The situation
Prince William County Public Schools (PWCS) are in the process of implementing a new program to teach K – 5 mathematics. The school division calls this new program, Math Investigations (MI). Here, for example, is how Marshall E.S. advertises the MI program.
The textbook series that the school division has chosen is called Investigations in Number, Data, and Space®. Its publisher is a nonprofit called TERC. As the TERC website notes:
Investigations in Number, Data, and Space was developed at TERC and funded in part by the National Science Foundation.
Change, whether good or bad (and it is usually a mix of the two) almost inevitably encounters resistance. In this case, the local resistance has a website, PWCTEACHMATHRIGHT.COM.
As I mentioned in this post, the learning theory upon which TERC based its Investigations series is called Constructivism. The point of this post is to gain an understanding of Constructivism and why TERC chose this method to teach mathematics.
Why change?
When one considers the prospect of teaching, one thing to keep in mind is that there is nothing necessarily sacred about tradition. Horace Mann is often credited with establishing many of the characteristics we associate with today’s American education system. He did not start his reforms until about 1837. What Mann helped to devise is our assembly line system of education. The input is skulls full of mush. The desired output is an educated citizen. At each grade level teachers serve as assembly line workers. Teachers instruct largely passive students in the knowledge assigned to their grade level.
To ensure students properly receive appropriate instruction, they are tested at each grade level. “Rejects” are held back and subjected again to the same instruction.
As a practical matter, what we consider traditional math only goes back about two or three generations. So we have not been doing traditional math very long. From our own perspective, traditional math is merely how we were taught math. Some attempts at innovation and some improvements should be expected.
What change?
TERC’s has based its approach to teaching innovation on Constructivism. On their website, TERC both describes what they believe Constructivism is (here) and what it is not (here). Since the TERC authors did not offer a simple, short definition, I went scrouging around for one that seem to match their approach. This website offers up a variety of definitions. I particularly enjoyed this one.
Constructivism
Constructivism is a philosophical position that views knowledge as the outcome of experience mediated by one’s own prior knowledge and the experience of others. In contrast to objectivism (e.g. Ayn Rand, 1957) which embraces a static reality that is independent of human cognition, constructivism (e.g. Immanuel Kant, 1781/1787) holds that the only reality we can know is that which is represented by human thought. Each new conception of the world is mediated by prior-constructed realities that we take for granted. Human cognitive development is a continually adaptive process of assimilation, accommodation, and correction (Piaget, 1968). Social constructivists (e.g. Berger and Luckmann, 1966) suggest that it is through the social process that reality takes on meaning and that our lives are formed and reformed through the dialectical process of socialization. A similar dialectical relationship informs our understanding of science (e.g. Bloor, 1976), and it shapes the technical artifacts that we invent and continually adapt to our changing realities (e.g. Bijker, 1995). Humans are shaped by their interactions with machines just as machines evolve and change in response to their use by humans. (Lemke, 1993).
However, the above is a bit difficult to digest. So I extracted a paragraph from another document that is a bit easier to read.
Constructivism’s central idea is that human learning is constructed, that learners build new knowledge upon the foundation of previous learning. This view of learning sharply contrasts with one in which learning is the passive transmission of information from one individual to another, a view in which reception, not construction, is key. (from here)
What Constructivists argue for is a basic theme change. Constructivist believe previous teaching methods focused too much on the presentation of information to the student; they insist that we must focus on how the student assimulates knowledge. Constructivists believe that when we learn we invent knowledge, building upon what we have previously learned.
I can only speak as a layman, but when I review the Constructivist approach, I think Constructivists are somewhat guilty of creating a strawman. We have known for some time that each person learns differently. Consider the reason we have focused on group presentations instead of how different individuals learn. Isn’t because previous generations of educators insisted on grouping children into classes based on their age and learning achievements? Don’t we still do it because this appears to be the most efficient use of a teacher’s time? Thus the Constructivist approach to instruction ultimately seems to be about how we individualize instruction at a reasonable cost without changing the public school organizational structure.
Wikipedia observes that constructivism does not suggest one particular pedagogy.
It is important to note that constructivism itself does not suggest one particular pedagogy. In fact, constructivism describes how learning should happen, regardless of whether learners are using their experiences to understand a lecture or attempting to design a model airplane. In both cases, the theory of constructivism suggests that learners construct knowledge. Constructivism as a description of human cognition is often associated with pedagogic approaches that promote active learning learning by doing. (from here)
Thus this statement on the TERC website.
Although references to constructivist approaches are pervasive, practical descriptions of such approaches have not been readily accessible. Therefore, to promote dialogue about instructional change, each “Research into Practice” column this year will illustrate how a constructivist approach to teaching might be taken for a specific topic in mathematics. (from here)
What change does Constructivism suggest?
