Thoughts on Numerical Integration: Index

I’m doing something that I should have done a long time ago: collecting a series of posts into one single post. The links below show my series on numerical integration.

Part 1 and Part 2: Introduction

Part 3: Derivation of left, right, and midpoint rules

Part 4: Derivation of Trapezoid Rule

Part 5: Derivation of Simpson’s Rule

Part 6: Connection between the Midpoint Rule, the Trapezoid Rule, and Simpson’s Rule

Part 7: Implementation of numerical integration using Microsoft Excel

Part 8, Part 9, Part 10, Part 11: Numerical exploration of error analysis

Part 12 and Part 13: Left endpoint rule and rate of convergence

Part 14 and Part 15: Right endpoint rule and rate of convergence

Part 16 and Part 17: Midpoint Rule and rate of convergence

Part 18 and Part 19: Trapezoid Rule and rate of convergence

Part 20 and Part 21: Simpson’s Rule and rate of convergence

Part 22: Comparison of these results to theorems found in textbooks

Part 23: Return to Part 2 and accuracy of normalcdf function on TI calculators

HyFlex Teaching During the Pandemic (and Beyond?)

I’m happy to say that an article I wrote teaching last spring — when I had to negotiate teaching both in-person students and students who were participating remotely — was published in this month’s issue of MAA FOCUS. I hope that some of these thoughts might be helpful to somebody else who might be in this position for the Fall 2021 semester.

Visualizing Vectors

From the Math Values blog of the Mathematical Association of America:

Anyone who has taught linear algebra knows how easy it is for students to get absorbed in performing matrix computations and memorizing theorems, losing the beauty of the structures in this foundational subject. James Factor and Susan Pustejovsky of Alverno College in Milwaukee, WI, bring back the visual beauty of linear algebra through their NSF-funded project Transforming Linear Algebra Education with GeoGebra Applets.

The applets are freely available in the GeoGebra book Transforming Linear Algebra Education Each topic is packaged with a video to show how the applets work, the applet, and learning activities.

Read more about it here:

YouTube’s Automatic Closed-Captioning of Mathematical Speech (Part 2)

Last semester, as I spend untold hours editing the closed captioning automatically generated by YouTube on the math videos on my YouTube channel, I got a crash course on the capabilities and limitations of this system. This crash course was perhaps not legally necessary but extra work that I took on because a student with a hearing impairment was enrolled in my class, and I wanted to ensure that the review videos that I provide to my students were accessible to him also.

I think the resources offered by my university are fairly typical to ensure that instructors are able to reach all students and not just those who don’t have audio/visual impairments. After discussions with the cognizant people at my university, I’ve made a few conclusions:

  • Mostly by accident, my videos are ADA compliant since I made the decision to both write out the solutions and also talking through the solutions.
  • While the automatic closed-captioning provided by YouTube may be minimally compliant with ADA, I’m not sure that a student with a hearing impairment could always follow the transcriptions due to a number of errors.
  • Aside from punctuation, capitalization, and the occasional homonym (e.g., right vs. write), YouTube does a pretty good job at transcribing ordinary speech.
  • Naturally, YouTube’s automated closed-captioning is not to blame when I don’t enunciate clearly, have a rabbit trail of thought but then have to backtrack, use poor grammar, make a outright mistake, etc.
  • However, YouTube seems to have a lot of difficulty providing automatic closed-captioning of mathematical speech.

Fixing these transcription errors took an awful lot of time. I don’t want to know how many hours I devoted to fixing the 120 or so videos (each video is about 3-10 minutes long) recorded so that my hearing-impaired student could have full access to my class. About halfway into this project of fixing the closed-captioning errors, I started writing down some of the closed-captioning errors. I wish I had thought to do this near the start, but oh well.

Phonetically, I can understand why most of these errors were made. But these mistakes really shouldn’t have happened. Here are my favorite howlers that I recorded, showing both what I said and what YouTube thought I said.

  • “931,147,496” became “930 1,000,000 147,000 496”
  • A \cap C,” pronounced “A intersect C,” became “A inner sexy”
  • “arithmetic” became “rhythm sick”
  • “capital X” became “Catholics”
  • “cardinality” became “carnality”
  • “divisible by 5” became “visited his wife live” (I have no idea how that happened)
  • e^x” became “eat ooh the x”
  • “for succinctness” became “force the sickness”
  • n \choose n,” pronounced “n choose n,” became “and shoes and”
  • “set containing” became “second taining”
  • \sqrt{2}” became “squirt tuna”
  • “two ways in” became “too wasted”
  • “what f(3),” pronounced “what f of 3,” became “whateva 3”
  • x \in B, pronounced “x is in B,” became “sexism be”
  • x \in B \cap C, pronounced “x is in B and C,” became “x is Indiana see”
  • x \in C, pronounced “x is in C,” became “excellency”

Here’s the complete list of howlers that I recorded for posterity. If I’ve learned nothing else, it’s that I need to be more proactive about ensuring the mathematical accuracy of closed-captioning for my YouTube videos.

