I think this definition is helpful, especially when you’re thinking about how this stuff might apply in a STEM context. So they say this to be trauma informed in any context is to understand how violence victimization and other traumatic experiences may have figured in the lives of individuals, and to apply that understanding to the provision of services and the design of systems so that they accommodate the needs and the vulnerabilities of trauma survivors. So I think this is a helpful place to start. And I’m going to come back to the in any context phrase. I think it’s helpful place to start because it frames trauma informed work as a way of thinking about how you design what you’re doing, you know, given the kinds of experiences people might be bringing with them. So in a teaching context, trauma, informed pedagogy refers to, you know, teaching practices that are designed in line with this goal. So the particular framework that we are working with with this course that I’ll be talking about. So this was built around five core principles. These are the usual kind of principles of a trauma informed pedagogy that you would see. So there’s safety, which is about protection from harm, there’s trustworthiness, which means kind of recognizing reliability and consistency and others, including the instructor, there’s choice, so students having the ability to make decisions. There’s collaboration, where individuals are kind of working together with a with a shared purpose and towards some common goal. And then there’s empowerment, which is all about discovering and developing your own capacities. So there’s a few classroom level strategies that people have developed in higher ed that try to, you know, put these principles into practice. So it’s been worked out in these various ways, but to explain why I think it’s helpful in a math class in particular, it’s I want to come back to that bit of the definition I highlighted so that in any context phrase. So most of the time when people are studying this framework, so this trauma informed pedagogy framework, they’re looking at situations which involve physical trauma. So some of the things that carello and Butler talked about in that definition, so, you know, you go and look up the definition of a traumatic event, it includes things like exposure to, you know, various kind of grim things like death or serious injury, sexual violence. These are typical examples. And if you look at the research on trauma, informed pedagogy, it’s kind of like you said in the beginning. It’s mostly focused on classroom environments where students have to engage with these kinds of experiences directly. So you see this work. Are kind of focused on counseling, disaster response, family violence, social work, religion and politics is one area and where these issues can come up kind of in the course content itself. And you know, the evidence from the research that we have is pretty clear. It’s something like teaching in a trauma informed way helps mitigate some of the risks that students face in those settings. So when they’re having to engage with material that’s kind of closely tied to physical harm, things like that. So like you said, that’s the context where these principles were really developed for and it’s where they’re most kind of naturally applied, I think. But the question I’m interested in is, what happens when you take that framework and you move it into a completely different context, so setting like a math classroom, where the content doesn’t really force students to have to confront those kinds of experiences in the classroom, and so to get a handle on why the trauma informed principles might be helpful in a math classroom, you want to be able to answer the following sort of questions. So something like, Well, what is it in that context? So, the context of a math classroom that plays a similar role in shaping how students engage with the material, given that they’re not being asked to confront topics like physical violence. And so the way I would put the answer to that question is that in a math classroom, what plays that role is something like how students interpret and respond to mistakes or to failure or to being wrong. So math, you know, this is a subject that really does depend on working through mistakes. So you try something, it fails, you figure out why, and then you might try again. But this process kind of only works if students are willing to stay with a problem, so when it becomes difficult, and if they’re willing to treat mistakes as kind of part of the learning process, rather than like being a signal to give up something like that. So, you know, one kind of word that kind of encapsulates this is something like intellectual tenacity. It’s this idea of not giving up in the face of obstacles. So you want to be able to persist with your reasoning even when it’s not going well, and even if you’re not sure how to fix it. But that said so a lot of students don’t experience math this way. So they might experience being wrong as as you know, almost something quite final that changes how they engage with the work. So they might become more cautious. They might be kind of hold back from committing to an approach to a problem, or they might disengage if things stop making sense. And so when that happens, the learning process that produces understanding in something like math can break down. And so what the principals are doing, at least the way I’ve approached things in this context, is they’re trying to help create conditions under which students can remain engaged in this process, this learning process, of working through obstacles and working through mistakes. So they make it more likely that students will carry a line of reasoning kind of far enough for the structure of the problem to become kind of visible to them. And then, you know, they might revise their approach in light of what they’ve learned. So all this is to say this is what allows them to move kind of towards correct solutions over time. So terms of the question, Why think trauma informed pedagogy is particularly helpful in a math class? So the answer is that math kind of places these strong demands on inquiry that’s driven by error, and if students can’t persist through that difficulty, then they can’t really participate in the subject. So what the principals do is support this kind of tenacity in a way that keeps students engaged with the reasoning itself. I guess one thing to add here, just to be kind of clear about how I’m thinking about the connection, so I’m not trying to draw a direct analogy here between a math classroom and the kind of contexts where trauma informed pedagogy is usually applied. So these situations, I think, are obviously quite different, but it’s more like what I’m trying to do is take some of the lessons from those contexts. So the usual contexts, especially around how people engage under pressure or in the face of obstacles, and. Apply those lessons in a setting where the obstacles might be different, but there’s still kind of significant barriers to learning.
