elementary students recognition of algebraic structure; not all tasks are created equal

7/27/2019 Elementary students recognition of algebraic structure; Not all tasks are created equal

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Elementary students recognition of algebraic structure: Not all tasks are created equal

Abstract

This paper reports results from a written assessment given to 290 third-, fourth-, and fifth-grade

students. We share and discuss students responses to items addressing their understanding of

equation structure and the meaning of the equal sign. We found that while an operational view ofthe equal sign was predominant, some students were able to recognize underlying structure in

arithmetic equations. The degree to which students were successful varied from task to task, with

extremely obvious tasks such as 5 + 3 = ___ + 3 being more apt to elicit structure-basedstrategies. Our findings can inform early algebra efforts by identifying tasks that have the

potential to help students begin to think about equations in a structural way.

Objectives and rationale

Algebra has historically served as a gateway to higher mathematics thatdue to high

failure rateshas been closed for many students. The National Council of Teachers of

Mathematics (2000) and others (e.g., Kaput, 1998, 1999) argue for the treatment of algebra as a

K-12 strand as a way to address this issue. Several mathematics education researchers (Blanton,2008; Brizuela & Earnest, 2008; Carpenter, Franke, & Levi, 2003) have investigated what this

might mean for the elementary grades. While the foci of such efforts vary, consensus exists thatalgebra in the elementary grades (i.e., early algebra) should not involve an exclusively symbolic

focus typical of a traditional eighth- or ninth-grade course but rather should introduce students to

algebraic forms of reasoning that are accessible to them now and that we believe will benefitthem when they begin a more formal study of algebra.

It is now largely accepted that instruction for elementary students should build from what

they already know (Carpenter, Fennema, & Franke, 1996). The success of Cognitively GuidedInstruction (Carpenter, Fennema, Franke, Levi, & Empson, 1999), for example, is based on a

fundamental understanding of student thinking in the area of whole number operations. The goalof this paper is to share findings from a written assessment about students understandings of the

equal sign and equation structure, ideas foundational to a study of algebra. We focus especially

on the role particular tasks played in eliciting these understandings.

Theoretical framework

When students are learning algebra, understanding the meaning of the equal sign and

holding a structural conception of equations are critical (Carpenter et al., 2003; (Knuth,Stephens, McNeil, & Alibali, 2006; Molina & Ambrose, 2008) and related ( Knuth, Alibali,

McNeil, Weinberg, & Stephens, 2005) issues. Kieran (2007) asserts that a structural

understanding of algebra comes from an ability to see abstract ideas hidden behind symbols.For students with a structural conception of equations, the symbols become transparent and

identifying allowable transformations is straightforward. Research suggests, however, that

students lack both a solid understanding of the equal sign and a strong structural sense ofequations. First, students tendency to view the equal sign as a stimulus to do something rather

than as a symbol expressing a relationship of equivalence ( Blanton, Levi, Crites, & Dougherty,

2011; Carpenter et al., 2003) is a misconception that often persists into high school (Kieran,

1981) and even adulthood (McNeil & Alibali, 2005). Second, students often have difficultyinterpreting and transforming equations, especially when several numerical terms are involved

(Kieran, 2007). Evidence exists, however, that early interventions focused on helping students


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develop a relational view of the equal sign and engage in structural thinking about equations can

be successful (Carpenter et al., 2003).A key component of any such intervention is the nature of the tasks employed (Hiebert et

al., 1997). True/false and open number sentences are particularly useful contexts for encouraging

students to discuss the meaning of the equal sign and confront their misconceptions (Molina &

Ambrose, 2006). Tasks that furthermore illustrate arithmetic properties (e.g., 12 + 11 = 11 + 12)are useful in helping students identify generalizations (Molina & Ambrose, 2008), and tasks that

include larger numbers such as 88 + 49 = __ + 48 are useful in stimulating students to look at an

equation structurally rather than rushing to compute (Carpenter et al., 2003). ). Tasks of thesetypes, as well as others, were employed in this study to examine students understandings. This

paper will focus specifically on the studys examination of the following questions:

1. What understandings and misconceptions do grades 3-5 students hold about themeaning of the equal sign and equation structure prior to early algebra instruction?

2. Do particular tasks encourage a relational understanding of the equal sign and a focuson equation structure more than others?

