# Ep 35. Preparation for university math

with Darja Barr and Dan Wolczuk

This transcript was created with speech-to-text software. It was reviewed before posting but may contain errors. Credit to Jazmin Boisclair.

You can listen to the episode here: Chalk & Talk Podcast.

Ep 35. Preparing for post-secondary math with Darja Barr and Dan Wolczuk

[00:00:05] Anna Stokke: Welcome to Chalk and Talk, a podcast about education and math. I'm Anna Stokke, a math professor, and your host.

You are listening to episode 35 of Chalk and Talk. My guests in this episode are two highly experienced university math instructors, Dr. Darja Barr from the University of Manitoba and Dan Wolczuk from the University of Waterloo. A common question that I get asked is, “What does it take to succeed in university math?”

High school teachers also often ask, “Which topics are most critical for success in university calculus?” We explore these questions in this episode, which I hope will be useful for teachers, parents, and students alike. We talk about the importance of algebraic fluency and other key concepts essential for success in calculus.

We examine the gap between high school preparation and university expectations, touching on grade inflation, the role of diagnostic tests and the value of high school exams with a discussion on the testing effect along the way. We highlight the importance of effective study habits, perseverance, and the ability to self-assess and seek help when needed.

We wrap up with practical advice for first-year students. Be sure to check out the resource page for this episode, which includes valuable information, such as a list of key high school topics essential for success in university math. I hope you enjoy this episode. Now, without further ado, let's get started.

I am very pleased to have two guests today. I have Dr. Darja Barr joining me from here in Winnipeg. She is a senior instructor in the Department of Mathematics at the University of Manitoba, she has a PhD in math education. She has an MSc in mathematical biology. She is the UManitoba Education Coordinator for PIMS, that's the Pacific Institute for the Mathematical Sciences.

She's won a number of awards from University of Manitoba for teaching excellence and outreach, including a merit award for promoting Indigenous achievement. I also have Dan Wolczuk joining me from Waterloo, where he is a lecturer in the Faculty of Mathematics at the University of Waterloo.

He has an MMath from the University of Waterloo. He is the 2022 recipient of Excellence in Teaching Award from the Canadian Mathematics Society. He has also won a number of teaching awards from the University of Waterloo, and he has developed several resources for helping students learn how to learn, including a public YouTube channel, which contains useful videos on learning.

Welcome to both of you. Welcome to my podcast.

[00:03:11] Darja Barr: Thanks for having us, Anna.

[00:03:12] Dan Wolcuzk: Thank you for having us.

[00:03:14] Anna Stokke: Let's first start by getting some background. So, I'm going to ask you both what courses you usually teach at your university. And this is going to be sort of a three-part question. So first, what classes do you typically teach? What is the typical size of say a first-year general calculus class? And the third question is what is the high school prerequisite for a general calculus class at your university? So, we'll start with Darja.

[00:03:42] Darja Barr: Usually I teach first-year classes, so students who are coming straight from high school and taking their math credit. Everything from applied finite math to linear algebra to calculus. Our first-year general calculus class ranges from about 150 to 200 students in one section in one classroom, and the prerequisite Is 60 percent or higher in their Grade 12 pre-calculus course.

[00:04:09] Anna Stokke: How about you, Dan?

[00:04:11] Dan Wolcuzk: I teach a wide range of courses. So, like Darja, I teach mostly first and second-year courses, although I do occasionally teach upper-year courses, and I regularly teach A course for our Master's in Mathematical Teaching. At Waterloo, we actually have seven different Calculus I courses. Six are for STEM disciplines and one for non-STEM majors. The three, including the one that I coordinate, usually have around a thousand students broken up into sections ranging from about a hundred to two hundred students.

All six of the STEM discipline-related calculus classes require students to have taken a high school calculus course.

[00:04:55] Anna Stokke: Okay, so high school calculus is the prerequisite at Waterloo and at the University of Manitoba, here in Manitoba, and it's the same at my university, which is the University of Winnipeg, the prerequisite is pre-calculus. Although, I would say that some students in our calculus classes do have some version of calculus from high school.

Now, the focus of today's episode is preparation for university math, and that's why I collected a couple of the best post-secondary math instructors in Canada to talk about this today. So, we discussed this previously, and we decided to focus on calculus and preparation for calculus. And Darja knows a lot about this.

Her PhD focused on student preparedness for university calculus. So Darja, I'll ask you this question. Why calculus? Why not discuss, say, some other university math course?

[00:05:53] Darja Barr: From talking to other educators across Canada and really globally, calculus is a big gateway course into the sciences and other disciplines as well. It's required by everything from chemistry, biology, computer science, to business, to engineering. In universities across Canada, it typically is the first-year math class with the largest enrollments but high failure and withdrawal rates.

Personally, I became interested in diving deeper into issues that students were having with first-year calculus because of the stories I was hearing from my students who were feeling like they were coming in very successful in high school pre-calculus and then hitting a big wall in calculus, and they weren't able to figure out why.

[00:06:48] Anna Stokke: Dan, did you want to add anything to that?

[00:06:50] Dan Wolcuzk: So I think what she said there is accurate. Like, why calculus is, I think a lot historical. As someone who has written a linear algebra textbook, I'm a big fan of linear algebra. And in our modern world, linear algebra is actually, in some sense, becoming more important mathematically than calculus is, more used widely.

That being said, calculus is really good in the sense of it is not too abstract, has lots of applications in a variety of fields, and it has some different entry points. So, it can be taught in a variety of different ways, which can help students latch onto it and understand it.

I also think calculus is really good for helping students learn problem-solving and analytical thinking skills. So, it's much easier to focus not entirely on learning the content but on helping students build those thinking and learning skills.

