Matemáticas y estadística

Matemáticas y estadística

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Stiffness of Ordinary Differential Equations
Stiffness of Ordinary Differential Equations
A problem is stiff if the solution being sought varies slowly, but there are nearby solutions that vary rapidly, so the numerical method must take small steps to obtain satisfactory results.
Nonstiff methods can solve stiff problems; they just take a long time to do it.
For truly stiff problems, a stiff solver can be orders of magnitude more efficient, while still achieving a given accuracy
You don't want to change the differential equation or the initial conditions, so you have to change the numerical method.
·blogs.mathworks.com·
Stiffness of Ordinary Differential Equations
The acquisition of problem solving competence
The acquisition of problem solving competence
Este artículo muestra la importancia de integrar y utilizar la matemática en otros cursos de ciencias con una orientación hacia la solución de problemas. Una deficiencia en matemáticas puede ser una consecuencia de esa mala integración.
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
·link.springer.com·
The acquisition of problem solving competence
Crisis de la educación matemática (video)
Crisis de la educación matemática (video)
El arrastre de deficiencias: ¿cómo seguir uno en algo si no se entiende nada anterior? Superando las deficiencias se tendrá más confianza en los temas futuros.
·youtube.com·
Crisis de la educación matemática (video)
Cursos de matemáticas diseñados junto con disciplinas STEM
Cursos de matemáticas diseñados junto con disciplinas STEM
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.
·scientificamerican.com·
Cursos de matemáticas diseñados junto con disciplinas STEM
Ley de Benford
Ley de Benford
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
d
La ley se aplica a conjuntos de datos que abarcan varios órdenes de magnitud que evolucionan a partir de ciertos tipos de procesos aleatorios.
Benford's law is the only leading digit distribution that is immune to such unit changes.
·scientificamerican.com·
Ley de Benford
Para mucha gente las matemáticas son una especie de amor imposible, porque no pueden vivir sin ellas pero cuando lo intentan no funciona
Para mucha gente las matemáticas son una especie de amor imposible, porque no pueden vivir sin ellas pero cuando lo intentan no funciona
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.
·bbc.com·
Para mucha gente las matemáticas son una especie de amor imposible, porque no pueden vivir sin ellas pero cuando lo intentan no funciona
t-Student distribution history
t-Student distribution history
The most common test of statistical significance originated from the Guinness brewery
the t-test, one of the most important statistical techniques in all of science
William Sealy Gosset, head experimental brewer at Guinness in the early 20th century, invented the t-test
William Sealy Gosset, el principal cervecero experimental de Guinness a principios del siglo XX, inventó la prueba t.
these plots re­­sem­ble the normal distribution in that they’re bell-shaped, but the curves of the bell don’t drop off as sharply
The self-taught statistician published his t-test under the pseudonym “Student” because Guinness didn’t want to tip off competitors to its research
El estadístico publicó su prueba t bajo el seudónimo de “Estudiante” porque Guinness no quería avisar a sus competidores sobre su investigación.
·scientificamerican.com·
t-Student distribution history
Why Probability Probably Doesn’t Exist (But It's Useful to Act Like It Does)
Why Probability Probably Doesn’t Exist (But It's Useful to Act Like It Does)
Uncertainty has been called the ‘conscious awareness of ignorance’
is not an objective property of the world, but a construction based on personal or collective judgements and (often doubtful) assumptions
it was not until the French mathematicians Blaise Pascal and Pierre de Fermat started corresponding in the 1650s that any rigorous analysis was made of ‘chance’ events
The situation has flipped from ‘aleatory’ uncertainty, about the future we cannot know, to ‘epistemic’ uncertainty, about what we currently do not know
any practical use of probability involves subjective judgements
It also depends on all of the assumptions in the statistical model
it would be appropriate to do extensive analysis of the model’s sensitivity to alternative assumptions.
