≥ to 30 years of Research on the Equals Sign

Mary Margaret Capraro, Robert M. Capraro, Shirley Matteson

Texas A&M University

Eric Knuth - University of Wisconsin

Cheryl Lubinski & Albert Otto - Illinois State University

Research Presession National Council of Teachers of Mathematics

Atlanta, GA March 21, 2007

Symposium Focus • (1) What changes have occurred in students’ conceptual

understanding of equality over the past 30 years?

• (2) What pedagogical and textbook issues remain problematic concerning students’ conceptual understanding of equality?

• (3) What challenges remain to be addressed, especially when extended to algebraic equivalence, and

• (4) What are the implications for teacher preparation programs?

• (5) Where do we go from here as mathematics educators?

Introduction

Teachers and researchers have long recognized that students tend to misunderstand the equal sign as an operator sign, that is, a signal for “doing something” rather than a relational symbol of equivalence or quantity sameness

(Behr, Erlwanger, & Nichols 1980; Sáenz-Ludlow & Walgamuth, 1998; Thompson & Babcock, 1978; National Council of Teachers of

Mathematics (NCTM), 2000).

Early studies

• Early studies investigated the placement of the operator before or after the relational symbol, and the position of the placeholder for the answer. (Weaver, 1971, 1973)

• 2nd and 3rd grade students were more proficient with open-ended sentences in which the operation is on the left (i.e., 5 + 8 = ; 5 + = 13; + 8 = 13) rather than the right (i.e., = 5 + 8; 13 = c + 8; 13 = 5 + ).

Historical Perspective

• Operational vs. relational

• Individualized understanding

• Cognitive development concerns

• International studies

• Older students

• The teacher’s role

• Methodological limitations

Common misconceptions concerning the visual appearance of number sentences• Students preferred the operator on the left and the

equal sign on the right (most common textbook format) (Weaver, 1971, 1973).

• Students frequently rewrote problems such as 12 = 4 + 8 so that the answer appeared after the equal sign (Baroody & Ginsburg, 1983; Behr et al., 1976, 1980).

• Students rejected mathematics problems such as 8 = 8 because no operation was required (Baroody & Ginsburg, 1983)

• Number sentences such as 4 + 3 = 5 + 2 were also rejected (Carpenter & Levi, 2000; Collis, 1974).

How do U.S. teachers’ edition textbooks address the issue of the equal sign?

Methodology• Textbook series (Grades 1-6)

1974-197619851991-19921998-1999

• Instrument

Analysis

Analysis• Symbols

• “=” - less than 1 page on average (.87)• “<”, “>” - 8 pages on average (8.04)

• Problem Types• traditional presentations (a + b = or a + b = some other

place holder)• graphical or countable problems (with objects or a pan

balance), and • complex problems (different operations on one side of the

equal sign as compared to the other, i.e., variations on X+Y= + Z, complex equalities, complex inequalities or other problem types.)

Discussion of Results

• Definitions of the equal sign • Directions to the student• Other relational symbols• Teaching guidelines

Bell (1999) provided the following:“Equals” as “can replace” or “means the same as” is an easy idea. But children who have been through several years of schooling have more difficulty than might be expected using the “=” symbol for that idea. Research studies show that most older children reject 5 = 5 (they may say there is no problem), 4 = 2 + 2 (they may say that the answer is on the wrong side), 4 + 3 = 5 + 2 (they may say that there are two problems but no answers). (Grade 1, p. 202)

• Six methods books examined• Cathcart, Pothier, Vance, & Bezuk, (2006)• Hatfield, Edwards, Bitter, & Morrow, (2005) • Reys, Linquist, Lamdbin, Smith, & Suydam,

(2004)• Smith, (2001)• Tucker, Singleston, & Weaver, (2006) • Van De Walle, (2004)

How do mathematics methods books in the U.S. present the equal sign?

Results• Strategies ranged from

• nothing at all (Smith, 2001)

• a single paragraph (Cathcart, Pothier, Vance, & Bezuk, 2006; Reys, Linquist, Lamdbin, Smith, & Suydam, 2004; Van De Walle, 2004)

• to an activity (Tucker, Singleton, & Weaver, 2006)

• Seemingly it may be assumed by the authors of these textbooks that preservice teachers understand the issues related to the equal sign and the implications for their students.

Cathcart et al. (2006)

• One paragraph suggests using the equal sign interchangeably with words like “makes” (addition) and “leaves” (subtraction)

• approach allows students to obtain correct answer to simple + & - problems : leads to misconceptions such as 2 + 6 = + 5

• verbiage also can lead students to misconceptions of the equal sign as an operator.

