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 Australian
pertaining to or originating in Australia.
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see Australian bat lyssavirus disease.
Australian cattle dog
a medium-sized, compact working dog used for control of cattle. Education Council, 1991; Even & Tirosh, 2002; Fennema, Carpenter, Franke Franke is a Swiss company involved primarily in the production of stainless steel and composite plastic sinks and taps. It is also involved in the making of kitchen systems such as cookers, kitchen accessories such as strainer bowls and food preparation platters. , Levi Levi (lē`vī), in the Bible.
1 Son of Jacob and Leah and eponymous ancestor of the Levites. His name appears infrequently—at his birth, when he and Simeon massacred the Shechemites out of revenge, when Jacob migrated to , Jacobs, & Empson, 1996; NCTM NCTM National Council of Teachers of Mathematics
NCTM Nationally Certified Teacher of Music
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NCTM National Capital Trolley Museum
NCTM Nationally Certified in Therapeutic Massage , 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 ped·a·gog·ic also ped·a·gog·i·cal
1. Of, relating to, or characteristic of pedagogy.
2. Characterized by pedantic formality: a haughty, pedagogic manner. issues. In this paper we describe our initial explorations of elementary school elementary school: see 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 diagnose /di·ag·nose/ (di´ag-nos) to identify or recognize a disease.
1. To distinguish or identify a disease by diagnosis.
2. learning difficulties and to direct the related remediation (see, for instance, Ashlock, 1990; Fischbein, 1987; Greeno, Collins & Resnick Resnick is a surname, and may refer to:
Process involved in finding a solution to a problem. Many animals routinely solve problems of locomotion, food finding, and shelter through trial and error. , and as a way to motivate reflection and inquiry about the nature of mathematics. Borasi pointed to the need to reconstruct re·con·struct
tr.v. re·con·struct·ed, re·con·struct·ing, re·con·structs
1. To construct again; rebuild.
2. 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 e·lic·it
tr.v. e·lic·it·ed, e·lic·it·ing, e·lic·its
a. To bring or draw out (something latent); educe.
b. To arrive at (a truth, for example) by logic.
2. incorrect responses) in mathematical classes. Moreover, he argued that the best way to address common mistakes is to intentionally in·ten·tion·al
1. Done deliberately; intended: an intentional slight. See Synonyms at voluntary.
2. Having to do with intention. introduce them and to encourage a mathematical exploration of the related definitions, theorems This is a list of theorems, by Wikipedia page. See also
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 in-service In-service training adjective Referring to any form of on-the-job training noun In-service training of an employee 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.
In 2002 a national, public committee examined the situation of mathematics education in Israel Education in Israel is an important part of life and culture in Israel. Israel has a developed and comprehensive education system, reformed over the years to adhere to secular trends in education. 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 spe·cial·ize
v. spe·cial·ized, spe·cial·iz·ing, spe·cial·iz·es
1. To pursue a special activity, occupation, or field of study.
2. mathematics teachers for elementary schools started in 30 institutes for higher education higher education
Study beyond the level of secondary education. Institutions of higher education include not only colleges and universities but also professional schools in such fields as law, theology, medicine, business, music, and art. 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 A group of characters or symbols representing a quantity or an operation. See arithmetic expression. NOT REPRODUCIBLE re·pro·duce
v. re·pro·duced, re·pro·duc·ing, re·pro·duc·es
1. To produce a counterpart, image, or copy of.
2. Biology To generate (offspring) by sexual or asexual means. IN ASCII ASCII or American Standard Code for Information Interchange, a set of codes used to represent letters, numbers, a few symbols, and control characters. Originally designed for teletype operations, it has found wide application in computers. ]). 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 erroneous adj. 1) in error, wrong. 2) not according to established law, particularly in a legal decision or court ruling. 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 in·ten·tion·al
1. Done deliberately; intended: an intentional slight. See Synonyms at voluntary.
2. Having to do with intention. 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.
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 ful·fill also ful·fil
tr.v. ful·filled, ful·fill·ing, ful·fills also ful·fils
1. To bring into actuality; effect: fulfilled their promises.
2. 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 HEIR, UNCONDITIONAL. A term used in the civil law, adopted by the Civil Code of Louisiana. Unconditional heirs are those who inherit without any reservation, or without making an inventory, whether their acceptance be express or tacit. Civ. Code of Lo. art. 878.
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 That which is unqualified or unconditional.
A categorical imperative is a rule, command, or moral obligation that is absolutely and universally binding.
Categorical is also used to describe programs limited to or designed for certain classes of people. objections. "These tasks should not be presented in elementary school classes" (I20), and "These tasks are too confusing con·fuse
v. con·fused, con·fus·ing, con·fus·es
a. To cause to be unable to think with clarity or act with intelligence or understanding; throw off.
b. for elementary school students" (I8). More hesitant hes·i·tant
Inclined or tending to hesitate.
hesi·tant·ly adv. 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 numerator
the upper part of a fraction.
see additive genetic relationship.
numerator Epidemiology The upper part of a fraction , then the expression in the denominator denominator
the bottom line of a fraction; the base population on which population rates such as birth and death rates are calculated.
denominator , and only then to divide ...". Several teachers recommended using these tasks as a means to elicit fruitful fruit·ful
a. Producing fruit.
b. Conducive to productivity; causing to bear in abundance: fruitful soil.
