Developing good responses to students' errors.

Abstract

This paper describes an activity involving pre-service elementary teachers. The exercise explored in this activity required pre-service teachers to reflect upon their own K-12 mathematics experiences and to determine what made them uncomfortable as mathematics learners. While the exercise revealed common negative mathematics experiences among this group, the brainstorming activity aimed at generating newer ways of correcting mathematics errors proved to be more difficult than first imagined by these pre-service teachers. What can be said about this activity is that it raised the level of awareness about the importance of teacher responses on the part of pre-service teachers.

Introduction

Among school subjects, mathematics is particularly associated with anxiety. In fact, mathematics anxiety has been a significant research topic for a long time. Although the research subjects vary from elementary school students to pre-service teachers (e.g., Newstead, 1998; Taylor & Fraser, 2003; Trujillo & Hadfield, 1999), the body of research mainly focuses on identifying possible causes and strategies to prevent or reduce high levels of mathematics angst. One of the repeatedly reported causes of mathematics anxiety is the pressure to find 'one correct answer using one correct procedure' (e.g., Baroody, 1987; Greenwood, 1984). This pressure prevents students from actively participating in class discussion and leads to drill-driven instruction and emotional tensions. Consequently, research studies suggest that teachers need to provide a safe environment conducive to learning by accepting multiple ways of solving a problem and encouraging students to focus on their thinking process rather than on a final answer (Faivilling, 2001; Hackworth, 1992; Hembree, 1990). This suggestion is also in line with the National Council of Teachers of Mathematics' (2000) emphasis on effective teaching, which urges teachers to "establish and nurture an environment conducive to learning mathematics" by encouraging discussion and collaboration (p.17).

The purpose of this paper is to describe an action research activity conducted in a pre-service elementary mathematics methods course. First, the instructor asked students to reflect upon their own K-12 experience in mathematics classrooms and to provide statements that outlined both positive and negative events. Each student then shared his or her set of former experiences, with special attention paid to those pre-service teachers considered to be negative. This information formed the basis for an in-depth classroom discussion on how teachers can reduce mathematics anxiety and create a classroom climate that helps to promote mathematics learning. The classroom discussion contained two instructional objectives: 1) For pre-service teachers to view their earlier K-12 mathematics experiences through the adult lens of a teacher. In other words, now that these pre-service teachers are about to embark on their own professional journey, how do they view their childhood mathematics experience from an adult perspective? 2) Once these pre-service teachers identified their own negative childhood mathematics experiences, how will these reflections change or transform their own teaching? The important question is to what extent can pre-service teachers" reflections improve student experiences in the mathematics classroom?

Why Elementary Pre-service Teachers?

According to McLeod (1992), the critical age for the development of attitudes and emotional reactions towards mathematics is between 9 and 11. Some unhappy experiences in this period turn into negative attitudes and anxiety, and they are difficult to change throughout the rest of the school life and may even persist into their adult life (Newstead, 1998). It is also reported that mathematically anxious teachers have a tendency to use more traditional teaching methods which focus on procedural aspects of mathematics, and their mathematics anxieties tend to be transmitted to their students (Buhlman & Young, 1982). Thus, it is imperative to prepare the elementary pre-service teachers to become a self-confident teacher who can possibly influence student attitudes towards mathematics.

Why Teachers' Responses?

Although many open-ended problems can be utilized in the mathematics instruction, there is a general assumption about mathematics--that is, the goal of a mathematics problem is to find a correct answer. In particular, when the problem is a typical computation, it is obvious whether the answer is right or not. However, it is embarrassing for the student who makes an error if the teacher's only response is his or her wrongdoing. This is the best way to kill students' interest, to stop conversation, and to create anxiety in the mathematics classroom. Thus, it is important for teachers to develop appropriate follow-up responses to students' errors which are conducive to the productive discussion.

Method

Participants A total of 21 elementary pre-service teachers who enrolled in a mathematics methods course participated in the activity. All of them completed four mathematics courses and three educational foundations courses prior to this course. Most of them were taking other methods courses at the same time and were planning to graduate within the following two semesters.

Context The activity lasted for one and a half hours and consisted of three parts: (1) reviewing students' computation errors, (2) sharing past school experiences and (3) developing responses.

(1) Reviewing students' computation errors Children's computation errors from previous research (Ashlock, 2001; VanLehn, 1990) and the instructor's work with elementary children were presented. All the examples were "systemic errors which appear to stem from consistent application of a faulty method, algorithm, or rule ... based on flawed knowledge" (VanLehn, 1990, pp. 14-15). Pre-service teachers were asked to identify the error patterns and to discuss the possible causes.

(2) Sharing past school experiences Pre-service teachers were asked to recall their teachers' memorable responses to computation errors, either positive or negative. Each student wrote an episode and shared it with the others.

(3) Developing responses The instructor showed an example of addition error patterns which resulted from unnecessary regrouping processes. See issue website http://rapidintellect.com/AEQweb/fal2005.htm Students were asked to generate as many various responses as they could other than "right-wrong" statements.