MethodParticipants

Participants were 290 elementary (104 third grade, 108 fourth grade, 78 fifth grade)students from two schools in southeastern Massachusetts. The school district in which these

schools reside is largely white (91%) and middle class, with 17% of students qualifying for free

or reduced lunch. The schools regular mathematics curriculumGrowing with Mathematics(Iron, 2003)does not include a specific focus on algebra. While participants were part of a

larger project that aimed to investigate the efficacy of early algebra in grades 3-5, the

assessments discussed in this paper were administered prior to our instructional intervention.Thus, we believe we can interpret our findings as representative of fairly typical elementary

students algebraic thinking (i.e., students with arithmetic-based mathematics experiences).

Data Collection

Students completed an assessment at the beginning of the school year designed to

measure their understanding of a variety of algebraic topics. We focus in this paper on studentperformance on four tasks (see Figure 1) that investigated students understanding of equation

structure and the meaning of the equal sign. Item 1 was designed to elicit students definitions of

the equal sign. Items 2 and 3 were designed to investigate students understandings of the equalsign in use as well as any recognition of underlying equation structure. Item 4 was designed to

investigate students understanding of the preservation of an equivalence relation.


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1. In the number sentence 3 + 4 = 7, what is the name of the symbol =?

What does the symbol = mean?

2. Fill in the blanks with the value that makes the following number sentencestrue. How did you get your answer?a) 7 + 3 = ____ + 4 Why?b) 5 + 3 = ____ + 3 Why?

3. Circle True or False and explain your choice.a) 57 + 22 = 58 + 21 How do you know?b) 39 + 121 = 121 + 39 How do you know?

4. The following number sentence is true: 15 + 8 = 23.Is 15 + 8 + 12 = 23 + 12 true or false? How do you know?

Figure 1. Equal sign and equation structure tasks.

Data AnalysisIn this section, we share information about the coding of each item. For all four items,

responses that students left blank, or for which they responded I dont know were grouped into

a no response category, while responses that were not sufficiently frequent to constitute theirown codes were placed into an othercategory. Student responses to Items 2-4 that included no

explanation were placed into an answer only category.

The first question in Item 1, What is the name of the symbol? was asked to preventstudents from using the name of the symbol in their response to the second prompt, What does

the symbol mean?. Student responses to this second prompt were coded as relational if theyexpressed the idea that the equal sign means the same as and as operational if they expressed

the idea that the equal sign means something like add the numbers. A third code, equals, was

used to categorize vague responses such as equal to or it means equals.Student responses to Items 2 and 3 were coded as correct if students provided the correct

number in the blanks in response to Item 2 or answered true in response to the equations in

Item 3. Student strategies for this item were coded as structural, computational, oroperational.The structural code was assigned when student rationales were based on underlying structure.

For example, a student might say that a 6 should be placed in the blank in 7 + 3 = ___ + 4

because 4 is one more than 3, so the number in the blank must be one less than 7. Responses

were assigned the computational code when there was evidence that students viewed the equalsign relationally but performed calculations. For example, given 57 + 22 = 58 + 21, students

might find 57 + 22 = 79 and 58 + 21 = 79 to determine the equation is true. Responses wereassigned to the operational category when students demonstrated an operational view of theequal sign. For example, some students stated that the blank in 5 + 3 = ___ + 3 must be 8

because 5 + 3 = 8. Some students even added all three given numbers and, in the case of this

item, placed an 11 in the blank. If a student used two strategies, the most sophisticated strategywas recorded. For example, in response to Item 3a, students occasionally computed the sum on

both sides of the equal sign but also provided an explanation indicating they noticed the structure

in the equation. In such a case, the response was assigned the structure code.


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Student responses to Item 4 were coded as correct if students responded that the second

equation is true. As was the case with Items 2 and 3, student strategies for Item 4 were coded asstructural, computational, oroperational. The structural code was assigned when students

showed understanding that adding 12 to both sides of the original equation preserves the

equivalence relation. Students were assigned the computational category if they determined the

second equation was true by computing the sum on each side. Finally, students were assigned theoperational code if they disregarded the first equation and stated the second could not be true

because 15 + 8 + 12 is not equal to 23.

Reliability was established by having a second coder separately assign codes to eachresponse. Agreement between coders was at least 99% for correctness and at least 88% on Items

1-3 and 77% on Item 4 for strategy use. Any discrepancies were discussed until full agreement

was reached.

