Ok, you want a way to reach and teach difficult kids, which is what I was trying to do in 2001/2 and 2003 when I wrote up a proposal for theater based lessons, and discovered that the Coalition for Essential Schools (Ted/Theodore Sizer I believe: the CES) had pretty much already done that, but only in a few expnsive private schools where the teachers were on a first name basis with the 10 students per class. (In DC in 2012 I was told that that idea is now being put into practice in more places...)
So, my 4 ideas that may help you:
1. Elmer Fudd + Theater Productions (done by the students: I'll find the doc and translate, as this backup copy from Puerto Morelos, MX, is all I can find at the moment, but I think the better copy is up on my Academia...) -> Constructivist/movement+art-based learning
2. Socratic Method based tutoring, which you've already seen: again, using "eliciting" instead of force-feeding the information to the kids, by asking questions and stimulating their curiosity. (I had an ADHD kid brilliant but never sat still, so I brought in a map of Aztec Empire and shot questions at him to go find places on the map during class, and throwing equations at him to solve at the same time: worked pretty well!)
3. Walking Tours of the local area with Freedom/Civil Rights songs: this is what I did w/adults in DC after I got turned down for funding to give classes outdoors, but the idea does work with willing students: start the lesson indoors in teh classroom, explain that we are going to find examples of what we are studying around us, and go outiside for a walk, talking about the lesson while we walk. Yes, harder to control a large class, but doable at least in theory, and certainly with a small class or one student.
4. Community Sing Alongs: again, adults came to sing Freedom and ethnic songs, but this can be worked into a lesson plan, if the teacher has the time and support of superiors. (yes, ok, a very big iff...)
Hope this helps, ask any and all questions, please,
1: Theater based teaching lesson plans from Puerto Morelos, Mexico: ( up on http://bath.academia.edu/
Teatral de Arachne
- © Siir (Xiir) Destinie Jones, (much removed due to formatting problems, but full doc is online...)
- Planes para profesores de matemáticas
- Día 1 y día 2: Coordinados Cartesianos
- Objetivo: El alumno aprenderá usar el sistema de coordinados rectangulares a través de ejemplo de tejer.
- Tiempo: 45 minutos, con actividad opcional para segundo día
- Resumen de actividad: 1. dibujar una línea recta horizontal, y una recta vertical, explicar como ejes X y Y. 2. encontrar el origen, varias puntos X y Y separados, y varias puntos (X,Y) como coordinados.
- 3. opcional: Construir un tejido con los hilos “warp” como el eje Y y los hilos “weft” como el eje X.
- Actividad1. Muestra una línea horizontal en el pisaron. Pide a un alumno que se acerca al pisaron para dibujar la línea recta. ---4--3--2---1--0--1---2---3---4-------
- La línea recta de dibujamos nos ayuda contar unidades desde 0 al lado positivo y al negativo. Estas unidades podrán extender hasta infinidad, así:
- | | | | | | |
- | | | | | | |
- -3 –2 –1 0 1 2 3
- | | | | | | |
- | | | | | | |
- por eso, esta recta es puede llamar el eje horizontal, o eje X.
- así podemos contar cuantas unidades al derecho o al izquierdo usando el eje X. Cuando hacemos otra recta vertical por 0, tendremos los ejes X y Y. Entonces podemos contar por derecho o izquierdo, y por arriba o abajo, verticalmente, por el eje Y.
- Actividad 2. ¿Dónde aparece el cero? Como cero está en inicios de todos lados, lo llamamos el punto en el centro “el origen” y empezamos a contar desde allá.
- Siempre ubicamos un punto así, con (X,Y) contando x unidades en el eje X, y y unidades en el eje Y. ¿Qué puntos ya tenemos dibujado? Por otros ejemplos, donde estará el punto (1,1)? (1,3)? (-2, 1)?
- Opcional: Actividad 3. Si sobra tiempo, pueden tejer algo y ubicar puntos allá…
Hale: (3,4) (-1,1) (0,3) (-5,1) (-2,-2), el origen, y (1,-4)…
Arachne matemáticas Planes para día 3: Distancia con Coordinados
Objetivo: El alumno empezará medir distancias en una línea con coordinados.
