Mathematics Constructivism
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Constructivist Theory and School Leadership

Constructivist Theory and School Leadership | Mathematics Constructivism | Scoop.it
Constructivism refers to the belief that knowledge is actually constructed in the mind of the learner. It is based on the theory that individuals actively and continuously build, or
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KMDI:

KMDI: | Mathematics Constructivism | Scoop.it

KMDI is a 21st century organisation and the University of Toronto's first virtual insitute. As a distributed network we are an experiment in living in a distributed environment. This maps well with our early and ongoing research focus on collaboration practices and collaboration technologies, and our long term association with the field of CSCW: computer-supported cooperative work.

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edtechopen

edtechopen | Mathematics Constructivism | Scoop.it
Welcome to our open community for educational technology using professionals

 

Our new Moodle site is open with a line-up of edtech related workshops we invite you to participate in.

Our first course of the new year starts this Friday with an in-depth look at engaging  online learners. The designer, Christopher Rozitis, has developed an exemplary course for you, not only sharing current research, but presenting it in an engaging manner employing techniques that will inspire you.

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GeoGebra

GeoGebra | Mathematics Constructivism | Scoop.it

Free mathematics software for learning and teaching

 

Interactive graphics, algebra and spreadsheet
From elementary school to university level
Free learning materials


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Daniel Thurmond's curator insight, April 13, 2014 12:16 PM

GeoGebra is a free resource that lets the user manipulate pre built and self constructed shapes so that algebraic and geometric properties can be observed and instructed. Depending on the level of use Geogebra can be used at any level in the TIMS, even transformation. Educators can use this to show geometric properties, allow the students to discover them, or use known properties to make connections to the real world. 

Nicolette Erkelens's curator insight, February 14, 2015 1:29 PM

#Wiskunde, #Natuurkunde

 

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Shapes – 3D Geometry Learning - AVATARGENERATION

Shapes – 3D Geometry Learning - AVATARGENERATION | Mathematics Constructivism | Scoop.it
SHAPES is a geometry app developed by Setapp, a technology company from Poland. SHAPES – 3D Geometry Learning allows kids to discover different three dimensional solids like prisms, pyramids, solids of revolution and Platonic solids. The app was verified and approved by the Faculty of Mathematics and Computer Science at Adam Mickiewicz University of Poznan. It was specifically designed to support teachers in the classroom, but can also be used as a tool for self-study. Check out their workshop for children, using the SHAPES app.

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From MOOCs to Learning Analytics: Scratching the surface of the 'visual'

From MOOCs to Learning Analytics: Scratching the surface of the 'visual' | Mathematics Constructivism | Scoop.it

"The two most prominent trends in education technology, for the moment, appear to be MOOCs and data analytics. While MOOCs are frequently accompanied by references to "disruptive innovation" [1], so-called "big data" in education, or "learning analytics" as it is often termed [2], is also cited in lists of imminent educational trends [3]."


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Vygotsky's constructivism - Emerging Perspectives on Learning, Teaching and Technology

In order to gain an understanding of Vygotsky's theories on cognitive development, one must understand two of the main principles of Vygotsky's work: the More Knowledgeable Other (MKO) and the Zone of Proximal Development (ZPD).


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Alexis Alexander's curator insight, September 22, 2014 1:41 PM

Not so recent article on cognivitism

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Vygotsky's Developmental and Educational Psychology - Peter Langford

Vygotsky's Developmental and Educational Psychology - Peter Langford | Mathematics Constructivism | Scoop.it
Here you can find where to get Vygotsky's Developmental and Educational Psychology - Peter Langford download.

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Leaders in Educational Thought: Mathematics K-12 - VIDEO collection and resources

Leaders in Educational Thought: Mathematics K-12 - VIDEO collection and resources | Mathematics Constructivism | Scoop.it
Conversations at The Ontario Association for Mathematics Education (OAME) 2014

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small-changes-big-returns - Math & Arithmetic

small-changes-big-returns - Math & Arithmetic | Mathematics Constructivism | Scoop.it

This is the math & arithmetic tools of my Small Changes; BIG RETURNS wiki. Enjoy!


