In-service elementary mathematics teachers' views of errors in the classroom.

Educators generally agree that familiarity with students' conceptions and ways of thinking about mathematical topics are essential for teachers (e.g., Australian Education Council, 1991; Even & Tirosh, 2002; Fennema, Carpenter, Franke, Levi, Jacobs, & Empson, 1996; NCTM, 1991; 2000). In much the same vein, it is important for teacher educators to be familiar with teachers' conceptions and ways of thinking about mathematical topics and about pedagogical issues. In this paper we describe our initial explorations of elementary school teachers' perspectives on one central, pedagogical issue: The role of students' mathematical errors in mathematics instruction.

Students' errors were traditionally perceived either as signals of the inefficiency of a particular sequence of instruction or as a powerful tool to diagnose learning difficulties and to direct the related remediation (see, for instance, Ashlock, 1990; Fischbein, 1987; Greeno, Collins & Resnick, 1996). Several researchers, including Avital (1980) and Borasi (1987, 1992, 1994) suggested an alternative approach to the role of errors in the learning endeavor. Borasi (1987) argued that errors could be used to foster a more complete understanding of mathematical content, as a means to encourage critical thinking about mathematical concepts, as springboards for problem solving, and as a way to motivate reflection and inquiry about the nature of mathematics. Borasi pointed to the need to reconstruct the role of errors in mathematics instruction in order to make full use of the educational potential of errors. Similar opinions were presented by Avital (1980). He recommended to present error-triggering tasks (i.e., tasks that are known to elicit incorrect responses) in mathematical classes. Moreover, he argued that the best way to address common mistakes is to intentionally introduce them and to encourage a mathematical exploration of the related definitions, theorems and concepts.

Avital (1980) and Borasi's (1987) challenging approaches to the role of errors in mathematics instruction are the ones that we aimed to encourage in our three years, in-service elementary school mathematics specializing project. We were aware that knowledge about the participating teachers' perspectives on using errors in mathematical instruction could significantly contribute to our attempts to promote this viewpoint. In this paper we describe and discuss our initial attempts in this direction.

The Setting

In 2002 a national, public committee examined the situation of mathematics education in Israel and recommended that mathematics should be taught only by mathematics specialists from Grade 1 on. In light of this recommendation a massive, three-year national program for specialized mathematics teachers for elementary schools started in 30 institutes for higher education in 2002. The principals of about one third of the elementary schools in Israel were asked to recommend three to five teachers that would participate in the course and become the mathematics specialists in their schools. This program is now in its second year.

Tel-Aviv University is one of the institutes that participate in this endeavor. The sessions (30 weekly meetings of four hours each year) were mainly devoted to enhancing the participants' own understanding of mathematical concepts and structures. Alongside the weekly sessions, we conducted individual interviews and small group meetings in which we discussed specific difficulties that individual participants faced and their views of various mathematical and pedagogical issues.

We developed, for this course, materials that were aimed to encourage the participants to come up with different solutions to the mathematical tasks and to examine them in light of the related concepts, rules and definitions. The tasks were designed to stimulate participants to pose questions such as: "Why is this so?", "How can we justify this?" "Is this explanation mathematically sound?" Participants were also given home assignments which they submitted every other week. The mathematical content that was addressed in a substantial number of sessions was fractions. This was a major reason for our choice to use this content to address the teachers' viewpoints regarding the role of errors in mathematics instruction.

The Simplifying-fraction Expressions Activity (SEA)

The activity that we developed for the purpose of this study included six parts, each of which explored some aspects of the teachers' views concerning the role of errors in mathematics instruction (see a brief description of the activity in the Appendix). The mathematics entities in SEA are fraction expressions (e.g., [6 + 4]/[12 + 8], [7 + 5]/[14 + 20]) and the mathematical tasks center around simplifying such expressions. We chose these entities and these types of tasks for five main reasons. First, discussions on simplifying fraction expressions and on the related errors are relevant to elementary school mathematics instruction because fractions and operations with fractions are among the central topics in the elementary school curriculum in Israel. Second, simplifying expressions are error-triggering tasks (Tsamir and Tirosh, 2003). We found that a substantial number of elementary school students tend to err on these tasks and that they use several inadequate strategies to simplify fraction expressions, (2) including addition-like (e.g., [7 + 5]/[14 + 20] = [7/14] + [5/20] = 1/2 + 1/4 = 3/4) and reduce-then-tops-by-bottoms (e.g., [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]). These and a number of other strategies that are commonly used by elementary school students could be used as springboards for rich mathematical explorations. Third, the chosen tasks illustrate cases where the application of erroneous considerations may yield correct solutions (e.g., in the case of [6 + 4]/[12 + 8], the application of reduce-then-tops-by-bottoms). Fourth, a substantial number of sessions in our course were devoted to fractions. Consequently, we assumed that most participants would correctly simplify the fraction expressions. Fifth, during the sessions on addition of fractions the teachers often volunteered remarks regarding their students' tendency to incorrectly "add the tops and add the bottoms". We therefore assumed that most teachers would feel that the tasks included in SEA are, indeed, error-triggering.

