iiinova — where your digital self keeps evolving

Jed @jedediyah.com · 27/08/2026

Definitely @oscarlevin.com's book, which I have used in my data science course. I have the 3rd edition which I am partial to (discrete.openmathbooks.org/dmoi3.html). Are there any particular topics? I think that there tend to be good texts on discrete topics with plenty of good exercises.

Short stack of books:
Combinatorics, by Daniel A. Marcus
Cryptological Mathematics, by Robert Lewand
Graphs and Applications: An Introductory Approach, by Aldous and Wilson
Combinatorics and Graph Theory, by Harris, Hirst, and Mossinghoff
Discrete Mathematics, by Oscar Levin

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Jed @jedediyah.com · 26/08/2026

Was the highlighted text on page 2 a pull quote? Because I'm not seeing where the authors wrote this sentence, and it sounded like the authors concluded that K-12 teachers *cannot* reliably prompt ChatGPT to create more ambitious, student-centered mathematics lessons...

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Jed @jedediyah.com · 26/08/2026

I was happy to see an honest review of lesson planning with "AI" in the cover article from this MTLT. I was thrown by the somewhat contradictory presentation in the journal. Maybe the partially completed progress bar is intended to imply that we are not there yet, despite the directive title.

Cover of Mathematics Teacher: Learning and Teaching PK 12. Cover photo contains a Gemini-inspired border around "Lesson Plan With AI", under which there is a progress part that is partially completed.

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Jed @jedediyah.com · 03/05/2026

This month marks the 20th anniversary of my entering the classroom as a teacher! I was just going through a folder from my first Computer Science course (2009) and am not surprised by myself. Class #1 was "Ethics" #TeamCompSci #ITeachCS #CompSci

Spreadhseet with columns Unit, Class #, Topic(s).
Under Topic(s), the first three classes are "The Course", "Ethics", "Hardware"

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Jed @jedediyah.com · 30/04/2026

I never met Dr. Nancy C. Dunigan, but I have a copy of the 1981 NCTM Yearbook that once belonged to her. I occasionally get a used math book and can find information about the previous owner. She sounds like a real swell lady and dedicated educator. www.dignitymemorial.com/obituaries/l...

Teaching Statistics and Probability
National Council of Teachers of Mathematics
1981 Yearbook

Table of Contents. Stamp across top of page reads "Property of Nancy C. Dunigan"

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Jed @jedediyah.com · 29/04/2026

It looks like there are some amazing presentations from teachers in the program for #NCTMDen26. I'll be presenting on the importance of computing in mathematics and how basic familiarity with a numerical computing platform can open up new worlds of beautiful mathematics. #ITeachMath #mtbos

The Missing Strand
Reviving NCTM's Vision for computing in Mathematics Education

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Jed @jedediyah.com · 04/04/2026

We recently looked at the very user-friendly Forced Expiratory Volume (FEV) data set as an exercise in data slicing and visualization: colab.research.google.com/drive/1u7qb6... Got an excited message from a student: "We think we can show that smoking is good for your lungs!" #ITeachmath #mtbos

Plot of FEV for Smokers and Non-Smokers.
Smokers appear to have a higher FEV which can be (erroneously) inferred as having "healthier" lungs.

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Jed @jedediyah.com · 02/04/2026

Another student sent an update on modeling the atmospheric CO_2 data, and was able to get the model: a sin(b(x +c)) + d + ex +f x^2 to fit quite nicely. #ITeachmath #mtbos

Pyplot figure of atmospheric CO2 data oscillating while increasing from 1958 to present. A model is well-fit to the data.

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Jed @jedediyah.com · 01/04/2026

This was a fun suggestion: if you take all of the pixels (in the 28x28 image) and let them fall down. This is really the sum of the columns. We did that for all digits 0-9.

Ten plots corresponding to column sums of images of the digits 0 through 9. The top left plot corresponds to zero, and has two distinct peaks corresponding to the left and right sides of the zero, suggesting people write zeros taller than wider. The next plot corresponds to one, and has a concentrated narrow peak.