Lillian Nave 15:08
For sure. Yeah, I can definitely think about classrooms, and also, well, about math classrooms, really, about any classroom for students before they’re getting to college or the university. And there can be, I can see various scenarios where students have a somewhat traumatic experience in Yeah, in that arena, where other they’re embarrassed, they’re Yeah, put on the spot. I hear about these all the time and and it really only takes one episode of of something, even as a younger child, for someone to then think, oh, math is no, I’m not going to touch that. I That’s my I hate that class. That’s the worst thing. And then it really becomes, you know, a really a self fulfilling prophecy because of maybe just one somewhat traumatic event that that happened to that child or adolescent or something like that, right?
Thomas Colclough 16:10
That’s right, that’s right.
Lillian Nave 16:12
Okay. So with that in mind, what sort of roadblocks or internal barriers do some students bring to a college math class.
Thomas Colclough 16:23
So the course wasn’t kind of a pure math course in the sense that people usually might think of it. So the course I’ll be talking about was an introductory logic course. So it was taught in the department of logic and philosophy of science. So it was taught at University of California Irvine, that department sits in the School of Social Sciences, as opposed to math, which sits in the School of Physical Sciences. That said, though the course did involve a substantial amount of mathematical material, so it was kind of an inductive logic course. So we covered probability theory, we covered some Bayesian reasoning, we covered risk we covered utility theory, so that most of the course was quite mathematical, but most of the students were not math majors, and this is, this is very important to the context. So they were largely social science students, humanities students, or non STEM majors, more generally, the course was offered as a general education requirement. So to fulfill this kind of quantitative requirement that it, that it fulfills, we get students from the, you know, the corresponding schools who take it. So the kinds of barriers that I was thinking about when I was putting this course together were probably somewhat specific to this population of students. And so I’m not kind of making the claim that you could just take this kind of approach and transplant it directly into, say, like a lower or upper division like algebra course over in the math department, and you should expect the same results. This is not the kind of thing I’m saying, but the students that entered this course, they kind of often had much less mathematical background than a math major. Would I guessed that they would have less confidence with math, because they don’t do it as often, and possibly, in some cases, had already formed kind of fairly settled views about themselves in relation to math as a subject. So this touches on your point earlier about these experiences that you have. Maybe you’re in elementary school or something like this awful experience in a math class, and you know that’s carried forward then, and you know it might contribute to avoiding the subject later on in life, when you get to college, something like this. So the kind of tentative guess was that be some students with these kind of experiences in the room. And so as far as barriers and or internal kind of roadblocks that they would bring in, I think there are some fairly general patterns here that show up a lot in college math classrooms. So I think students often come in with a fear of getting things wrong, especially publicly. Maybe they interpret mistakes in kind of a global, maybe kind of way. So then a failed attempt at a problem is kind of evidence to them that they’re bad at math in some deeper sense. So they’re not a math person or something like this. Quite a few students, I think, carry a kind of rigid picture of mathematical ability, so where being good at math means understanding things straight away. And you know, you don’t have to struggle. You just get it and you breeze through these classes, so that when difficulty appears or when they’re making mistakes, they. And students tend to read this as evidence that they don’t belong in the subject. Like I said, they’re not they’re not math people. And you know, you also see a lot of avoidance behaviors emerging from that. So students stop early when they’re unsure that they avoid committing to an answer or try to minimize visible mistakes, rather than, you know, writing down everything and trying to carry a piece of reasoning all the way through. And from the perspective of mathematics learning, I think this is a problem, because mathematical understanding, you know, as in my case, I think develops exactly when you’re working out why an attempt at a problem might have gone wrong or failed in some kind of way. There’s also an emotional dimension to this, especially among non math majors, or at least one might expect, especially among non math majors. So I think you kind of touched on this yourself. A lot of students come into these classes kind of tense around math, so they might associate the subject with with stress or embarrassment, these kinds of emotions. And once that kind of emotional pressure ramps up, it becomes, I think, harder for students to think clearly and monitor their reasoning and persist through problems. So the issue isn’t just that students, you know, lack confidence, necessarily. It’s more like their whole relationship to inquiry, mathematical inquiry becomes or is jeopardized in some kind of way. And I think this mattered quite a bit in the course that I taught. So the course I’m talking about, because logic and probability sit in this kind of interesting middle ground. So students, I think, came into the course assuming math was going to be kind of this cold, rigid thing. But at the same time, as I’ll kind of go on about shortly, they were being asked to reason about beliefs and about evidence and about, you know, things that they believed and uncertainty about beliefs that they had in kind of everyday decision making. So part of what I was trying to do was create these conditions where students could kind of stay with that reasoning process long enough so that, you know, they would come to discover that mathematical thinking could be something that they participated in, rather than, you know, something that’s kind of imposed on them from from the outside. So the barriers I had in mind are not deficiencies in this kind of sense that I’ve been talking about. It’s much more about students, relationship to error, relationship to difficulty, and about whether students could remain engaged in inquiry once those things enter the picture for them,
Lillian Nave 23:06
I see you’ve just given a really beautiful logical analysis of that effective part of learning. You know that the feelings that are so important for like things like safety that you’ve already what you’ve already stated. It just it makes me think how important it is that we say these things, that we really talk about them, and have other faculty understand that there are students that are coming into their spaces with these fixed mindsets or these roadblocks that we don’t really know and Well, the thing I love about Universal Design for Learning is that it’s all about taking down those roadblocks that maybe we didn’t know were there. Maybe the student doesn’t even know that they have that, like, their internal roadblock. But what a beautiful analysis of of this. Like, that’s such a logical way to explain to me as a very touchy feely. Like, Oh, I kind of feel this way. So, so thank you
Thomas Colclough 24:04
that that That’s right. And what, what you said at the end is, is, is dead on. They might not realize that they have these kinds of things or feelings inside. So, you know, part of the goal is to kind of bring those out in a way that they can reflect on it in this kind of metacognitive way and recognize the thing that might be holding them back in some kind of way.
