I will be giving a presentation on Saturday in the contributed paper sesson on Structures for Supporting Student Success in Entry-Level Mathematics. I’ll be reporting on the first two years of the HHMI Success in STEM initiative at Rice University to redesign our first-year calculus courses.
]]>I think it was overall a really valuable learning experience, and I definitely plan to continue it (with tweaks) in future semesters.
Special thanks to Hannah Turner and Nathan Willis, who shared their video/oral quiz materials from past semesters with me!
This was a small course of about 20 students, who were first year students interested in advanced mathemaics (mostly math double majors). Math 232 emphasizes both proofs and computations, and it is the second course in a sequence (coming after proof-based linear algebra).
I had the support of one graduate TA for this course, who primarily graded homework problems and helped to grade exams. I did not end up using the TA to help with grading oral quizzes, though I did consider it.
I wanted to help my students develop their (mathematical) communication and collaboration skills. In particular, I wanted my students to learn how to explain and defend their ideas.
For first-year honors math courses, I believe it’s important to show students what doing advanced mathematics is about (it is not taking timed closed note exams!). I also believe it’s important to teach them skills that they need for future math courses.
I assigned a total of 6 non-cumulative quizzes over the course of the semester, roughly occurring every two weeks.
However, to make the logistics manageable, for any given quiz (except the first one), roughly half the class would take an in-person oral quiz, and the other half would submit an online video quiz, instead.
Thus, each student would complete the following by the end of the semester:
Moreover, I arranged the schedule such that every student would take an oral quiz for both of the units in the course (differential multivariable calculus and integral multivariable calculus, respectively).
For both the video and oral quizzes, the quiz questions were based entirely on problems from past homework assignments.
Thus, the students were incentivized to take the HW very seriously. The HW questions were a mix of proof, conceptual, and computational questions. To avoid double-penalizing students, the subset of problems that were graded for accuracy on the HW was mostly disjoint from the subset of problems that appeared on the quiz.
Moreover, because the quiz questions came directly from past HW assignments, the primary focus of the quizzes was for students to explain and defend their ideas, rather than having them figure out completely new questions on the spot.
Their target audience for their explanations was a fellow student in Math 232.
The video quizzes were asynchronous, to be submitted within a 48 hour period. I assessed a late penalty for any quizzes submitted after the deadline.
For the video quiz, I would assign one particular homework problem, and ask students to submit a video (unedited, and no longer than 5 minutes) of them explaining their solution from scratch. I also required them to submmit an image of what they wrote during the quiz (in case their writing was not legible in the video).
For the video quizzes, I allowed them to occasionally reference notes, as long as it does not affect the flow of their presentation. (e.g. their presentation should not be read from a script).
The oral quizzes were in-person, without notes, during a recorded 30 minute appointment. Well-prepared students would complete the quiz within 10-15 minutes, but most students took 20-30 minutes to complete the quiz.
For the oral quiz, I would give them a selection of 4 problems from the homework. They would first choose one problem to present, and then I would choose a second problem for them to present. This allowed students agency in showing off what they understood, while also ensuring that there was full coverage in terms of both the topics and the types of problems (computational, conceptual, proofs).
During the oral quizzes, I would also ask them clarifying and/or follow-up questions to test their understanding. However, I emphasized that my role as questioner was observational, not antagonistic. I began every oral quiz with some version of this statement:
The way this differs from a video quiz is that this is a live conversation. So if I think you should explain something more, I’ll prompt you to do that. However, I’m not trying to trick you, I’m trying to gauge what you understand.
And if you get stuck, I’ll let you think about it for a little bit and get unstuck by yourself. But I’ll ask if you need me to give you a hint, I’ll give you a hint so that you can continue to show me what you know.
Grading oral quizzes was actually very fast - I had a good sense of what general letter grade they would receive based on our ~30 minute conversation, and I would even tell students immediately afterwards what general letter grade to expect.
After all the oral quizzes were finished, I would then double check my notes and/or the recordings to make sure that I was being consistent with grading, and to decide on plus/minus letter grades.
For video quizzes, the process was similar - I would watch each video once to take notes and assign an initial letter grade, and then go back and make sure that I was consistent with grading if necessary.