Tiempo: 40 minutos
Actividad: ¡Ubícate! Un juego para practicar nuestros coordinados.
Imaginase, alumno, que eres una araña en el centro (el “origen”) de tu telaraña. Caminas 4 pasos al derecho, y paras. ¿En cual punto estás ubicado? R.: (0,4)
Ahora, el alumno a tu lado seguirá como araña, caminando. Halle el punto en donde se para, y sigue el alumno vecino, etc.
Cada alumno tendrá que tener por lo menos tres turnos en el juego.
Tarea: Consigue un cuaderno o libreto para usar como bitácora de matemáticas. Por tarea de esta noche, escribe en tu bitácora de matemáticas como crees que podrías adivinar la distancia entre el origen y el punto (3,5). ¿Será la distancias diferente, desde el origen y (-3,-5)? ¿Por que?
día 9: repaso de la matemáticas en la obra “Arachne”
Objetivo: Alumnos dirán cuenta que ya saben ubicarse, medir distancias, e incluso dibujar un triangulo sin medida, todo por la obra “Arachne”.
Tiempo: 40 minutos
Actividad: 1. Elicitar de los alumnos sus recuerdos de coordinados, Valor Absoluto, y la Formula de Distancia antes de introducir el juego para mañana, que será dibujar un triangulo solo usando dos círculos.
En el pisaron, un alumno puede dibujar una gran telaraña, y un imagen de que piensa cuando se oye la palabra “distancia”.
¿Cómo se puede conectar los dos imagines?
Que imagen tienen cuando piensan en “navegar”?
A dibujar un barco de vela con triangulo, el la telaraña:
Para ubicar el barco en punto B, debemos saber los coordinados de punto B.
Para saber la distancia entre A y B, usamos, sabiendo los coordinados de punto A, la formula Distancia.
También se puede dibujar un triangulo con puntos A y B de un distancia fija, sin medirla. ¿Cómo? -Ya veremos mañana por la tarea!
Tarea: Busca en la biblioteca y escriben en sus bitácoras –quien era “Euclid”?
Fin: día 10, matemáticas después de la obra de Arachne
Objetivo: Alumnos aprenderán
dibujar un triangulo ABC con base de medida línea AB usando el teorema Primero de Euclid de Elementos de Euclid #1
los conexiones entre Euclid, Pitágoras, y Arachne.
Tiempo: 40 minutos
Actividad 1: 20 minutos
Dibujo barco de vela B, y punto a, con distancia AB (por línea AB).
Usando línea AB como radias, haz circulo A con centro A, y circulo B con centro B, con línea AB por radias común.
Sea la intersección de círculos A y B encima de línea AB el punto C. Ya tenemos triangulo ABC con base AB.
Actividad 2: 15 minutos
¿ De que tamaño creen (los alumnos) que sean línea AC y línea BC? ¿Por qué?
¿Qué tipo de líneas, o que parte del circulo, son línea AC y línea BC?
¿Qué relación tienen líneas AB , AC y BC?
¿Qué tipo de triangulo será ABC, entonces?
Sea CD la línea a AB, desde punto C, a medias de AB (se llama ese línea “el bisector perpendicular de AB”…)
Ahora tenemos dos triángulos rectangulares, y podemos medir la distancia entre punto B, donde está nuestro barco de vela, y cualquier otro punto, gracias al Teorema Pitagórica.
Actividad 3: 5 minutos
Discuten en la clase: ¿Piensen los alumnos que Euclid, Pitágoras y Arachne hubieran sido buen amigos?
Tarea: Escribe como habría podido cambiar su destino, Arachne.
Opcional extra: día de danza por Arachne
Objetivo: Repasar los coordinados, números en la historia, y geografía mientras aprendiendo un baile folklórico Griego
Tiempo: 45 minutos
Sabana con mapa del mediterráneo (con el origen en Atenas) (o del Grecia y Turquía)
Placa o letrero con los números griegos
Placa o letrero con los números Árabes (modernos)
disco de música de Tsamiko
un pañuelo blanco
con la sabana en el suelo y todos descalzos, el líder con el pañuelo blanco empieza en el origen (0,0) que sea Atenas y pasa al norte por Esparta, Macedonia, el mar Iónica, Cecilia, Ciprés, y Estambul e Izmir, Turquía (que fueron Constantinopulous y Esmyrne antes…).
llega un “Turco” quien va a reemplazar la placa de números griegos con la de números modernos
cambian de líder de la danza por el “Turco”
en fin de la danza, decimos gracias a los Turcos por traer los números árabes, que usamos hoy en día
localizamos Esparta, Macedonia, el mar Iónica, Cecilia, Ciprés, y Estambul, y Izmir, Turquía en el mapa, con sus coordinados.
recordamos que el Tsamiko es el baile de independencia de los griegos del emperio (Turco) Otomano.