Results

Throughout the review process, pre-service teachers noticed that a significant number of errors are associated with misunderstanding place value. Also, there was a consensus that it would be very hard to identify the error patterns and to offer interventions with one-shot standardized tests. Subsequently, pre-service teachers highlighted the need for sophisticated strategies that help teachers and students to communicate effectively with each other. The review of students' error patterns and the discussion on the need for sophisticated strategies led to the next question: "How were your computation errors handled by your teachers in your school life?" Only three out of 21 candidates reported positive episodes. The common theme in these positive episodes was teachers' patient attitudes and encouragement for further explanations as shown in the following excerpts:

* My 9th grade math teacher usually took the time to go over the problem again one on one no matter how many times it took.

* When I made a mistake in math class, my teacher always believed that I could fix it. She was really encouraging.

* Sometimes, I would solve problems without showing the work and made mistakes. The teacher explained to me about showing the process to prevent mistakes. It made me think about the process, not just the formula.

Five pre-service teachers responded that they could not recall any specific teachers' comments other than "correct" or "incorrect." The rest of pre-service teachers remembered unhappy events. Their experiences are categorized as follows: 1) unhelpful generic statements, 2) embarrassment, and 3) reinforcing procedures.

(1) Unhelpful generic statements

Three pre-service teachers recalled that their teachers were not angry about the incorrect answers, but their comments were not helpful at all in correcting the answer or relearning the concept. Those comments were generic ones that any layperson could say. For instance, one teacher candidate said, "I was really bad in geometry. I did badly on geometry problems. My teacher always said that I needed to study more. I knew that I was weak in math and needed to study more. She did not need to remind me of that. But I had no idea where I should start with what I had to study." Similar responses included: "You need to work more." "You need more practice."

(2) Embarrassment

Six pre-service teachers recollected embarrassing moments due to their teachers' inappropriate words or negative connotations. Some excerpts follow:

* I went to the board to complete an algebraic problem in my senior year. My answer was wrong. So, my teacher sent me back to the board to fix the problem. She looked angry. But I still did not understand. She sent another student to the board. I felt embarrassed and disappointed.

* In kindergarten, my teacher was teaching how to write our birthday in digits. I could not understand why June was a '6'. She became very frustrated with me and embarrassed me in front of the class.

* I was in third grade. I had to put a homework problem on the board. I was one of the last few kids who had to do it. Each kid in front of me kept giving the wrong answer, so the teacher was getting frustrated. When I went up, I miscalculated the problem. The teacher said, 'How many times do I have to tell you all how to do this?' That reaction was not good!

(3) Reinforcing procedures

Four pre-service teachers remembered their teachers' comments reminding them of algorithms, rules, formulas, or some mnemonic tricks. They stated that these comments helped them to fix the incorrect answer but not necessarily helped their understanding. The following are some excerpts:

* When learning how to tell time, I read the digital clock from right to left. So if it was '3:24,' I would say it was '4:237 The teacher just reminded me to read from left to right.

* I never understood 'Please Excuse My Dear Aunt Sally' [mnemonic device for the order of calculations--Parentheses, Exponents, Multiplication, Division, Addition, and Subtraction]. My teacher just told me to remember this phrase. Although we had some positive experiences, the majority of responses had negative aspects. One student commented on this result, saying:

* I shared one of the negative experiences. I cannot remember exactly, but I believe that many of my teachers utilized positive comments. However, somehow I remember negative experiences more clearly.

Developing Responses

Since pre-service teachers discussed the need for good responses while sharing their own experiences, the follow-up activity focused on avoiding negative comments. Each student created an average of 2-4 responses to the provided addition errors. Unlimited time was promised for this activity, but most of them put down their pencils in 10 minutes. Some students expressed the difficulties in making a variety of comments, questions, or other prompts for the error. Some common comments during this activity included: "It is more difficult than I thought." "I wrote two responses. I cannot think of any more." A total of 76 responses were classified into six categories: 1) suggesting retry, 2) praising the good part and suggesting retry, 3) asking others for help, 4) asking for explanations 5) asking for demonstrations using manipulatives, and 6) offering demonstrations or explanations. Pre-service teachers' intentions were to develop responses to reduce students' anxiety levels and to build the emotionally comfortable classroom climate. However, during the class discussion, they realized that some of their responses did not quite tit the original intention. The following section shows the examples of the pre-service teachers' responses and the follow-up discussions

(1) Suggesting retry

24 responses belong in this category. In these responses, the pre-service teachers suggested retrying or checking. Some examples include:

* Let's go over the problem. Always double check your work.

* Would you do it again?

* Try again. Is there anything wrong with the tens place of the first problem?

* Let's look at where you went wrong.

* Check out your answer again to make sure it is correct.

* Why don't you retry to fix the problem?

* Let's try again. Tell me, where did you start having trouble with the problem?