We chose to focus in this paper on two main issues: (1) the participants' views of the merit of intentional presentations of error-triggering tasks in elementary school classes, and (2) their perspectives regarding the soundness of the deliberate presentations of incorrect responses in such classes. Of the different types of tasks that are included in SEA, two are particularly relevant for examining these issues. The first type of task was aimed at revealing the participant teachers' views regarding the presentation of error-triggering tasks in elementary schools (Tasks 4.1b, 4.2b and 4.3b). The second attempts to explore the participants' viewpoints regarding the intentional presentation of errors in the classroom (Task 5.2 and 6.2). In the results section, we shall mainly refer to the participants' responses to these tasks.

Results

In this section we report on the participating teachers' perspectives regarding an intentional presentation of error-triggering tasks in elementary school classes and their attitudes towards a deliberate introduction of incorrect responses to such expressions. We should, however, first note that about half of the participants (14 out of 27) fulfill two requirements that we viewed as essential for a meaningful discussion on these issues: They solved all the simplifying tasks and expected that some students in elementary schools would err when simplifying these expressions. In fact, all but one of these teachers listed at least two types of errors that they expected students to make on these expressions (the other teacher wrote one common error to each expression). The two most prevalent types of errors that the teachers listed were indeed those that were most frequently made by elementary school students: Addition-like and Reduce-then-tops-by-bottoms (3). From now on, we shall report on the responses of these 14 participants.

Should Error-Triggering Expressions be Presented in Class?

The reactions of the 14 teachers expressed varied opinions, ranging from an unconditional recommendation for presenting the tasks in elementary school classes: "Yes!!! I'll try it tomorrow in my class" (I6); "Presenting such activity is a MUST" (I7) to categorical objections. "These tasks should not be presented in elementary school classes" (I20), and "These tasks are too confusing for elementary school students" (I8). More hesitant voices, both for and against the presentation of such tasks, could be perceived in between these two extremes: "Yes, but ..." or "No, unless...." As can be seen from Table 1, most teachers (8 out of 14) voted for presenting the expressions in elementary school classes, fewer (4) stated some conditions under which they would present the expressions, and two teachers objected to their presentation. The teachers used mathematically-oriented (i.e., explanations that related to the mathematical potential of these tasks), learner-oriented (i.e., explanations that addressed students' difficulties) and a mixture of mathematically-oriented and learner-oriented justifications to defend their positions.

These Error-Triggering expressions should be presented. Of the eight teachers who recommended to present simplifying fraction expressions in elementary school classes, six discussed their mathematical potential. Teacher 13, for instance, stated that "these expressions provide me, as a teacher, with an opportunity to discuss the role of the fraction-line, to explain that we should first solve the expression in the numerator, then the expression in the denominator, and only then to divide ...". Several teachers recommended using these tasks as a means to elicit fruitful discussions on the differences between addition and multiplication of fractions. Teacher 14, for instance, stated, "I shall use this task to talk about the differences between addition and multiplication of fractions and to distinguish between what is allowed in each of these operations. I shall emphasize that in multiplication it is OK but in addition it is forbidden ...." Teacher 16 expressed a more general stand on the issue of presenting error-triggering tasks in elementary classes: "I am all for presenting tasks that elicit mathematical discussions and debates." She concluded by stating, "I'll use it tomorrow in my class." Two of the teachers who recommended to present these tasks in elementary school related both to their mathematical potential and to the importance of presenting them from the learner's perspective. Teacher I10 explained, "Students tend to err on such expressions. They tend to reduce the numbers as if in multiplication. It is important to expose them [the students] to these expressions and to discuss the differences between addition and multiplication of fractions." Teacher I7 discussed the mathematical potential of the task and then further argued that: "I know, from my experience, that many students incorrectly separate the expressions and get 'two fractions', they also 'reduce' the fractions, and do other incorrect things. It is part of our responsibility, as teachers, to know where students tend to err and to assist our students in coping with these errors, to help preventing them."

