Unexpected Answers

Vibeke, a student teacher from Norway, plans to introduce multiplication. The teaching dilemma results from a teacher's inflexibility in dealing with unexpected student responses and the chaos that may result when the best laid schemes go awry.

Setting the stage

Vibeke is a student teacher who has been thoroughly socialized in child-centered education through her teacher training. She wants to promote the “whole child,” and she wants to create situations in the classroom where the students “experience success.” She puts the child at the center of her careful lesson preparation, and she expresses concern for “activity” and “experience” in her curriculum planning. She uses interactive teaching methods in all her classes.

The interactive teaching method is practiced by all child-centered educators. At the heart of this teaching method is a substantial contribution from the children to the development and progress of teachers’ explanations. In the hands of veteran child-centered educators it looks like an orderly and effortless process. It is, however, one of the most difficult teaching methods in the primary school, especially because of the uncertainty it can create in the classroom.

Vibeke, is on her last teaching practice (3 weeks) during her last year in teacher training college. She is concerned with whole-child development, that the children in her class feel they are “successful” and have something to “contribute.”

Planning to teach multiplication

Vivi has made a plan for the year, and Vibeke is at the point in her teaching practice when she should be introducing multiplication. When Vibeke makes plans she uses the textbook, the teachers’ guide, and other textbooks. She has had previous experience from her earlier teaching practices teaching second grade math and also multiplication. She does not rely on any notes that she had made in her earlier experiences. She relies on memory. Vibeke chooses to make a work sheet using ideas from the textbook she had previously used. Vibeke is clear in her planning, she puts the child in the center. She emphasizes child-centered activity, children’s experiences, and plans for a series of manipulative exercises. This is a new concept, and she wants to stimulate the children’s curiosity. She plans to gather the class of 22 children together on a carpet in the “cozy corner” of the classroom and demonstrate manipulative representations of multiplication. She explains: “I thought I could use the children and begin, for example, by asking two kids to come and stand by me to show the rest that each child has two eyes, and then they could write two plus two plus two and it is six… It is easier for them to understand when they are faced with concrete [representations], I think, and when they can use their own experiences like seeing apples in a shop and get to see it in front of them.”

Vibeke’s interest in getting children’s involvement in the lesson is the reason for her choosing an interactive teaching method. She values the method highly and plans all her teaching in such a way that she can engage the children in an interaction. More specifically she wants them to feel that they can make a contribution to the ideas and that they know “things.” She says: “It is to activate the children, and try to get them involved in as much as possible to reach an answer… Because if they just sit there and listen to me talk, they would get bored, and I know how easy it is to start thinking about something else and everything else becomes more interesting. [When using an interactive teaching method] you have to think and concentrate and be with it all the time.” The only thing Vibeke asks for advice about is communicating with the children. She specifically asks for Vivi to comment on this aspect of her teaching after she has finished her lesson. She does not ask for advice on any other aspect of teaching multiplication. She claims she feels secure with a careful lesson plan and that she is well prepared to face the unexpected.

Introducing multiplication

Vibeke introduces multiplication by reading a section from Pippi Longstockings, a popular children’s book by Astrid Lindgren. Pippi goes to school and learns about “plutification.” Next, Vibeke represents multiplication with manipulatives. She uses both an interactive teaching method and children as “manipulative objects” to represent multiplication. The class is sitting on the carpet in the cozy corner. Vibeke picks out three children. She makes them stand and face the rest of the class.

She tells the class that they have two eyes each. Per says that it is 2+2+2, and he is asked to write that on the board.

Olav raises his hand and says that it is “two times three.” Vibeke does not realize that Olav’s response is a mathematically correct answer for the problem she has represented. She tries to get him to say what she is after. Kari, at last, comes to her rescue and says that it is “three times two.” Kari is asked to write the problem on the blackboard. As Kari stands up and approaches the board, Vibeke says that “now we get to see what the times symbol looks like.” Kari writes the numbers correctly, but she writes the times symbol in the wrong place, it looks like a full stop.

Reflecting on the lesson

Afterwards when reviewing this episode on the video, Vibeke reflects: “I became so unsure when she placed the symbol so low down. How am I to tackle this without hurting her? How can I get them to say that it was right, but the symbol was in the wrong place … I expected that she would know how the writing was supposed to appear.” This expectation on the part of Vibeke makes the situation difficult. Several of the children want to correct it, but Vibeke says: “You can see that this is a times symbol.”

Vibeke returns to Olav who had said “two times three.” She uses pencils and children to explain “two times three.” Two more children are pulled out, and they get three pencils each. She explains her actions: “Because I said ‘three times two’ before Olav, I have to figure out a way to tell him that it is correct, but in a different way. At this point I was conscious of picking up the thread again in what he said.” Vibeke finds herself in a situation where she has to represent the commutative principle without having had the chance to think it through and plan. She believed she had to do it then and there because of Olav’s unexpected answer. Per gives the answer she was asking for, and he is asked to write his answer on the board. He does so, but puts the times symbol in the wrong place, just like Kari had done. Only after many comments from the children does Vibeke correct the mistakes.

Vibeke has the children write on the board several times and after a while it begins to look chaotic with numbers all over the place. She asks five new children to hold up their hands and show their fingers. She says: “Each kid has five fingers.” The children answer that they have ten fingers. Vibeke specifies that they have five fingers on each hand, and says “5+5+5+5+…” The children keep saying that they have ten fingers each. One says 10, another 20, while two say “five times ten” and “ten times five.” Vibeke observes herself on the video and reflects: “It is not surprising that I became confused… It didn’t work out what I had planned, and I became extremely surprised. What in the world am I doing, this is too much…they think so differently than I do.”

* This case is a slightly modified version of Nilssen, V., Gudmundsdottir, S. and Wangsmo-Cappelen, V. (1995). Unexpected answers: Case study of a student teacher derailing in a math lesson. Paper resented at the Annual Meeting of the American Educational Research Association, San Francisco, April.