Tarea: Muestra el Tsamiko a sus familias, los alumnos, con el cuento de cómo los Turcos trajeron los números árabes al Europa.
2: Socratic constructivist based teaching methods (Elmer Fudd):
17 April, 2002 Philosophy of Education The use of Constructivist methods in teaching mathematics “How Would Elmer Fudd Teach Math?” Elmer Fudd is a hunter. Most ADD (Attention Deficit Disorder) and ADHD (Attention Deficit Hyperactivity Disorder) diagnosed children are hunters, as defined by Thom Hartman in his series of books “Hunter in a Farmer’s World.” These children tend to change topics and interests often, moving their focus of attention, sometimes boisterously, from one activity to another rapidly. Standard/traditional teaching techniques, such as lecture, and rote memorization, have been shown not to suit these types of learners well. This paper will argue, on both pragmatic and philosophical grounds, in favor of the combining of constructivist teaching methods with traditional methods. Pragmatically, a person who is able to gain and use knowledge on his own, examining ideas critically and taking initiative, will be a more productive member of society, and more useful, in general. Philosophically, every person has the right and responsibility to take initiative both to care for herself , and also to contribute to the collective thoughts of society. In order to pursue either responsibility or freedom, knowledge of the available options, and how to increase those options, is necessary. John Dewey, in his essay on “The Child and the Curriculum” decried the evils of dumbing down material for all children, leading to dull-brained thinking, and passivity. Both hunters and farmers, to be responsible for their own lives, must be able to take initiative, think critically, and apply newly learned information. Traditional teaching is being shown to fall short with the vast majority of students in this regard, as well. Constructivism, which can be defined as the forming of a mental model in response to being placed in an environment that stimulates active wondering, is a useful alternative to the traditional style of education which also answers both of these objections. Note that the use of constructivist techniques is meant to be in addition to, not instead of the standard teaching methods. One suggestion is to devote two or three days per week to constructivist style teaching, with the remaining days devoted to standard lecture methods. Since all are generally familiar with the traditional style of teaching, usually defined by lectures, recitations, and memorization, little time will be spent on descriptions of that teaching format. This style of teaching will be defined, further, as the dissemination of information in verbal, written, or both formats, without interruptions or intermittent questions, or when all questions are saved for after the instructor has completed with giving out the bulk of information to the class. To summarize, lecture is defined here as the push of information from instructor to learner without substantial breaks during the lecture for questions, exchanges of information, or class participation. If, as social reproduction theorists agree, education is a primary element in perpetuating and creating the type of society we will have in the future, it is incumbent upon us to ensure that all of the talent available in our society is developed to the fullest. Education is the vehicle that will take us there. We are obligated to create a society in which all are truly free to participate, and this is only possible when all members of society are fully trained in critical thinking. Whether we are born with all knowledge, as Socrates believed, or must learn it afresh, questioning and initiative are crucial parts of participation in any free society. John Dewey, in his treatise “Democracy and Education,” pointed out that in order to truly learn something, the learner must absorb an idea, and take ownership of it. These concepts: ownership of an idea, putting information in context, and providing thought-provoking educational experiences, are at the heart of Dewey’s writings, and of the constructivist movement. Only by asking “why, and how, and from where,” can the learner fully internalize a piece of Information. He also felt that learning a particular subject in isolation from its context and the surrounding applications is not a complete way of learning the subject. This is in direct opposition to the traditional method of teaching each course as a subject unto itself. Geometry, as one example, is taught in complete isolation from other courses, and removed from its context. When geometry is taught in conjunction with art, or other applications, student understanding is enhanced. This context is, in fact, one part of how a teacher must, according to Dewey, provide learning experiences that encourage questioning, observation, and wondering, which leads to more thought, surrounding the subject to be learned. So how, then, does a mathematics teacher provide contextual and concrete experiences, when faced with such abstract topics as linear algebra, and matrix equations? How would Elmer Fudd, our hunter par excellence, teach them? Acting is a powerful teaching tool, particularly for learners who learn by moving around and using their bodies. Charlotte Perkins Gilman, in the novel Herland, advocated movement and play as the most effective means