A data analytics course can be taught as a tour of software. Here is the library, here is the function, here is the plot. Students leave able to call things, and unable to say what the things do or when they stop working.

This fall I am teaching the course the other way at UT San Antonio.MDC 4153, Data Analytics, cross-listed for undergraduates as MAT 4953 and for graduate students as MAT 5153, treats the data matrix as its central object and derives everything else from it. Students program the algorithms themselves rather than calling them from a toolbox. Every number in their reports is emitted by code that a stranger could run. And a large language model sits beside them for the entire semester, under a protocol that makes its output something to verify rather than something to paste.

The data matrix

Start with a matrix whose rows are records and whose columns are measurements. The singular value decomposition of that matrix gives principal components, derived twice, once as the directions of greatest variance and once as the best low-rank approximation. Least squares follows, with the conditioning of the normal equations as the first warning that a method can be right in theory and useless in floating point. The linear model is derived three ways. Generalized linear models add the variance function and the link. Robust, weighted, and regularized regression handle the cases the textbook examples avoid.

From there the course climbs: classification by linear and nonlinear discriminants, trees and boosting, clustering built from first principles with k-means appearing as a limit of expectation maximization, and neural networks reached by way of Newton’s method, gradient descent, and backpropagation as chain-rule bookkeeping.

The discrete Fourier transform gets the same treatment. It is derived as least squares on an orthogonal basis, programmed from scratch, and then made fast. Time series and dependence come with it, because real records arrive in order and the order carries information.

The semester closes with allocation: linear programming, duality, and the Karush-Kuhn-Tucker conditions as a certificate that a decision is the best one available under a budget. A concave-utility variant turns a single curvature parameter into an explicit equity dial, chosen in the open and defended.

Three objectives

The syllabus states each objective twice, as a goal and as the outcome by which it is assessed.

1. Public data and social phenomena. Records about education, crime, income, or births cannot be taken at face value. Students learn to ask about coverage, density, collection bias, and the past that produced a present measurement. The outcome: identify the biases in a public data set and introduce documented corrections.

2. Reproducible practice. Students learn what a technique is, use templates for how to do it, and learn where to find why it works. The outcome: choose the correct method for a problem and deliver an analysis a second analyst can rebuild from the recorded inputs.

3. Artificial intelligence for technical work. The outcome: evaluate what a language model produces, name the invalidation type when it is wrong, and correct it.

The Data Challenge

One team engagement runs the whole semester, on a multi-year extract of United States birth records, delivered in five milestones: the precommitted analysis plan, the data and feature configuration, the model ladder, allocation and explanation, and the report with a ten-minute defense.

Each milestone extends the previous one in place. The model ladder requires a naive baseline, ordinary least squares on the registered specification, a Gamma GLM or Huber regression, ridge, and a tree ensemble, all run by one orchestrator against a validation protocol fixed before any score is seen. The allocation milestone takes a fixed hypothetical budget and produces the linear program, its shadow prices, the equity variant, and a one-page explanation of one award written for the person receiving it.

The grading standard for the final report is the stranger-rebuild test. A competent stranger, given the repository alone, rebuilds the report and reaches the same numbers. Every table and figure is emitted by code. The environment file, execution order, seed policy, and the change log kept since the first milestone are part of the submission.

Augmented intelligence

This course requires the use of artificial intelligence, and it grades the part of the work these tools leave undone: the judgment that decides whether an output is correct, sufficient, and defensible for the decision it will inform.

Session 2 establishes the analyst’s ethical frame and a four-type taxonomy of invalidation, factual, logical, normative, and structural, each with the verification move its type requires. Students adopt a one-page governance protocol with a provenance log, an invalidation log, and a disclosure statement, and carry it through all five milestones. Every session names what is delegable to an assistant, what is not, and the failure most likely to arise in that session’s material.

Two rules follow. Work developed against a single model does not satisfy the requirement; each deliverable is developed against at least two independent models, with the provenance log recording where they agreed and where they parted, because models share training data and failure modes. And naive use earns no credit: accepting output because it reads well, prompting once and submitting, pasting code without running it, citing a reference the model invented. Expert use is an assessed outcome of the course.

Who should take it

The course is intended for Mathematics, Mathematics for Data and Computing, and Mathematics for Teaching majors. Graduate students in the MAT 5153 section work from their own set of assignments and tests. The catalog prerequisite is MAT 2253, or MAT 2233 and MAT 2213. Proficiency in a programming language helps and is not required. Students who have not completed the prerequisite may still enroll and receive extra assignments.

Course details

• Sections: MDC 4153.001, MAT 5153.001 (CRN 28088), MAT 4953.002 (CRN 30019)
• Term: Fall 2026
• When: Mondays and Wednesdays, 4:00 to 5:15 pm
• Where: McKinney Humanities Building, Room MH 3.03.10
• Format: In person
• Instructor: Juan B. Gutiérrez, Ph.D., Professor of Mathematics
• Questions juan.gutierrez3@utsa.edu

—

*Figures drawn from the course’s own examples: principal components of a point cloud (Lesson 5), least squares on a Fourier basis (Lesson 22), and gradient descent on a quadratic loss (Lesson 20).*