Lillian Nave 24:27
Yeah. Oh, absolutely. Okay, so, So practically, I wanted to ask you, what does it look like then to employ these principles into the into the design of this course, or of a course, and then what was the student experience after you implemented these principles?
Thomas Colclough 24:49
Good, so in terms of what this actually looked like in this course. And like I said, the context is very specific, but I think some of these techniques could apply. And. In other courses, more generally. So there were lots of lots of things that went into the design of the course, but what I really want to talk about are three of the major kind of assignments in the course, where the five trauma principles kind of showed up most clearly and were really intentionally embedded in there from the beginning. So at the start of the course, students completed, it was called a week one reflection assignment. So this was focused on their relationship to mathematics, their history of experiences with mathematics, and just what kind of sense they have of themselves as someone in a learning context. So that was the first one. Then later on in the class, there was a one on one exit interview with me. So this came after the final week. So the exit interview gave students a chance to reflect on whether any of the views they described in their week, one reflection had changed in any way over the course. So that was the second major kind of assignment, and then the third one. So throughout the class, there were these weekly they were called participation assignments. So these really were the kind of main day to day structure through which students engaged with the math itself. So they would do three or four of these kinds of activities every lecture. And there were a couple of lectures a week, so this was fairly frequent, and each of the assignments, so like I said, the each of those three kinds of assignments embodied these trauma informed principles in slightly different ways, depending on what they were there for in the course and what we were trying to achieve pedagogically, and the kind of engagement that I wanted to encourage from students so the and in, you know, let me go into a bit more detail about this, the reflection at the beginning. So the week one reflection, so students wrote about their relationship with mathematics. So I would ask them questions. I would prompt them to think about things like how they saw themselves as learners, what kinds of experiences they’d had with math in the past and factors which had shaped that over time. So whether this was school related factors, family related factors, anything that they thought was significant, and then the exit interview, so later on in the course, these were kind of framed as informal chats where we would revisit those same questions and talk about whether anything had shifted for them. So, you know, to go back to something that was raised earlier, my guess was that students, or, you know, at least some students in this course, would have, you might say, fairly negative views about themselves in relation to math going into the course. So, you know, the guess was, a lot of non math majors kind of carry around this idea that they’re bad at math, or they’re not math people, or that math is, you know, just a subject where they just fundamentally don’t belong. So they pursued something else a long time ago. So one of the first things I was thinking about was the principle of safety, in the sense of trying to create a learning environment where students could reflect honestly on those experiences without feeling, you know, exposed or judged or cornered in any kind of way. So part of this came down to how disclosure was handled, because I was asking them to disclose these, you know, personal experiences in this reflection. So there you want to be quite careful about boundaries and expectations. So I tried to be very transparent about what students were being asked to do, what counted, as, you know, appropriate disclosure and what didn’t. We also built in some choices around things like scheduling and so scheduling for the exit interviews and submission formats, just to make you know, these are small things, but you want students to feel like they have some agency or control over how they participate from the get go, right? So, you know, I was conscious of not, of not creating a situation where students felt pressured to, you know, just tell me something personal, because they thought that that’s what I wanted to hear. You know, I wanted the reflections to be honest and comfortable. So the other principle there that was important was so I talked about safety, empowerment was important here too. So I wanted students to spend some time thinking metacognitively about their own relationship with math. Lillian, because my guess would be that not many of them had done that kind of thing before. Like, you know, we’ve been talking about quite often. They just carry forward these kind of old assumptions about themselves as learners without having examined them very closely, kind of ever so the hope was, you know, these, these assignments, so the reflection at the beginning, the interview at the end, would give students a space to notice patterns in how they approach difficulty or failure, when they make mistakes, things like this, and start to see those patterns as something that they could work on, rather than something fixed about who they are. So the week one reflection was was partly about laying this groundwork for a different relationship to the learning process itself, and then the interview gives them a chance to look back. So what went on in between was these participation assignments. These were kind of fun to design. They were these. They consist of kind of activities on Canvas, discussion boards. So the whole course was online, so everything was through