of learning. Acting is play at its best, allowing both the actor and the audience to engage an idea actively, both consciously considering the idea, and subconsciously, through the artistic side of the brain, simultaneously. One application of Howard Gardner’s theory of multiple intelligences involves acting out, or becoming an equation. Mr. Fudd would probably use this technique to teach young hunters how to determine the trajectory of a bullet aimed for a rabbit, during rabbit season. Given the equation ‘X + 3 = 5’, two students stand for the variable X, another student for the plus sign, four other students each stand for the numbers one through four, and a student forms the equals sign, standing opposite the plus sign student. Five other students, each standing opposite a ‘number’ student, represent the numbers one through five. The evenly matched pairs of students show that the equation has been correctly solved.<SPAN style="mso-spacerun: yes"> </SPAN>There are many possible variations on this theme, leaving out the plus and equal signs, or the variable, for a more clear solution of the equation, or if fewer students wish to participate. Other uses of acting involve allowing one particularly gifted student to demonstrate a technique or concept, by becoming the concept. For example, an especially rambunctious pupil was having difficulty in one of my high school mathematics classes with the concept of reciprocals. After explaining the idea of inverse fractions several times, I asked him to do a handstand. To the delight of both the demonstrating student and the rest of the class, the concept became much clearer as I pointed to the inverted student, and explained that we were to do the exact same thing with our fraction! There are, of course, down sides to the use of acting as a teaching technique. One rather pointed example is the use of my overly athletic student to illustrate the concept of reciprocals. When I asked him to stand right side up, after completing my explanation of reciprocals, he promptly fell over, landing with a crash on the floor. While Elmer Fudd might have approved, the guidance counselor in the office next to my classroom did not. Although the student was not injured in his fall, the noise certainly did create a distraction, both for my class and for others in nearby rooms. This leads us to another pitfall of acting as a teaching technique. Acting can often be a noisy and fast-paced It is not easy to maintain proper teaching decorum over a classroom full of students, whether children or adults, even, when somewhere else in the classroom, one or more individual students are moving around, making noise, or even standing silently in a distracting pose –on one’s head, for example. There must be a focus on the idea to be learned, in order for the experience of acting to be of educational benefit, and that focus can easily be lost in the hustle and bustle of a group of actors showing off in front of a crowd. An additional concern with acting is that it does require imagination. Not everyone will benefit from acting out or watching the portrayal of a concept, since not everyone learns through movement or body language. Acting may thus be a waste of time for non-kinesthetic based learners. While they may enjoy the show as a form of entertainment, which is arguable valuable for education in itself, these students will miss the point of the actual lesson, unless non-acting based methods are employed, in addition to acting, to illustrate the concept being taught. Elmer Fudd would undoubtedly use acting at least occasionally, as one of the tools in his armory of young hunter training techniques. Beyond being enjoyable for restless young hunters, who are constantly on the lookout for rabbits and ducks to capture, acting as a teaching method can enhance the learning pleasure and effectiveness for young farmers as well. Mr. Fudd would be certain to remind all of the students to “be vewy verwy quiet,” and to be respectful of classmates in the entire building. To ensure that the point of the lesson is addressed in the skit, he would also be likely to give a short synopsis of the concept being illuminated by the skit, either before or after the performance. In addition to illustrating the pure mathematical concept under discussion, a skit can unobtrusively tie in the context, historical, social, or scientific, for which the math was developed. A group of students working on units of measure may take the opportunity of Patriot’s Day to enact a short skit on the Battle of Marathon, “running” the distance in miles, meters, and even cubits. This brings not only context, but passion and creativity into the classroom: two things that Jonathan Mooney and David Cole, co-authors Learning Outside the Lines,” point to as essentials for learning, and for life itself. Acting also provides a perfect methodology for team teaching. Teaming up with one or more teachers to combine several classes for a short time, with a specific purpose defined can work nicely, if planned out well beforehand. As pointed out by Theodore Sizer in the first book of his “Horace Trilogy,” The Dilemma of The American High School , team teaching can cause confusion and even be counterproductive, if a central focus and teacher coordination are not maintained. As an example, several students for a class that is studying arachnids in science, and cartesian coordinates in math, can act