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Math education plan worrisome, U of S prof says - CBC.ca

Math education plan worrisome, U of S prof says - CBC.ca | Mathematics Constructivism | Scoop.it
Math education plan worrisome, U of S prof saysCBC.caBeginning of Story Content A divisive debate about the place of mathematics in teacher education is heating up at the University of Saskatchewan, where some professors say aspiring educators...

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Information Design

Information Design | Mathematics Constructivism | Scoop.it
A Quick, No-Nonsense Guide to Basic Instructional Design Theory (A Quick, No-Nonsense Guide to Basic Instructional Design Theory http://t.co/QrJqdRaDii)...

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Lorraine Kasyan's curator insight, August 27, 2014 3:50 PM

I love the visual and succint descriptors. Great example of an education infographic too.

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Instructivism, constructivism or connectivism?

Instructivism, constructivism or connectivism? | Mathematics Constructivism | Scoop.it
Instructivism is dead. Gone are the days of an authoritarian teacher transmitting pre-defined information to passive students. In the 1990s, constructivism heralded a new dawn in instructional desi...

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systerwoody's curator insight, March 17, 2014 9:46 AM

Simple graphics help to understand basic concepts. :-)

Philippe-Didier Gauthier's curator insight, April 22, 2014 12:42 PM

#Apprenance  Presque vue comme une évolution des approches pédagogiques...

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The Eight Principles of Ethical Leadership in Education

The Eight Principles of Ethical Leadership in Education | Mathematics Constructivism | Scoop.it

While there has always been a requirement for ethics in leadership, the last hundred years have seen a shift in the paradigm of leadership ethics. Leadership for the 21st century is grounded in moral ...


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The shape of minds to come | Learning with 'e's

The shape of minds to come | Learning with 'e's | Mathematics Constructivism | Scoop.it

In this post, we will explore the work of Bärbel Inhelder on deductive reasoning. As usual, this is a simplified interpretation of the theory, so if you wish to learn more, please read the associated literature.

The theory

Swiss psychologist Bärbel Inhelder is perhaps the best known of Piaget's collaborators. She made some important contributions to his stages of cognitive development theory (which will be featured in greater depth in some upcoming articles on this blog). Inhelder was particularly interested in how children's minds develop to the point where they can reason for themselves. Her work with Jean Piaget led to the proposal that there is a 'formal operations' stage marking the transition from childhood to adolescence. They argued that when children reach the age of about 11 years old, they are capable of using deductive reasoning to make sense of the world around them.

How it can be applied in education

Inhelder's work with Piaget was instrumental in shaping the way schools are organised today and is a key influence on the design of curricula. The transition between primary (elementary) school and secondary (high) school is marked when children reach the age of 11 (or 12 in some countries such as Scotland).

It could be argued that these decisions were made because of Inhelder and Piaget's cognitive stages theory. The Formal Operations stage is where children are capable of higher order thinking such as abstract reasoning - imagining the outcome of their actions, and it is also the stage of development where they can develop their inferential reasoning skills. A good example of inferential reasoning in education is where the teacher presents students with puzzles or challenges as a part of their learning: 'If George is older than David, and David is older than Michael, who is the oldest?' Inferential reasoning skills can be developed over time as children learn about new concepts, how they compare, and how to make decisions. The ability to deduce from the general to the specific is the basis of all good science, and runs consistently through a number of disciplines such as mathematics and statistical analysis.

Deductive reasoning methods can therefore also be applied to good effect in just about any lesson on any subject. Students could be encouraged to ask 'what if?' hypothetical questions during physics or chemistry experiments, and then test out their predictions; or to predict the trajectory of a cricket ball in sport; or be asked to judge whether a statement is true or false, on the basis of evidence; or to detect grammatical errors according to 'the rules' of a language. Indeed, the entire secondary curriculum in schools is based on the premise that children between 11-16 years old have developed their higher level cognitive capabilities sufficiently enough to be able to think creatively, use abstract reasoning and perform numerical calculations.

It should be noted that many of the theories proposed by Inhelder and Piaget are contentious and have been challenged not only on the basis of their small sample size (he mainly used his own children as subjects in his experiments) and methods, but also due to alternative findings and interpretations carried out by a number of psychologists. Are there actually stages of cognitive development, and are they as Inhelder and Piaget claimed? And of course, the most difficult problem of them all - do all children develop through these stages at the same time and in the same way? For more details on these counter arguments see the work of Margaret Donaldson.

