What Do Constructivist Teachers Do?

Steve Rhine
Willamette University

Karen Smith
Linn-Benton Educational Service District

Think of a time in which you were most engaged in learning. Not necessarily a time in school, but a time in which you really wanted to learn . . . . . Many people describe that type of high engagement in learning as "focused", "interested", "active", "wanting to fill a gap", "challenging", etc. This is the kind of learning that we educators desire for our students. It is the type of learning that we often remember most vividly. While it is probably unrealistic to believe that students can be this highly engaged every moment of every day, it is certainly something to aim for. In this article we aim for the same goal, high engagment which results in rich learning, as we examine what a constructivist classroom might look like and what a teacher does that makes it a classroom environment that is sensitive to how students construct understanding.

As we begin, we're caught between a rock and a hard place for two reasons. First, constructivism is not a theory of how to teach, but a theory of how students learn. The theory of constructivism does not prescribe a particular way of teaching. So when we discuss constructivist teaching, we imply instructional decisions that are based upon a teachers' belief in how students' construct meaning. Second, given our belief in constructivist learning principles, it would be inappropriate for us to just tell you what we think constructivist teaching might be. If we, as teachers, want meaningul learning to take place, you must be actively involved in constructing that meaning, with the teacher making instructional decisions based upon how and what meanings you construct. This is not an easy task on paper, but we'll try!

Before discussing what a constructivist-minded classroom might look like, it is appropriate to develop some understanding of the meaning of constructivism. Although constructivism underlies much of the NCTM's Curriculum and Evaluation Standards for School Mathematics (1989) there is considerable debate among academics as to the definition. Three perspectives, therefore, might help us clarify the term. First, an excellent resource for teachers on this topic is Brooks and Brooks' (1993) The Case for Constructivist Classrooms which describes constructivism in terms of knowledge and learning:

Drawing on a synthesis of current work in cognitive psychology, philosophy, and anthropology, constructivist theory defines knowledge as: temporary, developmental, socially and culturally mediated, and non-objective. Learning is a self-regulated process of resolving inner cognitive conflicts that often become apparent through: concrete experience, collaborative discourse, and reflection. (p. vii)

From this constructivist perspective, students' learning is impacted by context and is developed through interactions with others and inner reflection to resolve conflicts with students' beliefs and their experiences.

On the other hand, my favorite explanation of constructivism comes from a Calvin and Hobbes cartoon in which Calvin is imagining that he is fearless Spaceman Spiff, flying through the universe in his spaceship. As he struggles to evade the aliens his spaceship is hit and he seems doomed for a certain crash (see Figure 1 for the exciting conclusion). You may have Calvin's in your classroom who are in their own world. While Calvin's example may be in jest, the point here is that no matter how a teacher teaches, students are constructing unique meaning of the situation based on their prior experiences, knowledge, and in Calvin's case, state of mind. When teaching from a strictly procedural approach, in which students become capable of performing operations upon numbers and symbols, little about the student's conceptual understanding of the mathematics involved in the task is incorporated into the learning experience. A constructivist perspective considers it essential for the teacher to understand how a student is constructing meaning so that the teacher can be most effective in their instructional intervention--addressing misconceptions or gaps in students thinking or challenging them to extend their thinking.

Figure 1 (permission not yet applied for)

Finally, some basic principles of constructivism are not necessarily new. An ancient Chinese proverb also speaks to how we might think about constructivism:

Tell me and I forget

Show me and I remember

Involve me and I understand

The NCTM Standard's goal is for students of mathematics to develop procedural capability with conceptual understanding. When students are actively involved in their learning they develop greater depth of meaning. Richer understanding of the problem and its context allows students to not only perform tasks that solve the problem but understand what mathematical tools are most effective in addressing the mathematical issues raised in the problem.

Given a sense of constructivism, how it might appear in the classroom? Take a moment to read the description of two different classrooms in Figure 2. Resist the temptation to label the teachers as good/bad, me/not me, or effective/ineffective. Consider each teacher as very capable and effective with the approach that they choose. Yet, they are accomplishing very different things. To facilitate your reflection on the cases, consider what each teacher believes about different components of schooling: curriculum (What is taught?), instruction (How is it taught?), assessment (How does the teacher know what they learned?), learning environment (What is the context for learning?), choice (What power do students have?), diversity (How are alternate viewpoints incorporated?), motivation (Why are they learning/working?), background knowledge (What do teachers/students need to know to teach this way?), and discipline (How is control maintained?). The answers to each one of these components are not necessarily readily available from the text, so you might need to extend the description in your mind to answer them adequately. . . . . . .

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Insert Figure 2 Here

(now located as an Appendix below)

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Now that you have read the cases and considered the components of schooling, what do constructivist teachers do? As we read the case studies, these are some issues that arose for us in relation to the choices each teacher made:

·Curriculum:

-How are skills or facts balanced with conceptual understanding or process?