Hatfield et al. (2005)

• Develops meaning of the four operation signs (add, subtract, multiply, and divide) but does not mention the equal sign

• section on multiplication: describe use of the “=” sign in one sentence and then substitute “are” for the “=” sign in the sentence immediately following implying a literal translation of “are” for “=”

Reys et al. (2004) and Van De Walle (2004)

• Both alert preservice teachers to the common misconception that the equal sign means, “the answer is coming”

• Both inform readers that using the calculator reinforces the equal sign misconception since the answer comes after the equal sign is pressed.

Tucker et al. (2006)

• This Equals That is presented as an activity to introduce the equal sign. Through the activity, the authors suggest saying to students that, “…we use the equal sign to tell how many blocks there are all together” (p. 98). Even though these authors do present an introduction to the equal sign, their explanation suggests that the answer follows the equal sign.

Counteractions to this Misconception

• A balance scale can help students develop the correct conceptual understanding of equality and the equal sign (Reys et al, 2004)

• Teachers should use the phrase “is the same as” (p. 139) instead of “equals” as students read number sentences (Van De Walle, 2004).

Comparisons Between Chinese and U.S. 6th Grade Students Concerning the

Concept of Equality

Cross-cultural comparisons lead to more explicit understanding of one’s own implicit theories about how children learn mathematics (Stigler & Perry, 1988).

Research Questions

A) Is the equal sign misconception present in China where instruction regarding the equal sign is presented as equivalence?

B) Is the misconception regarding the interpretation of the equal sign limited to the problem type 8 + 4 = + 7?

Do elementary children still interpret the “=” sign as an operator?

Participants• Randomly selected representative samples• U.S. sixth graders (N=105)• Chinese sixth graders (N=145)

Instruments

4 items• 6 + 9 =  + 4• + 8 = 12 + 5• + 3 = 5 + 7 = • 6 + 8 = 3 + 11 T or F

Pilot study• Interview for selecting two groups of students• Significant difference in the scores

Analysis

• For determining if item format was a source of confusion• used pair-wise correlations and a repeated

measures design

• For comparing U.S. and Chinese students’ performance • used a multivariate analysis

Results - Is the misconception limited

to the problem type 8 + 4 = + 5?

• Pairwise correlations among 4 items (except the 2nd box in “ + 3 = 5 + 7 = ”)• (R12 = 0.811, R13 = 0.688, R23= 0.706,

R14 = 0.523, R24= 0.528, R34= 0.485)

• It shows the fourth item is somewhat different.

Results - Is the misconception limited

to the problem type 8 + 4 = + 5?• Repeated measure (Bonferroni, alpha = .025)

• For first 3 questions: F (2, 208) = 1.799, p = 0.168 • For all 4 questions: F (3, 312) = 16.063, p < 0.001

• Therefore, U. S sixth graders’ misconception about the equal sign is not limited to the form 8 + 4 = + 5, the first three problems in our test reflected the same evidence of misunderstanding.

Results - Is the equal sign misconception present in China?

Question 1 Question 2 Question 3 Question 46 + 9 = + 4 + 8 = 12 + 5 + 3 = 5 + 7 = 6 + 8 = 3 + 11 T or F

1st Box 2nd Box

U.S. Grade 6 28.6 28.6 23.8 86.7 47.6 105Chinese Grade 6 98.6 96.6 98.6 97.9 91.7 145

Results - Is the equal sign misconception present in China?

• MANOVA was used to examine the effect of the outcomes of the four items (except the 2nd box in the 3rd item)

• Significant difference• Q1 (F = 138.628; p < 0.001; R2 = 0.471) • Q2 (F = 140.278; p < 0.001; R2 = 0.473)• Q3 (F = 121.592; p < 0.001; R2 = 0.438) • Q4 (F = 33.165; p < 0.001; R2 = 0.175)

Results - Is the equal sign misconception present in China?

A pairwise comparison test• Chinese sixth graders outperformed U. S.

sixth graders on all four questions (all p-values < 0.001)

Discussion

• Our findings are consistent with that of other researchers (28.6% vs. 31%, 32%). (Rittle-Johnson & Alibali,1999; Knuth, Stephens, McNeil, & Alibali, 2006).

• Our U.S. sixth grade sample and previous U.S. samples lag far behind Chinese sixth grade samples, which may be indicative of pedagogical issues and an answer for the disparate results.

This presentation is available at:

http://coe.tamu.edu/~rcapraro/2007%20presentations/