[00:07:47] Anna Stokke: So Dan, Waterloo is known to be one of the top institutions in Canada to study math, computer science, and engineering. So, can you say a bit more about some of the majors that require calculus at Waterloo?

[00:08:02] Dan Wolcuzk: Sure. At Waterloo, all students enrolled in the Faculty of Math and Engineering and most students enrolled in the Faculty of Science are required to take at least one calculus course. These include programs such as kinesiology, pharmacy, and psychology. Now, many people are surprised that students in such programs are required to take a University Calculus course.

But, as I just said, the reason for this is not that they necessarily need calculus for pharmacy or psychology, but it is about developing those analytical thinking skills.

[00:08:36] Anna Stokke: Yeah, exactly. So, a lot of students have to take calculus. In some cases, it's required for certain majors because the thinking skills that are developed through taking calculus are valued. And in other cases, calculus is absolutely necessary for the degree. It's certainly important for engineers and a lot of scientists, economists and data scientists, for example.

I should mention that I did have a conversation with Brian Conrad, who's a mathematician. on a previous episode about the uses of calculus and I'll include a link to that episode in the show notes. But the point is that calculus is a requirement for various disciplines. So, if we're thinking about keeping options open for students, calculus is a subject that we'd want students to be prepared to take, if possible.

So Darja, can you talk about some of the main topics from high school pre-calculus that students need to have a good handle on to succeed in calculus?

[00:09:37] Darja Barr: I think actually they need a good foundation in topics that go even further back, like algebra, specifically. Simplifying, working with fractions, solving equations, factoring, those are things that become integral parts of solving calculus problems along the way. Knowing basic graphs of functions that occur over and over again is important so that students can move between visual representations, numerical representations, and graphic representations.

I think that kind of fluency is critical. And then, of course, being able to do skills questions with things like trig, logarithms, and exponentials.

[00:10:25] Anna Stokke: Okay, that's a good summary. And I would, certainly echo algebra is a big issue, right? So, a lot of times, students struggle with algebra and I think often people don't realize just how important it is to be really good at algebra in order to do well in university math. Dan, are there any topics that you want to add?

[00:10:45] Dan Wolcuzk: Wow, it's hard to add to such a fantastic list. The one thing that I would add is just a general number sense. We see students are using calculators so much in high school that they don't have a sense of size of things. So, sometimes their answers are off by a large order of magnitude.

And we would like them to have this better number sense to recognize that, “Yeah, that answer does not make any sense.”

[00:11:11] Anna Stokke: Yeah, I would agree with you there. And I should mention that I asked you both before the episode about calculators, and we don't allow students to use calculators in calculus classes at the three universities that we teach at. I think that's fairly common in university calculus classes that students are expected to be able to do the work without a calculator.

So, following up on what you said, Dan, I've also noticed issues with number sense, like students thinking they need to reach for a calculator to figure out the cube root of eight, or not being able to reason something basic like the square root of two is between one and two.

And just to loop back to something Darja said earlier, I've also noticed that students often don't have basic graphs in their heads that would really help them in calculus, like say the graph of basic exponential or logarithmic functions, even the graphs of basic trig functions.

I suspect that may also be because of overreliance on calculators or software. I mean, if you don't have to think about it, you just get the software to draw the graph for you, and then you don't end up actually committing that to memory.

So, we discussed those topics that we think are really important to succeed in calculus. Now let's talk about whether there are particular pre-calculus topics that you found that your students in your calculus classes struggle with.

Do you want to say a little more about that Darja?

[00:12:33] Darja Barr: Sure, so actually the interesting thing I've found in my years of teaching calculus is that I don't actually think my students are struggling with calculus. I think they're struggling with the things that needed to build up to get to calculus. So, for example, we mentioned all three of us that algebraic fluency, being able to do algebra, like when solving a limit, quickly, confidently, catching your own small mistakes as you come back and look at your answers.

I want students who feel comfortable with functions. They struggle with, like, Dan said, the number sense. Being able to look at a graph and find a function's value at a certain spot and not catching themselves when they're saying something that's totally ludicrous.

I think working efficiently and without errors is perhaps something they don't get enough practice in. Repeat until you build that skill.

[00:13:37] Anna Stokke: So just to get a little more specific when we talk about algebra because we all mentioned that, we're talking about things like being able to factor, like even being able to factor a quadratic, being able to do polynomial long division, knowing how to factor a difference of cubes, or being able to simplify a complex algebraic expression.

I would also mention a lack of fluency with things like exponent rules. Those really should be automatic, but they're often not. Would you say that's what we're talking about?

[00:14:09] Darja Barr: Exactly. Knowing how to cancel things properly in a rational expression. I often feel like students panic, and then they just start doing things on paper that perhaps their gut even tells them is not correct, but they plug forward because they aren't confident with the algebra.

[00:14:29] Anna Stokke: Okay, Dan, we've talked a lot about algebra. Do you want to add to this? Perhaps you can say something about, say, trigonometry or exponential and logarithmic functions if you think it's relevant.

[00:14:40] Dan Wolcuzk: Sure. First off, what I think, when we're saying algebra, what I think we're talking a lot about here is actually like a high cognitive load. So, if the students don't understand all of those properties or if they just have to spend that much cognitive effort thinking about them, that just makes everything else so much more difficult.

So having their algebra skills and those other skills and knowledge memorized well, will actually really help them be able to focus on the new concepts. And so related to that, the one topic that we always find students struggle the most with is trigonometry. And again, I think this is not surprising as trigonometry imposes a high intrinsic cognitive load.