Canvas, and they were they happened during each live online lecture. And they were designed to consolidate course material and foster reflection. So there were three main kinds of these, the participation assignments. So the first kind students chose one or more problems to solve out of a list, and then they reflected on their performance and kind of evaluated how they’ve done. So they would write up their solutions. I had uploaded my own solutions as well to this discussion board. And then the idea was they would have to submit, and once they submitted, they would be able to see my solutions. They’d also be able to see everyone else’s solutions. So there’s kind of, you know, you’re putting yourself out there through these assignments, and then they could kind of evaluate how they did. So that was the first one. So this element of choice was built in there, and they could pick and choose which questions they wanted to have a go at. The second kind of participation assignment was designed to kind of reverse some classroom roles a little bit, so I would give students a question together with a solution that I’d written up, which was kind of totally wrong. And their job was to act as the professor in the situation. So they had to grade my solution, had to explain what had gone wrong, you know, critique my reasoning. And you know what I hoped would be gentle kind of ways. And then the third kind, which we only did once, because it’s a lot of work for the instructor. I asked them all to just make up probability questions for me to solve. So they would post all their questions again on canvas. I would, you know, answer each question in this public kind of forum, and then students would reply to my solutions and get me some feedback. So the difference between the second and third one is, you know, the this in this third kind of assignment, the the role reversal is supposed to be completely authentic and genuine. So I wasn’t pretending to make mistakes like I had been for the second one. I was just, you know, you just respond in real time to problems that they were making for me. So what did these things build in? This is where kind of the rest of the principles come in. So choice. I mentioned choice. This was probably the clearest example. So in the first kind of assignment, students chose which problems they wanted to do. This should matter, because a lot of students experience math as something where they maybe have very little control or agency, something like this. So they, you know, they’re confronted with a question, maybe expected to solve it, you know, straight away, and then evaluate it on the outcome, but giving students some control over that and how they approach. The work was designed to change the atmosphere a little bit and hopefully made it easy for them to engage with problems they found difficult without immediately feeling pressured to do so. So that was choice, trust or trustworthiness. So this showed up in a few ways, at the kind of basic level, the grading policies and the expectations were very transparent, so students knew exactly what counted as complete participation and what the goals of each assignment were. But you know, there’s also this more social side to it. So like I said, Once students submitted their work, everybody’s work became visible to everybody else. So over time, the idea was that the discussion board would become a space where students could see other people making mistakes, and then, you know, revising how they thought about things, and eventually figuring things. Out. And this was important because I wanted the whole idea of making mistakes to be a focus here. So become, you know, something that we could reflect on and discuss, rather than something that we’re trying to hide, maybe, or something like that. And the, you know, the role reversal assignments were important for a similar reason. So having students, you know, critique my wrong solutions kind of was designed to shift this usual instructor student dynamic a little bit so students aren’t sat there waiting for you know, you as the authority to tell them whether they’re right or they’re wrong. Rather, they had to reason through things themselves and evaluate things themselves. And like I said, the last kind of assignment, that role reversal was was more genuine. And then so the last principle, so empowerment, kind of runs through all of this, right? So I wanted students to become more aware of what they were capable of mathematically when you take some of that pressure out of the situation, especially students who might have already decided that they weren’t math people. So a lot of these assignments asked students to reflect on what they understood, be able to describe or articulate, where they got stuck, and what kinds of things they were trying and how that thinking was changing over time. So, you know, overall goal here was to kind of build this structure where students could remain engaged with mathematical thinking long enough to develop confidence in their own ability to kind of work through these difficult problems.
Lillian Nave 36:40
That’s great. So I have a little follow up question. When you were talking about kind of the Q and A section like a discussion board, where you would put in the students would put in their answer, and then they would get to see yours, and they get to see everybody’s. And that was a way to make mistakes, be okay. What would happen if they didn’t do it right? And how were they graded? Was it graded on participation, like they tried their best or and then they had the chance to fix it? Or, you know, traditionally, and this doesn’t sound like what this is at all, is they got the question wrong on the discussion board, so they didn’t get credit. So how did that work that helped to create that trustworthiness and that sense of safety too, when they could make mistakes good?
