to 30 years of Research on the Equals Sign

1
to 30 years of Research on the Equals Sign

• Mary Margaret Capraro, Robert M. Capraro, Shirley
  Matteson
• Texas AM 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

2
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?

3
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).

4
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 ?).

5
Historical Perspective

• Operational vs. relational
• Individualized understanding
• Cognitive development concerns
• International studies
• Older students
• The teachers role
• Methodological limitations

6
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).

7
How do U.S. teachers edition textbooks address
the issue of the equal sign?

• Methodology
• Textbook series (Grades 1-6)
• 1974-1976
• 1985
• 1991-1992
• 1998-1999
• Instrument

8
Analysis

• Analysis
• Symbols
• - less than 1 page on average (.87)
• lt, gt - 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 XY ? Z, complex
  equalities, complex inequalities or other problem
  types.)

9
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)

10
How do mathematics methods books in the U.S.
present the equal sign?

• 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)

11
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.

12
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.

13
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

14
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.

15
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.

16
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).

17
Comparisons Between Chinese and U.S. 6th Grade
Students Concerning the Concept of Equality

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

18
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?

19
Do elementary children still interpret the
sign as an operator?

• Participants
• Randomly selected representative samples
• U.S. sixth graders (N105)
• Chinese sixth graders (N145)

20
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

21
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

22
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.

23
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 lt
  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.

24
Results - Is the equal sign misconception present
in China?

• Question 1 Question 2 Question 3
  Question 4
• 6 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 47.6 105
• Chinese Grade 6 98.6 96.6 98.6 97.9 91.7 145

25
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 lt 0.001 R2 0.471)
• Q2 (F 140.278 p lt 0.001 R2 0.473)
• Q3 (F 121.592 p lt 0.001 R2 0.438)
• Q4 (F 33.165 p lt 0.001 R2 0.175)

26
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 lt
  0.001)

27
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.

28

• This presentation is available at
• http//coe.tamu.edu/rcapraro/200720presentations
  /