In this study, we explored second and fifth graders’ noticing of negative signs and incorporation of them into their strategies when solving integer addition problems. Fiftyone out of 102 second graders and 90 out of 102 fifth graders read or used negative signs at least once across the 11 problems. Among second graders, one of their most common strategies was subtracting numbers using their absolute values, which aligned with students’ whole number knowledgepieces and knowledgestructure. They sometimes preserved the order of numbers and changed the placement of the negative sign (e.g., −9 + 2 as 9 – 2) and sometimes did the opposite (e.g., −1 + 8 as 8 – 1). Among fifth graders, one of the most common strategies reflected use of integer knowledgepieces within a wholenumber knowledgestructure, as they added numbers’ absolute values using whole number addition and appended the negative sign to their total. For both grade levels, the order of the numerals, the location of the negative signs, and also the numbers’ absolute values in the problems played a role in students’ strategies used. Fifth graders’ greater strategy variability often reflected strategic use of the meanings of the minus sign. Our findings provide insights into students’ problem interpretation and solution strategies for integer addition problems and supports a blended theory of conceptual change. Adding to prior findings, we found that entrenchment of previously learned patterns can be useful in unlikely ways, which should be taken up in instruction.
The theoretical approach that guides our work is one that builds on Robbie
The knowledgeinstructures perspective of viewing students’ conceptions as being organized into theories or frameworks is supported, according to scholars, through robust patterns of student responses to conceptual questions: responses that are commonly portrayed as unitary misconceptions. By contrast, the knowledgeinpieces perspective of perceiving students’ conceptions to be a loose assemblage of fragmented knowledge elements cobbled together for each context is seen to be supported by the contextuality of students’ responses…In blending these opposing positions, a new understanding of a knowledge system emerges: the elements in a knowledge system are seen as independent in the sense that they are not statically connected to other knowledge elements; however, they also clump into structures that are dynamically formed and maintained (
When working with integers, one complication is to negotiate the multiple meanings of minus
Even if students interpret minus signs as negative signs in integer addition problems, they may not change their entrenched interpretation of numbers’ values, depending on the extent to which they rely on their CCSN (
The other knowledgepiece students must negotiate with integer arithmetic are the revised meaning of addition and subtraction. Students who interpret addition as
Based on their experiences with whole numbers, students learn that saying the next number in the counting sequence corresponds to getting one more (or one less if counting backward), which corresponds to adding one (or subtracting one). Further, they learn to map number words to numerals and operations to operation signs (
As children transition to learning integers, they must rework their entrenched understanding of the elements of the CCSN and how the elements are put together; students must make sense of multiple meanings of the minus sign (
In other situations, students’ reasoning breaks down and they have to reinterpret the meaning of one of the knowledgepieces in order to reason productively (e.g., −5 + 6 = 1 does not necessarily help one solve 5 + −6 = −1). During their reinterpretation process, when reasoning about integer values, students will blend absolute value language with linear value language, vaguely suggesting that −3 is larger than −5 but a smaller number, without articulating that they are referring to linear versus absolute value, respectively (
In this study, we compared how secondgrade and fifthgrade students interpreted and solved a series of integer addition problems. According to Common Core State Standards for Mathematics (
We presented students with a sequence of problems with different feature patterns (i.e., the location of minus sign(s), whether the negative or positive number had a larger absolute value, whether the positive or negative number was first, etc.) that could prime their knowledgepieces and knowledgestructures differently. We investigated the interaction of their knowledgepieces and knowledgestructures, as reflected in their strategies. For example, the two signs next to each other in 3 + −3 with the placement of negative signs between two numbers might prime students’ whole number knowledge. Then, seeing −1 + −7 might encourage students to question the role of the negative sign before the numeral one. We anticipated that students’ strategies might differ, depending on where the negative signs were located and how students interpreted them in relation to their knowledgestructures. Therefore, we investigated the following research questions:
What name do second and fifth graders give to negative signs?
To what extent do second and fifth graders read and use the negative signs?
We hypothesized that fifth graders would read or use the negative signs more because they would have had more opportunity for exposure to negative number concepts than second graders.
When do students use the negative signs in a series of integer addition problems?
We hypothesized that fifth graders would be more likely to use the negative sign consistently because they are more likely to know that they designate negative numbers; on the other hand, we expected second graders would be more likely to use the negative sign intermittently because they would more likely use them when the problems looked closer to wholenumber subtraction feature patterns.
How do students’ knowledgestructures interact with their knowledgepieces as shown through their interpretation of problem features and strategies for solving integer addition problems?
We collected data from 102 second graders (7to8yearolds) and 102 fifth graders (10to11yearolds) across three public elementary schools in the Midwest, United States.
Participant Data 
SchoolLevel Data 


School / Grade Levels  Gender  ELLs  Free or reducedprice meals^{a}  Ethnicity 
School A ( 

2^{nd} 
42% Male 
36.5%  92.7%  49% White 
5^{th} 
44% Male 
10.6%  91.2%  51.3% White 
School B ( 