Thomas Colclough 37:31
So there was they were not graded on the correctness of the problems. This was very important, and you know, that’s part of this whole design, to take the pressure out of the situation, but the criteria for completeness were supposed to be, you know, quite rigorous in a different way. So the kind of effort that they put into the problems, I was grading on completion. So, you know, I wasn’t expecting just, I don’t know how to do this problem. I tried, you know, that’s not going to cut it. So there were multiple parts to these. So they, they’d submit their solutions, then they could see mine and everyone else’s, and then they there was a reflective portion. So it was, you know, did you go wrong? If so, where and you know, how might you approach this differently, or did things make sense after seeing everyone else’s solution? So as long as there was substance in what they wrote in those reflections, that this, this also counted for the completeness element as well. So yeah, there’s, there’s a fine balance here. It’s, you know all very well and good saying, Oh, this, this kind of class, I claim, you know, can help students feel more confident in working through mistakes. And then, you know, someone might turn around and say, Well, you didn’t grade them on correctness. So of course, they felt more confident. There was, there was no pressure at all, but I think there was still some pressure, right? It’s, it’s an uncomfortable situation to be asked to do math and then show everyone your work, especially if you couldn’t do the questions. So I you know, the pressure isn’t gone entirely, and it was a question of, what kind of balance do we want to strike in this class?
Lillian Nave 39:18
Yeah, that’s great. I I’ve done things like that too in my classes, not math classes, but, like, the whole idea is to, like, show off your best failure, you know, and what did you learn from it? But I love that you’ve really made that the whole process of the class, like, that’s every week there. It’s really about it’s more effort and persistence, and that’s what you’re grading them on, rather than the rightness of their answer. And I think I agree with you, that’s how we learn. You know, it’s not just, did you get it right the first time?
Thomas Colclough 39:53
Very right. Well, this, this idea of persistence was, you know, very much the focus here and having them. Do this again and again and again and again, just week by week, the same process, you know, that was designed with this idea of fostering persistence in mind, for sure.
Lillian Nave 40:11
Thank you. Can you tell me more about the experience of the students too?
Thomas Colclough 40:16
Yes. So the students experiences kind of came through in these exit interviews. That’s where we really tried to figure out what had changed, if anything had changed, and what had helped change those views. So we kind of maybe, I’ll talk about two kind of, maybe prototype experiences. So I’m going to group the principles into a couple of different, couple of different groups and talk about kind of the general experience of both of these groups of principles. So the first group was safety together with choice together with empowerment. So these three principles. Now, one student who really stood out here was a student who I will call Arya for the purposes of, you know, anonymity, although she wasn’t the only one who said this kind of thing. So quite a few students said things in their exit interviews that touched on similar things as Arya did, and pointed in similar directions. So ARIA was not a math major, so she was a Criminology, Law and Society major. She told me in the interview she’d enrolled to just satisfy this general education requirement. So, you know, the it’s, it’s kind of good, in a way, that the the guesses we were making about maybe students backgrounds coming into the course did show up in the course. So right at the beginning in a week one reflection, she started talking about herself very strongly as not a math person. So she used this phrase. She talked about math as you know, fundamentally something that she struggled with. She framed ability in math and ability and writing as two opposite kind of categories. So she had this sense that people are either, you know, math people or English people. It’s what what she kept saying, and she’d always seen herself, you know, firmly in that second category. And what was interesting to me was how deeply this shaped the way Arya approached math. So she described a lot of frustration with math as something that reinforced these beliefs that she had about herself. So when something didn’t work, the conclusion she drew wasn’t, you know, just I made a mistake, but it’s okay. It was like, Oh no, this this confirms that I’m not a math person. This confirms I’m bad at math. She also talked about some experiences that kind of reinforced that picture. So, you know, falling behind early on in high school, that kind of thing. These had lasting effects that were, you know, relevant to how she saw herself in relation to the subject at the time this course was taught. So, you know, she’d come to think of herself as someone who was strong in the humanities, strong in writing. And that was tied to this idea that math was, was just not a space where, where she could succeed. So, you know, started the course, you have this settled picture in place. But what became interesting over the class was that she didn’t suddenly start describing herself as fantastic at math. She didn’t do that. And, you know, I think it’s unreasonable to expect that kind of thing, and she would say this in the exit interview, but the structure of the course and the assignments seemed to change the way she engaged with mathematical difficulty. And so this came through in the way she spoke about the participation assignments, especially the ones the first kind where students chose some problems from a list that they wanted to work on and then reflected on how they did afterwards. So Arya kept bringing up this role of choice. So being able to, you know, pick questions, change the way she approached the work, because it gave her some sense of control over the process. So she talked explicitly about this reducing some of the anxiety she carried that was associated with mathematics. But more importantly, you know, with respect to the focus of the course, it kept her engaged with these problems for longer, so she didn’t just give up at the point where something became difficult. So choice was a big thing. And then she also talked a lot about the reflective components of the assignments. So she said things like she’d never previously been asked to think about why she got a. Math problem wrong or right. So usually it’s just, you know, presented to her as a matter of you’ve got to arrive at the correct answer. But once she started kind of reflecting on her reasoning, she found, or at least she said she found, she could approach math in a way that connected kind of more naturally with the kinds of thinking dispositions that she associated with writing. So reflection is one example of this. So this was, you know, one of the more striking kind of shifts in thinking changes. So early on, you know, you got this picture of math and writing as mutually exclusive, almost alternatives. But by the end, this division hadn’t gone away, but it had softened, I would say. So she started talking about different ways of