Constructivism
After a few courses into my MET degree, I was tantalized by glimpses of constructivism seen in discussions with other students who were farther along in the program. In ETEC 512: Applications of Learning Theory to Instruction, I was able to study some of the theory behind constructivism, and in ETEC 530: Constructivist Strategies for e-Learning, I delved into constructivist theory and practical applications in depth.
Constructivist strategies involve collaboration, communication and active learning. These have allowed me to expand my vision of learning to create learning environments for students that involve group work, discussion and negotiation, and hands-on real life activities.
A new view of constructivism in the mathematics classroom
While in ETEC 530: Constructivist Strategies for e-Learning I wrote a paper entitled Constructivism in the Mathematics Classroom: Towards a New Approach. This paper examines three research articles about constructivist teaching techniques in the mathematics classroom, and then proposes an alternative constructivist strategy for early math classes based on work by V.V. Davydov. The three research studies, which looked a math anxiety levels in preservice teachers, achievement in the acquisition of basic math skills, and use with low-achieving students, seem to indicate that constructivist teaching techniques did not result in any gains over traditional teaching techniques. However, the work by Davydov shows that even very young students - as early as third grade - can grasp algebraic concepts when introduced to constructivist math teaching techniques in the first grade.
Looking back
One of the studies looked at the anxiety levels of preservice elementary teachers when taught math in either a constructivist or traditional manner. Constructivist strategies improved the levels of anxiety in the review course, but worsened them in the course where many new math concepts were being taught.
I have seen the anxiety with math students in my own classes. "Why don't you just tell us?" one student asked me. In a way, it's like a big secret that the instructor is keeping from the students, because the students know that you know the answer. Students who were used to "being instructed" resented this approach, and some became downright hostile. Another thing that adult students worry about is learning the "wrong" answer. In the math class that I taught in a one-year Teacher Assistant program, the students were always under a time crunch. Many of them had additional obligations such as caring for children or part-time work. Understandably, they didn't want to spend a lot of time looking for wrong answers or answers they might never find when the ultimate resource tool was standing right in front of them.
Why do we push constructivist techniques in math and science classes? Anecdotal evidence seems to indicate that students a generation ago were more well-prepared for the higher level courses that they were taking. Some faculty who participated in a study of technology in higher education noted that students were "poorly prepared (reflecting declining standards and a lack of 'basic skills')" (McGraw-Hill Ryerson, 2007, p. 30). Yet students a generation ago were most likely being taught in a transmissive manner. What has changed? I believe part of the reason is that years ago, you either got it or you didn't. Only the brilliant, the ones who didn't really need the teachers, went on to pursue further studies in mathematics or the sciences. Nowadays I think we place a greater value on more students understanding the concepts better rather than a few students excelling.
There are other reasons to use constructivist methods in our classes. The world outside of the school is changing, and teachers and instructors must keep up with it. If students are multi-tasking more rather than sitting and concentrating on one thing for a long period of time, then our instructional strategies need to adapt to this new reality. Students need to be more involved and engaged in their studies, which can be difficult when just sitting and listening. Constructivist teaching techniques can assist with these and other issues.
We are starting to place greater value on teamwork, rather than having a few people know all the answers. With the internet age and the ready availability of massive amounts of information, life becomes more a matter of knowing what to do with the information rather than having it all stored in one's head. There is a greater emphasis on problem-solving, both in school and in real life. In many cases this requires collaboration and negotiation skills which can be developed during one's stint in the educational system with constructivist techniques. Authentic learning, including project-based, problem-based, and case-based learning, becomes important in order to prepare students for real-life situations.
Looking forward
Constructivism is not a single theory but a set of theories about how people learn; it has been interpreted in many different ways. I want to continue to promote collaborative learning and using scaffolding in my classes. Learners have much to offer each other, and I believe that students will rise to the task when challenged if the necessary supports are put in place to help them succeed. However, I still agree with Schmittau (2004) that students must be guided toward the accepted algorithms. Even if they don't use them, they need to know what they are. Whether using a top-down or bottom-up approach, I believe that it is the instructor's responsibility to guide students toward a culturally accepted way of accomplishing a task, and this is the vision of a constructivist learning environment that I want to pursue.