“all models are wrong, but some are useful”
we would still need to define what an objective probability actually is
Attempts to define probability are often rather ambiguous
the probability of an event on certain evidence is the proportion of cases in which that event may be expected to happen given that evidence
In this view, if the probability of rain is judged to be 70%, this places it in the set of occasions in which the forecaster assigns a 70% probability. The event itself is expected to occur in 70% of such occasions
In our everyday world, probability probably does not exist — but it is often useful to act as if it does
starting from a specific, but purely subjective, expression of convictions, we should act as if events were driven by objective chances.
·scientificamerican.com·
Why Probability Probably Doesn’t Exist (But It's Useful to Act Like It Does)
Brutal math test stumps AI not human experts
Brutal math test stumps AI not human experts
yet AI has hardly touched frontier research in math, an indication that its test-taking prowess does not reflect real mathematical skill.
Leading models correctly answered fewer than 2% of the questions,
AI models will catch up to the new benchmark sooner or later
current math benchmarks are mostly pitched to high school or undergraduate-level math—a far cry from research-level math,
they often get to peek at solutions to similar questions, a problem known as data contamination
Despite the exhortations, no model scored above 2% on the test
the models often provided wrong answers, reflecting their usual misguided confidence.
we have some objective test to [gauge] predictions regarding mathematicians becoming obsolete
Some are optimistic that AI will be more of a companion than a competitor.
·science.org·
Brutal math test stumps AI not human experts
Research in University Mathematics Education
Research in University Mathematics Education
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)
·worldscientific.com·
Research in University Mathematics Education
Conceptualisation of the Role of Competencies, Knowing and Knowledge in Mathematics Education Research
Conceptualisation of the Role of Competencies, Knowing and Knowledge in Mathematics Education Research
What does it mean to master mathematics?
What does it mean to possess knowledge of mathematics?
concepts, definitions, rules, theorems, formulae, methods, and historical facts
challenges the curiosity of his students by setting them problems proportionate to their knowledge and helps them to solve their problems with stimulating questions
definitions, concepts, theorems, and theoretical structures
problem solving
mathematical thinking
they learn to value mathematics
mastering mathematics goes beyond possessing mathematical content knowledge and procedural skills
role of attitudinal, dispositional and volitional aspects
Mathematical literacy is an individual’s capacity to formulate, employ, and interpret mathematics in a variety of contexts.
teachers are not always provided with the professional competencies and didactic-pedagogical resources needed to create classroom cultures in which systematic work to develop students’ mathematical competencies is the norm
The lack of a unified conceptual and theoretical framework for competencies, proficiency, processes, practices etc. tends to impede the possibilities of overcoming the challenges identified
·link.springer.com·
Conceptualisation of the Role of Competencies, Knowing and Knowledge in Mathematics Education Research
Lessons and Future Directions for Improving Students’ Learning
Lessons and Future Directions for Improving Students’ Learning
we discuss four of the many lessons we can learn from international comparative studies for improving students’ learning
Lesson 1: Promoting Students’ Mathematical Literacy
ability to use mathematical knowledge in situations that are likely to arise in the lives and work of citizens in the modern world.
Formulating situations mathematically
Employing mathematical concepts, facts, procedures, and reasoning
Interpreting, applying, and evaluating mathematical outcomes
Lesson 2: Understanding Students’ Thinking
44% of the Chinese students and 1% of the U.S. students used abstract strategies
different students can use different strategies to obtain the same score
Lesson 3: Changing Classroom Instruction
Shanghai lessons: correcting errors, encouraging students to think further
German lessons: questioning to stimulate student mathematical thought
Japanese lessons: eliciting students’ mistakes, their puzzlement, and their opposing solutions; pointing out different solutions or difficulties and giving explanations
all of the countries except Japan used more “using procedure” problems than “making connections” problems.
·link.springer.com·
Lessons and Future Directions for Improving Students’ Learning
What is Mathematics
What is Mathematics
there is no agreement about the definition of mathematics
unnatural separation into the classical, pure mathematics, and the useful, applied mathematics
Some students might be motivated to learn mathematics because it is beautiful, because it is so logical, because it is sometimes surprising
If the typical mathematician is viewed as an “old, white, male, middle-class nerd,” then why should a gifted 16-year old girl come to think “that’s what I want to be when I grow up”
Showing applications of mathematics is a good way
transforming mathematicians into humans can make science more tangible
stories can make mathematics more sticky
By stories, we do not only mean something like biographies, but also the way of how mathematics is created or discovered
Telling how research is done opens another issue.