thinking mathematically and different kinds of math and different ways of approaching problems. So at one point, I’ve got a few of her quotes here, she said the math in the course showed her that math isn’t always terrible. Quote, yeah, it sounds funny, but, you know, there’s something important going on there. So the subject had become something she could remain engaged with, rather than, you know, something that she experienced as kind of closed off to her in the way described. So what mattered for me was that she was staying with this reasoning process longer. So she was really reflecting on her mistakes, instead of treating them as this, be all, end all confirmation kind of thing. So she demonstrated more of this idea of tenacity that the course was was trying to cultivate. So the principles, the trauma informed principles mattered here because they helped, I think, create conditions under which that became possible. So, you know, the three I mentioned, safety mattered because, you know, mistakes no longer carried the kind of immediate sense of finality for him, choice mattered. She had some agency in how she approached all the questions and empowerment mattered because she was repeatedly being asked to reflect on what she’d understood and how she changed her thinking over time. So, you know, overall effect here, math didn’t become easy, but she was able to stick with the inquiry long enough for her relationship to the subject to start to change in these, you know, incremental ways. So that was one group of principles that was safety, choice and empowerment. So the other two trustworthiness and collaboration. So this was, this was super interesting to hear this experience. So this was from another student. I’ll call her Bridget, and I should say up front, she was, I would say, a bit less representative of the class as a whole. So in her exit interview, and even in her week one reflection, not many students kind of articulated views about math in quite the way Bridget did. But you know, precisely because of that, I think her interview was was really revealing, particularly with respect to these two principles, so trust and collaboration. So Bridget was a philosophy major. She was taking the course because she wanted to prepare for the LSAT. So some you know goal in mind here in her week one reflection, she described math in in I’d say, a very kind of elevated way. So she talks about math as this kind of foundation of objectivity. It’s this thing that guarantees access to the truth. She described math or mathematical laws as indisputable, and she contrasted math with philosophy, which is her major, by saying that philosophers often work in areas where there may not be, like a single verifiable answer in the same way as there would be in math. So what I found interesting was how the participation assignments, especially the collaborative discussion board type assignments, and those, those two kind of role reversal type assignments, seem to shift her relationship to mistakes and to, you know, what I’d call the authority of mathematics that she had in mind. So one thing that she said in the interview was that reading other students responses was really helpful, especially when other students had made mistakes similar to the ones she might have made. So she said sometimes she didn’t, you know, she didn’t even know exactly what she didn’t understand until she saw somebody else, kind of RE. And through the same problem, incorrectly in a similar way. So this collaborative element really, really seemed to help here. And I thought this was interesting and important, because it meant that mistakes themselves had maybe started to function as opportunities for for reflection, rather than just, you know, being a sign of failure, something like that. So she talks about, you know, learning, both from students who got the problems right, because, you know, then she can see what she was maybe supposed to have done, but also from these students who’d also got them wrong. So an interesting perspective on that. And I think, a place where trust and collaboration were working together there. So the structure of these assignments, you know, created a kind of setting where you can see other people thinking through the problems publicly and over time that seemed to make the errors feel less maybe isolating and more more productive. So there was one of the moments that she talked about in the interview that I found very kind of striking, which involved the assignments where students made the questions for me to solve. So one group of students created a probability question involving a Rubik’s Cube. Probability of a face of the cube coming up all of one color, I think, was the question. And my attempt, you know, shame, to say at solving it was wrong. So the students then pointed out this mistake, you know, I acknowledged it in this, this canvas thread, and we, we worked through the correct way to do this problem in this discussion thread. So Bridget talked about this moment a lot in the exit interview she had, and what stood out to her was not just that I’d got something wrong as the instructor, but that the mistake became part of this collaborative process of inquiry. So, you know, we worked through it to get to the right answer. She said it made the math feel more accessible. So she used that word, and she also said it changed the way she thought about professors and mathematical expertise more generally. So she talked about there being this quote, intimidating level of separation between students and professors. So you know, professors are perceived as people who just arrive at the truth kind of immediately and without any problems, but watching an instructor make a mistake and then revise it and then continue reasoning in this public way changed that dynamic for her a little bit. So she said the whole process made this feel more human. So I think this mattered, you know, epistemically speaking, because once mathematical reasoning becomes something human beings do, rather than something that’s, you know, delivered from above, or something like that, then mistakes, you know, they stop functioning in this way, are talked about as evidence that you don’t belong. So they become part of the process through which understanding can develop. So, you know, what was especially interesting about this is that Bridget she retained her strong beliefs in things like the objectivity of math. So, you know, these things didn’t go away by the time of the exit interview, and that that wasn’t the goal. She still talked about math being this indisputable thing. So the shift wasn’t that she stopped respecting, you know, rigor or correctness here, the shift was in how she understood participation in this rigorous inquiry, particularly with when mistakes are made. So by the end of the course, you know, she was talking much more about revision, collaboration, accessibility. She said that seeing mistakes revised publicly in the way that happened in the course made things made getting things wrong feel less discouraging. And so this was, you know, super interesting to hear, because that discouragement is often the point where, you know, inquiry can stop. So in Bridget’s case, trust and collaboration seem to create these conditions where students could remain engaged with reasoning, without treating mistakes as a sign of, you know, exclusion from the practice or a confirmation that you don’t belong.
