Matemáticas y estadística

Pitágoras se dejó morir de hambre hacia el año 495 a.C.

todo lo relacionado con él está envuelto en un halo de misterio y leyenda

Semicírculo de Pitágoras»

Samos

muy dotado para la amistad

gran capacidad de persuasión

Estos discípulos recibían el nombre de esotéricos, es decir, «los de dentro [del velo]» o matemáticos

el orden del cosmos descansa en los números

descubrió las proporciones numéricas que rigen las escalas musicales.

Creó melodías que apaciguaban los padecimientos del alma

Fueron muchas las señales de la divinidad de Pitágoras

filósofo». Él inventó esa palabra

Los adeptos se reconocían por medio de signos secretos. Conocemos un signo llamado pentalfa,

Pitágoras ayunó durante 40 días en el templo de las Musas de Metaponto hasta morir de hambre.

Como el maestro no había dejado nada escrito y había impuesto el secreto más absoluto sobre sus enseñanzas, fue inevitable la pérdida de valiosos conocimientos y la confusión general sobre sus doctrinas.

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

preliminary reports on flipped classrooms are troubling. Students perceive a disconnect between out-of class components and in-class components

-class activities may fail to address student misconceptions

there is a need to create a theoretical understructure to support the implementation of the model.

We propose two types of coherence that impact flipped classroom models: instructional coherence (cohesion and coordination among instructional materials) and curricular coherence (the extent to which mathematics content is logically, cognitively, and epistemologically sequenced).

synthesis of theories of mathematical thinking that allowed us to articulate a particular flipped classroom model.

The main contributions are a theoretically-based model for designing flipped classroom instruction and extension of conceptual analysis techniques (Thompson 2002, 2008) to the differential equations domain and to blended learning environments

The flipped classroom students performed better on the three exams during the semester, but similarly to the traditional classroom students on the final exam.

it allows for more individualized instruction and for students to set their own educational pace

the systematic investigation of the impact of flipped classrooms on student learning is still necessary and important.

students tend to react well to the flipped classroom in terms of enthusiasm and likability

the flipped classroom seems to be a source of incoherence in students’ understanding of content.

coherent curriculum as one that is “marked by effective, logical progressions from earlier, less sophisticated topics into later, more sophisticated ones

define curricular coherence as the logical sequencing of mathematical content.

instructional coherence as alignment among in-class materials, out-of-class materials, and target content.

is perhaps more productive to think about a derivative as a limit of difference quotients

Other difficulties in implementing a flipped classroom model have been reported, such as: (1) failure to address student misconceptions, (2) overuse of low cognitive-level activities that required only recall of facts, and (3) an emerging disconnect between lecture materials and active-learning in-class components

explicitly focused on exposing and addressing cognitive obstacles in order to surmount the difficulties in connecting in-class and out-of-class learning component

mathematics teaching focuses too much on moving students toward “translucent symbolism” (p. 46) and away from the core ideas of mathematics concepts.

Mathematics-in-use was developed to explore from an epistemological and cognitive standpoint, and with the support of extant mathematics education literature, how mathematical concepts and procedures might come together to address a mathematical problem.

we offer an example of how the in-class and out-of-class instructional materials were designed to work together to promote curricular and instructional coherence

in most cases, it is not appropriate to examine the rate of change according to one variable

without an understanding of covariation, individuals could not develop the ability to reason meaningfully about rate of change.

The course used a mathematical modeling approach to differential equations

solving canonical engineering problems

complexity of the engineering situations

The students enrolled were in general highly motivated

flipped classroom as a source of incoherence

ts of a math emporium (ME), a model for offering introductory level college mathematics courses through the use of software and computer laborato

gate 1) whether the emporium serves certain popu- lations differently than ot

rs, 2) the nature of mathematical learning that occurs in the ME,

found that the emporium ap- peared to best serve students with higher math achievement, who enjoyed mathematics, and who spent more time taking their

but had limited impact in terms of developing meaning for symbols or flexibility in solving unfamiliar

nt cost. The ME gains efficiency by outsourcing specific tasks to technological and human resources other t

ver, we believe that the research base on mathematics emporia leaves several significant gaps that need to be ad

on the final, the emporium best served students with higher math achievement, who enjoyed math- ematics, and who believed less strongly that mathematics is about m

er. In terms of mathematical learning, the emporium appeared to improve students ’ ability to recall and use formulas for familiar problem types, but had limited impact in terms of developing meaning for symbols or flexibility in solving B transfer task

ed that some were able to exploit these features to obtain higher scores on quizzes by memorizing specific problem types covered on the quizze

at students were generally not thinking of th

that the ME did not improve dispo- sitions towards mathematics on a significa

thematics. Future research on MEs should take into account not only passing rates, exam scores, and cost savings, and but also should explore the nature of the mathematical reasoning it promotes and how students interact with and perceive their expe- riences in the M

there might be a link between “planes-coordination” and “long-term-prediction” difficulties.

understanding differential equations is usually conducted with the help of visualization

solution curves, stable/unstable solutions, and direction-field

many students had difficulties about prediction of long-term behavior

The Standard Method

two planes; [y, f(y)] and [t, y(t)]

three planes [y, f(y)], [t, y(t)] and phase-line

Dynamic Method

five planes [y, f(y)], sign-chart, [t, y(t)], phase-line, and long-term-prediction

students made progress in the Dynamic class

evidence for the Dynamic method as another way of looking at teaching differential equations

By using the Dynamic method, students have interpreted the three coordination items visually, and this is an important observation about the implementation of the Dynamic method in ODE classes

“understandings of the entire space of solutions”

“the Dynamic method makes the geometrical approach meaningful”

using the Dynamic method is effective to overcome the long-term prediction difficulties

in the Dynamic method, planes are coordinated by the horizontal dotted lines that makes it easy to move flexibly between them

visualization strategies that were introduced in the Dynamic method have the potential to develop “visual proving” in the class

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.

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

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.

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

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.

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

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

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.

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.

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)

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

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.

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

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!

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.

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 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

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

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

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

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

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

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