For each problem, I used a grading rubric based on 3 major categories:
The general sentiment of the rubric was that the conceptual understanding and details categories accounted for the main letter grade (e.g. A, B, C, etc.), and the mathematical fluency category would be the plus/minus adjustment. If you want to see the exact rubric I used, send me an email!
When I run oral quizzes again, I would make some changes to how I implemented my rubric. For example, I would add some quantitative elements to the rubrics, such as:
Oral quizzes (as I’ve designed them) are not a replacement for in-person exams, because students are not asked to figure out how to apply their knowledge to new questions.
However, I think the oral quizzes prepare students well for oral exams, and I did use oral exams for makeup exams (and an optional redemption assignment).
My oral exams were similar to the oral quizzes, but the exams were held in-person during an hour-long appointment, and I allowed students the use of a notecard (I also allow a notecard for written exams).
For the oral exam, I gave students 1 proof-based question, 1 conceptual question, and their choice of 2 out of 3 computational questions. Just like the oral quizzes, I would also ask them clarifying and/or follow-up questions to test their understanding.
]]>If you are interested in taking Math 232, you should send me an email. You are welcome to attend lectures as a guest before the add/drop deadline, and you should also talk to me before/after class to be added to the course Canvas page.
You can find the course syllabus here.
What tools do we have to calculate an integral like $\int_0^1 xe^{-x^2} \ dx$ or $\int_0^1 e^{-x^2} \ dx$?
When your calculator tells you that the value of $\int_0^1 e^{-x^2} \ dx \approx 0.746824$, how do you know it’s correct? How accurate is your calculator? Is it possible for an infinite region to have finite area? Can we calculate the area of a fractal?
You’ve previously seen in Math 101 that calculus is an important and powerful tool that allows us to describe the physical world around us. In Math 102, we will push the notion of integration to its utmost limits. In this class, we will build our toolbox for integration; we will study the behavior of the infinite, and we will learn how to quantify how accurate our approximations are.
In this course, you will develop the critical thinking and questioning skills needed to answer these complex questions. Moreover, you will become fluent in precisely communicating your ideas through the mathematical language of calculus.
You can find the course syllabus here.
How can we describe the physical world mathematically? What changes, and what stays the same when we move from single variable calculus to multivariable calculus? What does it mean to take a derivative of a multivariable function? What does it mean to integrate a multivariable function? What regions can we integrate over? How can we generalize the Fundamental Theorem of Calculus?
Multivariable calculus is the mathematical language that allows us to describe the geometry of the physical world around us, such as the motion of planets in orbit, the path of steepest ascent through the hills of Los Angeles, or how to calculate the amount of electricity flowing along a curve or through a surface. In this course, you will develop the reasoning and questioning skills needed to explore these geometric concepts and apply them to real-life situations. Moreover, you will become fluent in communicating your ideas through the mathematical language of multivariable calculus.
The course MATH 232 differs from MATH 212 in that it covers the topics of multivariable calculus with more mathematical rigor (e.g. proofs). That is, in addition to learning how to perform standard computations, we will also learn how to grapple with and prove statements about complex mathematical concepts. Moreover, this course builds the foundation for more advanced topics, such as real analysis, complex analysis, and differential geometry. This course differs from MATH 222 in that we will not discuss general manifolds (only curves and surfaces), nor differential forms.
This course is recommended for students interested in learning about advanced mathematics.
]]>This post is a guide to the packages, commands, and other tweaks that I have compiled from across various sources to make my existing TeX files compile correctly as tagged pdfs. This post should be thought of as a supplement to the Tagging Project’s Guidelines for using LaTeX to generate accessible PDF.
Sample tex files and materials are included below, and I have checked that they are parsed correctly by VoiceOver (Apple’s built-in screen reader), and also pass the tests on my LMS (Canvas, Ally Course Accessibility Report).
DISCLAIMER: I am not an expert on accessibility nor an expert in the inner workings of TeX, and this post is not legal advice, nor it is a guarantee that your documents will automatically satisfy any or all accessibility standards or legal regulations.
If you are interested in learning more about digital accessibility in regards to mathematics, a good reference to get acquainted is Accessibility for the Working Mathematician by Julius Ross.
The tl;dr is that making documents accessible takes work, beyond the challenge of writing appropriate alt text (see sections 3.2-3.4). It may also require philosophical changes in how you structure your documents (see section 3.5). You may also need to remove or replace certain classes or packages (for example, beamer cannot create tagged pdfs, and will need to be replaced by ltx-talk).