4 for
857 a 50 7
1232 1230 two
4761 4760 1
19,999 19,000 999
46,376 40 6376
123,552 120 3,552
5,565,120 five million 565,000 120
931,147,496 930 1,000,000 147,000 496
(2,\emptyset) 2d sent
(20,8) 28
[1,2] one too
12 \choose 4 12 juice 4
16 \choose 8 16 choosing
3 + 1 = 4 surplus one mix for
4 \choose 0 4 2 0
4 \choose k four twos k
49 \choose 5 49 she’s 5
50 \choose 6 52 six
8 \choose 2 a choose to
A \cap C A inner sexy
A \cap D a intersecting
A \cup B a you be
A \cup C a UNC
A \cup C a you will see
a proof approved
A^c a compliment
a_i asa by
all multiples of almost visit
an element of A known the debate
an element of A normal today
and divisible and as above
and positive 50 + + 50
and tens intense
and would let this be 3 andrew lippa p3
arithmetic earth to
arithmetic rhythm sick
As ace
B but not C be but not si
B \cap C b in a sexy
B if beef
bijection bi CH action
bijection bite jection
bijection by dejection
bijection by ejection
bijection by jection
bijection by Junction
both sets both says
capital X Catholics
cardinality carnality
Cartesian car to shull
codomain code Amin
coordinate cordon
coordinate court
coordinates corners
coordinates have cort in sap
cosine cosign
disjoint destroyed
divisible by 5 visited his wife live
e^x eat ooh the x
element of A illness of A
element of A mellow today
element x that Windex
elements of us
empty MQ
\emptyset descent
\emptyset intercept
equal able
exponent x1
factored acted
factorial fact welders
fill in film
flipping four coins philippine for coins
for succinctness force the sickness
hence in Hanson
i eye
i aye
If I divide by 15 If I / 15
in A nae
in there a bear
infinite if an
infinite imp an
infinite infant
into five in 2 5
is ice
j \choose r j choose arms
kth cave
kth kate
likewise lakh wise
n \choose n and shoes and
nth row nth throw
one-to-one 121
onto on 2
r \choose r our shoes are
r to art at
r to already
\mathbb{R}^2 are too
\mathbb{R}^2 our too
r‘s hours
same row samro
second coordinate sec cornered
set containing second inning
set containing second taining
set containing seconds hanging
set containing secretary
set containing 1 second anyone
since A has say has
sixth one six-month
square swear
\sqrt{2} score 2
\sqrt{2} squirt of tuna
team A teammate
term in it terminate
than zero gloves are off
that’s chosen that’s Showzen
then x the next
therefore there for
this entry in the century plus
to the k decay
two are to are
two ways in too wasted
union you need
up here pier
what f(3) whateva 3
will be 4 will before
with n=4 finials 4
would subtract was attract
writing riding
x is extras
x is in exiting
x is in A x as a native
x is in A x is nay
x is in B sexism be
x is in B and C x is Indiana see
x is in C excellency
x is in C X’s and see
x_2 next to
x_2 text too
x-coordinate export
y why
y wine
y is greater than or wider
ys wise

YouTube’s Automatic Closed-Captioning of Mathematical Speech (Part 1)

The biggest change that I’ve made to my teaching in the past ten years has been posting review videos for my students as they prepare for exams. The playlists that I post for my students can be found at my YouTube channel. The production quality of the videos is definitely low-budget: I just placed a ruler along the top of two textbooks of equal height, balanced a webcam on the ruler to point downward, and then recorded myself as I wrote out and talked through the solutions of the review problems. I’m not going for high production value in my videos, unlike excellent sites like Physics Girl or Numberphile, since my target audience is deliberately narrow (the students in my classes and, more recently, in some of my colleagues’ classes) and not worldwide.

For what it’s worth, I have recorded roughly 650 videos, each usually between 3 and 10 minutes long, which have collectively amassed over 200,000 views since I started recording them in 2011. Not bad for your friendly neighborhood mathematician.