Mathematics I: A collection of basic tools
Mathematics II: A field of knowledge with a long history, which is a part of our culture and an art
Mathematics III: An introduction to mathematics as a science
·link.springer.com·
What is Mathematics
Making Sense of Mathematics Achievement in East Asia: Does Culture Really Matter?
Making Sense of Mathematics Achievement in East Asia: Does Culture Really Matter?
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!
·link.springer.com·
Making Sense of Mathematics Achievement in East Asia: Does Culture Really Matter?
Three Mathematical Cultures
Three Mathematical Cultures
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.
·mathvoices.ams.org·
Three Mathematical Cultures
A Mathematician Explains Why Driving at 140km/h on a Highway Makes No Sense
A Mathematician Explains Why Driving at 140km/h on a Highway Makes No Sense
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
·todoalicante.es·
A Mathematician Explains Why Driving at 140km/h on a Highway Makes No Sense
Why Physics Is Unreasonably Good at Creating New Math
Why Physics Is Unreasonably Good at Creating New Math
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
·nautil.us·
Why Physics Is Unreasonably Good at Creating New Math
Myths of Mathematics Teaching
Myths of Mathematics Teaching
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
·ams.org·
Myths of Mathematics Teaching
Integration of Engineering Application Examples in Mathematics Courses
Integration of Engineering Application Examples in Mathematics Courses
https://ieeexplore.ieee.org/document/10365385
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
·ieeexplore-ieee-org.ezproxy.eafit.edu.co·
Integration of Engineering Application Examples in Mathematics Courses
Design Principles for Using Rubrics in Engineering Mathematics
Design Principles for Using Rubrics in Engineering Mathematics
Link: https://ieeexplore.ieee.org/document/10339269
A general rubric is presented, aligned to six mathematical competencies
design of rubrics to assess engineering mathematics tasks
Well-designed assessment should provide students with opportunities for self-reflection and metacognition
Rubrics offer a structured, concise, fair and transparent [9], [10] way to guide and evaluate student work that can reduce student queries about grades
constructing valid arguments,” criteria
The six competencies of the MCRF are: [4, p. 111-112] problem solving competency; reasoning competency; procedural competency; representation competency; connection competency; communication competency.
Provides a clear and concise plan.
Demonstrates logical and systematic thinking.
Identifies all relevant information.
Shows correct application of procedures or algorithms.
Uses appropriate maths notation
Communicates concepts clearly and effectively using written or spoken words/pictures/tables.
Makes clear and accurate connections between maths concepts and real world
Information is organized logically
Ideas are clearly communicated
·ieeexplore-ieee-org.ezproxy.eafit.edu.co·
Design Principles for Using Rubrics in Engineering Mathematics
Assessing mathematical competencies: an analysis of Swedish national mathematics tests
Assessing mathematical competencies: an analysis of Swedish national mathematics tests
Link: https://www.tandfonline.com/doi/full/10.1080/00313831.2016.1212256#d1e238
Mathematical competence then means the ability to understand, judge, do, and use mathematics in a variety of intra- and extra-mathematical contexts and situations in which mathematics plays or could play a role
Problem Solving Competency
Reasoning Competency
Procedural Competency
Representation Competency
Connection Competency
Communication Competency
a mathematical procedure is defined as a sequence of mathematical actions that is an accepted way of solving a common task type (e.g., to solve a linear equation)
define arguments to be mathematically founded if, in the terms used by Lithner ( Citation 2008), they justify why the conclusions are true or plausible and are anchored in intrinsic properties of the mathematical components
A representation is therefore defined to be the concrete mental or real replacement (substitute) of an abstract mathematical entity
to use something that makes a link between two things, for example, a relationship in fact or a causal or logical relation or sequence
information is exchanged between individuals through a common system of symbols, signs, or behaviour
·www-tandfonline-com.ezproxy.eafit.edu.co·
Assessing mathematical competencies: an analysis of Swedish national mathematics tests
Pensamiento y procesos matemáticos
Pensamiento y procesos matemáticos
Tener un buen desempeño en matemáticas significa resolver problemas, formular nuevas preguntas y plantear nuevos problemas en diferentes contextos (internos y externos a la matemática)​, considerando que la matemática es útil y valiosa, aplicando (i) notación simbólica y conceptos matemáticos pertinentes, (ii) diferentes tipos de pensamiento matemático y lógico, y (iii) procesos adecuados (solución de problemas, procedimientos y algoritmos, modelación, razonamiento y argumentación, comunicación).