One of the discussion issues for this category was overt or covert implications. Many examples in this category implied that the answer was wrong. Some responses apparently provided such cues like "trouble", "wrong", or "fix" and other responses also strongly implied that 'you have to do it again because your answer is not correct.'

(2) Praising the good part and suggesting retry

6 responses started with praises and ended up with suggesting retry. Examples are the following:

* Good job! By the way, this part is unclear for me. Could you do it again?

* You did a good job trying! But I see something wrong. Let's see if you can fix it.

* Nice try. But let's try again.

During the discussion, some concerns were expressed. Simply saying, "Good job!" is not an appropriate praise. It could make students more contused. When praising, teachers need to be specific about what aspects of the solution were good.

(3) Asking others for help

This category has 8 responses that looked for someone who had different opinions. Examples are as follows:

* Can someone come up and help you check your answer?

* Did anyone get another answer?

The majority of pre-service teachers evaluated that this type of response was good because it encourages cooperative learning. However, it is necessary to build the good discussion culture first so that no one gets embarrassed when his or her answer turns out to be incorrect.

(4) Asking for explanations

25 responses asked the student to explain or justify the process. The following are some examples:

* Let's look at the problem and explain how you came up with this answer.

* How did you decide what needed to be regrouped?

* Can you tell me why you need to regroup in the tens place?

* Could you explain to me how you get your answer?

Some responses asked students to explain general process while others suggested students explain the specific procedures or concepts. Overall evaluation for this category was good. However, it was recommended for pre-service teachers to have strong subject matter knowledge in order to provide various specific prompts.

(5) Asking for demonstrations using manipulatives

In 5 responses, pre-service teachers offered manipulatives and asked the student to explain the process. Examples are as follows:

* Let's use the chip trading game to explain the regrouping process.

* Let's use the popsicle sticks to check the answer.

Since they were in the middle of taking a methods course, many of them were not familiar with the variety of manipulatives available. Pre-service teachers agreed that concrete materials would help students to think and express more comfortably and that teachers' understanding of concepts and the use of manipulatives are required on the teacher's part to utilize effective questions or other responses.

(6) Offering demonstrations or explanations In 8 responses, pre-service teachers attempted to re-teach or to remind about algorithms. The following are some examples:

* You don't always have to regroup every place value. Only use regrouping when needed.

* If the answer is more than 10 in each place, you can regroup. If not, just add the numbers and record.

* You added the first two numbers correctly. However, you regrouped too many times. The tens place did not need to be regrouped.

Some critiques pointed out the lack of students' involvement. Although the teacher could re-teach the concept, it possibly prevented the student from participating in the discussion since this is another way to say, "You are wrong." It was recommended to use two-way communication strategies before re-teaching.

Concluding Remarks

The body of research on mathematics education clearly shows that students at all levels suffer from "math anxiety" to some degree and urges educators to search for strategies to improve mathematics learning. In response to this need, the activity described in this paper intended to initiate pre-service teachers into the art of reflective practice, which in turn will help to transform mathematics classroom from one that is anxiety ridden to one that promotes "safe" learning. The challenge for all teachers is to create a classroom climate conducive to mathematics learning and one in which students feel free to make mistakes, ask questions, and try out their problem solving abilities without tear of embarrassment or feelings of failure. In the end, teacher reflection can help to reduce the sturm und drang found in mathematics classrooms, which in turn may help to improve student attitudes toward this important subject.

References

Ashlock, R. (2001). Error patterns in computations: Using error patterns to improve instruction. Upper Saddle River, N J: Prentice Hall.

Baroody, A.J. (1987). Children's mathematical thinking: A developmental framework for preschool, primary, and special education teachers. New York: Teachers College Press.

Buhlman, B.J., & Young, D.M. (1982). On the transmission of mathematics anxiety. Arithmetic Teacher, 30(3), 55-56.

Faivillig, J. (2001). Strategies for advancing children's mathematical thinking. Teaching Children Mathematics, 7 (8), 454-459.

Greenwood, J. (1984). My anxieties about math anxiety, Mathematics Teacher 77, 662-663.

Hackworth, R. D. (1992). Math anxiety reduction. Clearwater, FL: H & H Publishing. Hembree, R. (1990). The nature, effects, and relief of mathematics anxiety. Journal for Research in Mathematics Education, 21,36-46.

National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author.

Newstead, K. (1998). Aspects of children's mathematics anxiety. Educational Studies in mathematics, 36, 53-71.

Taylor, B.A., & Fraser, B.J.(2003). The influence of classroom environment on high school students' mathematics anxiety. (ERIC Document Reproduction Service No. ED 476644)

Trujillo, K.M., & Hadfield, O.D. (1999). Tracing the roots of mathematics anxiety. College Student Journal, 33 (2), 219-232.

VanLehn, K. (1990). Mind bugs: The origins of procedural misconceptions. Cambridge, MA: The MIT Press.

Ji-Eun Lee, Auburn University-Montgomery

Ji-Eun Lee, Ed. D., is an Assistant Professor of Early Childhood, Elementary, and Reading Education at Auburn University-Montgomery.