And I think this is a really good example of where a greater emphasis on the memorization of things like facts would greatly benefit students, and I really do think that students struggling with trigonometry is a big problem as trigonometry occurs frequently in the applications of mathematics.

[00:15:44] Anna Stokke: So, Dan, what kind of facts are you talking about that students should have memorized?

[00:15:49] Dan Wolcuzk: So, well, a) all the basic, trigonometric, ratios, the graphs. I am actually finding a lot now that students coming into university cannot draw the basic graphs of sine, cosine, and tangent. They're not familiar anymore with sine and cosine or wave functions. They just think of them as SOHCAHTOA, as ratios.

But more than that, there's many trig identities, and I know students hate trigonometric identities, but there are a lot of them, and they're very useful. And again, this is where this high cognitive load comes in, is that students need to be able to use a trigonometric identity in amongst a calculus problem, or a linear algebra problem, or many other areas.

And so, if you don't have those well memorized, then it just imposes that high cognitive load, which really hampers problem-solving.

[00:16:46] Anna Stokke: Okay, so you're talking about things like they should have the basic trig identities memorized, the basic trig values memorized, graphs of basic trig functions in their heads, the basic trig ratios, that's going to make things a lot easier. And I agree with you; it's going to make it a lot easier to solve problems when you're working on integrals and things like that.

I must admit, Dan, I'm a little bit surprised that you're seeing some of this because all of your students actually have calculus from high school. Am I right?

[00:17:20] Dan Wolcuzk: Yes, so all of my students that I teach have high school calculus. But as Darja already mentioned, they're actually better at calculus than they are at pre-calculus. Now, I think this is actually not that surprising when you think about it, in that for our students they take their pre-calculus in the fall, and then their calculus in what we, what we call the winter term.

So, it's actually, their calculus is more recent than their pre-calculus and a lot of the calculus they learn is basic limits taking derivatives, which are just sort of memorizing some basic facts. It doesn't have that same intrinsic cognitive load that trigonometry does. And so, with the time delay and the complexity of the material, the fact that memory, well, yeah, memorization wasn't a focus of their initial training, I think really hampers their pre-calculus knowledge.

[00:18:19] Darja Barr: I just totally agree with what Dan is talking about when he's talking about cognitive load, and how it's really impossible to dive into a problem that's even one step higher than just basic, plug into a formula when you don't have those facts memorized and easily accessible in your mind.

Automatic; you know what sine of zero is, you know, what cos of pi is. You're not taking time drawing things. And even, you know, taking time isn't that bad, but students who don't even know how to begin with something like that have no chance.

[00:18:58] Anna Stokke: Yeah, exactly. And what about things like log rules?

[00:19:03] Darja Barr: Exactly, same thing. So, when you're solving a derivative, and you have to use logarithmic differentiation, and you're looking at something and you have to simplify it before you dive into taking the derivative, you need to know your log rules.

Or solving a limit that has an exponential function where you're actually not going to do any factoring and cancelling or rationalizing or any of those things. You just need to remember the basic graph of a log or an exponential function. It's this kind of fluency with the basics that I think forms a big mental block for a lot of students in calculus.

[00:19:40] Dan Wolcuzk: Let me jump in there, in that, I think it's not just about the trigonometry and the log rules, but even something, and I hate to use the word simple here, but I'm going to, as simple as adding fractions. If the students aren't comfortable with that, then that can actually take a lot of cognitive load. We actually see this when teaching class.

Is if you're going through an example carefully, step by step, when you're trying to teach them this new method, this new algorithm, if they don't understand that how you added that fraction step, then they're not going to understand all the new material that you're teaching them.

And so all of those basics, the better they understand them, the more focus they can put on the new content.

[00:20:28] Anna Stokke: Absolutely. And I'll just sort of say something at this point that we're not intending to have this discussion to be critical of high school teachers or anything like that. We know that high school teachers are working really hard to teach the content. And, you know, and, and we know that they've taught the content.

It's hopefully there somewhere in the students and we can pull it out. But I think what maybe is happening is oftentimes students aren't getting enough practice in some of that fundamental content. To the point where, you know, you should practice enough so that it's automatic. You shouldn't have to think about the quadratic formula, right?

You should just know it. And a lot of these things just need a lot of practice to get really good at.

[00:21:12] Dan Wolcuzk: I'll add to that in that, I would like to see, yeah, that emphasis on memorization. So, it's not just about teaching the students how to add fractions, but in some sense, telling the students that you'll need to know how to do this really, really well when you get to university. So not just teaching the contents, but actually trying to express to the students how important it is for them to not only learn, but to remember this.

[00:21:38] Anna Stokke: Right, so making it clear to students that university math will be difficult if they haven't practiced the pre-calculus topics until they're completely fluent, meaning they can, say, solve an algebraic equation easily or they can quickly see that ‘x’ times the square root of ‘x’ is ‘x’ to the three halves, and, you know, they can recall and work with trig identities and log rules effortlessly, for example.

[00:22:05] Darja Barr: And giving them a chance to diagnose themselves. Do they actually have this automatically in their heads or not? I think there's not enough places for students to have those moments of like, “Oh, I'm actually not as fast as I thought I was. I don't have this memorized.” And then working through it until they get to that spot that they need to be at.

[00:22:30] Anna Stokke: Speaking of which, I'd like to ask if you have placement tests or diagnostic tests for calculus at your university, and if so, how are they used and what have the results been like?

So, I'll start with you, Darja.

[00:22:45] Darja Barr: We do have placement - it's not so much a placement test because a true placement test would place students into calculus or into another course, but more of, like you said, a diagnostic test. We've had one in one form or another for the past 10 years or so.