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Jed @jedediyah.com · 01/04/2026

In #MathsToday we started our feature-engineering project with 0s and 1s from MNIST Given a spreadsheet or DataFrame where each row contains 784 pixel values representing an image of a hand-written digit, how can we distinguish a 0 from a 1? #ITeachMath #mtbos

Image of a hand-written 0, composed of pixel values, next to an image of a 1 also composed of pixel values.

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Jed @jedediyah.com · 27/03/2026

In #MathsToday we modeled CO2 concentration as reported from gml.noaa.gov/webdata/ccgg... One student got quite close to capturing the seasonal fluctuation with: a sin(bx +c) + d + ex +fx^2

Linear regression on CO2 data displays slope of 1.66

Quadratic regression on CO2 data with quadratic term 0.01346

Atmospheric CO2 data with a model plotted that nearly matches the data. There is noticeable phase offset in the center of the plot.

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Jed @jedediyah.com · 19/03/2026

I searched @nctm.org pub archives for 'computing', 'programming', 'monte carlo', and some others. Then I scrolled a few thousand pages looking for articles that contain code in them. This is what I've found so far. #ITeachMath #mtbos

Bar plot of NCTM articles per year containing code. There is a surge in the 60s and huge surge in the 80s.

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Jed @jedediyah.com · 15/03/2026

In #mathstoday (last week) we investigated several datasets. Students generated data questions, including one about what makes YouTube videos more popular. This plot made for a lot of great discussions about causation! #ITeachMath #mtbos

Scatter plot of trending YouTube videos, Number of Views vs. Character Count of Title. There are some distinct peaks around 35, 50, and 65.

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Jed @jedediyah.com · 08/03/2026

In #MathsToday (tomorrow) we are looking at the results of the 2025 Rubik’s WCA World Championship. All of my Data Science students know how to solve a Rubik's cube (it was our first assignment), so they emotionally invested in these tournaments 🤣. We'll look at rankings using different averages.

The Rubik’s WCA World Championship 2025 took place at the Seattle Convention Center on July 6, 2025. The sixteen finalists are listed in the DataFrame below, along with their country and their times from five rounds of solving.

A code block displays the creation of a Pandas DataFrame.

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Jed @jedediyah.com · 27/02/2026

In #MathsToday we read Chapter 3: Arms Race - Going to College, from Weapons of Math Destruction by Cathy O'Neil. Students annotated, discussed in groups, and will share some reflections in writing.

Book cover of "Weapons of Math Destruction" by Cathy O'Neil.

Weapons of Math Destruction, by Cathy O’Neil
Chapter 3 - Arms Race, Going to College
1. What are the pros and cons of college rankings, and do they ultimately offer a net benefit?
2. What is measured and what is not?
• What variables did U.S. News include in its college ranking formula?
• What aspects of educational quality were excluded or difficult to measure?
• What are the consequences of building a model that accounts only for what can be easily mea-
sured?
3. O’Neil writes about systems that can be gamed. Who benefits from these systems and who is harmed?
4. The major theme from first semester was feedback loops.
• What feedback loops are created by college rankings. Model these loops with a system diagram.
• What consequences of rankings are due specifically to feedback loops?
• Are the looping systems of college rankings chaotic, or do they converge toward cycles or fixed
points?
5. The output of algorithms is often presented as neutral. There are numerous tropes about how “the
numbers don’t lie”, “the data speaks for itself”, and ”you can’t argue with the math”. However, O’Neil
describes how those creating the models have great power. To what extent is a model a representation
of the values of the modeler? Or, what choices is the modeler in a position to make that influence the
model?
6. You are more than your data, yet your grades, applications, resume, etc. are variables in algorithms.
What aspects of being human are systematically excluded from data-driven systems?

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Jed @jedediyah.com · 22/02/2026

I had one of these beautiful shapes hanging in my old classroom for years! Tangent: this picture was the data from a pendulum lab. I ask students to: 1. Measure the period of a pendulum 2. Measure 10 swings then divide by 10 3. Measure 10 swings again, but this time do a good job Never fails.

A white board with experimental data from a pendulum lab. Columns include: 1-swing, 10-swings, 10-careful swings. Above the white board are several student made posters, an origami piece, and a street sign that reads Williams Ln.

