kitchen table math, the sequel: found: NCTM unicorn

Wednesday, February 28, 2007

found: NCTM unicorn

RE: NCTM & constructivism

Constructivist Mathematics and Unicorns

by Lee V. Stiff

Constructivist math is a term coined by critics of Standards-based mathematics who promote confusion about the relationships among content, pedagogy, and how students learn mathematics....

Like unicorns, "constructivist math" does not exist.
Apparently someone didn't get the memo.


Instructivist said...

The late physicist and science prof Alan Cromer (author of Connected Knowledge) says this about constructivism:

"Constructivism is a postmodern antiscience
philosophy that is based upon Piaget’s work on
how children construct concepts and conceptual
relations and on the philosophy of two early
nineteenth-century opponents of the Scientific
Revolution, Giambattista Vico and George
Berkeley.... It’s a form of subjective empiricism
that puts its emphasis on the thoughts of the
knower and views the search for truth as an illusion….
Such an ideology would be of no interest
to scientists and science educators were it
not, in effect, the official ideology of the reform
movements in the United States and elsewhere….
But when push comes to shove, no one
knows how students are to construct their own
theories of atoms and electrons, of stars and
galaxies, of DNA and genetics…”

Instructivist said...

From a review of Connected Knowledge:

In Connected Knowledge, physicist Alan Cromer offers a way to bridge the chasm, with a lively, lucid account of scientific thinking and a provocative new agenda for American education. Science, Cromer argues, is anything but common sense: It requires a particular habit of mind that does not come naturally. For example, something as simple as buoyancy can only be explained through Archimedes' principle--that a body in a fluid is subject to an upward force equal to the weight of fluid it displaces--yet few scientists could arrive at this ancient concept by trial and error.

School children, however, are often given a ball and a tank of water, and asked to explain buoyancy any way they can. Today's de-emphasis on teaching pupils necessary facts and principles, he argues, "far from empowering them, makes them slaves of their own subjective opinions."

This movement in education, known as Constructivism, has close ties to postmodern critics (such as the editors of Social Text) who question the objectivity of science, and with it the existence of an objective reality. Cromer offers a ringing defense of the knowability of the world, both as an objective reality and as a finite landscape of discovery.

The advance of scientific knowledge, he argues, is not unlike the mapping of the continents; at this point, we have found them all. He shows how the advent of quantum mechanics, rather than making knowledge less certain, actually offers a more precise understanding of the behavior of atoms and electrons.

Turning from philosophy to education, he argues that instead of allowing students to flounder, however creatively, schools should follow a progressive curriculum that returns theoretical knowledge to the classroom.

Instructivist said...

A reader at my site left an explanation of what he thinks constructivism is. It's pretty long but I think it might stimulate discussion. The theory also goes by many other names, so I think he is wrong to deny its prevalence in ed school just because that term is not always used specifically.

Here is the comment with apologies for its length:

In response to your post on arrested development I felt that an explanation of what constructivism is may be of the most use. Constructivism like any theory or philosophy can be thought of as falling on a continuum. On one end there is radical constructivism which would be the philosophy that children will just naturally learn given the proper learning environment and that no direct teaching is required. On the more temperate end of constructivism is the thought that children need to be taught certain things and that the learners will than use this foundation to build more complex thoughts. Or if a text book definition suits your fancy than perhaps constructivism could be thought of as a perspective that suggests that learner create a body of knowledge from their experiences-knowledge that may or may not be an accurate representation of external reality (Ormrod, 2003). It is with the second part of the definition that you appear to take issue with and I would agree could be problematic.
I would argue that children in the early years of school can have plenty of false beliefs and it is important that a teacher give the student information to dispel these false beliefs. To leave a child to create all their own knowledge would be a mistake and I have not seen any evidence that would indicate that schools of education are advocating this. My current textbook on learning tells the reader that information needs taught in order to build on the student’s level of understanding (Bransford, 2000). However I would also argue that direct teaching may not necessarily convey the information intended by the teacher because such a practice ignores students’ experiences and preexisting knowledge. In contrast to constructivism, direct teaching leaves the teacher leaves the teacher ignorant of what the student is thinking because children are not given the opportunity to speak. Therefore preconceived notions held by the child never have an opportunity to be corrected. Now that a clearer understanding of what constructivism is has been created a discussion of its touted theorists may be in order.
Piaget would be a good place to start in that he would appear to have the least to do with formal education. It would seem that Piaget did indeed encourage learning in the natural environment and in so much as he focused on the learning that was naturally taking place with his theory. However the idea that Piaget never spoke of development beyond the sensorimotor stage is not true. The formal operational stage found in children age 11 and up is defined by the concept of understanding abstract concepts. In other words a learner that is in the sensorimotor stage thinks beyond themselves and that which is black and white. However I would agree that Piaget’s stages of Cognitive development are rather lacking when it comes to the teen age child. I would also agree that it seems that children are arrested in their development but not at the age of 2 such that you suggest but in adolescence. However one could argue that constructivism is taking these stages of development and adding to them to create deeper understanding.
Vygotsky is another matter entirely. His ideas of zone of proximal development seem not only to fit in with constructivism but give it the support it needs. In Vygotsky’s model a learner knows only so much and it is just beyond this firm grasp of knowledge that the educator should set the bar. In a group learning environment learners of similar ability can help each other in that they understand the areas in which a child of that age may be having problems. It is by knowing what a child is misunderstanding that we can help them understand it differently and thus reach this higher bench mark. With a model such as Vygotsky’s a teacher must know what the learner thinks they know and work with them from there.
I believe where one could get side tracked with the constructivist theory is when one reads the views of Von Glaserfeld. One could argue that Von Glaserfeld would belong to the radical constructivist camp. It would not be entirely accurate to attribute his views to all constructivists. Even in the most modern schools of education teachers are still being taught how to teach. If the philosophy was that students do not require the knowledge dissemination of teachers than teachers would no longer need to be educated. However again I would challenge the notion that he is entirely wrong. I would hate to think of myself as a robot mindlessly receiving information and storing it never questioning it or thinking on it further. If it were truly the case that a learner never constructs their own knowledge than we the human race would still be stuck in the dark ages. Inventions are not directly taught but the pieces that the learner needed to create them was. It is at this point that I again must depart from the constructivist camp and align myself with more traditional views.
There are some children for which constructivist teaching may not work. It does no good to pretend otherwise. Group work is not for everyone. For instance an above average child might grow bored waiting for the rest of the group to catch up to his/her level of enlightenment. Some students may be entirely motivated by grades and thinking beyond how to ace the test is not even on their radar. It is for these types of learners that constructivism may be like that pair of shoes that is really cute but really painful. However I would challenge anyone to find a method of teaching that challenges all students at their current level of mastery and encourages further comprehension. It would be more worthwhile to work with what we have to create the best learners than to argue that the current system stinks. Although, building on or constructing new ways of teaching would also be helpful.
In conclusion I would agree that constructivism is hard to define. Even in all the education text books I currently own approximately 3 pages are devoted to the topic. These three pages and one lecture I received appear to contradict your assertion that constructivism reigns supreme in schools of education. The reason that constructivism is not taught more or explained further might be because constructivism is not a way of teaching but a theory about how one learns. It was my sincere wish to enlighten you with this response and distil any worries you have about arrested development and to this end I hope I was successful.

David said...

I was astonished by the statement that direct instruction does not permit students to speak! If, as he says, constructivism is a continuum, then isn't direct instruction also a continuum?

Barry Garelick said...

I agree with David. Direct instruction is at one end of the spectrum or continuum; pure discovery at the other. Vygotsky lies in the middle and probably falls in the "guided discovery" category, and is on the right track when he says social interaction between student and teacher is important because of the teacher imparting knowledge of "previously discovered knowledge" to the student. He goes off the deep end when he suggests that student to student interaction is a necessary ingredient. Yes and no. This is where we get the "work in small groups" BS that pervades schools, and where students are expected to teach other students based on their "innate understanding" and "construction of knowledge".

What is important to note is that psychologists generally agree that all knowledge is ultimately constructed as a part of being "learned", but Anderson, Reder, Simon (1998) also point out that such construction occurs whether knowledge is imparted passively OR actively. That is, knowledge is constructed by the "learner" even if imparted directly (direct instruction) as opposed to obtained actively (working in small groups with hands on activities and cognitive overload--pardon the cynicism and bias).

Some good empirical studies have been done that show that discovery learning has some good effects for those whose knowledge domain is already developed. But for novices (like grade school and high school students, experiencing a topic for the first time), a direct instruction approach provides better results.