The way it works is that students who are already registered in calculus at our university take the diagnostic at the very beginning of the course, and it makes up about 5% of their grade, so they have to score a certain amount on the diagnostic in order to earn that 5%.

If they don't on their first attempt, we direct them to some remediation that they can do and then they can have a second attempt and then they can earn part or all of that 5%.

[00:23:29] Anna Stokke: That sounds like a good idea. Dan?

[00:23:32] Dan Wolcuzk: So, I use a diagnostic test at the beginning of term in my calculus classes, although I think I'm the only one at UW, at least that I know of, who's doing that. I use the test in a couple ways. So first, as Darja just said, is to help students identify what gaps they have in their required prior knowledge.

And I provide students with resources so that they can build up their foundational knowledge. And this is very important to me. I have seen time and time again that one of the main reasons that students struggle in university math is they simply don't remember their high school mathematics well enough.

The other way that I use a diagnostic test is to inform my course design. For example, I found that, as we already talked about, students have better calculus knowledge than pre-calculus knowledge coming into my course. And so, I redesigned my course so that we can actually meet the needs of the students.

So don't re-teach what they know, focus on building up those foundational skills and going beyond that.

[00:24:40] Anna Stokke: And so, at my university, I teach calculus as well. We just kind of assume that students likely are going to have to brush up on their pre-calculus skills. And so, we actually give them, you know, tests on that, similar to what Darja is talking about. And we provide them with resources and it counts for a certain part of their grade.

And so, you know, we talked about a few things. We talked about the types of things that are important for students to know, like the topics that students need to know to do well in University calculus. And actually, high school teachers often ask me about this. So, you know, I think this will be really helpful.

And I am going to post some resources on the resource page. I'll post something from the University of Toronto because they've compiled a list. I'll also post something that Brian Conrad from Stanford University gave me. He and his colleagues came up with a list of topics and concepts that are important for students to know to succeed in university math.

And I mean, it's the same everywhere, right? Like calculus is the same no matter where you go. So, I'll post those things. And then if you have some things that you'd like me to post on the resource page that you think would be helpful, I'll do that as well. Okay?

[00:25:52] Darja Barr: Yeah, we actually went through one year a bunch of old calculus midterms and final exams and pulled the calculus out of them and made what we call a boot camp resource out of what was left. So, it's all from whatever you need from kindergarten up to Grade 12 focused for calculus. So, I can certainly share that as well.

[00:26:14] Anna Stokke: Okay, so let's shift a bit and I'd like to talk about high school grades and preparedness for university. So, I believe that generally, in Canada, high school grades are used as the main predictor for student readiness for university math. It may be different at some universities in the United States, but I'd like to talk about that.

Whether high school grades actually are the best measure and if there might be better ways to do this. And Darja, your PhD actually focused on this. So, you analyzed the relationships between Grade 12 pre-calculus grades and first-year university calculus grades. This is a local study, but my guess is that the results are likely similar in other jurisdictions.

So, can you talk a bit about that? Like, what did you find with your research?

[00:27:09] Darja Barr: Yes, so I looked at students in our calculus course who are coming from Manitoba high schools and over 15 years took data that correlated the grade they finished with in high school pre-calculus and then the grade they earned in our first-year calculus class. And I looked at all sort of demographic variables as well, how were the mature students, what was their gender, etc.

And what I found was that actually, not shockingly, but disappointingly, there was only a moderate correlation between high school pre-calculus grade and university calculus grade. And that was especially crushing to me given that our Manitoba curriculum documents specifically describe pre-calculus as a course that's aimed at preparing students who plan to take university calculus.

So, it should be the pre-calculus course as it is named. There should be a strong correlation. Though, of course, there are lots of other factors that come into play. What I found was that in Manitoba, our average calculus grades over the years range between 50% - 60% final grades, with about a 20% to 50% failure and withdrawal rate.

Whereas the average incoming pre-calculus grade has actually been increasing over the years from about 80% to closer to 85%. So, we would expect students to have a bit of a drop in grade going from high school where things are very, they're small classrooms and student teachers are kind of knowing everybody's name and holding their hands to university where you're small fish in a big pond.

But that is quite a significant drop. And though that drop might be expected, what was less expected was the lack of predictability. You would expect that the better students who did better in pre-calculus would do better in calculus, even if there was a drop in that grade. But what I found that was the most shocking, at least to me, was that someone coming in with an A in pre-calculus was equally likely to finish calculus with an A as they are with a C or a D.

So essentially students coming in with high grades feel very prepared, but the data shows that that A doesn't really mean much in terms of whether they're going to succeed in calculus or not. That's really important for us as instructors to know, but also for the students to know.

[00:30:03] Anna Stokke: Yeah, it is alarming. And like you say, it is particularly worrisome because students have the false impression that they're prepared when perhaps they're not,

[00:30:15] Darja Barr: Exactly.

[00:30:16] Anna Stokke: And like I said, I don't think this is just a local phenomenon. I have certainly read a lot of articles coming out of the United States about grade inflation. And in the United States, they actually collect, often, a lot more data on student performance than we do here in Canada.

But Dan, what about your experience at Waterloo? Would you say there's a disconnect between the grade a student received in high school and how prepared they are for calculus?

[00:30:47] Dan Wolcuzk: So, absolutely. Although I haven't done as rigorous of a study as Darja, I'm jealous. I have done some of my own little studies. For example, like I do use my diagnostic tests, and I compare that whenever I can to the student's incoming grade from their calculus or pre-calculus courses.

And like Darja, found there, there's no correlation. A student could be coming in with a very high grade. And in fact, to get in at math at Waterloo, typically, the grade cut off averages in the high 80s or even low 90s. So, we know these students are coming in with very high grades in their calculus and pre-calculus.