5^{th} 
36% Male 
27.6%  82.9%  37.5% White 
School C ( 

2^{nd} 
44% Male 
30.1%  70.9%  31.2% White 
^{a}This data is used as a proxy for socioeconomic status.
The data for this paper is a part of a larger experimental, intervention study where students took a pretest, engaged in small group sessions and a wholeclass lesson, and completed a posttest. In this paper, we present a series of comparisons between fifth and second graders’ reading and use of the negative signs; we then, through a multiplecase study (
First, we conducted a wholeclass written pretest (paperandpencilbased) involving integer order and comparing integer value questions (circling a number that is most positive/most negative/least positive/least negative among three integers, e.g., −1, −9, −8). Second, for the taskbased interviews, we individually interviewed and recorded each student solving 12 integer addition (one was only with positive numbers), 17 subtraction, and 10 transfer problems (i.e., threeaddend integer and missing integer addition and subtraction problems). For each problem, we individually asked students to read the problem aloud and explain their strategy after solving each by asking, “How did you get the answer?” or “How did you solve this problem?” and further explored their explanations asking, “What part of the problem tells you to
We did not randomize the order of the problems because we knew the negative signs in some problem types were more likely to stand out based on our prior work. For instance, in problems of the form of x + y = z, where both x and y are negative, students are likely to use the negative sign even if they do not know about negative numbers (
According to
To answer the first research question, we identified if students
To answer the second and third research questions, we employed the results of the first research question to report on when and where students read (or did not read) or used (or did not use) the negative signs; this then allowed us to determine patterns in when students used the negative signs. Further, because of our categorical data, we used KruskalWallis
To answer the fourth research question, besides exploring the
Strategy  Description  Example(s)  References 

No use of negative sign ( 
Treating numbers as their absolute value and adding them.  −4 + −3 → 4 + 3 = 7  
Binary or subtraction meaning 
Treating the numbers as their absolute value and subtracting them as larger minus smaller (Larger – Smaller) or smaller minus larger (Smaller – Larger).  −9 + 2 → 9 – 2 = 7 (Larger – Smaller) 

Negative numbers equal to zero 
Negative numbers are worth nothing or they are a subtraction from themselves.  −1 + 8 → (1 – 1) + 8 = 0 + 8  
Use addition and binary meaning ( 
Incorporating both the negative sign as the subtraction sign and plus sign as addition.  3 + −3 → (3 + 3) – 3 = 3  
Symmetric meaning ( 
Adding the numbers’ absolute values and appending the negative sign to the total.  −1 + −7 → 1 + 7 = 8 → −8 

Binary and symmetric meaning ( 
Using a combination of the Binary (Larger – Smaller) and Symmetric strategies above.  4 + −6 → 6 – 4 = 2 → −2  
Unary meaning ( 
Counting right (or up on a number line) from a negative number  −9 + 2 → starting at −9 and counting up/right: −8, −7. 

Counting left (or down on a number line) from a negative number  −2 + 3 → starting at −2 and counting down/left: −3, −4, −5. 

Unary and symmetric meaning ( 
Decomposing integers and making zero pairs (or additive inverses)  −1 + 8 → 8 is 1 + 7 → 1 and −1 is 0 → 0 + 7 = 7  
Unary meaning ( 
Using additive identity property rule.  0 + −9 → “zero is nothing” → the answer is −9.  
( 
Using a strategy that does not fall into any of the above descriptions.  “I guessed.” or 
Not Applicable 
Almost half of second graders (47/102) and 12 fifth graders (12/102) did not read or use the negative sign in any of their solutions for the integer addition problems, meaning they ignored the negative signs and added the remaining whole numbers. The students who read negative signs differed in their use of terms; 34 second graders and 16 fifth graders read them as
Negative Signs  Integer Addition Problems 


−9 + 2  3 + −3  −1 + −7  −8 + 8  4 + −6  0 + −9  7 + −3  −1 + 8  1 + −3  −4 + −3  −2 + 3  
Both read and used  
5^{th}  74  76  76  63  71  77  67  67  67  73  68 
2^{nd}  27  25  29  19  24  16  22  22  22  19  22 
Used only  
5^{th}  8  9  13  20  13  9  14  15  16  14  16 
2^{nd}  4  4  2  5  4  7  5  3  4  5  4 
Read only  
5^{th}  6  5  1  5  5  3  7  7  7  3  5 
2^{nd}  12  11  9  10  8  9  7  4  8  8  7 
Neither read nor used  
5^{th}  14  12  12  14  13  13  14  13  12  12  13 
2^{nd}  59  62  62  68  66  70  68  73  68  70  69 
Total Used  
5^{th}  82  85  89  83  84  86  81  82  83  87  84 
2^{nd}  31  29  31  24  28  23  27  25  26  24  26 
Total Read  
5^{th}  80  81  77  68  76  80  74  74  74  76  73 
2^{nd}  39  36  38  29  32  25  29  26  30  27  29 
Across both grade levels, students had the greatest overall use of the negative signs on −1 + −7 (tied with −9 + 2 for second graders). Interestingly, fifth graders had the lowest number of students using the negative sign on 7 + −3, although this corresponded to one of the problems with their highest number of students
As seen in
Negative Signs  Integer Addition Problems 