 

 


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Oh, the Fractions You’ll See!

Oh, the Fractions You’ll See! | Mathematics Constructivism | Scoop.it
A quick quiz: How many fractions are there?
Fractions like 13/37 and 1/100 rarely appear in elementary mathematics curricula.
This may sound like an absurd question, but in the context of elementary mathematics curricula, it makes a lot of sense.

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Deep thinking is not reserved for geniuses but is for everyone. Here's why.

Deep thinking is not reserved for geniuses but is for everyone. Here's why. | Mathematics Constructivism | Scoop.it
What is intelligence? What is the nature of creativity, thinking, learning, and even of mind?
In his latest book, Deep Thinking: What Mathematics Can Teach Us About the Mind, mathematician William Byers claims that all of these processes have a common root: Deep Thinking, something that we know as insight, that is, discontinuous "aha! thinking."
What is new in Byers' approach is his insistence that deep thinking is not exceptional and reserved for geniuses but the default condition of the mind. It is to be found whenever a problem is resolved by discovering a new approach. This happens when a child grasps a new conceptual system or a scientist develops a new paradigm.
Deep thinking can be difficult to access because it is more elementary than, and lives upstream of, rule-based, algorithmic thought. Thus creative insight, which is a paradigm of deep thinking, produces, but cannot be produced by, logical thought. Deep thinking is what you must rely on if you want to produce something that is truly original.
Evidence for this thesis comes from many disparate sources: the conceptual development of children, the history of mathematics and science, studies of the creative process, a reconsideration of the nature of learning, and the neurobiology of the brain.
The implications are enormous. To take just one example that is emphasized in the book: if real learning involves deep thinking, that is creative reframing, rather than merely accumulating and analyzing data, then teaching must be transformed so as to make this its primary goal.
A reconsideration of the nature of creative thought is vitally important in this day and age. There is a huge difference between human thought with its creative potential and the derivative and limited possibilities of machine thought. There is a trend in Western thinking that is very popular today, which maintains that the computer is potentially capable of just about anything and the human mind and brain are merely computational devices.
This book indicates where the fallacy lies in this approach. Deep thinking gets you out of the box; algorithmic processes, on the other hand, can only reveal new features of the box. In general any activity that is truly original is based on deep thinking; the child's mastery of the counting numbers is identical in kind to the thinking that produced Einstein's Theory of Relativity, Tolstoy's War and Peace, or Beethoven's Ninth Symphony.
More information:
phys.org/news/2015-01-robots-k… y-cooking-video.html
Provided by World Scientific Publishing
This Phys.org Science News Wire page contains a press release issued by an organization mentioned above and is provided to you “as is” with little or no review from Phys.Org staff.

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This Course at MIT | Introduction to Probability and Statistics | Mathematics | MIT OpenCourseWare

This Course at MIT | Introduction to Probability and Statistics | Mathematics | MIT OpenCourseWare | Mathematics Constructivism | Scoop.it
This page focuses on the course 18.05 Introduction to Probability and Statistics as it was taught by Dr. Jeremy Orloff and Dr. Jonathan Bloom in Spring 2014.
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What Is Learning? 12 Principles of Peer-Led, Connected, Interactive Education

What Is Learning?  12 Principles of Peer-Led, Connected, Interactive Education | Mathematics Constructivism | Scoop.it
Here are a baker's dozen of the main principles of connected learning. As you will see, they form an “ecosystem,” where each component influences and changes the others. These apply in any field (although differently in each field). These principles draw from constructivist, engaged “public educators” (Stuart Hall’s term)  going back as far as Lev Vygotsky and John Dewey and including Howard Gardner, Franz Fanon, Jacques Rancière, and digital pedagogy theorists including Yochai Benkler, Howard Rhinegold,  Mizuko Ito, and many others.

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David Hain's curator insight, December 17, 2014 3:03 AM

Sustainable learning is the Holy Grail to stay current - and, maybe, survive!  Some ideas here...