2. discussions on the differences between addition and multiplication multiplication, fundamental operation in arithmetic and algebra. Multiplication by a whole number can be interpreted as successive addition. For example, a number N multiplied by 3 is N + N + N. 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 In mathematics, specifically model theory, a class K of models for a first-order language L is an elementary class if there is some sentence : "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 In arithmetic and algebra, when a number or expression is both preceded and followed by a binary operation, a rule is required for which operation should be applied first. From the earliest use of mathematical notation, multiplication took precedence over addition, whichever side of a and with the fraction line as parentheses See parenthesis.
parentheses - See left parenthesis, right parenthesis. . It is too complicated for elementary school students; it might confuse con·fuse
v. con·fused, con·fus·ing, con·fus·es
a. To cause to be unable to think with clarity or act with intelligence or understanding; throw off.
b. them with addition." Teacher I20 analyzed an·a·lyze
tr.v. an·a·lyzed, an·a·lyz·ing, an·a·lyz·es
1. To examine methodically by separating into parts and studying their interrelations.
2. Chemistry To make a chemical analysis of.
3. 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 ful·fill also ful·fil
tr.v. ful·filled, ful·fill·ing, ful·fills also ful·fils
1. To bring into actuality; effect: fulfilled their promises.
2. 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 sharp·en
tr. & intr.v. sharp·ened, sharp·en·ing, sharp·ens
To make or become sharp or sharper.
sharp and deepen deep·en
tr. & intr.v. deep·ened, deep·en·ing, deep·ens
To make or become deep or deeper.
to make or become deeper or more intense
Verb 1. 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 relating to relate prep → concernant
relating to relate prep → bezüglich +gen, mit Bezug auf +acc 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 upon the side of; favorable to; for the advantage of.
See also: favor 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 Adj. 1. summative - of or relating to a summation or produced by summation
additive - characterized or produced by addition; "an additive process" 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.
Ashlock, B. (1990). Error patterns in computation Computation is a general term for any type of information processing that can be represented mathematically. This includes phenomena ranging from simple calculations to human thinking. . 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
This article is about reference works. For the subnotebook computer, see .
Fischbein, E. (1987). Intuition intuition, in philosophy, way of knowing directly; immediate apprehension. The Greeks understood intuition to be the grasp of universal principles by the intelligence (nous), as distinguished from the fleeting impressions of the senses. in science and mathematics. Dordrecht, The Netherlands: Reidel.
Fennema, E., Carpenter, T., Frankie, M., Levi, L., Jacobs, V., & Empson, S. (1996). A longitudinal study longitudinal study
a chronological study in epidemiology which attempts to establish a relationship between an antecedent cause and a subsequent effect. See also cohort 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 cognition
Act or process of knowing. Cognition includes every mental process that may be described as an experience of knowing (including perceiving, recognizing, conceiving, and reasoning), as distinguished from an experience of feeling or of willing. and learning. In D. Berli'er & R. Calfee (Eds.), Handbook of educational psychology The Handbook of Educational Psychology has been published in two editions, appearing in 1996 and 2006 respectively. Produced by Division 15 of the American Psychological Association (APA), the handbook broadly presents the theories, evidence and methodologies of educational (pp. 15-46). New York New York, state, United States
New York, Middle Atlantic state of the United States. It is bordered by Vermont, Massachusetts, Connecticut, and the Atlantic Ocean (E), New Jersey and Pennsylvania (S), Lakes Erie and Ontario and the Canadian province of : Macmillan.
National Council of Teachers of Mathematics The National Council of Teachers of Mathematics (NCTM) was founded in 1920. It has grown to be the world's largest organization concerned with mathematics education, having close to 100,000 members across the USA and Canada, and internationally. [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 manuscript, a handwritten work as distinguished from printing. The oldest manuscripts, those found in Egyptian tombs, were written on papyrus; the earliest dates from c.3500 B.C. [In Hebrew].
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.
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? ______
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? ______
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?
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?
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?
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 Tel Aviv University (TAU, אוניברסיטת תל־אביב, את"א) is Israel's largest on-site university.
(1) A previous version of this paper was presented in the International Symposium symposium
In ancient Greece, an aristocratic banquet at which men met to discuss philosophical and political issues and recite poetry. It began as a warrior feast. Rooms were designed specifically for the proceedings. : Elementary Mathematics Elementary mathematics consists of mathematics topics frequently taught at the primary and secondary school levels. The most basic are arithmetic and geometry. The next level is probability and statistics, then algebra, then (usually) trigonometry and pre-calculus. Teaching, August 2003, Prague.
(2) The term "strategy" denotes the way or the algorithm algorithm (ăl`gərĭth'əm) or algorism (–rĭz'əm) [for Al-Khowarizmi], a clearly defined procedure for obtaining the solution to a general type of problem, often numerical. 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.