Lillian Nave 54:36
Wow, you have made me think a lot about the process that we hope, that I hope our students go through in college, and that is, when they come into college, they often have an understanding of how knowledge, of what the basis of knowledge is, or how knowledge is produced and Bridget thought. And many of our students think. That knowledge is gathered from authority, thinking the professor or a book, right, or the internet, right, and that construction of knowledge is very typical, or that thought about how we gain knowledge is very typical for 18 to 20 year olds or those going into college. And I ran across this with the reflective judgment model by King and Kitchener, and that I came across through Ken Bain’s book What the Best College students do. And there is a movement as you move up or through that model, from what we think of as how knowledge is created, or how how knowledge is constructed. And one is that it just exists absolutely or maybe it’s somewhat hidden, and we have to go find it. So we find it in a book, or we go to some authority figure, and then the next, and those that’s called pre reflective judgment, so you’re not thinking about it, you’re not really investigating that. And then the next one is a quasi reflective judgment, when you find out, Oh, maybe there’s multiple answers to a question, or multiple ways of doing things. Maybe there’s not one right answer, and you think, maybe everybody has the right answer, or maybe nobody has the right answer, and it’s you don’t, you can’t find the right answer. And then the final bit is the reflective judgment model. Is when you realize that you’ve you can construct that knowledge like you are the one you’re not looking for somebody else to do it, and it’s what I hope. But that our students gain in four years, or three years, or whatever, or somehow, whatever they do, is that they come to that realization. And it looks like your class did that for Bridget and in a I never would have thought it would be a math class that would do that, to have that very nuanced and deep understanding of the construction of knowledge.
Thomas Colclough 56:57
That’s right. And one thing that strikes me about that model that you talked about. So, you know, we’ve talked about rigor and correctness, and this idea of, you know, getting at the truth, something Bridget spoke about. And then there’s this idea of knowledge, and you know what knowledge is and how you construct it. And truth and knowledge are, you know, two different things. And it sounded to me, and this is something that you know, was baked into the philosophy of the design of this class was, was knowledge requires something else. It’s not just a matter of being, you know, fed random truths or or, you know, even non random truths, important truths, or something like this. So what is it that makes knowledge different? And I think for me, in this course, it was, you know, you want students to be motivated to get to that right answer. And part of that whole process was being asked to reflect on where they went wrong, which is, you know, something that prevents them from getting to the the right answer, something like this. But once they once they have that realization, that reflective element, that that kind of motivation that’s related to when they really start learning, and that’s that’s where knowledge kind of enters the picture. And I think that lines up with, you know, that that move from, I can’t remember the terms pre reflective to quasi reflective
Lillian Nave 58:20
to reflective judgment, yes,
Thomas Colclough 58:22
Reflective judgment, yeah. So that, yeah, that element of this, there’s something kind of meta going on that you’ve you can look at yourself in relation to the this, this idea of truth. And when you’ve got that, then that’s where knowledge can, can kind of start entering the picture there.
Lillian Nave 58:39
Wow, yeah, okay, so great. And I will add, we’re going to put the resources that you’ve given me, you know, and the notes for this podcast. And I’ll add the king and Kitchener reflector judgment model and Ken Bain’s book that brought it to my attention. But before I let you go, I’ve got that one final question about the syllabus, because you were also able to show the use of trauma informed pedagogy to the students via the syllabus. How did that work?