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The Maker Movement and the Rebirth of Constructionism

The Maker Movement and the Rebirth of Constructionism | Mathematics Constructivism | Scoop.it

"Educational theory and practice have begun to appear more frequently in the popular press. Terms such ascollaborative learning, project-based learning, metacognition, inquiry-based learning, and so on, might be new to some audiences, but they have a relatively long and well-documented history for many educators. The most widely-known and promising pedagogical approach is constructivism grounded on the work of Piaget, Vygotsky, and Bruner. Given how it has transformed my own understanding of pedagogy, teaching, and learning, constructionism seems ripe for a similar resurgence — like a phoenix rising from the ashes of Taylorization and standardized testing. Constructionism brings creativity, tinkering, exploring, building, and presentation to the forefront of the learning process. As I searched for the theoretical vocabulary by which to explain what I was doing, I gravitated towards the works of John Dewey, Jean Piaget, and Lev Vygotsky. However, it wasn’t until I encountered the ideas of Seymour Papert, one of the founders of MIT Media Lab and the first proponent of constructionist pedagogy, that I realized there was a clear theoretical and research basis for my pedagogical practices. At its heart constructionism argues for what I had been trying to articulate: learning happens best when learners construct their understanding through a process of constructing things to share with others. Kafai, Peppler, & Chapman explain in their book The Computer Clubhouse: Constructionism and Creativity in Youth Communities." | by Jonan Donaldson

 
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Teaching Students to Think: All Our Students Thinking

Teaching Students to Think: All Our Students Thinking | Mathematics Constructivism | Scoop.it

February 2008 | Volume 65 | Number 5
Teaching Students to Think Pages 8-13

All Our Students Thinking

Nel Noddings

Any subject—be it physics, art, or auto repair—can promote critical thinking as long as teachers teach in intellectually challenging ways.

One stated aim of almost all schools today is to promote critical thinking. But how do we teach critical thinking? What do we mean by thinking?

In an earlier issue on the whole child (September 2005), Educational Leadership made it clear that education is rightly considered a multipurpose enterprise. Schools should encourage the development of all aspects of whole persons: their intellectual, moral, social, aesthetic, emotional, physical, and spiritual capacities. In this issue, I am primarily concerned with intellectual development, in particular, with teaching students to think. However, as we address this important aim, we need to ask how it fits with other important aims, how our choice of specific goals and objectives may affect the aim of thinking, and whether current practices enhance or impede this aim.

Thinking and Intellect

Writers often distinguish among such thinking categories as critical thinking, reflective thinking, creative thinking, and higher-order thinking. Here, I consider thinking as the sort of mental activity that uses facts to plan, order, and work toward an end; seeks meaning or an explanation; is self-reflective; and uses reason to question claims and make judgments. This seems to be what most teachers have in mind when they talk about thinking.

For centuries, many people have assumed that the study of certain subjects—such as algebra, Latin, and physics—has a desirable effect on the development of intellect. These subjects, it was thought, develop the mind, much as physical activity develops the muscles. John Dewey (1933/1971) rejected this view, writing, "It is desirable to expel … the notion that some subjects are inherently 'intellectual,' and hence possessed of an almost magical power to train the faculty of thought" (p. 46). Dewey argued, on the contrary, that

any subject, from Greek to cooking, and from drawing to mathematics, is intellectual, if intellectual at all, not in its fixed inner structure, but in its function—in its power to start and direct significant inquiry and reflection. What geometry does for one, the manipulation of laboratory apparatus, the mastery of a musical composition, or the conduct of a business affair, may do for another. (pp. 46–47)

 

More recently, Mike Rose has shown convincingly not only that thinking is required in physical work (2005), but also that nonacademic subjects can be taught in intellectually challenging ways (1995). We do our students and society a disservice when we suppose that there is no intellectual worth in such subjects as homemaking, parenting, getting along with others, living with plants and animals, and understanding advertising and propaganda (Noddings, 2005, 2006). The point is to appreciate the topics that matter in real life and encourage thinking in each area. This is not accomplished by first teaching everyone algebra—thus developing mental muscle—and then applying that muscle to everyday matters.

