All posts by Webmaster
Music Posts
UPR – Selected Posts
Videos
Mathematics Posts
Sitemap
Pages
- Acids and Bases
- Home
- In Praise of Lectures
- Math 4995 Graph Theory II
- Matroid Theory
- Math for Elementary Teachers
- Generic Course Objectives
- Classroom Behaviour
- Math 3018 (3171-72) Precalculus
- Greek Alphabet
- Math 3026 Statistics
- Math 3151 Calculus I
- Math 3152 Calculus II
- Math 3153 Calculus III
- Math 3325 – Discrete Mathematics
- Math 4031 Linear Algebra
- Math 4032 Abstract Algebra I
- Math 4021 Mathematical Logic I
- Math 4009 Ordinary Differential Equations
- Math 4019 Metric Differential Geometry
- Math 4033 Abstract Algebra II
- Math 5100 Applied Combinatorics
- Math 5021 Advanced Calculus I
- Graph Theory I
- Math 5037 Complex Analysis
- Site Information
Posts
Blog
- A Comment on the Movie ‘Churchill’
- A Nobel Prize For Inanity
- An Warning concerning Computer Instruction
- Beware the Slithy Troll
- Brain Death – The Hoax That Won’t Die
- Censorship of Prolife Images
- Covid Vaccines cause Abortion
- Education – Selected Posts
- Infanticide in Puerto Rico and now New York
- Marcha por la Vida, San Juan, Puerto Rico, 2026
- Obama – Infanticide President
- On the carbon footprint of cranberry sauce
- ON THE INSTRUCTION OF CALCULUS
- Open Letter to America
- Our Composite President
- Physicist blames Christians for global warming
- Puerto Ricans are Dying to be Vaccinated
- Puerto Rico Demographic Winter
- Puerto Rico – 40 Days for Life
- Puerto Rico Abortion Rates
- Puerto Rico NAEP Math Scores
- Puerto Rico NAEP Results -Why the wheels of education don’t turn
- Reunion del Claustro de Profesores – El 5 de Mayo 2003
- The Andalusian Illusion – Book Review
- The cost of human eggs – Warning to Puerto Rico students
- The Dictatorship of (Thermodynamic) Relativism
- The Logical Case against Climate Panic
- The Puerto Rico Mathematics Curriculum Framework
- The Thoughts of Chairman Grok
- U.S. Citizenship Naturalization Test
- UN – Warning to Donors
- Unmasking the Strategy to Legalize Abortion throughout Latin America
- UPR: Fuente de Adoctrinación
- Vandals set fire to University of Puerto Rico Library
Climate
- A Fable of Global Warming
- A Nobel Prize For Inanity
- Concerning the Measurement of Global Mean Surface Temperature
- Harmonic Analysis of Global Temperature
- On the carbon footprint of cranberry sauce
- Physical Applications of the Stevens’ Classification
- Physicist blames Christians for global warming
- The Dictatorship of (Thermodynamic) Relativism
- The Logical Case against Climate Panic
Education
- An Early Warning of the Death of Geometry
- An Warning concerning Computer Instruction
- Cognition, Logic, and Instruction
- Dr. Duke Pesta and Dr. Sandra Stotsky Discuss Failed Colleges of Education
- Dr. Duke Pesta on the Dangers of Government Schools
- ESSA and the Vanishing Puerto Rico Content Standards
- ON THE INSTRUCTION OF CALCULUS
- Professing at the podium
- Puerto Rico NAEP Math Scores
- Puerto Rico NAEP Results -Why the wheels of education don’t turn
- The Puerto Rico Mathematics Curriculum Framework
- The Story Behind Project Follow Through
- The Thoughts of Chairman Grok
- Unidad 1: Fundamentos de Geometría
- Virus of Constructivism Threatens Puerto Rican Catholic Education
Mathematics
- A Mathematical Introduction to Melody
- Commutative Diagrams and the Construction of Musical Scales
- Concerning the Measurement of Global Mean Surface Temperature
- Condom Roulette
- Graduate Courses in Discrete Mathematics
- Guitar Chord Sequences Ordered by Pitch
- Mathematical Challenges to Darwin’s Theory of Evolution
- Mathematical Physics
- Mathematics Courses
- Physical Applications of the Stevens’ Classification
Music
- A Mathematical Introduction to Melody
- Commutative Diagrams and the Construction of Musical Scales
- Guitar Chord Sequences Ordered by Pitch
- Modes
- Music Papers
- Pitch Equivalent Fingerings of Specific Guitar Chords
- Tables of Guitar Chord Triads – Ordered by Pitch
Politics
- Dear Mr. President: A Quiz
- Obama – Infanticide President
- Obama Care
- Puerto Ricans are Dying to be Vaccinated
- Puerto Rico Election Guide
- The Democrats: Party of Infanticide
Post Set
- Climate – Selected Posts
- Mathematics Posts
- Music Posts
- Political Posts
- Pro Life – Selected Posts
- Recent Posts
- Religion and Philosophy Posts
- Sitemap
- UPR – Selected Posts
Pro-Life
- A Critique of Jarvis Thomson’s Violinist Argument for Abortion
- ACCIÓN PROVIDA URGENTE PR
- Brain Death – The Hoax That Won’t Die
- CBR Saves Irish Babies
- Censorship of Prolife Images
- Condom Roulette
- Fetal Development
- Infanticide in Puerto Rico and now New York
- Marcha por la Vida, San Juan, Puerto Rico, 2026
- Puerto Rico Demographic Winter
- Puerto Rico – 40 Days for Life
- Puerto Rico Abortion Rates
- The cost of human eggs – Warning to Puerto Rico students
- The Democrats: Party of Infanticide
- The Science of Fetal Pain
- Unmasking the Strategy to Legalize Abortion throughout Latin America
- ¿Por Qué el Verdadero Amor Espera?