Here is list of basic tenets written in a form a teacher could conceivably put into practice as guiding principles. (from here)
- Encouragement and acceptance of student autonomy and initiative.
- Utilization of raw data and primary sources along with manipulative, interactive, and physical materials.
- When planning, teachers use cognitive terminology such as “classify”, “analyze”, and “create.”
- Allowance of student responses to drive lessons, shift instructional strategies, and alter content.
- Inquiry concerning students’ understanding of concept before sharing their own understanding of those concepts.
- Encouragement of students to engage in dialogue, both with the teacher and with one another.
- Encouragement of student inquiry by asking thoughtful, open-ended questions and encouraging students to ask questions of each other.
- Pursuit of elaboration of students’ initial responses.
- Engagement of students in experiences that might engender contradictions to their initial hypotheses and then encourage discussion.
- Allowances for wait time after posing questions.
- Providing time for students to construct relationships and create metaphors.
- Nurturing students’ natural curiosity through frequent use of the learning cycle model.
Unfortunately, the above list is vague. It is not a lesson plan, but that is the point of the Investigations in Number, Data, and Space® textbook series. It is suppose provide sufficient materials to implement a constructivist approach to math instruction. Here (vickis-new-math2) is a powerpoint presentation from Marshall E.S. that provides a brief description of the MI program. Note this statement from the presentation.
Myth
Students learn by remembering what they are taught.
Reality
Students construct meaning as they learn mathematics. They use what they are taught to modify their prior beliefs and behavior, not simply to record what they are told. It is students’ acts of construction and invention that build their mathematical power and enable them to solve problems they have never seen before.
When the Constructivist approach is properly applied, students should become more involved in their own learning. Group exercises and problem investigations can be fun. Children undoubtly enjoy the diversion of playing with manipulatives. Thus, I can see why the School Board and the school administrative staff might consider the MI program promising. Even though Investigations in Number, Data, and Space® does not cover as many of the Standards of Learning as other textbooks, if using it helps to motivate students to learn math, that easily could compensate for the trouble of generating supplemental materials to cover the remaining SOLs.
Change is never easy
Investigations in Number, Data, and Space® probably provides less coverage of the SOLs because TERC is still figuring out how to put Constructivist theory into practice. Even without the desire to implement a new educational approach, writing textbooks for a K – 5 program would be a daunting task. However, TERC also tried to come up with a program that encourages children to learn math. So TERC had to devise games, group activities, manipulatives and so forth as well writing textbooks. In addition, there are tradeoffs to consider. How much time can be allowed and is needed for each game, group exercise, manipulative activity? How many ways should a child be taught to solve the same problem before moving on to another lesson? What is the right balance?
There is also the problem of the teacher. The teacher requires a new skill set.
According to the social constructivist approach, instructors have to adapt to the role of facilitators and not teachers (Bauersfeld, 1995). Where a teacher gives a didactic lecture which covers the subject matter, a facilitator helps the learner to get to his or her own understanding of the content. In the former scenario the learner plays a passive role and in the latter scenario the learner plays an active role in the learning process. The emphasis thus turns away from the instructor and the content, and towards the learner (Gamoran, Secada, & Marrett, 1998). This dramatic change of role implies that a facilitator needs to display a totally different set of skills than a teacher (Brownstein 2001). A teacher tells, a facilitator asks; a teacher lectures from the front, a facilitator supports from the back; a teacher gives answers according to a set curriculum, a facilitator provides guidelines and creates the environment for the learner to arrive at his or her own conclusions; a teacher mostly gives a monologue, a facilitator is in continuous dialogue with the learners (Rhodes and Bellamy, 1999). A facilitator should also be able to adapt the learning experience ‘in mid-air’ by using his or her own initiative in order to steer the learning experience to where the learners want to create value. (from here)
Undoubtedly, some teachers have had difficulty making the switch. There is increased complexity. Instead of all the students focusing the teacher, the teacher must focus on each student. Learning facilitation does not make the teacher’s role any less. What it means is that the teacher must work harder to adapt instruction for each student.
Should we change?
The problems involved in evaluating and accepting innovation can easily be underestimated. The proponents too often promise more than they can deliver. That does not mean the proponents are lying, but they may be too optimistic.
The private market handles the difficulties of innovation by giving the consumers a choice. Only those consumers who are willing to suffer the pains normally associated early adoption suffer those pains. If not enough consumers like an innovation, the innovation is abandoned. If enough consumers like an innovation, then eventually the bugs are worked out and the innovation may be more widely accepted.
Unfortunately, PWCS suffers the normal defect of any public school monopoly. PWCS does not offer its consumers the MI program as a choice. Instead, PWCS offers the MI program as the only choice. So the MI program has some grumpy consumers. Without real consumer choice, it is even difficult to determine whether the program has any real merit. If the School Board really wants to find out what parents think of the MI Program, the School Board will have to give parents a choice.