-What might be the role of the texts?

-What is the nature of the classroom tasks?

·Instruction

-What is the role of students' practice or homework?

-Who is talking when and why?

-What is the role of mathematical tools such as calculators or manipulatives?

-What kinds of questions does the teacher ask?

·Assessment

-When are students assessed? How?

-What are students assessed on?

·Learning environment

-What are students' relationships to each other and the teacher?

-How comfortable are students in taking risks--developing and communicating their hypotheses about mathematics?

·Choice

-Who has the power over the direction of the class? How? Why?

-How are tangents/students' suppositions valued by the teacher/other students?

·Diversity

-What might be the impact of students' culture on instruction?

-How are multiple perspectives/methods treated?

-Do all students have equal access to the mathematics?

·Motivation

-Are students intrinsically or extrinsically motivated to do the mathematics? Why?

·Background knowledge

-How are students' experiences and prior knowledge utilized?

-What do teachers need to know in order to teach in each way?

·Discipline

-How might classroom management issues be impacted by each style of teaching?

Hiebert (1997) describes the classroom as an integrated system consisting of multiple interrelated elements that work together to create a particular classroom environment for learning. Taken as a whole, they are much more than their individual parts. So, it is important to reflect upon each of the components above while considering their impact on other elements. For instance, if a teacher chooses to value multiple perspectives in the class, the types of problems, assessment, and her questions would probably change to create opportunities for students to develop and communicate those perspectives. The relationships between students would invariably change as they begin to see each other as people to learn from as well as the teacher or the text.

From a constructivist viewpoint, each person who reads the above cases will build unique meaning about what constructivist teachers do based on their prior experiences, knowledge, and attention to aspects of the case. However, Brooks and Brooks (1993) present five principles of teaching from a constructivist perspective: 1) Pose problems of emerging relevance to students: Teachers don't have to do rap everyday, but should consider problems that will gain relevance for students. For instance, I have done problems in classes in which I was not interested when I walked in the door, but were very engaging to me by the time I walked out. 2) Structure learning around primary concepts: The quest for essence: Conceptual understanding is the goal, although mathematical skills are essential to make use of that understanding appropriately Skills can be developed through a focus on conceptual understanding. 3) Seek and value students' points of view. As with Calvin, students may have very different interpretations of the mathematics that you do. Find out what they are thinking and make instructional decisions accordingly. 4) Adapt curriculum to address students' suppositions: One of the arts of teaching is determining when students' suppositions are worth them or the class exploring. Facilitating development of students' suppositions often means accomplishing the curriculum in a non-linear fashion. 5) Assess student learning in the context of teaching: Don't wait until the unit test to find out if students understand. Also, assessment within the context of teaching creates circumstances for instructional intervention at "teachable moments". In order for teachers to assess in the context of teaching, instruction must include opportunities for the teacher to interact with individuals.

Some justified criticism has been leveled at teachers who adopt constructivist minded teaching. There are those who say that students are now engaged and having fun in mathematics classrooms without the mathematics. The Third International Mathematics and Science Study (TIMSS) results support this impression. When identifying characteristics of reform minded teaching, over eighty percent of the teachers in the TIMSS study referred to something other than a focus on thinking. Only nineteen percent considered conceptual understanding as the goal of the reforms (Peak, 1996, p. 46). Similarly, Ball (1992) describes classrooms in which teachers have "magical hopes" that manipulatives and activities will teach students mathematics. Burrill (1997) adds that "You can have students in cooperative groups working on trivial tasks. You can use manipulatives to do rote, meaningless procedures. A teacher can walk around encouraging students but never check their work or their thinking." It is important to capture the interest of students and provide them with opportunties for communication and active learning. However, the most important question to keep in mind is what Burrill (1997) goes on to ask, "Where is the mathematics? In contrast to mathematically 'empty' instruction referred to above, constructivist teaching practices are those in which students' conceptual understanding is the heart of instruction. Teaching becomes the art of helping students bring the important mathematics out of their active engagement with challenging problems.