Thomas Colclough 59:07
Yeah, so one thing that I probably should have led with in this chat, but I didn’t design this course, you know, entirely from scratch. So all the syllabus entirely from scratch. So this course syllabus, and you know, I’m happy to share it with you, if that would be, you know, something useful. A lot of this was based on a religion and politics syllabus that had been developed at Irvine’s Center for Teaching and Learning. So it’s called the division of teaching excellence and innovation, with some colleagues at the time that I had in the CTL. So three of the shout outs here go to misbehida, Ashley Hooper and Danny Mann. So these people had already put a lot of thoughts into how trauma informed principles could show up in course design, more generally. And. And they were involved in the design of this religion and politics course. And therefore, you know, when I applied these ideas to a logic course, they were also, you know, I owe a debt of gratitude to them for for how this course was shaped. So, you know, a lot of the broader ideas came out of these, these conversations and this, these collaborations. I do think you know that the principles came alive in the Course Assignments we’ve spoken about this and the syllabus can only describe these activities in, you know, a terse, kind of general, fairly high level. You don’t really get a sense as a student of what it feels like to do the assignments just by reading the syllabus and but we did try to build a lot of smaller things into the syllabus, which also helped lay some groundwork for the kind of environment that we were trying to foster. So in that sense, the syllabus was maybe doing some more of this same preparatory work as the week one reflection assignment. So the goal was, you know, to establish from the beginning the the expectations that would make it easier for students to persist through difficulty in the course. So the way this came across in the course description, we would emphasize things like collaborative activities, reflecting on your own views about math to reflections. You know, big part of this course thinking about how reasoning connects to your own beliefs, your own experiences. So the idea was to signal to students early on that we were expecting them to actively engage with the material and reflect on their own experiences and participate in it with one of them another throughout the course. And there were also a lot of smaller structural kind of choices that we built in to make the course feel predictable and feel manageable. So it was an online course, like I said, we offered a choice of kind of modality in that students could participate synchronously or asynchronously. So there’d always be asynchronous options if they couldn’t make the live lectures, different media were available to submit assignments in, so their reflections could be submitted in written or audio, or they could do videos and students, you know, they covered that range. So this was seemed like a useful option to offer. We built in what we call Muddiest Point sessions. It was kind of like an office hour, but it was something I hosted on Zoom, where students could just come and ask questions in like a low pressure settings if they didn’t want to ask the question on Canvas in front of everyone else, or that kind of thing. And you know these things, they might sound fairly minor on their own, but together, the idea was we’re communicating that confusion or revision or asking questions are ordinary parts of this course. So transparency as well was also very important. We were very explicit in the syllabus about grading, expectations, participation, what that you know what completion meant in this course, and what you’re actually being evaluated on. And, you know, part of the reason for that was predictability matters. If you want students to stick through with difficulties, and you know, if students feel unsure about how they’re being evaluated, then the idea was that they would become more cautious and less willing to take these, these kind of risks in the course, if you like. And you know, aside from that, there were also these smaller collaborative elements that we built into the syllabus. These were scattered throughout. So we offered a range of discussion boards. We had a big student introduction discussion board, so that was, you know, foregrounded in the syllabus, Q and A spaces on canvas where students could ask questions and answer each other’s questions, and even little community building. Things like the syllabus asked all the students to contribute a piece of music to the class playlist. So this is how the how the live zoom session would open someone’s song playing. You have to keep these things PG,
Lillian Nave 1:04:25
yeah, right.
Thomas Colclough 1:04:26
So there was some selectiveness there. So, you know, none of these things are individually, you know, profound, necessarily, but together, they helped, or we hoped they would help, establish the sense that the class is this shared space where students are expected to participate and reflect and ask questions and you know, above all, work through these mistakes openly.
Lillian Nave 1:04:52
That’s great. Yeah, it really is making that whole experience much more human, that you’re really paying attention. And to your students that they are full and complete humans. They have this effective part of their brain, you know, we, I shouldn’t say they. We all do. We have all of those feelings that are really important to learning. And here we are in a math class, a logic class that’s paying attention to that, and that’s what I think is really unique about it. So thank you so much.
Thomas Colclough 1:05:24
Well, thank you very much. Lillian, it’s been a it’s been fun to talk about.
Lillian Nave 1:05:27
Yes, thank you. And I really appreciate you being on the podcast, and I think this is really helpful for a lot of people. So thanks for sharing.
Thomas Colclough 1:05:35
Oh, it’s been a pleasure, no problem.
Lillian Nave 1:05:40
Thank you for listening to this episode of The think UDL podcast. New episodes are posted on social media, on LinkedIn, Facebook, X and blue sky. You can find transcripts and resources pertaining to each episode on our website. ThinkUDL.org, the music in each episode is created by the Oddyssey quartet. Oddyssey is spelled with two D’s, by the way, comprised of Rex Shepard, David Pate, Bill Folwell and Jose Cochez. I’m your host, Lillian Nave, and I want to thank Appalachian State University for helping to support this podcast. And if you call it Appalayshun, I’ll throw an apple at you. Thank you for joining. I’m your host. Lillian Nave, thanks for listening to the Think UDL podcast.