And yet, on my basic high school math diagnostic test, we have students score extremely badly. More broadly, in Canada, this is a known problem of we're seeing high school grade averages steadily increase while our PISA scores are plummeting, and so this is a really good indication that the high school grades are not reflective of how prepared students really are.

And I definitely want to jump in there and say that, like, I agree that, in addition to making it difficult for admissions to pick candidates who are the best prepared, to me, the big problem is how it's affecting the students. So, as you said, these students are coming and always having gotten really high grades, and then all of a sudden, they're really struggling.

And it's not totally their fault. They are simply not prepared. I have seen some of these students who could be really excellent engineering, science, or math students, but they're being made to believe that they are not good at math simply because they don't. Well, they're coming in not actually prepared.

[00:32:36] Darja Barr: Echoing what Dan said, when I looked at our diagnostic test results and thinking about that in the context of students coming in from pre-calc with 85% as an average, we were seeing when we did the placement test in a proctored way, so students are being invigilated, they can't use other resources, the average on that diagnostic test, which is just straight up pre-calculus material, was from 15% to 30%.

When we started allowing them to do it online, at home, it went up to about 50%, but nowhere near the grades that they're coming in with.

[00:33:22] Anna Stokke: Yeah, and I know Dan's kind of laughing because I actually know he's done some research on, actually, physical invigilation versus letting students do tests online. So, did you want to add something, Dan?

[00:33:37] Dan Wolcuzk: Yeah. I'll add a couple things to that. So, yeah, I'm really trying to do all my diagnostic tests in particular not online as a proctored test so that we can ensure that not only we are seeing what the students know, but the students are seeing what they know. They're not cheating on tests, they're getting the help that they think they can use.

And so, they're not necessarily identifying the areas that they are actually weak in which is a problem. The other thing I actually really wanted to mention is that there is one other problem that I thought of to these students coming in underprepared. And that's actually for us as instructors.

The wider the range we have of students of prerequisite knowledge, especially in our classes of 100 or more students, the much more difficult is for us to provide every student with the instruction that they actually require.

[00:34:32] Anna Stokke: Absolutely. And so Darja, you did this study, and basically, the high school grades have been going up over time, but the university calculus grades - and you haven't changed the calculus class too much I don't think - have been going down. And so, we're seeing a big discrepancy between the high school grade and the university calculus grade.

Did you look at reasons for that discrepancy or do you have any ideas why this is going on?

[00:35:01] Darja Barr: Yes, for sure. I mean, that's really the meat on the bones there. Why is this happening? And there's so many factors. There are huge culture shifts from high school to university. Like I said, whereas every teacher knows every student's name in their classroom, and is kind of on top of whether they're doing their homework, whether they're there for tests.

At university, especially at these large research institutions, that's not the case. I am not learning all 200 students’ names in my class. I don't know if they've done even the midterm. I have no idea, really, what's going on other than sort of aggregate pictures of my classroom, as much as you try. So, that culture shift, I think, is a real shock for students.

Where we can tell them it's important to be on top of things, it's important to do this, but without having somebody actually checking in and having all this freedom all of a sudden, I think it's difficult for students to make that mind shift. So, some of that is to be expected and it's natural and it's part of growing up and becoming an adult.

But I think some of the finer things, some of the things we have more control over that actually make a big difference are things like content emphasis. When I look at the Grade 12 content in the pre-calculus course, and what is emphasized, and how much time they spend on certain topics, like the one that always grinds my gears, is how much time they spend on translations of graphs, shifting, stretching it's shocking.

And of course, when you emphasize one thing, that's at the expense of all of the other things. How much time do we spend on that, or how much do I actually think it's useful for students to do that? Perhaps one question on one lab somewhere could be assisted if you could translate it rather than do something else to it.

But really, I would throw that completely out and add a refresher on algebra into Grade 12, or something along those lines, so that content emphasis. Looking at the Grade 12 pre-calculus provincial exam or final exam, in my opinion, that should look exactly like our calculus diagnostic test. Then we know that we've got a seamless link between those two things that are supposed to be bridges, two sides of a bridge.

Assessment is another big one. The style of assessments that you can run in a 200 person, 2000 student class in one term are very different than what students are getting in high school. We're not doing projects that have to be graded. We simply don't have the resources and time for that. And then there's things like teaching style, classroom style, and just the expectations of accountability I think is a really big one.

[00:38:07] Anna Stokke: Okay, so you mentioned a few things. I'll try to recap. So, you mentioned just work ethic and the culture, university classes versus high school classes. There seems to be a disconnect between the content that we want students or that we would need students to be good at in order to succeed in calculus, and then the content that's perhaps focused on sometimes in high school.

And I'll add to that, if you're focusing on content that maybe isn't as useful for success later, students are getting a lot of practice on the wrong thing. That practice piece is really important. You talked about assessment, and that's a big one that students are going to have to write exams when they get to university. There'll be a lot of emphasis on exams And I think that kind of sums it up, right?

[00:38:56] Darja Barr: I would agree.

[00:38:58] Dan Wolcuzk: There's one other aspect of this that I think is definitely worth talking about, and I think it's that the teachers here, again, are not to blame. They're under a lot of pressure. As I mentioned, like, at the University of Waterloo, in our math and engineering in particular, computer science, we have these very high cut-off averages.

And so, if some other school is inflating their grades and you want your students to be able to make it into University, you kind of have no choice. So, teachers and schools are in some sense being forced into these grade inflations and it's becoming this this war that this keeps bumping up trying to get their students into universities.