Posting these videos have spurred some immediate changes to my pedagogical practice. First, I no longer give review lectures in class immediately before my exams. Instead, I ask students to take a shot at completing the review problems on their own, asking them to watch the videos only to check their work or else to get the answer if they get stuck. Students are still welcome to come to me for help during office hours or by appointment, but they’re expected to watch the videos first. In my end-of-semester evaluations, my students seem to really appreciate having these videos. They tell me that they like having after-hours help while studying for their exams and that, unlike a regular review lecture, they can rewind the video and start over again if they need to hear a concept repeated.

Another positive development is that eliminating the review lectures have given me three or four hours of extra contact time each semester with my students. Rather than add new material or cram in extra examples, I’ve mostly used these extra hours to slow down the pace of my lectures and to include group activities and other forms of student engagement during class time. I’m particularly happy that I have three dedicated days in my Discrete Mathematics class when my students can practice their new (for them) techniques of writing mathematical proofs. If they get stuck, I’m around to answer questions about the mechanics of proof-writing. If they don’t need help, they can get immediate affirmation from me about whether or not their proofs are correctly written. Discrete Mathematics is our math majors’ first introduction to writing mathematical proofs, and that my students have their initial struggles with this technique in class as opposed to when they do their homework on their own time.

So I intend to maintain this practice for the rest of my career.

However, there’s been one complication that I should have foreseen in 2011 but didn’t: the Americans with Disabilities Act. This had been mostly a potential problem for me that I hid away in the deep recesses of my mind until last semester, when a student with a hearing impairment was enrolled in my class.

In my next post, I’ll discuss some humorous examples of erroneous closed-captioning of mathematical speech which were automatically generated by YouTube.

Engaging students: Box and whisker plots

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This submission comes from my former student Chris Brown. His topic: how to engage students when teaching box and whisker plots.

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How could you as a teacher create an activity or project that involves your topic?

My all-time favorite TV show as a child was Pokémon. This show is still a staple amongst the young and even adult generation of today. The activity that I have created, was designed to take place after a formal lesson over how to create Box and Whisker plots. For this activity, students will be given a labeled bar graph of the Pokémon Type Distribution for generations 1 through 6 of Pokémon, which I have listed an online data source below. The students will be tasked with identifying the top 7 Pokémon types and creating a Box and Whiskers plots for each of those types. They will then go through and analyze the consistency of the creation of Pokémon for that specific type and then compare contrast this same box plot to any other box plot of their choice. The students will then make predications for the number of Pokémon for each of the top 7 Pokémon types, for generation 7 and base their reasoning in the box plots they created. Then the student will finally research the type distributions for the 7th generation of Pokémon, and discuss how the actual number compares to their prediction.


This is the online source for the type distributions for generations 1 – 6:

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How does this topic extend what your students should have learned in previous courses?


From my experience, Box Plots are first taught in the early middle school years, in 6th or 7th grade. When constructing box plots by hand, in its essence, box plots require knowledge of how to order sets of numbers from least to greatest; an understanding and ability to find the maximum, minimum, median, and mean of a data set; and lastly, critical thinking and analytic skills developed from general course content. Box plots allow students to combine each of these skills to effectively analyze data sets with ease and compare different data sets with precision and accuracy. If any or all of these skills are not quite up to par, students will have an opportunity to develop them through box plots as they spend time creating them. For all students no matter their level, they will still gain better insight on how to properly analyze data and grow as analytical thinkers as they take the represented data and turn it into meaningful interpretations.


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How can technology be used to effectively engage students with this topic?


In a classroom, I personally believe that Desmos is a wonderful online tool that can aid students in the understanding of how box and whisker plots function, and also a great place to check their work. Desmos, which is linked below, gives students the ability to list as many data points as they need to, and concurrently creates a box plot as they do so. In this way, students are able to see how singular data points can skew the data in significant and insignificant amounts. What I also love about Desmos is that, the list of data points does not have to be in any kind of order, so students do not have to worry about that tedious step! Desmos also lists the 5-point summary in two different places, on the box plot itself, and also on a drop-down menu, which is super convenient. Lastly, I love how Desmos also displays the mean of the data set as well, students can calculate the skew of the data, and definitively determine how it is skewed. This is a super visual, and interactive tool that will allow the student to manipulate box plots so seamlessly they will not be focused on the tediousness of the setup and solely on the concept.


The link to the Desmos setup is here:



Engaging students: Vectors in two dimensions

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Sarah McCall. Her topic, from Precalculus: vectors in two dimensions.