Religion
- Advertencia a los niños
- Diálogo Ininteligente
- Eucharistic Prayer Of St Thomas Aquinas
- Is Their an Afterlife?
- Love
- Physicist blames Christians for global warming
- Rock Music from a Religious Perspective
- Student’s Prayer by Thomas Aquinas
- The Andalusian Illusion – Book Review
- The Dictatorship of (Thermodynamic) Relativism
- The wise men’s gift
- Virus of Constructivism Threatens Puerto Rican Catholic Education
- ¿Por Qué el Verdadero Amor Espera?
UPR
- Abu María y Abu Pepe en la UPR
- Apelación – Junta de Síndicos
- Apelación a La Junta Universitaria
- Astrología y Socialismo en la UPR
- Aviso Importante
- Boletín — marzo de 2003
- Boletín — mayo de 2003
- Boletín — septiembre 2005
- Boletín —27 de septiembre de 2004
- Boletín al Claustro
- Boletín de agosto 2005
- Boletín de febrero de 2003
- Boletín — 26 de agosto de 2003
- Boletín—Febrero de 2004
- BS in Mathematics
- CISE – a “Virtual” Ph.D.
- Claustro Rechaza el “Nuevo Bachillerato”
- Cronología
- El Debate
- El «nuevo bachillerato para el 2000» y el ajuste neoliberal de la universidad
- Escuela de Comunicación
- Filosofía
- First Year Seminar
- La Utopía Defectuosa
- Nuevo Bachillerato Universitario: ¡Andy Warhol vive!
- Observaciones
- Philosophical Turn
- Primer día de clases en la UPR
- Quantitative Reasoning Report
- Reunion del Claustro de Profesores – El 5 de Mayo 2003
- The UPR Curriculum War
- UPR: Fuente de Adoctrinación
- Vandals set fire to University of Puerto Rico Library
- ¿Reconceptualización igual a reducción?
Video
- A Comment on the Movie ‘Churchill’
- Dr. Duke Pesta and Dr. Sandra Stotsky Discuss Failed Colleges of Education
- Fetal Development
- Harmonic Analysis of Global Temperature
- How Donald Trump Changed Boxer William Campudoni’s Life
- Is Their an Afterlife?
- Kreeft: How to Win the Culture War
- Mathematical Challenges to Darwin’s Theory of Evolution
- Obama – Infanticide President
- Spem In Alium
- Video: Don’t Cry For Me Sudan
- Video: Pleebo – Liberia
- Video: Puerto Rico Se Levanta
- Video: The Myth of the Spanish Inquisition
- Videos
A Mathematical Introduction to Melody
Algebraic music theory rests on the observations that the set I of intervals forms a Z-module (or free abelian group) under concatenation and that the set of pitches forms an affine space (or torsor) over this module. A change of key is simply a shift of origin.
Within this framework a melody is a walk in the affine space subject to welldefined local constraints (conjunct motion, leap compensation, tendency-tone resolution, diatonic membership, contour, range, etc.). A chord is expressed, via Chasles’ identity, as a vector sum of intervals; the same language yields a clean description of closed and open voicings, pitch-equivalent fingerings, and the affine subspaces that correspond to whole-tone scales, augmented triads, diminished-seventh chords and tritones.
The paper develop these ideas in elementary terms, supply the necessary multiplication and inversion tables, give explicit fretboard realisations of the principal intervals and dyads, and illustrate the theory with a detailed analysis of the traditional melody and harmony of Greensleeves. An appendix collects interval-stack constructions for the most common triads, seventh chords and ninth chords.
Music Papers
- Philip Pennance, Commutative Diagrams and the Construction of Musical Scales. Sept. 12, 2025.
- Philip Pennance, Guitar Chord Sequences Totally Ordered by Pitch, May 2026.
- Philip Pennance, A Mathematical Introduction to Melody, Aug. 11, 2026.
Tables of Guitar Chords
- Philip Pennance, Guitar Chord Triads, Ordered by Pitch, May 2026.
- Philip Pennance, Pitch Equivalent Fingerings of Specific Guitar Chords, June 2025.
The Puerto Rico Mathematics Curriculum Framework
Brutal Analysis by Grok
Ah, the 2022 Marco Curricular de Matemáticas from Puerto Rico’s Departamento de Educación. A bureaucratic masterpiece that takes the noble subject of mathematics — the one discipline built on irrefutable logic, precise definitions, deductive proof, and universal abstraction — and buries it alive under a mountain of mandated social, ethical, environmental, and ideological fluff.
In the section “Los temas transversales e integradores en el currículo de Matemáticas,” the document waxes philosophical about how these “ejes transversales” must “penetrar e influir toda la práctica educativa.” They supposedly integrate “ser, saber, hacer y convivir” through concepts, procedures, values, and attitudes. They are to be handled at theoretical, personal, and social levels, via “temas generadores” per grade, contextualized problems, PBL, real-life situations, and all the usual edu-speak. The goal? Transform education, connect to “preocupaciones sociales más actuales y urgentes,” align with 21st-century competencies and the UN’s Sustainable Development Goals, and turn every math class into a micro-laboratory of social engineering. The glorious list of irrelevancies. The document (and its grade-specific generators) serves up a greatest-hits compilation of modern curriculum fashion:
- Identidad cultural e interculturalidad
- Equidad y respeto entre todos los seres humanos
- Educación cívica y ética
- Educación para la paz
- Educación ambiental / concienciación ambiental y ecológica
- Emprendimiento e innovación
- Promoción de la salud
- Tecnologías de la Información y la Comunicación
- Educación para el trabajo / educación y tecnología
Then the per-grade “temas generadores” pile on even more: autogestión, convivencia, diversidad cultural/funcional, ética en la vida diaria, finanzas, responsabilidad social, toma de decisiones, disciplina digital, cuidado del medio ambiente, antirracismo (explicitly in sexto grado), solidaridad, sensibilidad por la naturaleza, capital social, empresarismo, etc.
These are not optional flavor text. They are to be woven into the actual mathematics standards and expectations through “problemas en contexto,” actividades interdisciplinarias, and reflexión crítica. Math teachers are expected to treat quadratic equations as vehicles for discussing equitable resource distribution, use data analysis to promote health or environmental awareness, tie geometry to cultural identity, and turn budgeting problems into entrepreneurship-and-innovation sermons.
Their total, magnificent irrelevance to teaching math
Mathematics is not a delivery mechanism for your favorite social causes. It is the study of patterns, structure, quantity, space, and change using rigorous reasoning that does not require (and is actively hindered by) constant detours into “how does this make you feel about equity?” or “let’s reflect on our responsibility to the planet while solving for x.”
When you force a 6th grader learning ratios or statistics to also process “antirracismo” or “diversidad funcional,” you are not enhancing mathematical understanding — you are hijacking limited instructional time for amateur activism. When you turn fraction lessons into meditations on “equidad y respeto entre todos los seres humanos,” you replace deliberate practice with performative discussion. The math suffers. The social theme is usually treated superficially because the teacher’s actual expertise is in mathematics, not in running a sensitivity workshop.
The document claims this integration makes learning “significativo” and connects math to “la vida.” In reality, it does the opposite for serious mathematics. Real mathematical power comes from abstraction and transferability. A student who can manipulate variables fluently can later analyze any dataset — environmental, financial, health-related, or otherwise — without needing the curriculum to spoon-feed the “correct” social interpretation every single time. Forcing the transversal themes assumes students are too stupid to apply math to real problems unless the teacher explicitly lectures them on the approved values while doing it.
This is classic edu-babble cargo cultism: dress up vague attitudinal goals in fancy language (“ejes transversales,” “temas generadores,” “triple perspectiva”), mandate their presence in every subject, and pretend you’ve improved education. What you’ve actually done is dilute the one subject that most reliably builds clear, independent thinking — precisely the kind of thinking that might eventually allow students to evaluate social claims critically rather than recite them.
Pythagoras did not need to reflect on cultural identity or ecological awareness to discover his theorem. Gauss did not pause his work on number theory to discuss entrepreneurship or digital discipline. The mathematics itself is the point. Everything else is either a distraction or, at best, an occasional, teacher-chosen illustration that does not require bureaucratic enforcement across the entire syllabus.
The transversal themes in this document are not “integrated” with mathematics in any meaningful pedagogical sense. They are imposed upon it. They are irrelevant to the actual teaching of math because math does not need them to be rigorous, beautiful, or useful. Their presence reveals far more about the priorities of curriculum writers than about what children need to learn in order to think mathematically.
In short: stop turning math class into a mandatory TED Talk on whatever social issue is fashionable this decade. Teach the damn math. The rest can wait for subjects actually equipped to handle it.
References
- Mathematics Standards of Puerto Rico – Analysis and Recommendations, Philip Pennance, 2002.
- Marco Curricular Matemáticas. Departamento de Educación de Puerto Rico. 2022. url: https://dedigital.dde.pr/pluginfile.php/147953/mod_resource/ content/6/Marco%20Curricular%20Matematicas.pdf (visited on 06/28/2026).

































