Combining constructivism with online learning
I wrote Introduction to the Slopes of Lines in Coordinate Geometry: A Constructivist Lesson as the online constructivist lesson for my final project in ETEC 530: Constructivist Strategies for E-learning. It is based on the grade 10 principles of math stream in British Columbia.
From the introduction to the lesson, I noted:
This lesson uses White and Gunstone’s (1992) Prediction – Observation – Explanation (POE) model, as well as adhering to the stages of constructivist teaching described by Driver and Oldham (1986; in Matthews, 1994). The stages are Orientation; Elicitation; Restructuring of Ideas; Application of Ideas; and Review; ...
At the beginning of the lesson, the students look at some pre-drawn graphs and make predictions about their slopes. They then review how to graph ordered pairs and graph some lines running through the origin. Working in small groups, they find the equations of the lines and then try to find the relationship between the equations and the slopes. The application section involves designing playground equipment (a slide) for children of various ages. At the end of the lesson the students are asked to evaluate their and their groupmates' contributions and problem-solving strategies and analyze the lesson in terms of difficulty. Evaluation of the student occurs throughout the lesson and includes marks for participation, achievement, and analysis.
Looking back
As I was writing this paper, I was doubtful that it would ever be a realistic lesson for reasons of the synchronous vs. asynchronous nature of online courses and the timing of the lesson. I believe that collaborative work is a very powerful motivator for many people, and it may mean the difference between a student finishing a distance education course or not. However, the reason that many people take courses through distance education is that they can be flexible with their schedules and not have to be in a certain place at a certain time. Scheduling synchronous sessions in an online course can be a nightmare. To be part of a successful online math course, this lesson would most likely have to be adapted to take advantage of the asynchronous discussion areas rather than being posted as a synchronous lesson.
With regard to the timing of the lesson, I knew as I was writing it that the timing was tight and probably not very realistic given the variances in the students' abilities and available technology. A more realistic estimate of time for this lesson would require about double the time frame given.
I really enjoyed writing this lesson as this is the level of math that I like teaching. Also, because constructivism can seem to be such a nebulous topic, I appreciated having the POE and Driver and Oldham frameworks to follow as I developed the lesson. (This seems to be a constant theme with me: I do well when I have a framework to follow but struggle when no framework is given or I have to develop the framework myself, as noted in the November 9 entry of my learning journal.)
Looking forward
At first glance, math and science topics don't really lend themselves to constructivism. In many cases there is only one right answer, and there can be no negotiation. However, the path to that right answer is not so clear, and this is where constructivist teaching techniques can be used to engage and involve students in their own learning. Consider these two scenarios:
One: Each student sits silently at his or her own desk copying notes as the teacher stands up at the board explaining a math concept orally while writing formulae and drawing diagrams.
Two: Students create graphs on their graphing calculators, then move to meet with other students and compare graphs while trying to discover an explanation for what they are seeing on their calculators.
Although I believe good teachers use many strategies, the math class in my vision for learning looks much more like the second scenario than the first. The students get to move around purposefully in the process of doing their work; they get to analyze and discuss the math with their peers; they get to explain things rather than have it explained to them.
While my vision of my own future is cloudy at this point, I would be very interested in either teaching math at the grade 10 or 11 level or developing and teaching an online math course. With the constructivist teaching and learning strategies that I have begun to adopt and plan to explore further, the future looks very interesting.
References
Matthews, M. R. (1994). Constructivism and science education (Chapter 7). Science Teaching: The Role of History and Philosophy of Science. New York: Routledge.
McGraw-Hill Ryerson Ltd. (2007). Technology and student success in higher education. Retrieved November 27, 2006, from http://www.utorinto.ca/ota/Technolgyand StudentSuccess.pdf
Schmittau, J. (2004). Vygotskian theory and mathematics education: Resolving the conceptual-procedural dichotomy. European Journal of Psychology of Education, 19 (1), 19-43. Retrieved February 22, 2006, from Academic Search Premier database.
White, R. & Gunstone, R. (1992). Prediction – Observation – Explanation (Chapter 3). Probing Understanding. Philadelphia, PA: The Falmer Press.