I've read reports of students having 96% averages and not getting into the program that they want. And so, this is leading teachers and students and schools to be grade-focused. And we know that grades are not the same as learning. And so, the more that there's this war on who can have the highest grade so their students can get into their desired programs the worst it's getting.

[00:40:11] Anna Stokke: And there's also the pressure for scholarships. So, for example, in our province, students qualify for different tiers of university scholarships based on their high school percentage average, with students above 95 percent average, say, getting the better entrance scholarship.

So, there's a lot of pressure, I think, from parents and students to make sure that the high school grades would qualify students for scholarships. So that might also be an issue.

[00:40:38] Darja Barr: And there's also pressure on those teachers. Especially the ones who know what's coming next in calculus and know what they should be emphasizing to prepare their students. But then they have to, they also have to prepare their students for the provincial exam that's coming, that's provincially mandated, they don't write it themselves. Every student writes the same one.

So even though they know that this topic may not be important for calculus and they wish they could focus on something else, their students are going to be writing this provincial exam where that topic is going to appear on multiple questions.

So even the good intentions there of the teachers are being thwarted in terms of grades, by pressure to align to other curriculums or assessments that are coming, pressure from administrators, pressure from parents.

[00:41:29] Anna Stokke: I mean, maybe there's a way around this though, right? Around the whole grade inflation and what's been going on. I mean, maybe we should be using some measure other than high school grades to determine whether students are ready for calculus, so what do you think about that? And what could we use?

[00:41:47] Darja Barr: For sure. I think high school grades are an important part of the picture because they are longitudinal. Right? Something like a placement test or a diagnostic test happens at one snapshot in time. A student could have a bad day. But, all of the research out there shows that students are most successful when they're placed properly in courses that align with their skill set.

And that placement should be done using multiple measures. So, high school grades as part of the picture and other measures such as placement tests.

[00:42:20] Dan Wolcuzk: So, I agree completely. I would really like to see some sort of nationalized, standardized testing. Now, I mean, as Darja mentioned, there's flaws and cons to standardized testing and making sure your standardized test is aligned with what we desire our students to have coming into university would be fantastic.

But I do see two main benefits of standardized tests. Again, it would really help admissions identify the individuals with stronger prerequisite knowledge. It's like Darja just said, it would help not only us, but the students place themselves into a course that's going to meet their needs, which is going to help them, well, help them succeed, build their confidence, etc.

But the other, and I've heard this of some of my colleagues I talked to from the UK, where they typically use standardized tests. And that is a great thing about standardized tests is that the teacher is no longer the nasty test creator and marker. The teacher is the student's ally who is helping prepare them for the standardized test.

[00:43:28] Anna Stokke: Yes. In fact, Dylan Wiliam mentioned that. So, he advised many schools in the UK for many years and he just expressed this exactly the same way as you did. So yeah, I agree. So, Dan, you can be the advocate at the national level for some sort of standardized exam like that.

But speaking of which, let's actually talk about exams. You know, there's actually a lot of talk across Canada about cancelling high school exams. And this has been coming up again and again. In fact, more so since COVID, when exams did get cancelled for that period of time. And some schools would like to keep up with that.

So Darja, what do you think about that? Will this result in students that are even less prepared or what are the benefits of high school exams?

[00:44:18] Darja Barr: The world is full of stressful situations, and I know that's kind of a fallback that we say sometimes, and I understand that high-stakes exams can be stressful for sure. Nobody's arguing that they're not. But the reality is that students see all kinds of high-stress situations like accreditation exams. Students who go into engineering, medicine, law; they need to process huge amounts of information and then they need to show what they know in high-stakes scenarios.

I mean, think about doctors and the stakes that they're dealing with there. There's deadlines in the work world that you need to meet or you lose your job. High stakes is part of our reality and I think we're doing students a big disservice by not exposing them to some of those things.

They need to understand what that feels like so that they can build skills like resilience, perseverance, learn how to prepare themselves and get those important life skills in their tool belt as they go up and grow up through school.

[00:45:29] Anna Stokke: Alright, so Darja, those are some great points about taking exams. Dan, do you have anything to add to that why it's important for students to take exams and maybe you could say something about the testing effect. I know you know a lot about that research.

[00:45:44] Dan Wolcuzk: Sure. So yeah, the testing effect is a very robust finding in psychology, which says that learning is improved when we retrieve information from memory. Essentially, I think of learning as a two-way street. It's not just about getting information into our brains; we need practice getting it back out. I always tell my students is that it doesn't matter what gets in there.

If you can't get it back out, that is not helpful. So, I actually believe that students would really benefit more from a lot of low-weight testing. Yeah, I mean those great big, stressful, 90 percent of your grade exam - no, no, that's bad. But, lots of low-weight testing not only improves learning but there's actually lots of growing research that shows that lots of this, lots of low-weight testing reduces test anxiety.

There's a really nice paper from 2023 by Yang and colleagues that analyzed that really well. And I've also done some of my own research here at the University of Waterloo, which through student surveys says the same thing. Is that, essentially, if you get lots of practice at writing tests then you're of course going to be more confident with it.

But one thing I think that is missed is that the reason that a lot of students have such high test anxiety is because of assignments. So, they're used to having all this time, getting all the help they need, and so they have very high averages on these assignments. Like, yeah, 90% average in all your assignments.

And then, they get a midterm. And the midterm they have to write it on their own, they have limited time, they don't get all that help. And so all of a sudden, they're getting 60 percent and they blame it on the test. When it's really not the test that was too hard is that the assignments weren't preparing them for it.

I was giving them false expectations. It was leading to an illusion of competence. So, by having these low weight tests, not only the students get practice in testing, but they get to set their expectations to be much better. In my studies at the University of Waterloo I've looked at this correlation between assignment grades and test grades, and there was very little correlation.