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Jed @jedediyah.com · 21/02/2026

A text that is great for #GuessTheYear! "The notion that we understand how mental capacities develop well enough to dare to control their development is not just an exaggeration; it is a pack of nonsense, and we rely on it at our students' peril." #mtbos #ITeachMath #NCTM

because the Skinnerians do not.) There are two fundamental and intractable troubles with CAI and PI. Every conception of pedagogy is based on a concept of knowledge, and CAI and PI are based on a falsified concept: that knowledge is the ability to respond in the expected way. For reasons I dis- cussed earlier, the use of this idea produces cognitive knowledge-at least for the moment-at the price of psychic intelligence. When advocates of CAI or PI are exuberant, they claim that their methods supersede other teaching styles. When challenged, they sometimes retreat to the position that their methods replace various kinds of drill that do not deserve the attention of a human teacher; but these retreats are instances of the Law of

the Conservation of Bologna under Thin Slicing. The trouble is that CAI and PI are based on naive misconceptions of the psychology of learning and
retention.
Every artificial teaching scheme is predicated on the idea that learning processes work best when they are carefully designed and controlled. This might be true if learning processes were understood, but they are not. The notion that we understand how mental capacities develop well enough to dare to control their development is not just an exaggeration; it is a pack of nonsense, and we rely on it at our students' peril.
It is equally naive to suppose that people understand and remember the things that are explained or told to them clearly. In practice, people under-

But some very prevalent teaching schemes bear strong resemblances to CAI and PI and have much the same vices. One of their distinguishing features is that they neither require nor produce functional literacy. The literate may skim a paragraph or a page to get its general drift. The question is whether they expect to get more than the general drift. When they do not understand exactly what the paragraph says, do they reexamine it until it comes into sharp focus? If so, they have half of functional literacy (and evidently this half is largely a trait of temperament rather than a purely cognitive skill); the other half is the ability to write exactly what one means. Any education that does not include functional literacy is deficient. The trouble is that once formal education is over, we have nobody to feed us knowledge out of spoons, and so we must read with care to add to our knowledge or even to preserve it. Moreover, inarticulate knowledge is adequate only for people who work in isolation (if, indeed, for them); and in

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Jed @jedediyah.com · 18/02/2026

Really stretching hard to make the math jokes work in this chapter on Number...

Then your friend Chelsea, also from twenty-eight thousand years ago, notices another connection: sometimes she can combine two arrangements from Pattern 2 to get a bigger arrangement also in Pattern 2, but it doesn’t always work. She finds this one:

9 with 16 makes 25

Pattern 3
1 rock, 8 rocks, 27 rocks

Chelsea, who is rightly proud of her discovery about combinations in Pattern 2, searches for similar combinations in Pattern 3. She wants to find two piles in Pattern 3 that can combine together to get another pile in Pattern 3, but this last format is challenging and it will be quite a wiles before someone figures it out.

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Jed @jedediyah.com · 03/02/2026

In #MathsToday we found sums by creating sequences with NumPy.

"Find the following sums" followed by four sigma notation expressions.

Python print statements containing numpy summations.

"Write the most concise NumPy expressions you can to evaluate the summations of the following sequences" followed by a set of sequences.

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Jed @jedediyah.com · 02/02/2026

In #MathsToday we explored the palmer penguins data set from allisonhorst.github.io/palmerpengui... and did some manual feature engineering.

Scatter plot of beak depth vs beak length for three species of penguin.

Scatter plot of beak depth vs beak length for three species of penguin. Decision boundaries separate the points.

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Jed @jedediyah.com · 01/02/2026

Speak of the devil...

LinkedIn post from CSTA referring to Microsoft as "our partners".

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Jed @jedediyah.com · 01/02/2026

Yes! Lattice problems, combinations, permutations, AND, OR, NOT, if "kg" in units, A∩B, x ∈ S, ... Every one of the problems in the image is the SAME problem, and they all reduce to a binary word.

Screen capture of several problems about sets, binary words, permutations, and graphs.