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What interesting (i.e., uncontrived) word problems using this topic can your students do now?

For such an applicable topic, I believe that it is beneficial to have students see how this might apply to their lives and to real world problems. I selected the following word problems because they are challenging, but I think it is necessary for students to be a little frustrated initially so that they are able to learn well and remember what they’ve learned.

1. A DC-10 jumbo jet maintains an airspeed of 550 mph in a southwesterly direction. The velocity of the jet stream is a constant 80 mph from the west. Find the actual speed and direction of the aircraft.

2. The pilot of an aircraft wishes to head directly east, but is faced with a wind speed of 40 mph from the northwest. If the pilot maintains an airspeed of 250 mph, what compass heading should be maintained? What is the actual speed of the aircraft?

3. A river has a constant current of 3 kph. At what angle to a boat dock should a motorboat, capable of maintaining a constant speed of 20 kph, be headed in order to reach a point directly opposite the dock? If the river is ½ a kilometer wide, how long will it take to cross?

Because these problems are difficult, students would be instructed to work together to complete them. This would alleviate some frustrations and “stuck” feelings by allowing them to ask for help. Ultimately, talking through what they are doing and successfully completing challenging problems will take students to a deeper level of involvement with their own learning.



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How could you as a teacher create an activity or project that involves your topic?

I believe vectors are fairly easy to teach because there are so many real life applications of vectors. However, it can be difficult to get students initially engaged. For this activity, I would have students work in groups to complete a project inspired by Khan Academy’s videos on vector word problems. Students would split off into groups and watch each of the three videos on Khan Academy that have to do with applications of vectors in two dimensions. Using these videos as an example, students will be instructed to come up with a short presentation or video that teaches other students about vectors in two dimensions using real world applications and examples.




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How has this topic appeared in pop culture (movies, TV, current music, video games, etc.)?

Immediately when I see vectors, I think of one specific movie quote from my late childhood that I’ll always remember. The villain named Vector from Despicable Me who “commits crimes with both direction AND magnitude” is a fellow math nerd and is therefore one of my favorite Disney villains of all time. So of course, I had to find the clip (linked below) because I think it is absolutely perfect for engaging students in a lesson about vectors as soon as they walk in the door, and it is memorable and educational. I would refer back to this video several times throughout the lesson and in future lessons because it is a catchy way to remember the two components to vectors. This would also be great to kick off a unit on scalars and vectors, because it would get kids laughing and therefore engaged, plus they will always remember the difference between a scalar and a vector (direction AND magnitude!).  



My Favorite One-Liners: Part 9

In this series, I’m compiling some of the quips and one-liners that I’ll use with my students to hopefully make my lessons more memorable for them.

Today, I’d like to discuss a common mistake students make in trigonometry… as well as the one-liner that I use to (hopefully) help students not make this mistake in the future.

Question. Find all solutions (rounded to the nearest tenth of a degree) of \sin x = 0.8.

Erroneous Solution. Plugging into a calculator, we find that x \approx 53.1^o.


The student correctly found the unique angle x between -90^o and 90^o so that \sin x = 0.8. That’s the definition of the arcsine function. However, there are plenty of other angles whose sine is equal to 0.7. This can happen in two ways.

First, if $\sin x > 0$, then the angle x could be in either the first quadrant or the second quadrant (thanks to the mnemonic All Scholars Take Calculus). So x could be (accurate to one decimal place) equal to either 53.1^o or else 180^o - 53.1^o = 126.9^o. Students can visualize this by drawing a picture, talking through each step of its construction (first black, then red, then brown, then green, then blue).arcsin45

However, most students don’t really believe that there’s a second angle that works until they see the results of a calculator.


Second, any angle that’s coterminal with either of these two angles also works. This can be drawn into the above picture and, as before, confirmed with a calculator.

So the complete answer (again, approximate to one decimal place) should be 53.1^{\circ} + 360n^o and 126.9 + 360n^{\circ}, where n is an integer. Since integers can be negative, there’s no need to write \pm in the solution.

Therefore, the student who simply answers 53.1^o has missed infinitely many solutions. The student has missed every nontrivial angle that’s coterminal with 53.1^o and also every angle in the second quadrant that also works.

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Here’s my one-liner — which never fails to get an embarrassed laugh — that hopefully helps students remember that merely using the arcsine function is not enough for solving problems such as this one.

You’ve forgotten infinitely many solutions. So how many points should I take off?

For further reading, here’s my series on inverse functions.