Nor is it accomplished by simply adding thinking to the set of objectives for each disciplinary course. More than 20 years ago, educators and policymakers advocated greater emphasis on thinking as an aim of education. Commenting on this popular demand, Matthew Lipman (1991), one of the founders of the modern Philosophy for Children movement, remarked,

School administrators are calling for ways of "infusing thinking into the curriculum," apparently on the understanding that thinking can be added to the existing courses of studies as easily as we add vitamins to our diet. (p. 2)

 

But thinking cannot be formulated as a lesson objective—as something to teach, learn, and evaluate on Thursday morning. How, then, do we go about it?

Learning as Exploration

A few years ago, I watched a teenager whom I'll call Margie struggle with courses that discouraged thinking. In her U.S. history course, students were required to learn a list of facts for each unit of study. Margie had to memorize a set of 40 responses (names, places, and dates) for the unit on the American Revolutionary War and the postwar period. Conscientiously, she memorized the material and got a good grade on the test. When I talked with her, however, it was clear that she had not been asked to think and would soon forget the memorized facts. None of it meant anything to her; passing the test was her only objective.

Suppose, instead, that the teacher had asked students to consider such questions as these:

What happened to the Tories during and after the war?Why was Thomas Paine honored as a hero for his tract Common Sense but reviled for his book The Age of Reason?Why might we be surprised (and dismayed) that John Adams signed the Alien and Sedition Acts?

 

Such questions would encourage students to read, write, argue, and consider the implications for current political life—all important aims of education. How many Tories left the United States? Where did they go? Where do refugees go today? Discussing the question on Thomas Paine could lead to a critical discussion of both nationalism and religion centered on Paine's statement, "My country is the world; my religion is to do good." Who reviled Paine and why? After reading biographical material on John Adams, students might indeed be amazed that he signed the Alien and Sedition Acts. What lesson might we take from this story about the effects of fear and distrust on even highly intelligent people?

Algebra for Some

When I first met with Margie, she was taking algebra. Looking through her textbook, I thought the course would be wonderful. The textbook was loaded with real-world applications and exercises that invited genuine thinking. But the teacher did not assign even one of these exercises. Not one! The following year, in geometry, Margie was never asked to do a proof. These algebra and geometry classes were composed of kids who, had they had a choice in the matter, would not have chosen courses in academic mathematics. Today, in the name of equality of opportunity, we force nearly all students into courses called Algebra and Geometry, but the courses often do not deserve their names because they lack genuine intellectual content. This practice is little short of pedagogical fraud. Many of Margie's classmates (and Margie, too) would have been better served by good career and technical education courses that would challenge them to think about the world of work for which they were preparing.

I am not suggesting that we go back to a system in which students are tested, sorted, and assigned either to academic courses or dead-end tracks in which they are treated with neglect, sometimes even with contempt. But the present practice of forcing everyone into academic courses is not working well. We would do better to design excellent career and technical education courses—very like the job-oriented programs provided in two-year colleges—and allow students to choose their own course of study. Students should not be forced into or excluded from academic courses, but they should be able to choose a nonacademic program with pride and confidence. Such programs are available in many Western countries, such as Germany and the Scandinavian countries. Programs like these might offer courses to prepare machinists, film technicians, office managers, retail salespersons, food preparation and service workers, mechanics, and other skilled workers. Recent studies have shown that the United States actually has an oversupply of engineers and scientists but badly needs workers with high technical skills (Monastersky, 2007).

We can give students opportunities to think well in any course we offer, provided the students are interested in the subjects discussed. Algebra can be taught thoughtfully or stupidly. So can drafting, cooking, or parenting. The key is to give students opportunities to think and to make an effort to connect one subject area to other subject areas in the curriculum and to everyday life.

Consider the ongoing debate over popular science versus "real science." Many critics scorn popular science courses (for a powerful criticism of the critics, see Windschitl, 2006). They would prefer to enroll all students in science courses that would prepare them—through emphasis on vocabulary and abstract concepts—for the next science course. According to this view, practical or popular science has little value and should certainly not carry credits toward college preparation. But intelligent, well-educated nonscientists depend on popular (or popularized) science for a lifetime of essential information. Nonscientists like myself cannot run our own experiments and verify everything that comes through the science pipeline. Instead, we read widely and consider the credentials of those making various claims. High school courses should prepare not only future specialists but also all students for membership in this circle of thoughtful readers.