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Jed @jedediyah.com · 29/01/2026

In #MathsToday, we continued solving linear systems both at whiteboards and with NumPy, and there were several wonderful breakthroughs in understanding! We saw the first steps of Gaussian elimination (without yet naming it) and one great reaction was "wait you can just solve any matrix you want"

>>> import numpy as np
>>> r = np.random.rand(3,4)
>>> r
array([[0.18487845, 0.57661705, 0.04339225, 0.34596879],
[0.71953505, 0.9111707 , 0.62555752, 0.91202757],
[0.91937155, 0.69415961, 0.17472662, 0.47850455]])
>>> r[0] = r[0] / r[0][0]
>>> r
array([[1. , 3.11889819, 0.23470691, 1.87133115],
[0.71953505, 0.9111707 , 0.62555752, 0.91202757],
[0.91937155, 0.69415961, 0.17472662, 0.47850455]])
>>> r[1] = r[1] - r[1][0]*r[0]
>>> r
array([[ 1. , 3.11889819, 0.23470691, 1.87133115],
[ 0. , -1.33298588, 0.45667767, -0.43446079],
[ 0.91937155, 0.69415961, 0.17472662, 0.47850455]])
>>> r[2] = r[2] - r[2][0]*r[0]
>>> r
array([[ 1. , 3.11889819, 0.23470691, 1.87133115],
[ 0. , -1.33298588, 0.45667767, -0.43446079],
[ 0. , -2.17326664, -0.04105624, -1.24194406]])
>>>

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Jed @jedediyah.com · 28/01/2026

In #MathsToday we solved systems of equations with row operations in NumPy. I did this in MATLAB for many years and don't like the NumPy syntax... but it went well! #ITeachMath #mtbos #ComputingInMath

Let's use row operations to solve the system:
-x-y-3z =-9
3x+0y+z=-1
1x-5y+z=23

A Python script demonstrates row operations on a numpy array.

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Jed @jedediyah.com · 14/01/2026

The number of players on @chess.com who abandon games without resigning has skyrocketed in the past few weeks. Tried to leave feedback but got the chat bot that told me to clear my browser cache. Thanks AI!

Screenshot of the Chess.com AI chatbot. A user prompt reads:
A HUGE number of my opponents disconnect without resigning. I NEED chess.com to be more active about consequences for abandoning games.

The bot then responds to an unrelated concern.

Continuation of previous screenshot.
The bot continues to respond to an unrelated concern.

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Jed @jedediyah.com · 11/01/2026

Consider we have a sense of quantity but don't have numbers. We have stones that we can arrange in piles and patterns. A natural way to arrange stones is in rectangles *if you can*. I use this with 3-5 year olds and very quickly they are exploring and finding factorizations and identifying primes.

Rock configurations of numbers; rectangular representations of 1, 2, and 3.

Rock configurations of numbers; rectangular representations of 4 and 5.

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Jed @jedediyah.com · 07/01/2026

In #MathsToday we modeled a tide. Part (e) led to some amazing passionate mathematical arguments! (e) Our model has four parameters. Could you still model the tide well enough with only three?

Here is a a dataset of water depths: Depths

a. Copy and paste the data into Desmos to create a scatter plot. This should create a table.
b. We are going to adjust, by hand, the parameters of a sinusoidal model (do not do an automated regression):
y = beta_0 + beta_1 sin(beta_2 t + beta_3)
Add this model to Desmos as shown (you can type "beta" to create a
).
c. Add the sliders and adjust them until your model appears to fit the data. Record your best hand-fit parameter values.
d. Interpret each of the parameters in the context of the model. For example, represents a particular property of the tide which you can describe.
e. Our model has four parameters. Could you still model the tide well enough with only three?

Scatter plot of sinusoidal data with a sine wave roughly fitting the data.

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Jed @jedediyah.com · 29/12/2025

I gave a talk in 2024 called "Sensitive Dependence on Initial Conditions: How AI Predictably Fails Unpredictably". A dope title! jedediyah.github.io/aapt2024/ In examples of AI failing where we shouldn't be using AI, I highlighted a behavior nobody should be surprised by: AI encouraging suicide.