Results and Discussion

Research has repeatedly shown that many elementary students hold an operational view

of the equal sign (e.g., Alibali, 1999; Falkner, Levi, & Carpenter, 1999) and lack an

understanding of equation structure (e.g., Linchevski & Livneh, 1999). We likewise found that tobe the case, beginning with the results from Item 1. Of all of the tasks discussed in the

literature designed to examine students understandings of the equal sign, asking students toproduce an appropriate definition is considered among the most difficult (Rittle-Johnson,

Matthews, Taylor, & McEldoon, 2011). This proved to be the case in our study as well, with

only six students (one third grader, one fourth grader, and four fifth graders) providing arelational definition of the equal sign.

This tendency to treat the equal sign as an operational symbol extended to our other tasks

as well. We found that many students responded to 5 + 3 = ___ + 3 by placing an 8 or an 11 inthe box and said that 57 + 22 = 58 + 21 was false because 57 + 22 is not equal to 58. Rather than

focus on such well-established results, we focus in this paper on the student responses that didillustrate some attention to underlying equation structure. We report how differences in tasks

related to differences in student responses. We pay particular attention to tasks that encouraged

recognitioneven with just a small number of studentsof underlying equation structure.

Item 2: Open number sentences

Figure 2 presents the results of students performance on items 2a and 2b as a function of

grade. As illustrated in the graph, success (i.e., placing the correct number6 or 5in theblank) was nearly identical across these items. This was a little surprising to us given the

obvious nature of Item 2b. Only when we examined students strategies did we see differences

in the ways in which students were thinking about the items. In particular, note the differences instudents use of the structural strategy (see Figure 3). Item 2b elicited in some fourth- and fifth-

grade students an explanation that suggested recognition of the underlying structure rather than a

need to compute. One fifth-grade student stated, for example, that 5 + 3 is equal to 5 + 3because it is the same exact problem.


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Figure 2. Proportion of students giving correct response to Items 2a and 2b.

Figure 3. Proportion of students giving structural explanation in response to Items 2a and 2b.

Item 3: True/False number sentences

Figure 4 presents the results of students performance on Items 3a and 3b as a function of

grade. As was the case with Items 2a and 2b, success on these items (i.e., answering true) wasvery similar. It is often suggested that items such as 3a be used to encourage students to look for

relationships across the equal sign (e.g., Carpenter et al., 2003). Figure 5, however, illustratesthat very few students responded to this task by providing an explanation focused on the

equations structure. The more obvious 39 + 121 = 121 + 39 (Item 3b) was much more apt to

elicit such a response. One fourth-grade student stated, for example, that the equation was true

because its turned around, so its the same.


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Figure 4. Proportion of students giving correct response to Items 3a and 3b.

Figure 5. Proportion of students giving structural explanation in response to Items 3a and 3b.

Item 4: Equivalent equations

Item 4 was designed to take students understandings of the equal sign a step further. Torecognize that 15 + 8 = 23 implies 15 + 8 + 12 = 23 + 12 and that no computation is necessaryrequires more advanced thinking about relationships across the equal sign. A student with a

structural understanding of equations would not need to perform any computations to recognize

the transformation maintains the equivalence relation. However, as Figure 6 illustrates, very few

studentseven by grade 5demonstrated such an understanding.


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Figure 6. Proportion of students giving structural explanation in response to Item 4.

ConclusionOur results are consistent with well-documented findings that students in the elementary

grades tend to view the equal sign operationally and that very few develop a strong structural

sense of algebraic equations. We focused in this paper, however, on the structural understandingsthat were demonstrated by some students and how these varied by task.

Our findings suggest that task selection matters. Even among tasks of the same formdesigned to encourage structural sense, variation existed in students recognition of equation

structure. Several students who defined the equal sign operationally were in fact able to shift

their thinking when confronted with very obvious tasks such as 5 + 3 = ___ + 3 and 39 + 121 =

121 + 39, suggesting that such tasks might provide an entryway into discussions about algebraicstructure and the meaning of the equal sign for students at the very beginning of their early

algebra experiences.

Endnote

The research reported here was supported in part by the National Science Foundation under

DRK-12 Award #1207945. Any opinions, findings, and conclusions or recommendations

expressed in this paper are those of the authors and do not necessarily reflect the views of theNational Science Foundation.


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elementary students recognition of algebraic structure; not all tasks are created equal

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