out the myth of Arachne’s contest with Athena. A history or social studies class could even join in, if enough room is available. Each student can take turns at the loom, and keep samples of the weaving. The geography, language, attitudes, and clothing of ancient Greece can be taught through this skit, as well as the grid coordinate system, of course, using a real cloth example. Latitude and longitude lines can be compared to the X and Y axis, referring to the warp and weft that the students created with their own hands. Not to mention the unfortunate Arachnid. Another well-respected constructivist technique that Elmer Fudd would likely have occasion to use is that of building things. It is generally acknowledged that if one is able to build a working item, of almost any kind, then that individual has mastered the principles involved in its making. While this may sometimes be up for debate, it is undeniable that to build a thing is to involve some practical application of at least a few concepts. Practical application is often the best way to understand a concept, and also gives the satisfaction of having produced a tangible object when completed. Vocational schools are often popular for this very reason –they allow students the opportunity to see results built by their own hands very soon. The shorter time frame between learning concepts and putting those concepts to use can be a great help and motivator for a young person (or an adult) who is apt to ask “why are we learning this?” Theodore Sizer, in his chapter on agreement (between teacher and those taught) in Horace’s Compromise: The Dilemma of The American High School, argues that sometimes letting students discuss what interest them, and then pointing out the curricular application in that topic, can be more effective than doggedly sticking to the prepared lecture. If that happens to be building an electronic circuit, as it was in my Algebra1A class, one day, then building a hands-on model for display can be more instructive than any textbook work, or lecture. As it happened, on this particular day, we were actually reviewing graphs and charts. A student interrupted my lecture to comment about his heartbeat, so I took the opportunity to return to the topic of the day by explaining how to graph a heartbeat in terms of beats per minute. I then asked the class to draw a series of graphs, from flatliners to 70 beats per minute. The gregarious student, stymied that I had redirected his comment, began to talk about his electronics project with several of his classmates. I used this conversation as an opportunity to review the solution of single step equations, using Ohm’s Law as a starting point. At least for that particular student, this proved to be more interesting, and he came back after class for several days in a row to work out the equations needed to determine how to build his circuit. Rousseau and Elmer Fudd would very likely agree on one thing –Emile, like Mr. Fudd’s students, will learn best by doing, and experimenting, and building. From tree-stands to bows, arrows and quivers, and maybe even muskets and balls, young hunters under either of these two hands-on teachers would learn by doing and building. This project used laptops and GPS (Global Positioning System) units. Students discussed and were taught the general elements of cartography, then provided equipment and one adult guide for each group of students, and encouraged to discover for themselves the challenges of mapping out an area. This is a wonderful idea, but how many school districts will realistically be able to implement such a project, given the expense of a laptop, GPS unit, and even a simple topographical map? Any one of these items may be beyond the reach of a school district, particularly in an inner city struggling for basic funding of any kind. Even in cases where money is not an issue, many schools face the problem of limited space. At least one Middle-High school in New Hampshire uses trailers for temporary classroom space, and even shares space with a neighboring school. Given constrains like these, it may be difficult to find the room needed to spread out enough to build individual projects, store them, or even manage to transport them through the halls, crowded as they generally are. While it is important to cover all of the required material, it is equally, if not more important to help learners absorb what is being covered. Rousseau would have argued that less is better, and that anything covered must always be done through building. As with his example of Robinson Crusoe, whatever is taught must be taught through experience and practical experimentation. Elmer Fudd might have to remind Emile, though, that the consequences of firing a musket improperly could be rather permanent, and so, learning to read is a necessity in order to avoid fatal experimentation. Thus, not everything is best taught by hands-on methods. Reading the directions can be both more efficient, and even life-saving. Keeping that in mind, Elmer Fudd would have to balance the impatience of young hunters against the cautiousness of young farmers. Mr. Fudd would also remember to balance the need to inspire passion in both groups against the need to cover all of the requisite mathematics to be able to count the number of days from duck season to rabbit season. Most days, Mr. Fudd would likely cover the standard math, using lecture format. He