Title Slide:
Sensitive dependence on initial conditions
How AI predictably fails unpredictably
Jedediyah Williams
AAPT Summer Conference 2024
AI in K-12 Classrooms
jedediyah.github.io/aapt2024

Examples of AI responses to Google queries. In the right pane, a query "i'm feeling depressed" resulted in a suggestion of "jumping off the Golden Gate Bridge"

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Jed @jedediyah.com · 18/12/2025

Of course we validated our models with data splitting!

Model Validation
A flow chart depicts splitting data into Training and Testing sets. Modeling uses the training set, and the testing set is reserved until the very end where it will be used for validating the model.

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Jed @jedediyah.com · 18/12/2025

We use three of those points to find the equation of that quadratic, and then identify the vertex which corresponds to the minimum error at our optimal model parameter. Two students volunteered to share their beautiful work!

Hand written data tables and plots in a student's notebook.

Hand-written data and plots in a student's notebook. They have done quite a lot of algebra!

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Jed @jedediyah.com · 18/12/2025

Instead of getting a magically-fit regression from software, we find it ourselves! For several values of the slope, we find the sum of squared errors for our model. This turns out to generate an ideal quadratic (even from noisy real data!).

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Jed @jedediyah.com · 18/12/2025

Recognizing that a ball dropped from height 0 should bounce 0 cm high, we choose the a simple linear model.

How high does a bouncy ball bounce?
Y = B_1 X
A graph of bounce height vs drop height shows a line closely fitting a red scatter plot with a moderate positive slope.

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Jed @jedediyah.com · 18/12/2025

In #MathsToday we did one of my favorite activities: Regression by Hand. Our goal is to model the bounce height h_bounce of a bouncy ball dropped from height h_drop. #mtbos #ITeachMath

How high does a bouncy ball bounce?
A table shows columns h_drop and h_bounce. An image depicts four students next to a whiteboard. One holds a meter stick and another has dropped a bouncy ball. One looks on as another takes a video.

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Jed @jedediyah.com · 10/12/2025

In #MathsToday we are playing and analyzing Hexapawn as an introduction to machine learning. You can play and have a machine play itself here: jedediyah.github.io/hexapawn/ There are two problem sets here: williams-bhs.github.io/docs/hexapaw...

A game tree corresponding to several games of Hexapawn. A Hexapawn board is displayed in the top left; it is a three by three board with three white pawns and three opposing black pawns.

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Jed @jedediyah.com · 28/11/2025

The top functions are known. I don't know the function(s) that fit just the blue section. If you can find them, you'll have the solution! Since we can integrate the known functions - the unknown ones.

A labeled version of the previous graph containing the quartic function definitions.

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Jed @jedediyah.com · 28/11/2025

Here is a zoomed in version. 1. A tetrahedron is not possible if f is longer than 1/3. 2. If f < 1/5, a tetrahedron is necessarily possible. 3. The function that separates the orange points from the blue points is piece-wise quartic, and the probability of a tetrahedron is therefore rational!

A zoomed-in portion of the previous graph. A normalized histogram approximating a probability density function. An orange area corresponds to successful tetrahedra constructions.

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Jed @jedediyah.com · 28/11/2025

Particularly I focused on the the *longest piece f*. Here is a stacked normalized histogram to approximate the probability density function (which is known!) for just the longest piece. Again, the orange corresponds to successful tetrahedra.

A normalized histogram approximating a probability density function. There is a small area on the left that corresponds to successful tetrahedra constructions.

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Jed @jedediyah.com · 28/11/2025

Consider breaking the stick into six pieces ordered by length: a,b,c,d,e,f. I generated and shared this cross plot of the six ordered pieces. Orange points correspond to positive tetrahedra configurations, and blue are negative. There is a lot that is interesting!

A cross plot of six by six variables. The main diagonal has probability distributions while other cells contain blue and orange points corresponding to failed or successful tetrahedra constructions for the lengths of ordered pairs a through f.

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Jed @jedediyah.com · 28/11/2025

Afterwards, I shared the 3D version of this problem I know from @benjamindickman.bsky.social: Given a stick broken randomly at 5 places, what is the probability that a tetrahedron can be formed from the six pieces.