Deference to the formal disciplines sometimes actually impedes student thinking. A few years ago, it was recommended that math courses should teach students how to think like a mathematician. In science courses, they were to think like a scientist; in history, like a historian, and so on. But aside from the possibility that there may be more than one way to think like a mathematician, education efforts might better be aimed at showing students how to use mathematics to think about their own purposes. For example, carpenters don't need to think like mathematicians, but they do need to think about and use mathematics in their work.

Modeling Open-Ended Thinking

It may be useful, however, for students to see and hear their teachers thinking as mathematicians, historians, or artists. When I was studying for my master's degree in mathematics, I had one professor who frequently came to class unprepared. His fumbling about was often annoying; he wasted time. But sometimes his lack of preparedness led to eye-opening episodes. He would share aloud his thinking, working his way through a problem. Sometimes he would stop short and say, "This isn't going to work," and he'd explain why it wouldn't work. At other times, he'd say, "Ah, look, we're going great! What should we do next?" He modeled mathematical thinking for us, and I found it quite wonderful. The process was messy, uneven, time-consuming, and thrilling. That's the way real thinking is.

I am not recommending that teachers come to class unprepared, but we should at least occasionally tackle problems or ideas that we have not worked out beforehand. In doing so, we model thinking and demonstrate both the obstacles that we encounter and our successes.

Too often, we state beforehand exactly what we will teach and exactly what our students should know or do as a result. This is the right approach for some objectives. There is a place for automatic response in student learning; we do want students to carry out some operations automatically, without thinking. That sort of skill frees us to think about the real problems on which we should concentrate.

In today's schools, however, too much of what we teach is cast in terms of specific objectives or standards. Margie was told the 40 things she was expected to know about the American Revolutionary War. Some educators even argue that it is only fair to tell students exactly what they must know or do. But such full disclosure may foreclose learning to think. Thinking involves planning, ordering, creating structural outlines, deciding what is important, and reflecting on one's own activity. If all this is done for students—Cliffs Notes for everything—they may pass tests on material they have memorized, but they will not learn to think, and they will quickly forget most of the memorized material.

Encouraging Teachers to Think

Our focus thus far has been on students. But what about teachers? Are they encouraged to think? Unfortunately, many teachers are told what topics to teach and how to teach them. In too many cases, they are even compelled to use scripted lessons. Ready-made lessons should be available for teachers who want to use them or for special purposes, but professional teachers should be allowed—even encouraged—to use their professional judgment in planning lessons and sequences of lessons.

If teachers want to teach students to think, they must think about what they themselves are doing. Critics both inside and outside the United States have characterized the U.S. curriculum as "a mile wide and an inch deep." The pressure to cover mandated material can lead to hasty and superficial instruction that favors correct responses to multiple-choice questions over thinking. Countless teachers have told me that they can't spend time on real-life applications of mathematics or the kinds of questions I suggested for Margie's history class. If they were to do so, they tell me, they wouldn't get through the required curriculum. But what is the point of getting through a huge body of material if students will soon forget it? How can we claim to educate our students if they do not acquire the intellectual habits of mind associated with thinking?

Teachers should also be willing to think critically about education theory and about what we might call education propaganda. Slogans are mouthed freely in education circles, and too few teachers challenge them (Noddings, 2007). For example, it is easy and politically correct to say, "All children can learn," but what does that mean? Can all children learn, say, algebra? If we answer a qualified no to this, are we demeaning the ability of some children (perhaps many), or might our answer be a respectful recognition that children differ and exhibit a wide range of talents and needs?

What Competing Really Means

Even if we believe that all children can learn algebra, we too seldom ask the question, Why should they? When we do ask it, the answer is usually that we live in an information age and that if students (and the United States) are to compete in a worldwide economy, they must know far more mathematics than previous generations did. We need, they say, more college-educated citizens.