could then periodically remind his students that once they have learned enough of the required math, they would be able to more effectively go on their planned hunting expeditions. In the meantime, as an optional homework project, individual students could be allowed to research and build model rabbits or ducks to show off to their classmates, and explain the various uses rabbit and duck parts could be put to after their expeditions. Mr. Fudd would always make sure to point out the various mathematical topics and principles that were used in the creation of these models, and tie them into the ongoing classwork. That would give the students a context into which to put both the previous, current, and upcoming classwork and homework. He would also allow the students to help planning the expeditions, which would keep all of the students engaged in and looking forward to both the upcoming trips, and the ongoing classwork which is in preparation for those trips. That way a smaller number of projects could be stretched across more lecture format classes, while holding the attention of the young hunters in the classroom. Manipulatives of any kind will certainly cost more money than simply drawing on the board would cost. Then there is the additional custodial cost of cleaning up after the class that used manipulatives, quite often. Does Not Show Graphics... If we know either C or D, we can find the other. This is an extra credit assignment for which you should collaborate with your classmates. You may put on a skit during class next Friday, that will serve as a review of circumference, area of a circle, and the meaning of Pi. You must decide who will be actors, who will build the set, what sort of scenery needs to be drawn, and what music to use. All of this must relate to and help explain the uses of Pi, area, and circumference as you would use them in your own lives. Have fun, and Good Hunting! Constructivist teaching methods strive to supplement lecture methods by filling in the gaps that lecture leaves open, such as body-kinesthetic and interpersonal learning. Constructivist techniques also emphasize critical thinking and learning how to find and interpret information based on a broad range of connections. Matthew Miltich, in his recent article for the NEA Higher Education Journal entitled “All the Fish in the River: An Essay on Assessment,” likens ideas and knowledge to fish to be caught. He defines the educator’s job as that of helping the learners to learn how to catch those fish for themselves. As asserted by Theodore Sizer in his section on teachers in Horace’s Compromise: The Dilemma of The American High School, one needs a broad base of knowledge both to teach and to learn effectively. The fish require a broad net. As our society becomes more completely industrialized, and moves into the post-modern information age, a larger and larger percentage of our population will have to be well educated to provide a workforce that will allow our businesses to continue to function. Even from this strictly Machiavellian point of view, we can no longer allow the large numbers of our learners to slip through the educational cracks. It costs too much to import trained workers. That requires us to adopt new techniques in educating our learners to the minimum level necessary (which continues to rise, as the technological complexity and business requirements rise) to contribute to the workplace. From the more idealistic standpoint, ours is a democracy, and to be a full participant in a democratic society, one must be able to analyze and debate the issues, which requires training in critical thinking and analysis.Also required to function in a democracy, is the ability to draw connections between even pieces of information that may seem only remotely related to one another.As Jack Dewey points out in Burned Out: A Teacher Speaks Out, both learners and teachers must be exposed to a wide variety of topics within a subject. Good critical thinkers must also be able to draw upon and make for themselves the connections between traditionally separate concepts, much in the same way as connections must be inferred between such traditionally separate subjects as mathematics and history and science. The connections are there, but are made unapparent by the strict division of subjects in modern schools. While Jack Dewey may or may not be correct in arguing that cross-disciplinary certifications is the answer to the connections problem, there are certainly connections between each of the various subjects that are taught in schools, and there is certainly room for both traditional and constructivist methods in math teaching. 3: Walking Tours with songs adapted to or taken from the communities that lived in the local area. Use local landmarks or significant locations to elicit questions and discussion or debate to stimulate and increase natural curiosity and critical thinking. 4: The idea from the Community Sing Alongs and Cooperative Values Discussions was similarly to stimulate discussion on ways to build cooperation based on shared values and shared cultural connections. These same ideas can be used to teach any subject, given sufficient creativity, time, and support from other teachers and staff/administrators. The use of maths journals and math portfolios or construction projects for evaluation is an adaptation of this idea.
Read, Write, Run, Teach !
13 February, 12016 HE