Six examples of tetrahedra formed from a virtual broken stick. Each piece is colored based on its length.

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Jed @jedediyah.com · 21/11/2025

Shared a Thanksgiving gift with my data science students today! I spend a couple months collecting used copies. Next month we will read chapter 3 Arms Race and reflect on feedback systems and the susceptibility and sensitivity of data models to manipulation.

A stack of copies of "Weapons of Math Destruction" by Cathy O'Neil

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Jed @jedediyah.com · 21/11/2025

Last week we worked on combinatorics and I asked about the weight of a binary word. In #MathsToday we worked on random walks and I asked what is the probability of ending where we started. Some of the students used the weight solution to find the probability! 🥳 #mtbos #ITeachMath

Let the weight of a binary string be the sum of its bits (or the number of 1s). So the string 101001 has weight 3.
a. How many binary words of length 3 have weight 1?
b. How many binary words of length 5 have weight 3?
c. How many binary words of length n have weight k?

For a random walk of length n, what is the probability that you end at x=0?

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Jed @jedediyah.com · 14/11/2025

In #MathsToday we rolled a lot of dice and found estimates of probability mass functions for sums of n=2,3 4 dice. In #MathsNextWeek we start to explore broken stick problems! Draft of next week's guide: williams-bhs.github.io/docs/Broken_...

Desmos plot of three histograms that look like triangles.

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Jed @jedediyah.com · 13/11/2025

In #MathsToday we played with the "Cat Machine" and talked about theoretical probabilities, experimental probability, and confidence in our experimental probabilities.

The Cat Machine has a random probability of generating a cat.
Try the machine and then guess P(cat)!

A button labeled "Cat Machine" with an arrow pointing to ten pictures, 8 of which are cats.

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Jed @jedediyah.com · 04/11/2025

Was really curious to see how these problems would go over in #MathsToday. It was great! We talked about how to map all of the problems onto the last problem and the usefulness of knowing "how to solve it" when we can recognize and abstract problems to known strategies.

A series of combinatorics problems. First problem asks: On a 2x2 grid of squares, in how many ways can we color the squares black or white?

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Jed @jedediyah.com · 02/11/2025

I once gave a talk called "AI is not what you think it is unless you think it is hot garbage" I've just enjoyed and highly recommend @chrissimon.au's "AI is a hype-fuelled dumpster fire that will burn down the planet and get you thrown in jail" www.youtube.com/watch?v=0bF_...

AI is not what you think it is unless you think it is hot garbage.
Four images depict robots and trash bins on fire.
Jedediyah Williams, PhD
Belmont High School
ATMIM 2024 Spring Conference: All Students Deserve a Piece of Pi
March 14, 2024

AI is a hype-fuelled dumpster fire that will burn down the planet and get you thrown in jail
Chris Simon

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Jed @jedediyah.com · 29/10/2025

In #MathsToday, trying to talk about state machines without talking about state machines. Also debated with students whether we should fully represent all 12 states or reduce the number of bits to the minimum required (depending on number of states used).

Four faces of a 4-way stop light labeled L1, L2, L3, L4. Greens on L1 and L3, reds on L2 and L4.

There are twelve total lights that can be On (1) or Off (0), and we can represent the current state of those lights with a -bit binary word.

For example, with green lights both On in one direction while the other direction has red lights, we can write:
001100001100
which means that G1 and G3 are On while and R2 are R4 on.

Create a directed graph to represent the behavior of the stop lights as they transition from one state to the next.

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Jed @jedediyah.com · 28/10/2025

In #MathsToday we continued working on our graph theory exercises.

Fred, along with the five people from the previous problem need to split into two groups of three for a project. However, there is drama! Alejandro refuses to work with Camille or Elias; Dalia "just can't even" with Ben or Camille; Ben and Camille both had a falling out with Fred; "aint no way" Elias will work with Dalia.

a. Define and draw a graph W, with six vertices, to represent who CANNOT work with whom.
b. Show that it is possible to create two groups of three by partitioning
into a bipartite graph with three vertices on each side.
c. Find W', the complement of W. What does W' represent?

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