Finally, Airasian (1997) offers five cautions for those of us thinking about constructivist teaching: 1) A constructivist orientation will not make the same demands on teaching time as a nonconstructivist orientation. It takes time to explore students' thinking and allow them to construct deeper understanding of a concept. One criticism of American education that is coming out of the TIMMS international study of mathematics is that our curriculum is a mile wide and an inch deep. Perhaps discuss with your departments what you can leave out so you can spend more thoughtful time on certain concepts. 2) Constructivist instructional techniques do not provide the sole means by which students construct meanings. Students construct meaning whether you are lecturing or doing small groups--possibly an understanding (or lack thereof) completely different than the one you intended. How do you adjust your instruction with that understanding? 3) Recognize the difference between a theory of learning and a well-thought-out and manageable instructional approach for implementing it. There is no "constructivist teaching", only suggestions for facilitating students' building of conceptual understanding that is richer and more thoughtful, and instruction that is based upon what the students are thinking and doing. 4) The opposite of "one-right-answer" reductionism is not "anything-goes" constructivism. Right answers do exist. The question constructivism raises is what instructional impact is there in telling someone the right answer? Research has shown that not only is there little or no learning, but there can be negative impacts. Instead, the focus of instruction becomes what justification students have for their answers, particularly when multiple answers are possible. These justifications are compared to others' perspectives to determine the most effective processes and plausible answers--if not the correct answer. Multiple methods, however, are respected and encouraged.

A last thought is that evolution of instruction takes one curriculum piece at a time. There are things you can do in tomorrow's class that will make it more of a constructivist minded environment--such as questioning techniques and valuing students' perspectives. However, to change a classroom system takes focused energy on one unit at a time. Think about what you do already, give yourself time to evolve, and identify small steps you can take today to develop students' conceptual understanding.

References

Airaisan, P. & Walsh, M. (1997). Constructivist Cautions. Phi Delta Kappan, 78 (6), 444-449.

Ball, D. (1992). Magical Hopes: Manipulatives and the Reform of Math Education. American Educator, Summer. 14-18 (46-47).

Brooks, J. & Brooks, M. (1993). The Case for Constructivist Classrooms. Alexandria, VA: Association for Supervision and Curriculum Development

Burrill, G. (1997, April). President's Message: Show Me the Math! NCTM News Bulletin, 3.

National Council of Teachers of Mathematics (1989). Curriculum and Evaluation Standards for School Mathematics. Reston, Va.: The Council.

Peak, L. (1996). Pursuing Excellence: A Study of U.S. Eighth-Grade Mathematics and Science Teaching, Learning, Curriculum, and Achievement in International Context. (National Center for Education Statistics 97-198). Washington, DC: U.S. Government Printing Office.

Smith, K. (19··). Developmentally Appropriate Practices in Mathematics.

Figure 2

TEACHER A

When you visit a typical high school you overhear a teacher in a classroom introduce a lesson like this: "Get out your homework. Anyone have any questions? (Students call out numbers, the teacher or students do them on the board). Okay, today we are going to learn about graphing. There are graphs all around us, in newspapers, in your books, on television. They are used to determine how much your car insurance should be, how much you pay in taxes, and whether it is worth it for you to have surgery. They are part of almost everything we do. I'm going to make a graph. I will start with an equation, such as y = 3x + 4. 4 is called the "y-intercept" and 3 is called the "slope". I will plug in numbers for "x" and get a number for "y" and I will keep track of them in a chart. For example, when I plug in 1 for x I get 8 for y. When I plug in -2 for x I get -2 for y. You should plug in about 5 numbers, including 0, and usually some positive and negative numbers. Each of the pairs of numbers are called "coordinate pairs" and you just connect the points to get your graph of a line. Now I will do y = 2x - 1. (T shows class how to fill out the chart to get the points, then graphs the points, then draws the line.) Okay, turn to page 127 in your books and graph problems one through 10."

TEACHER B

In another high school classroom you might observe a teacher engaging the students in this way: "Each of you have brought examples of graphs in with you today and a reflection on how that was mathematics. Discuss with your neighbor the similarities and differences of the graphs you brought. Paste a graph into your math journal and write down answers to the following questions: What does the graph mean? What information can you get from it?" (Teacher walks around to monitor the conversations.) "Who would like to explain one of their graphs to the class?" (Students take turns explaining their graphs.) "What is something that is the same about each of the graphs presented?" (T leads discussion to make point that the graphs usually have two pieces of information, an x and y axis.) "There are many types of graphs. However, the type we will focus upon today is a line graph, similar to the stock market graph that Roberto brought in. Discuss with your partner how you think that graph was made. (T leads discussion of how graph was made, again focusing on the two pieces of information needed to draw the graph. T then draws the graph of y = x + 1 on the board.) "Read this graph for me. Tell me something about the information you get from this graph." (T leads discussion focusing on the fact that y is always one more than x. T then writes the equation y = x + 1 on the board. T then draws graph of y = x + 2 on the board.) "Again, read this graph for me. What information can you get from this graph?" ( T repeats process with y = 2x, y = 3x, and y = 4. T then draws graph of y = 3x + 4 on the board.) "With a partner, figure out the equation for this graph. When you are confident of your answer, compare with another pair and defend your choice. When you are confident of your choice, create a graph and challenge the other pair to discover the equation for the graph. Be sure you have evidence to support your equation matching your graph."