Pensamiento numérico y sistemas numéricos
Pensamiento espacial y sistemas geométricos
Pensamiento métrico y los sistemas métricos o de medidas
El pensamiento aleatorio y los sistemas de datos
El pensamiento variacional y los sistemas algebraicos y analíticos
solución de problemas
procedimientos
Modelación
Razonamiento,
Comunicación
·siscontexto.blogspot.com·
Pensamiento y procesos matemáticos
Is there a switch for “making sense” ? |
Is there a switch for “making sense” ? |
While they did not suddenly became great at math, their mental activity and learning efforts are much more productive, since they are consciously directed towards comprehension and expressing their ideas verbally with a degree of precision.
students who emphatically told me how this course entirely changed the way they view and approach math
math is communicated through meaningful statements
Both exclusively formal processing of math tasks and making sense of math tasks are learned, eventually habitual, behaviors.
Effective learning of mathematics does not happen until mathematical communication is perceived as meaningful statements.
A dedicated computation-free and writing-intensive class which stays away from problems that may suggest formal manipulation can turn on the “making sense” switch.
The class should be writing intensive
The course should be light on content and big on thought, allowing sufficient time to think and write about problems.
if they could not solve the problem, they should write down their attempts, for example, how they used  problem-solving strategies discussed in class
·blogs.ams.org·
Is there a switch for “making sense” ? |
Teorema del límite central (estadística)
Teorema del límite central (estadística)
El teorema del límite central establece que, en general y dado un conjunto de muestras aleatorias independientes e idénticamente distribuidas con una función de distribución arbitraria, si se calcula la media a partir de un subconjunto de muestras, entonces la función de distribución de dichas medias tiende a una función de distribución gaussiana
·siscontexto.blogspot.com·
Teorema del límite central (estadística)
Método de bootstrapping
Método de bootstrapping
El bootstrapping (con difícil traducción al español) es un método de remuestreo para el cálculo simple del intervalo de confianza y otras estadísticas de los parámetros de un modelo a partir de un solo conjunto de datos experimentales y sin necesidad de fórmulas matemáticas.
en el bootstrapping residual se asume que el modelo es correcto y los residuos son variables normales idéntica e independientemente distribuidas con media cero
·siscontexto.blogspot.com·
Método de bootstrapping
Robust misinterpretation of confidence intervals (CI)
Robust misinterpretation of confidence intervals (CI)
  • Our findings suggest that many researchers do not know the correct interpretation of a CI.
  • The misunderstandings surrounding pvalues and CIs are particularly unfortunate.
  • Null hypothesis significance testing (NHST) has been criticized for many reasons, including its inability to provide the answers that researchers are interested in.
  • Within the frequentist framework, a popular alternative to NHST is inference by CIs. CIs are often claimed to be a better and more useful alternative to NHST.
  • CIs provide information on any hypothesis, whereas NHST is informative only about the null.
  • CIs give direct insight into the precision of the procedure and can therefore be used as an alternative to power calculations.
  • The findings above show that people interpret data differently depending on whether these data are presented through NHST or CIs.
  • A CI is a numerical interval constructed around the estimate of a parameter. Such an interval does not, however, directly indicate a property of the parameter; instead, it indicates a property of the procedure, as is typical for a frequentist technique.
  • It is incorrect to interpret a CI as the probability that the true value is within the interval
  • Correct interpretation: If we were to repeat the experiment over and over, then 95 % of the time the confidence intervals contain the true mean.
·ejwagenmakers.com·
Robust misinterpretation of confidence intervals (CI)