My understanding is that if you were to start from scratch, many experts are recommending how to compose documents using a system that can create both HTML and pdf (and other) outputs such as Ximera or PreTeXt.
I will probably learn PreTeXt over the summer, and I definitely see the value in writing HTML output. However, I think there is value in having accessible pdfs because they are documents that can be downloaded and used by people without a stable or reliable internet connection (or when Canvas goes down).
Below are the modifications you will need to create digitally accessible pdfs from your TeX files. You can see sample TeX files below.
LuaLaTeX is recommended by the tagging project; but the packages and commands detailed below should work with pdfLaTeX. After switching to using LuaLaTeX, I saw no difference in compile times for the documents I am working on.
You should also make sure that you have the latest version of TeX installed (currently LaTeX 2025-11-01 as of this post). This is important for new classes and packages being developed, such as ltx-talk.
If you use Overleaf, here are instructions on how to change compiler and TeX version.
Note: the distribution of TeX used by overleaf (TexLive 2025) is not as up to date as the latest distribution of TeX (e.g. it does not yet have support for ltx-talk, but you can opt into the Overleaf Labs program for access to newer versions of TeX).
To create tagged pdfs from your TeX files, you should include this code in the preamble (e.g. before \begin{document}):
\DocumentMetadata{
tagging=on,
tagging-setup={math/setup=mathml-SE,math/alt/use}
}
The tagging-setup={math/setup=mathml-SE,math/alt/use} option turns on the necessary code for screenreaders to read mathematics (see below).
This command has other options - you can set the language (which is US English by default), the PDF standard, etc.
See the documentation for \DocumentMetadata here.
To set the title and author of the PDF, you should include this code in the preamble:
\usepackage{hyperref}
and you should include this code in the body of the document (e.g. immediately after \begin{document})
\hypersetup{
pdftitle={Spring 2026 Math 102 Syllabus},
pdfauthor={Richard Wong},
pdfdisplaydoctitle%
}
The tagging project renders math formulas and equations using MathML, but some screenreaders (e.g. VoiceOver, Ally) treats them like images which require alt text.
The option tagging-setup={math/setup=mathml-SE,math/alt/use} in the \DocumentMetaData command described above will automatically generate the alt text via the latex commands used. For example, the formula $\displaystyle \int_0^1 xe^{-x^2} \ dx$ has alt text
Latex formula starts
\begin{math} \int_0^1 xe^{-x^2} \ dx \end{math}Latex formula ends
This setup also allows screen readers to “dig in” and navigate these formulas (e.g. one can move back and forth between the bounds of integration, or different parts of the integrand).
Note: Because the alt text currently repeats verbatim the latex commands used, you should be careful about the macros that you use.
See the documentation for math content here.
Adding alt text to images is easy - simply use the alt-text options in \includegraphics:
\includegraphics[alt={A picture of me and Ernie}]{ernie_mueller2020.jpg}
However, in my opinion, writing appropriate alt text is the most difficult part of making your documents accessible.
Especially in the context of teaching, writing alt text requires you to think carefully about the pedagogical intent of the image. (see Ross 3.2-3.4)
THe AMS, EMS, LMS, and SIAM also recently published Author Guidelines for Preparing Accessible Mathematics Content, which set standards for labeling complex math diagrams (with examples).
To indicate the headers of the table, you can include this code immediately before the relevant table:
\tagpdfsetup{table/header-rows={1}}
See the documentation for tables here.
This is another challenge of making accessible documents - you should make sure that your existing documents use headings (e.g. \section or \subsection, etc.) appropriately.
That is, you should make sure your document does not use purely visual commands (e.g. \textbf, \Large, etc.) to indicate headings or sections.
Otherwise, the command \DocumentMetadata described above will take care of the tagging.
If you are still encountering errors (which is likely), you should check package compatibility with the Tagging Project’s compatibility tracker.
A lot of packages are currently supported, but some packages are not supported yet (for example, packages like titlesec).
Other packages will never be supported due to the way the package works (e.g. beamer). In that case, you should look to alternatives (e.g. ltx-talk).
A worksheet for the second day of Calc II:
A syllabus for Calc II:
]]>This semester, I will be having weekly lunches at West Servery on Thursdays from 12-1pm. If you see me there, feel free to sit down and chat!