These Error-Triggering expressions should not be presented Two teachers argued that these expressions should not be presented in elementary classes. They related to both the complexity of these expressions and to the students' abilities. Teacher 18 wrote, "It is inappropriate to present such expressions. These tasks include a fraction line; it has to do with the order of operations and with the fraction line as parentheses. It is too complicated for elementary school students; it might confuse them with addition." Teacher I20 analyzed the tasks in much the same way and concluded, "It is too demanding for fifth or sixth graders."

These Error-Triggering expressions should be presented only if ... Two of the four teachers in this group listed several mathematical topics that they viewed as preconditions for presenting the tasks. Teacher 116 recommended "to give such expressions only after the students know that the fraction line is actually division," and Teacher I9 stated, "Yes, but only after the students have finished their studies of operations with simple fractions and have shown mastery in this respect. Otherwise it would cause confusion, frustration and uncertainty." The two other teachers in this group differentiated between more advanced and less advanced learners and recommended to present these tasks only to the more advanced ones. Teacher I12 stated, "Only to the good ones, because they know the rules and it will not confuse them" and Teacher I18 said, "It is good for those that are really good in mathematics. It is really bad for those that are not doing so well, they will be confused."

Should Errors be Presented in Class?

Only five teachers unconditionally advocated presenting the assignments that included several incorrect strategies for simplifying fraction expressions in elementary school classes. Six teachers recommended presenting such assignments only if certain conditions are fulfilled and three strongly objected to such presentations (see Table 1). We shall briefly describe how these teachers explained these responses.

Teachers should intentionally present such errors. The five teachers in this group described the mathematical potential of presenting incorrect responses to simplifying fraction expression tasks (e.g., it provides an extremely challenging opportunity to discuss the role and the meaning of the fraction line, the differences between addition and multiplication, the meaning of the numerator and the denominator, the conditions for reducing a fraction). Two teachers (I7, I10) described the mathematical potential of these tasks and then related to the learner's perspective. Teacher I10 noted, "This activity could lead to a mathematical discussion that would sharpen and deepen the students' knowledge of the role and the meaning of the fraction line and of operations with fractions. I can learn from it about the conceptions of each of the students in my class and to help each of them." Teacher I7 expressed a quite unique position: "It is important to show the students, right from the start, that this [using the addition-like strategy] is incorrect. In fact, I have been doing this, and my experience shows that it is especially important for the weak students because they actually make such mistakes. If I won't show the students these erroneous ways, they might think that such solutions are correct, that it is possible to solve tasks in these ways." Teacher I7 was the only teacher who advocated presenting this task for reasons related to "weak" students.

Teachers should not present such errors. Of the three teachers who expressed strong opinions against using the activity that included erroneous strategies for simplifying fraction expressions, one stated that "this activity is very difficult for elementary school students (I8). Another teacher (I20) also stated that this activity is "too demanding" and went on to suggesting an easier version of the task, starting with addition of fractions: "I prefer to ask the students if 1/2 + 1/2 = [1 + 1]/[2 + 2] = 2/4 = 1/2." The third teacher (13) expressed a more general position: "I do not teach mistakes. If a student makes mistakes, we discuss them."

Teachers should present such errors, only if ... The six teachers in this group expressed less definite positions than those in the other two groups. Five advocated to present these activities only to the more advanced students (I4, I9, I12, I16 and I18) and one teacher (I14) recommended to "use this activity only after ensuring that the students have mastered the operations with fractions."

Summing Up, Looking Ahead

In this paper we separately addressed two questions relating to using errors in the classrooms: that of introducing error-triggering tasks in elementary school classrooms and that of deliberately introducing errors in these classes. A more general look at the teachers' responses to these two questions suggests that those that were in favor of presenting such tasks described the mathematical merit of the tasks while those that were against such presentations related to the difficulties that students are likely to face when attempting to solve the tasks (see Table 1). These differences in the reasoning given by those who were "for" and those who were "against" are clearly one of the issues that we would like to pursue in our future work, e.g., Is there a similar trend among middle school and high school students? Is this a general phenomenon or is it specific to the content and/or to the structure of SEA?