And between my quizzes and the test grades, there's almost too high of a correlation. I would like students to be spending more time learning from their mistakes on the quizzes and getting higher grades on the test.

[00:48:22] Anna Stokke: Okay, so what you're saying is that tests, low stakes tests actually help students learn better, and they also prepare students for higher stakes tests. They reduce test anxiety because students get the experience taking these tests. And again, it's about giving students the correct interpretation of where they're at. And quizzes are better at doing that, and tests and exams are better at doing that than assignments.

I would agree with that. That has been my observation as well. I've actually moved away from giving as many assignments. Even in my third-year class, actually, I'm putting more weight on quizzes now.

And actually, it was advice I got from Dan because I emailed you about it and asked you what you thought and you suggested give more quizzes. And actually, I think that's really effective.

[00:49:14] Dan Wolcuzk: I mean, I found it extremely effective. it really seems to be beneficial, and the students, once they get used to it, and it takes the adaptability from students a bit, but then they really appreciate it in the long run. I get a lot of comments on my student evaluations of how they love how the quizzes help them keep up with the material and show them what they know, and they don't know.

The one comment I commonly get from instructors who are reluctant to use quizzes is they feel that they can't ask those same deeper, longer questions they could on a homework assignment. And my answer to that is that my practice quizzes, or practice problems prepared for the quizzes, are those old homework assignments with those deeper questions.

But now, instead of students doing them to get marks, so making sure they have the right answer, they're actually spending much more time and effort on learning how to solve the problem, learning the content in case that will be asked. So in office hours, students aren't asking me for the solution, they're asking me the thought process and how to solve the problems.

And so even those old homework assignment problems, the students are now using them for learning. over just trying to get a high grade.

[00:50:33] Anna Stokke: Okay, makes good sense. So Darja, we've talked quite a bit about the issues we're seeing sometimes with incoming students not having the skills that they need to succeed in calculus. So, let's talk about some things that we're trying to do at the university level to help students succeed because we have some responsibility to do that.

So Darja, do you want to talk about some of the things you're doing to help students succeed?

[00:51:00] Darja Barr: Yeah, I mean, we don't have a lot of control over the content. Like you said, calculus is kind of the same everywhere, and there's certain things that need to be in there. But what I do personally is a lot of talking about expectations and a lot of talking about self-diagnosis. and what it looks like to be a good student in terms of study quality over study quantity.

When you reach a problem that you can't solve, you don't go to the solutions manual, look at the solution, say, “Oh yeah, I understand that solution. Therefore, I know how to solve that problem.” You stop, you pause, you ask a friend, you ask your TA, you ask your professor. Only then have you solved that problem.

Then you try another one of the same type. Really giving them a recipe for what it looks like to practice in a smart way. Test taking tips, tips for what to do once you get your test back, right? Don't just crumple it up and throw it away, but use it as a diagnostic tool. So, on my end, I think there's a lot of support in terms of what Dan does as well, helping students learn how to learn the math content.

[00:52:17] Anna Stokke: Absolutely. And I will add, you know, we talked about your research on student preparedness for university calculus and how that didn't necessarily match up with high school grades. And I do want to add that I know that you have worked with school divisions in the province and that you work with teachers to sort of try to smooth that transition.

So, you're not just here complaining, you're actually working with teachers and trying to help smooth the transition from high school to university, correct?

[00:52:49] Darja Barr: Correct, yes.

[00:52:52] Anna Stokke: Okay. And Dan, you know, Darja talked a bit about discussing with students how they can succeed in the math, in her math classes. And I know you've done quite a bit of work on educating students about learning how to learn.

So, can you talk a bit about that and do you think this is one of the keys? Like should we be working harder to educate students about how best to learn math?

[00:53:18] Dan Wolcuzk: Absolutely! I actually believe that teaching students how to learn is the key. So many times I've heard instructors say something like, “They don't need to worry about their best students. They will learn no matter what you do, what you do in class. They are dedicated to learning, they love learning, they know how to learn. You don't need to worry about them.”

Now, just imagine if all of our students were like that. Isn't that the goal? When I read university goals, a big common one is to create lifelong learners. Well, if we're going to create lifelong learners, then teach them how to learn. So, my goal is to turn all of my students into my best students.

I honestly believe that the only thing that will revolutionize education is to help our students become better learners. So, I put a lot of effort into teaching learning skills and strategies both inside and outside of class. So not only do I provide my students with resources, videos and articles about learning skills and learning strategies, but I talk about these inside of class.

And when teaching content, I don't just teach the content. “Here's the fundamental theorem of calculus.” I will give my students strategies of how they can go about learning this. Or let them know the level that they need to learn this. I'll tell my students things like, “You really need to have all of these derivatives well memorized. They need to be instantaneous to reduce your cognitive load later.”

So, by providing students with the overall strategies, but even specific strategies for specific content, we can really make students all the best students. And in the end, whatever we do in class, if, well, it depends on what the student does, is if students don't go to class, there's nothing we can do to help.

But, if they are strong learners, and I've, every once in a while, I have a student who gets 100 percent in my course, and I never see them in class, and I think to myself, “Well, you probably, you don't need to be in class if you can learn on your own.” And, when students go beyond university, or even into grad studies, they need to learn how to learn on their own.

So that, I really think that is the most important skill that we can teach.

[00:55:49] Anna Stokke: Yeah, I agree. And even just. Just something like making students aware that just reading a solution, as Darja said, because I actually see that a lot. So, students think, if they read a solution, then they know how to do the question or they get help from a tutor and the tutor shows them how to do it and then they think they know how to do it.