]]>I will also be coordinating Math 102; and I am helping organize the undergraduate math colloquium at Rice.
You can find the course syllabus here.
How can we describe the physical world mathematically? How can we use mathematics to describe phenomena in physics, biology, chemistry, or other STEM fields?
Calculus is the mathematical language that allows us to describe and model the behavior of the physical world around us, such as the speed and acceleration at which we travel, as well as our distance and displacement; or how a population grows and changes over time; or the rate at which chemicals react or move towards equilibrium.
In this course, you will develop the reasoning and questioning skills needed to explore these concepts mathematically. Moreover, you will become fluent in communicating your ideas through the mathematical language of calculus.
You can find the course syllabus here.
What tools do we have to calculate an integral like $\int_0^1 xe^{-x^2} \ dx$ or $\int_0^1 e^{-x^2} \ dx$?
When your calculator tells you that the value of $\int_0^1 e^{-x^2} \ dx \approx 0.746824$, how do you know it’s correct? How accurate is your calculator? Is it possible for an infinite region to have finite area? Can we calculate the area of a fractal?
You’ve previously seen in Math 101 that calculus is an important and powerful tool that allows us to describe the physical world around us. In Math 102, we will push the notion of integration to its utmost limits. In this class, we will build our toolbox for integration; we will study the behavior of the infinite, and we will learn how to quantify how accurate our approximations are.
In this course, you will develop the critical thinking and questioning skills needed to answer these complex questions. Moreover, you will become fluent in precisely communicating your ideas through the mathematical language of calculus.
]]>I will be giving a presentation on Midweek Math Training at Rice.
I (along with José Pastrana) will also be presenting a poster on Year 1 of the HHMI Driving Change efforts in redesigning Calc I and II.
]]>I will also be coordinating Math 102; and I am helping organize the undergraduate math colloquium at Rice.
You can find the course syllabus here.
What tools do we have to calculate an integral like $\int_0^1 xe^{-x^2} \ dx$ or $\int_0^1 e^{-x^2} \ dx$?
When your calculator tells you that the value of $\int_0^1 e^{-x^2} \ dx \approx 0.746824$, how do you know it’s correct? How accurate is your calculator? Is it possible for an infinite region to have finite area? Can we calculate the area of a fractal?
You’ve previously seen in Math 101 that calculus is an important and powerful tool that allows us to describe the physical world around us. In Math 102, we will push the notion of integration to its utmost limits. In this class, we will build our toolbox for integration; we will study the behavior of the infinite, and we will learn how to quantify how accurate our approximations are.
In this course, you will develop the critical thinking and questioning skills needed to answer these complex questions. Moreover, you will become fluent in precisely communicating your ideas through the mathematical language of calculus.
]]>I will also be coordinating Math 102; and I am helping organize the undergraduate math colloquium at Rice.
You can find the course syllabus here.
How can we describe the physical world mathematically? How can we use mathematics to describe phenomena in physics, biology, chemistry, or other STEM fields?
Calculus is the mathematical language that allows us to describe and model the behavior of the physical world around us, such as the speed and acceleration at which we travel, as well as our distance and displacement; or how a population grows and changes over time; or the rate at which chemicals react or move towards equilibrium.
In this course, you will develop the reasoning and questioning skills needed to explore these concepts mathematically. Moreover, you will become fluent in communicating your ideas through the mathematical language of calculus.
You can find the course syllabus here.
What tools do we have to calculate an integral like $\int_0^1 xe^{-x^2} \ dx$ or $\int_0^1 e^{-x^2} \ dx$?
When your calculator tells you that the value of $\int_0^1 e^{-x^2} \ dx \approx 0.746824$, how do you know it’s correct? How accurate is your calculator? Is it possible for an infinite region to have finite area? Can we calculate the area of a fractal?
You’ve previously seen in Math 101 that calculus is an important and powerful tool that allows us to describe the physical world around us. In Math 102, we will push the notion of integration to its utmost limits. In this class, we will build our toolbox for integration; we will study the behavior of the infinite, and we will learn how to quantify how accurate our approximations are.
In this course, you will develop the critical thinking and questioning skills needed to answer these complex questions. Moreover, you will become fluent in precisely communicating your ideas through the mathematical language of calculus.
]]>