Our results show that the participating teachers expressed different opinions on the central issue that was addressed in our paper, namely: Should teachers present error-triggering tasks and errors in their elementary school classes or shouldn't they? Two other, less salient issues on which the teachers' expressed different opinions were: Who are the students that would benefit from presenting such tasks (e.g., all students, only the most advanced students, only the less advanced students), and what might be an ideal timing for presenting such activities (e.g., before teaching addition of fractions, as a summative activity)? We surveyed the mathematics education literature in an attempt to find research-based answers to these issues, and got the impression that these issues have so far hardly been addressed. This led to the question that we posed in the title of our paper, namely: What do we, researchers and teachers, know about using errors in the classrooms? And finally we call for more research on ways of making full use of the educational potential of errors in mathematics classes.

References

Ashlock, B. (1990). Error patterns in computation. Columbus, OH: Merrill.

Australian Education Council. (1991). A national statement on mathematics for Australian schools. Melbourne: Curriculum Corporation.

Avital, S. (1980). What can be done with students' errors? Sevavim (15). [In Hebrew].

Borasi, R. (1987). Exploring mathematics through the analysis of errors. For the Learning of Mathematics, 7, 2-8.

Borasi, R. (1992). Learning mathematics through inquiry. Portsmouth, NH: Heinemann Educational Books.

Borasi, R. (1994). Capitalizing on errors as "springboards for inquiry": A teaching experiment. Journal for Research in Mathematics Education, 25, 166-208.

Even, R., & Tirosh, D. (2002). Teacher knowledge and understanding of students' mathematical learning. Handbook of international research in mathematics education (pp. 219-240). Hillsdale, NJ: Erlbaum.

Fischbein, E. (1987). Intuition in science and mathematics. Dordrecht, The Netherlands: Reidel.

Fennema, E., Carpenter, T., Frankie, M., Levi, L., Jacobs, V., & Empson, S. (1996). A longitudinal study of learning to use children's thinking in mathematics instruction. Journal for Research in Mathematics Education, 27, 403-434.

Greeno, G., Collins, M., & Resnick, L. (1996). Cognition and learning. In D. Berli'er & R. Calfee (Eds.), Handbook of educational psychology (pp. 15-46). New York: Macmillan.

National Council of Teachers of Mathematics [NCTM]. (1991). Professional standards for teaching mathematics. Reston, VA. Author

National Council of Teachers of Mathematics [NCTM]. (2000). Principals and standards for school mathematics. Reston, VA. Author

Tsamir, P., & Tirosh, D. (2003). Elementary school students' thinking about fractions. Unpublished manuscript [In Hebrew].

Appendix

The following parts of the questionnaire were given to the teachers one at a time, so that only after handing in one part did they receive the next.

Part 1

1.1 Solve in different ways: [6 + 4]/[12 + 8]

1.2 Here is a solution given to this expression by some students: [6 + 4]/[12 + 8] = 1/2 In your opinion, what are the way/ways that was used by those who gave this solution?

1.3. This expression was given in a national, 6th grade test. In your opinion, what percentage of the students in your class (or in a class of 6th graders in your school) answered correctly? ______

Part 2

2.1. Solve in different ways: [7 + 5]/[14 + 20]

2.2. Here is a solution given to this expression by some students: [7 + 5]/[14 + 20] = 1/3 In your opinion, what are the way/ways that was used by those who gave this solution?

2.3. This expression was given in a national, 6th grade test. In your opinion, what percentage of the students in your class (or in a class of 6th graders in your school) answered correctly? ______

Part 3

3.1. Solve in different ways: [3 + 50]/[6 + 100]

3.2. Here is a solution given to this expression by some students: [3 + 50]/[6 + 100] = 1/2 In your opinion, what are the way/ways that was used by those who gave this solution?

3.3. This expression was given in a national, 6th grade test. In your opinion, what percentage of the students in your class (or in a class of 6th graders in your school) answered correctly? ______

3.4. In your opinion, what can one learn from Parts 1,2 and 3?

Part 4

4.1. Consider the expression: [6 + 4]/[12 + 8]

4.1.a. In your opinion, what are the common errors that students will make when solving this task?

