jedediyah.github.io/nctm2023


Beauty, Utility, and Consequence
in Mathematics


Jedediyah Williams, PhD
Belmont High School

October, 2023

jedediyah.github.io/nctm2023





About Jed

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Outline


I. Beauty
II. Utility
III. Consequence
IV. Our Teaching



I. Beauty
   
II. Utility
III. Consequence


please share your conceptions or experiences of mathematical beauty

You can turn and talk or/and add to this Google Doc: https://docs.google.com/document/d/1g1yIbBIhOo3H5e2ujZyJpHFhf2XWYXh7gOJl63z2EH4/edit?usp=sharing





I. Beauty
   
II. Utility
III. Consequence






\[ e^{i \pi} + 1 = 0 \]

G.H. Hardy. "A Mathematician's Apology"
I. Beauty
II. Utility
III. Consequence

WSJ algorithmic rhyme scheme analysis.

I. Beauty
II. Utility
III. Consequence


There is tremendous beauty in mathematics, and the sources of beauty are diverse in nature and mode.

Beauty often involves acts of creativity that satisfy constraints.




I. Beauty
   
II. Utility
III. Consequence


please share your conceptions or experiences of mathematical utility

You can turn and talk or/and add to this Google Doc: https://docs.google.com/document/d/1g1yIbBIhOo3H5e2ujZyJpHFhf2XWYXh7gOJl63z2EH4/edit?usp=sharing





I. Beauty
II. Utility
III. Consequence






I. Beauty
II. Utility
III. Consequence


The justification for teaching mathematics is often tied to utility in the solving of real problems.





I. Beauty
II. Utility
III. Consequence


The justification for teaching mathematics is often tied to utility in the solving of real problems.


I recently asked my students what real-world problems they had worked on in their math classes.





I. Beauty
II. Utility
III. Consequence

Much of the history of and motivation for mathematics has deep cultural connections.

Tallying, astronomy, calendar, navigation, trade, bookkeeping, surveying, measuring (the Earth), enumeration, physics, statistics, engineering, communication, computability, encryption, analysis, analytics,

"
Davis and Hersh. "The Mathematical Experience" p. 129

I. Beauty
   
II. Utility
III. Consequence


please share your conceptions or experiences of mathematical consequence

You can turn and talk or/and add to this Google Doc: https://docs.google.com/document/d/1g1yIbBIhOo3H5e2ujZyJpHFhf2XWYXh7gOJl63z2EH4/edit?usp=sharing





I. Beauty
   
II. Utility
III. Consequence






Engima Keys
The "Bombe"

IBM's support of Nazi Germany with compute power for census,
statistics, and logistics calculations.

Abraham Wald and the Statistical Research Group at Columbia.

You could build an entire course on the mathematics of World War II

  • Encryption
  • Cryptanalysis
  • Computation
  • Statistical Research Group
  • Ballistics
  • Fluid dynamics
  • Warcraft engineering
  • Manhattan project
  • Operation Paperclip
Recommended reading: "Crypto" by Steven Levy
I. Beauty
   
II. Utility
III. Consequence



"Essentially, all models are wrong, but some are useful."
- George Box




I. Beauty
   
II. Utility
III. Consequence
What are some consequences of data technologies?
↓

Some of the more well known harms













https://www.nytimes.com/2019/08/16/technology/ai-humans.html
Anatomy of an AI system, Crawford and Joler
Adversarial attack
Algorithms are brittle - Melanie Mitchell
Lack of oversight or auditing
The act, by those in power, of making decisions for us is a display of the imbalance of power.
- Sun-ha Hong, Prediction as Extraction of Discretion
We are being surveilled.
We are being experimented on.
"Our success, happiness, and wellbeing are never fully of our own making. Others' decisions can profoundly affect the course of our lives...

Arbitrary, inconsistent, or faulty decision-making thus raises serious concerns..."

- Fairness and Machine Learning, Barocas, Hardt, and Narayanan

  • Math-based technologies are ubiquitous, often do not work, and are capable of broad and arbitrary harm.
  • Are we training the next generation of phrenologists?
  • Are we training the next generation of eugenecists?
  • To what extent should ethics be incorporated into mathematics classrooms?
"Yes, train these young people to get these skills, but integrate into that not only the technical capacity but the critical capacity to question what they're doing and what's happening. To me, it is not true empowerment unless people can have the power to question how these skills are going to be used."
IV. Our classrooms


Data Modeling Process


Data
→
Preprocess
→
Explore
→
Model
→
Communicate



Data Modeling Process


Environment

→
Data
→
Preprocess
→
Explore
→
Model
→
Communicate
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A framework for critical analysis

Data
 • Harmful data collection, lack of consent, insecure / lack of privacy, historical, representational, or measurement bias, ...

Preprocess
 • Labor exploitation, labeling by non-experts, incorrect labeling, trauma experienced by labelers, ...

Explore
 • Feature selection bias, bias in interpretation of data visualization, data manipulation, feature hacking, ...

Model
 • Bias in model choice, model-amplified bias, environmental impact, learning bias, evaluation bias, peripheral modeling, ...

Communicate
 • Biased model interpretation, ignoring variance, rejecting model, deploying harmful products, deployment bias, ...

Meta
 • "Pernicious feedback loops", runaway homogeneity, susceptability to adversarial attack, lack of oversight or auditing, ...

Critical Questions:
  • What are the motivations for the project?
  • What is the intended use?
  • What is the unintended use or misuse?
  • Where does the data come from?
  • Who collects the data?
  • Who owns the data?
  • How is the data collected?
  • How is the data stored?
  • How old is the data?
  • When will the data expire?
  • How will the data be secured?
  • What happens with the data when the company is sold?
  • Who does the labeling?
  • What labels will they decide to use?
  • Are the labelers experts?
  • Are the labels accurate?
  • What biases are represented in the data?
  • How is data included or excluded?
  • How are outliers addressed?
  • What subpopulations are represented?
  • What subpopulations are over- or underrepresented?
  • What portions of the data are inspected?
  • What features are selected for modeling?
  • What model is chosen?
  • What features do we think are being modeled?
  • What latent features are actually being modeled?
  • What is the domain of the model?
  • What are the consequences of error?
  • What decisions will be made with the model?
  • What biases are perpetuated?
  • Where will the model be deployed?
  • What could go wrong?
  • Who is responsible when things go wrong?
  • How can issues be reported?
  • Will new data be fed back in to update the model?
Students should witness math fail. They should experience models performing poorly.
  • Data and all of its questions
      • Privacy, security, power, surveillance, consent, access, stereotyping, fairness,...
  • Encryption
  • Military
  • The pitfalls of our predecesors
  • Misuse
  • Notation is convention
  • See models break
  • Bias and variance
  • Environment
  • Complexity, incompleteness, decidability, the unfinished state of mathematics
  • Accountability
  • Justice
  • ...
When handing over the tools of mathematics, we are responsible as educators for teaching their responsible use.

It is a sin of omission when we fail to acknowledge the consequences of the content we teach; Consequences which include ethical and technical pitfalls.
To what extent are the three themes of
I. Beauty
   
II. Utility
III. Consequence
experienced in our classrooms?

Resources

Education: