Matemáticas

Yet in an uncanny number of real-world data sets, an astonishing 30.1 percent of the entries begin with a 1, 17.6 percent begin with a 2, and so on. This phenomenon is known as Benford's law. The law holds even when you change the units of your data.

Physicist Frank Benford made the same observation in 1938 and popularized the law, compiling more than 20,000 data points to demonstrate its universality.

the expected distribution of leading digits and therefore were probably fabricated

many data sets do not conform to Benford's law

The law is more likely to apply to data sets spanning several orders of magnitude that evolve from certain types of random processes

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Benford's law is the only leading digit distribution that is immune to such unit changes.

Por mucho rechazo que puedan suscitar, las matemáticas no solo son necesarias para resolver situaciones prácticas, sino también para procesar la información que recibimos diariamente y para tomar decisiones con sentido crítico

“El infinito placer de las matemáticas", Maccarrone

no le importe reconocer abiertamente su falta de conocimiento matemático

placer de buscar patrones

se ha promovido una enseñanza de las matemáticas muy basada en la resolución de algoritmos

vamos a analizar un problema, a ver todas las maneras que se nos ocurren de afrontarlo y a partir de los errores vamos a entender por qué eso es un error

matemáticas mucho más basadas en darle sentido a los conceptos matemáticos, en su comprensión, en la indagación

son útiles para procesar toda la gran cantidad de información que recibimos

interpretar correctamente las gráficas, fijarse en todos los detalles, entender el significado de los parámetros

entender un poco más la ciencia y la tecnología y cómo moldeamos con ellas el mundo actual

¿Y cuál es la motivación? El placer.

hay mucha belleza matemática a nuestro alrededor en patrones, en formas, en regularidades, pero que solo apreciamos a fondo si tenemos este conocimiento matemático

belleza de los razonamientos

El poder entender algo complicado de una manera muy sencilla y muy directa esconde también mucha belleza.

no hay nada más humano, más esencialmente humano, que la capacidad de razonar. Y no hay razonamiento más libre que el de las matemáticas.

Si los matemáticos expertos han rediseñado el currículo, ¿por qué los resultados no son mejores?

los expertos no están tomando en cuenta las etapas de desarrollo de la mayoría de los estudiantes, y porque realmente no son conscientes de los problemas a los que se enfrentan la mayoría de los maestros.

el nuevo plan de estudios de matemáticas no está logrando estos objetivos

Dado que cada estudiante tiene un perfil único de lo que comprende o no comprende, este es el origen del "plan de estudios en espiral", donde cada año se introducen muchos temas, y cada año, los textos de matemáticas profundizan un poco más en cada tema

el currículo actual introduce tantos temas que pocos se dominan realmente

Aquellos que no se vuelven competentes en el cálculo pierden confianza en sí mismos y ciertamente son aún MENOS propensos a estar abiertos a cualquier discusión de "comprensión".

es una pérdida de tiempo precioso de clase a esa edad dedicar mucho tiempo al POR QUÉ

más estudiantes que nunca no dominan los procedimientos básicos.

Este problema se debía a que los maestros de la escuela no enseñaban a los niños a traducir entre palabras en inglés y lenguaje matemático.

estudiantes estarían mucho mejor servidos aprendiendo a calcular y teniendo PRÁCTICA GUIADA DIARIA en tipos particulares de problemas de historias, tanto para reconocer tipos de problemas, como para poder entender fácilmente cómo traducir el idioma inglés al lenguaje matemático.

Lo que los "expertos" en matemáticas que diseñan el currículo no se están dando cuenta es que mostrar a los estudiantes todas las diferentes formas posibles de resolver todo tipo de problemas matemáticos NO crea los pensadores innovadores "divergentes" que están buscando.

El primer requisito para convertirse en un pensador divergente es la confianza en las propias habilidades

Lo principal es DOMINAR al menos un método.

El currículo que obliga a los estudiantes a calcular por muchos métodos diferentes fatiga a muchos estudiantes y, de hecho, los desmotiva para seguir aprendiendo por sí mismos.

encuesta nacional para medir percepciones sobre la educación.

La importancia que le asignan a la educación es alta

las competencias más importantes para la vida son las matemáticas, seguidas por el inglés. Las matemáticas duplican a la lectura crítica y triplican a las habilidades emocionales y las “competencias blandas”.

Los docentes son destacados como el factor más importante para el éxito en el aprendizaje

Es hora de que los profesores de STEM prioricen la colaboración entre disciplinas para transformar las clases de matemáticas de mecanismos de eliminación a terreno fértil para cultivar una generación diversa de investigadores y profesionales de STEM.

enfatiza su aplicación en un contexto

los estudiantes de las nuevas clases terminaron con "<a href="https://www.lifescied.org/doi/10.1187/cbe.20-11-0252" _istranslated="1">calificaciones significativamente más altas</a>" en los cursos posteriores de física, química y ciencias de la vida que los estudiantes del curso de cálculo tradicional

El aprendizaje de las matemáticas es fundamental para todos los campos STEM, pero también parece ser cierto lo contrario: los campos STEM pueden ser fundamentales para que el aprendizaje de las matemáticas sea eficaz para más estudiantes. Involucrar a otras disciplinas STEM en el rediseño de las clases de matemáticas es una forma clave de garantizar que esas clases ofrezcan rampas de acceso atractivas e inclusivas a STEM.

sometimes we have other reasons for teaching students to solve certain kinds of problems

we shouldn’t simply ask ourselves the question, “Do we want our students to be able to solve related rates problems?” A better question to ask is, “What are the educational benefits of teaching related rates problems?”

Confusion needs to be managed and responded to, not avoided at all costs.

Presenting something that confuses your students can sometimes create an opportunity to confront and clear up the students’ confusion.

Rigor makes the rules of the game of mathematics clearer, even if it makes the game harder to win.

Recognizing when a technique isn’t working is much easier when reasoning is held to high standards of rigor

Teaching rigor and precision, provided it is done without the veil of complexity interfering, burns away the fog, leaving everything crisp and clear and making it possible to drive faster and to enter uncharted lands.

Learning to detect these flaws is one of the greatest challenges of learning mathematics

Everyone needs mathematics. It is the heavy industry of scientific development, the unseen basis on which the more spectacular advances in science, in technology, and in medicine are often built.

mathematics is cheap

mathematics transcends culture

high level of mathematical competence

creativity

passion

The teacher is respected as a source of knowledge

something is often perceived to be lacking

quest for creativity

there is little room for students to create their own knowledge or to invent new ways to reorganize the knowledge

How can the student know something the teacher doesn’t?

teachers who promote creativity see results in achievement

But mathematicians were relatively free.

Mathematics departments and classrooms became centers for a silent rejection of totalitarian values.

There are certainly creative Asians, passionate Americans, and schools in Eastern Europe that reach all their students with deep and important mathematics.

language is the medium and tool for mathematics learning

reading vocabulary, reading comprehension, mechanics of language and spelling have higher correlations with arithmetic reasoning than with arithmetic fundamentals at all elementary grade levels

. Some features of English make it even more difficult to see the underlying tens and ones structures

The results show that Chinese-speaking children significantly outperformed the English-speaking children on general visual perceptual abilities.

In contrast, native Chinese speakers tended to engage a visuo-premotor association network for the same task.

cannot be explained completely by the differences in languages per se

Confucian Heritage Culture

“Does culture really matter?” the answer is still: “Probably”. However, there is more evidence today that it is probable than there was 20 years ago!

International Journal of Research in Undergraduate Mathematics Education

average examination scores improved by about 6% in active learning sections

very positive feedback from the students on the use of pen-enabled tablets

There are many endeavors making service courses more helpful and relevant to students by implementing or strengthen the discipline related perspective

five areas in which the field has made significant progress (Theoretical Perspectives, Instructional Practices, Professional Development of University Teachers, Digital Technology, and Service-Courses in University Mathematics Education)

seven areas are in need or further development (Theories and Methods, Linking Research and Practice, Professional Development of University Teachers, Digital Technology, Curriculum, Higher Years, and Interdisciplinarity)

math and science competence significantly contribute to problem solving

math-science coherence is significantly related to problem solving competence

low problem solving scores seem a result of an impeded transfer of subjectspecific knowledge and skills

problem solving competence is defined as the ability to solve cross-disciplinary and real-world problems by applying cognitive skills such as reasoning and logical thinking

there is a conceptual overlap between the problem solving models in these two domains

Understanding and characterizing the problem, representing the problem, solving the problem, reflecting and communicating the problem solution

Scientific literacy has been defined within a multidimensional framework, differentiating between three main cognitive processes, namely describing, explaining, and predicting scientific phenomena, understanding scientific investigations, and interpreting scientific evidence and conclusions

mathematical literacy refers to students’ competence to utilise mathematical modelling and mathematics in problem-solving situations

we found that in most countries, math and science competence significantly contributed to students’ performance in analytical problem solving.

math competence was a stronger predictor of problem solving competence.

reading ability can be regarded as another, shared demand of solving the items

there are a number of skills that can be found in math, science, and problem solving: information retrieval and processing, knowledge application, and evaluation of results

math-science coherence, which reflects the degree to which math and science education are harmonized

higher levels of coordination between math and science education has beneficial effects on the development of cross-curricular problem-solving competence

mathematical explanation to demonstrate that increasing speed does not result in significant time savings

when increasing speed from 120 km/h to 140 km/h, the reduction in travel time is practically insignificant

The article describes an innovative concept to create authentic application examples for mathematics courses in an engineering degree program in order to illustrate the relevance of mathematics for engineering

application problems have to be authentic

a problem actually associated with the subject, with values which are used in practice or at least theoretically plausible to occur.

The problem should have the potential to serve as a bridge for other problems of similar type

All tools (engineering or mathematical) needed to solve the problem should be available as known knowledge

podríamos decir que la nuestra es una economía basada en las matemáticas, ya que son responsables directas de nada menos que el 10'1% del PIB español

El impacto de las matemáticas en sus economías es cercano al 15% del PIB

las matemáticas están en el núcleo de los programas informáticos que soportan la actual sociedad de la información

aplicación de la matemática abstracta al incremento de la velocidad de escáneres de resonancia magnética

Las matemáticas han producido en los últimos años una revolución silenciosa en todos los sectores productivos, que está moldeando la economía mundial.

The experience of mathematical beauty excites the same parts of the brain as beautiful music, art, or poetry.

physicists can happen across new, powerful mathematical concepts and associations, to which mathematicians can return, to try and justify (or disprove) them.

There’s an intimate connection between empirical science and mathematics

sistema indoarábigo decimal que utilizamos en la actualidad.

al-Juarizmi

Brahmagupta, matemático y astrónomo hindú del siglo VII que fue el verdadero artífice de este sistema.

Esta no fue la única aportación de al-Juarizmi, quien en su Libro de cálculo de restauración [al-jabr] y oposición estableció las bases de la matemática árabe y se convirtió en el fundador del álgebra (palabra que proviene justamente del término al-jabr

Philosophers cannot agree on whether mathematical objects exist or are pure fictions

mathematicians agree to a remarkable degree on whether a statement is true or false, but they cannot agree on what exactly the statement is about.

Was mathematics discovered by humans, or did we invent it?

Before proving a new theorem, therefore, a mathematician needs to watch the play unfold.

This gives the process of doing mathematics three stages: invention, discovery and proof.

a group of mathematicians and philosophers began to wonder what holds up this heavy pyramid of mathematics.

Some mathematicians hoped to solve the foundational crisis by producing a relatively simple collection of axioms from which all mathematical truths can be derived.

Gödel showed that any reasonable candidate system of axioms will be incomplete: mathematical statements exist that the system can neither prove nor disprove.

discovery of a system of basic axioms, known as Zermelo-Fraenkel set theory, from which one can derive most of the interesting and relevant mathematics.

mathematical knowledge is cumulative. Old theories can be neglected, but they are rarely invalidated

Christian Goldbach hypothesized that every even number greater than 2 is the sum of two primes.

this evidence is not enough for mathematicians to declare Goldbach's conjecture correct.

The Goldbach conjecture illustrates a crucial distinction between the discovery stage of mathematics and the proof stage.

math feels both invented and discovered.

The process of mathematics therefore seems to require that mathematical objects be simultaneously viewed as real and invented

Mathematical realism is the philosophical position that seems to hold during the discovery stage: the objects of mathematical study—from circles and prime numbers to matrices and manifolds—are real and exist independently of human minds.

This explains why people across temporal, geographical and cultural differences generally agree about mathematical facts—they are all referencing the same fixed objects.

That is the difficulty with realism—it fails to explain how we know facts about abstract mathematical objects.

If math is simply made up, how can it be such a necessary part of science?

the burden of scientific description is placed exclusively on mathematics, which distinguishes it from other games or fictions.

mathematicians are incredibly effective at producing disciplinary consensus.

math is for the most part an internal process, hidden from view

He wanted to further interrogate — and help other people understand — how mathematicians think about and practice their craft

you’re constantly doing math — and that you’re capable of expanding your mathematical abilities far beyond what you think possible

they became such powerful mathematicians because they were willing to constantly question and refine their intuitions

dialogue between the two: between reason and instinct, between language and abstraction.

something that can be improved through training

everyone can, and should, try to improve their mathematical thinking — not necessarily to solve math problems, but as a general self-help technique

Genius is not an essence. It’s a state. It’s a state that you build by doing a certain job.

Math is a journey. It’s about plasticity.

high school students are often unhappy with math, because they think it requires some innate things that they don’t have. But that’s not true; really it relies on the same type of intuition we use every day.

Whenever you spot a disconnect between what your gut is telling you and what is supposed to be rational, it’s an important opportunity to understand something new

Joy, clarity and self-confidence

Look at what you can do if you don’t give up on your intuition

When you do math, you’re exposed to the human thought process in a way that is really pure

It’s very good training for creativity

los conceptos matemáticos tienen aplicabilidad más allá del contexto en que son originalmente desarrollados.

El milagro de la adecuación del lenguaje matemático para la formulación de las leyes de la física es un regalo maravilloso que no entendemos ni tampoco merecemos.

la estructura matemática de una teoría física a menudo señala el camino para futuros avances en aquella teoría o incluso en predicciones empíricas.