Is this true? The information world is certainly growing, but in addition to its own growth, it has generated an enormous service world, and people in this world should also learn to think. The Bureau of Labor Statistics provides charts showing that, of the 10 occupations with the most openings in the next decade, only one or two require a college education. Occupations such as food preparation and service worker, retail salesperson, customer service representative, cashier, office clerk, and laborer and material mover will employ about five times more people than the computer/high-tech fields requiring a college education (see www.bls.gov/emp/home.htm for employment projections). No matter what we do in schools, most of our high school graduates will work at such jobs.

We live in an interdependent society, and one of our education aims is to prepare students for democratic citizenship. As part of that task, we should help students develop an appreciation for the wide range of essential work that must be done in our complex society. In the future, not everyone will need to have a traditional college education to experience occupational success, although postsecondary education or training will frequently enhance that success. Rather, occupational success will require flexibility, a willingness to continue learning, an ability to work in teams, patience and skill in problem solving, intellectual and personal honesty, and a well-developed capacity to think. Success in personal life requires many of the same qualities.

Even for those who go on to college and postgraduate education, the intellectual demands of the future are moving away from a narrow disciplinary emphasis. The biologist E. O. Wilson (2006) has commented on the new demands:

The trajectory of world events suggests that educated people should be far better able than before to address the great issues courageously and analytically by undertaking a traverse of disciplines. We are into the age of synthesis, with a real empirical bite to it. Therefore, sapere aude. Dare to think on your own. (p. 137)

 

That's good advice for both teachers and students.

References

Dewey, J. (1933/1971). How we think. Chicago: Henry Regnery. (Original work published 1933)

Lipman, M. (1991). Thinking in education. Cambridge, UK: Cambridge University Press.

Monastersky, R. (2007, November 16). Researchers dispute notion that America lacks scientists and engineers. The Chronicle of Higher Education, 54(12), A14–15.

Noddings, N. (2005). The challenge to care in schools (2nd ed.). New York: Teachers College Press.

Noddings, N. (2006). Critical lessons: What our schools should teach. New York: Cambridge University Press.

Noddings, N. (2007). When school reform goes wrong. New York: Teachers College Press.

Rose, M. (1995). Possible lives: The promise of public education in America. Boston: Houghton Mifflin.

Rose, M. (2005). The mind at work: Valuing the intelligence of the American worker. New York: Penguin.

Wilson, E. O. (2006). The creation: An appeal to save life on earth. New York: Norton.

Windschitl, M. (2006). Why we can't talk to one another about science education reform. Phi Delta Kappan, 87(5), 348–355.

 

Nel Noddings is Lee L. Jacks Professor of Education, Emerita, at Stanford University, Stanford, California; noddings@stanford.edu.


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George Siemens: ‘Students need to take ownership of their learning’

George Siemens: ‘Students need to take ownership of their learning’ | Mathematics Constructivism | Scoop.it
George Siemens

George Siemens is an internationally renowned and highly respected professor and researcher of technology, networks, analytics, and openness
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Too many teachers can't do math, let alone teach it - Globe and Mail

Too many teachers can't do math, let alone teach it - Globe and Mail | Mathematics Constructivism | Scoop.it
Too many teachers can't do math, let alone teach itGlobe and Mail“If you don't know math, you can't teach math,” says Anne Stokke, a math professor at the University of Winnipeg who has launched a petition to raise the standards.
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MOOC Research Learning Curves | Higher Ed Beta @insidehighered

MOOC Research Learning Curves | Higher Ed Beta @insidehighered | Mathematics Constructivism | Scoop.it

At HarvardX we’re increasingly interested in how MOOC data can guide us towards better teaching. In fact, we aim to learn from research on every MOOC that we build to continuously improve our teaching on campus and online. But to understand what helps students learn and what doesn’t, to get a handle on cause and effect, we have to shift from observational studies to building experimentation into the very DNA of our courses.

 

This necessitates increasing collaboration between the faculty, professional learning researchers, and instructional designers. For course teams and professors with little experience in educational research, the task seems daunting: Where do I even begin?


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Sense-making and Multiplication

Sense-making and Multiplication | Mathematics Constructivism | Scoop.it
Enthusiastic ways to teach multiplication and math to students in grades 3, 4 and 5. Using patterns and other learning tip can help elementary school students to learn multiplication faster and allow them to apply it to other problems.
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