4.1.b. In your opinion, should this expression be given to students in a class that is studying fractions? Why?

4.2. Consider the expression: [7 + 5]/[14 + 20]

4.2.a. In your opinion, what are the common errors that students will make when solving it?

4.2.b. In your opinion, should this expression be given to students in a class that is studying fractions? Why?

4.3. Consider the expression: [3 + 50]/[6 + 100]

4.3.a. In your opinion, what are the common errors that students will make when solving it?

4.3.b. In your opinion, should this expression be given to students in a class that is studying fractions? Why?

Part 5

5.1 Consider each solution, determine if it is right/wrong and explain why:

5.1a. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] Correct/ Incorrect: If incorrect--what is the mistake?

5.1b. [7 + 5]/[14 + 20] = 12/34 = 6/17 Correct/ Incorrect: If incorrect--what is the mistake?

5.1c. [7 + 5]/[14 + 20] = 7/14 + 5/20 = 1/2 + 1/4 = 3/4 Correct/ Incorrect: If incorrect--what is the mistake?

5.1d. [7 + 5]/[14 + 20] = [7/[14 + 20]] + [5/[14 + 20]] = 7/34 + 5/34 = 12/34 = 6/17 Correct/ Incorrect: If incorrect--what is the mistake?

5.1e. [7 + 5]/[14 + 20] = [[7 + 5]/14] + [[7 + 5]/20] = 12/14 + 12/20 = 6/7 + 3/5 = 9/12 = 3/4 Correct/ Incorrect: If incorrect--what is the mistake?

5.2 In your opinion, should this Part be given to students in a class that is studying fractions? Why?

Part 6

6.1 Consider each solution, determine if it is right/wrong and explain why:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

6.1a. Correct/Incorrect: If incorrect--what is the mistake?

[3 + 5]/[6 + 10] = 8/16 = 1/2

6.1b. Correct/Incorrect. If incorrect--what is the mistake?

3 + 5/6 + 10 = 3/6 + 5/10 = 1/2 + 1/2 = 1

6.1c. Correct/Incorrect: If incorrect--what is the mistake?

[3 + 5]/[6 + 10] = [3/[6 + 10]] + [5/[6 + 10]] = 3/16 + 5/16 = 8/16 = 1/2

6.1d. Correct/Incorrect: If incorrect--what is the mistake?

[3 + 5]/[6 + 10] = [[3 + 5]/6] + [[3 + 5]/10] = 8/6 + 8/10 = 4/3 + 4/5 = 8/8 = 1

6.1e. Correct/Incorrect: If incorrect--what is the mistake?

6.2 In your opinion, should this Part be given to students in a class that is studying fractions? Why?

Pessia Tsamir and Dina Tirosh

School of Education, Tel Aviv University

(1) A previous version of this paper was presented in the International Symposium: Elementary Mathematics Teaching, August 2003, Prague.

(2) The term "strategy" denotes the way or the algorithm that is used to simplify the expression, "answer" designates the result reached by applying this strategy, and "solution" stands for both the strategy and the answer.

(3) Note that this strategy may yield correct answers in some cases (see, for instance, Task 1.1).

Table 1 Responses to Present Error-triggering-expressions and to Present
Errors

Present Error-Triggering Expressions Present Errors

YES                           8 Teachers               5 Teachers
  Mathematical perspective    I2, I3, I4, I5, I6, I14  I2, I5, I6
  Learner perspective         --                       --
  Mathematics & Learners      I7, I10                  I7, I10

ONLY IF ...                   4 Teachers               6 Teachers
  Mathematical preconditions  I9, I16                  I14
  Learner perspective         I12, I18                 I4, I9, I12, I16,
                                                         I18
  Mathematics & Learners      --                       --

NO                            2 Teachers               3 Teachers
  Mathematical perspective    --                       --
  Learner perspective         --                       I3, I8, I20
  Mathematics & Learners      I8, I20                  --

Note. The table presents (a) the numbers of teachers whose answers were:
"Yes", "Only if" and "No", (b) the identity of the teachers (like 12)
who used mathematical perspective, learners' perspective, or both, in
their justifications.