But they don't, right? Like, they actually just read a solution or they understood someone when they told them how to do it. The student actually has to do it themselves. So even just telling students those types of things so that they realize that what they think is an effective study strategy actually isn't, is really helpful for students to know.

[00:56:28] Dan Wolcuzk: Yeah, I'll echo this going back to what we're talking about quizzes is so again, not only do students learn this way by not knowing how to solve the problem and this reading the solution, but then they're getting, they're doing this on the homework assignments, getting these high grades, they see all the check marks. So, this is reinforcing that illusion of competence.

And so, this is where, again, I think the quizzes really help students recognize the, the, “Oh, I thought I really did know that, but I clearly don't.” And so I, again, I'll use this to reemphasize how wonderful quizzes are.

[00:57:06] Anna Stokke: Okay. Excellent. So Darja, what are the main characteristics of a successful university math student?

[00:57:14] Darja Barr: You know, it's funny. I often ask my students this question. What do you think are the characteristics of a successful calculus student? And inevitably the answer is almost always focus on content. Somebody who can do this, somebody who can do that.

But in my experience the students that are the most successful not just in calculus, but in all the math courses I find are the ones who know how to figure out when and where they need help and then they seek out that help.

So, it's exactly what Dan said. We need to give them opportunities to figure out where they need help. They need to know how to do that, and we can help them along the way with that. And then they need to seek out the help and they know they have to know where that help is available. That's what I think is actually the number one trait of a successful student.

[00:58:07] Anna Stokke: And how about hard work and doing lots of practice?

[00:58:11] Darja Barr: For sure, those are 1.1 and 1.2.

[00:58:14] Anna Stokke: Absolutely.

[00:58:15] Dan Wolcuzk: I will echo that the getting help is, I think, the vital piece. In the end, I tell my students that to think about Olympic athletes. So, an Olympic athlete does not get to where they're at by doing it themselves. The reason they succeed is because they get all the help they need. They have coaches, and trainers, and sports psychologists, and physiotherapists and their peers, and their family, and they have so many avenues of support that they get and use.

The key to success, I don't think just at university, but in life, is to use all the available help that's there. At university, we provide all sorts of help. It is about recognizing when you need it, and going and getting that help. We don't succeed on our own, we succeed as a group. I always describe my course to students as it is a game. It is a cooperative game, where you're trying to get the largest score possible.

The other piece I would add to that is perseverance grit. So, this idea of being adaptable when something goes wrong and inevitably in university and life, things will go wrong, it's about those metacognitive skills of identifying what went wrong and getting the help to try to correct it and moving forward.

[00:59:43] Anna Stokke: So, is there anything else you want to add to the discussion today? Is there anything we missed?

[00:59:48] Darja Barr: I would add maybe some advice to students in their first year of university. I think coming from high school, it's natural that when these failures happen that Dan talked about, when you hit the wall, when you fall, the natural instinct is to push the blame outward. “The test was too hard,” “The teacher didn't cover that content.”

The advice to students is that those are things you can't change. And putting the onus there, you're crippling yourself. Turn it around, focus it inward. What did I do that resulted in this? What could I do better? And if you need help laying that out, seek out the resources, study resources, counselling resources, content resources from your instructors to make that happen.

Keep pushing, stick to it. You can do it, but it's all about that accountability turning inward.

[01:00:46] Anna Stokke: Excellent advice.

[01:00:47] Dan Wolcuzk: I would say that do not underestimate how fast-paced university can be, so really try not to get behind. A former student of mine sent me an email of how she really learned and worked hard this term to try to stay at least a week ahead in her classes. And she got really sick for a week, and if she hadn't been ahead, she would have been in trouble.

But she said like, yeah, so you've learned that things happen, and so by getting ahead, there's that buffer for when those bad things happen. So, really, try to get ahead, and if you only stay on pace, that is much better than ever falling behind. If you fall behind, it'll be really hard to catch up.

And the other comment I would say is, when I talk to a lot of students as they're first incoming to university, they don't think that these things apply to them. They think, “Well, no, I mean, I've heard this in high school, but I always got high 90s, so, so this doesn't apply to me.” And so, my comment to them, or my comment for teachers to tell their students is that, it's not necessarily about giving them all these things, all this information right up front, but it's letting them know they're there.

So, telling the students that if you encounter difficulties, here are some resources you can go to look at for help. I do find that after the midterm, students are way more likely to listen to learning advice and strategies than they are when they're fresh out of high school, where they feel they have good strategies and have evidence for that.

[01:02:31] Anna Stokke: So, we'll end on that note. So, thank you so much to both of you for sharing your time and your expertise with me today. I think it was a really great conversation, and it will be really useful to teachers and university instructors and students. So, thank you very much.

[01:02:51] Darja Barr: Thank you again for having us.

[01:02:53] Dan Wolcuzk: Yes, thank you very much. It was an honour to be on here. I'm a big fan of your podcast.

[01:02:58] Darja Barr: Ditto.

[01:02:58] Anna Stokke: Thank you.

As always, we've included a resource page that has links to articles and books mentioned in the episode.

If you enjoy this podcast, please consider showing your support by leaving a five-star review on Spotify or Apple Podcasts. Chalk and Talk is produced by me, Anna Stokke, transcript and resource page by Jazmin Boisclair, social media images by Nicole Maylem Gutierrez.

Subscribe on your favourite podcast app to get new episodes delivered as they become available. You can follow me on X for notifications or check out my website, annastokke.com, for more information. This podcast received funding through a University of Winnipeg Knowledge Mobilization and Community Impact grant funded through the Anthony Swaity Knowledge Impact Fund.