−9 + 2  3 + −3  −1 + −7  −8 + 8  4 + −6  0 + −9  7 + −3  −1 + 8  1 + −3  −4 + −3  −2 + 3  
First read  
5^{th}  80  6  0  1  0  0  0  0  0  0  0 
2^{nd}  39  6  3  0  1  1  0  1  0  0  0 
First used  
5^{th}  82  5  2  0  0  0  0  0  0  0  0 
2^{nd}  31  8  0  0  0  2  0  2  0  0  0 
Of the 55 second graders and 90 fifth graders who used or read the negative sign, the majority of fifth graders (82%) used the negative sign across all problems in some way as opposed to only 20% of second graders making use of negative signs in all of their strategies. Overall, the differences in number of times that these second graders versus fifth graders used the negative sign was significant (
To test if the distributions of patterns in using the negative sign were the same between second and fifth graders (see
Grade  Always Used  Used on Initial; Intermittent Use  Delayed Use  Used on Initial Only  Read, Never Used  Never Read or Used 

5^{th}  73  9  7  0  1  12 
2^{nd}  11  16  12  4  12  47 
Excluding those who never read or used the negative signs, fifth graders were significantly more likely to always use the negative sign,
To better understand why students might have different patterns in their use of the negative signs, we further investigated their strategies for solving the integer addition problems.
Grade  WholeNumber Strategies^{a} 


Absolute Value  L – S or S – L  Both signs  Neg = 0  
19%  49%  6%  4%  
80%  60%  16%  25%  
IntegerStrategies^{a} 

Subtract Make Negative  Add Make Negative  Count Left  Count Right  Identity  Zero Pair  Other  
38%  77%  40%  27%  60%  9%  17%  
2%  16%  5%  5%  9%  4%  7% 
^{a}See
Overall, fifth graders were more likely to use strategies that indicate some acceptance of the unary meaning of the minus sign (i.e., they show acknowledgment that negative numbers exist), more aligned with an integer knowledgestructure. They frequently solved problems by adding or subtracting the two numbers’ absolute values and then appending a negative sign to the remaining value, or they started at a negative number and counted along the number sequence. Second graders, on the other hand, primarily used strategies that indicate their interpretation of the negative signs aligned with wholenumber knowledgestructures. They frequently solved problems by either ignoring the negative signs or by using them as subtraction signs.
We further explored students’ strategies by investigating how their strategies changed across the problems with their varied problem feature patterns.
Students who relied on their whole number knowledgestructure to solve the integer problems either used an absolute value strategy or interpreted minus signs as subtraction signs. At each grade level, of the students who started with the
Among the second graders, one of the most common minus sign interpretations was as a subtraction sign or having a binary meaning. Second graders treated the negative numbers as their absolute values and subtracted the smaller number from the larger one, which sometimes required switching the order of the numbers, as when solving −1 + 8 as 8 – 1 or 1 + −3 as 3 – 1, aligning the problems with common feature patterns for wholenumber subtraction where the minus sign appears inbetween two numbers. For example, one secondgrade student (4.D11) read, “Four plus minus six” for 4 + −6 and said, “[starting at 6] I counted down to four, then I got two.” Six fifth graders also incorrectly used subtraction to solve 4 + −6 as 6 – 4, demonstrating that some of them, like the second graders, also relied on a wholenumber knowledge structure to solve the problems.
Interestingly, some second graders did not follow the pattern of always starting with the larger absolute value. Instead, some adjusted problems such as 4 + −6 and 1 + −3 to 4 – 6 and 1 – 3 and answered 0. One second grader (4.D06) read 1 + −3 and explained, “One plus take away three [equals zero] because three is the biggest number, and you only have one [to start]. So, if you take away three, it’s zero.” Exploring the relations among the problem structures and use of the
A handful of students in both grade levels, rather than being conflicted about which operation sign to use (the plus sign or the minus sign), used both signs; they used this strategy more often when both signs occurred between the two addends, similar to standard feature patterns for whole number addition (e.g., 3 + −3, 4 + −6, 1 + −3). Interestingly, fifth graders were more likely to use the both signs strategy consistently across problems; whereas, second graders were less likely to use this strategy, especially when the minus sign appeared first in the problem (e.g., −9 + 2). Rather, second graders were more likely to interpret an initial minus sign as a signal to subtract the first number and interpret the resulting problem as 0 + −2, using the
For example, the three fifth graders who used the
Overall, second graders who started off with strategies that indicate a whole number knowledgestructure did not go beyond interpreting negatives as worth zero and did not use any of the strategies that indicate knowledge of negatives. In fact, for 0 + −9, the second graders who had not acknowledged the negative sign on any of the prior problems, continued to ignore it while also using the rule that zero does not change the answer. For example, 4.B15 actually read the problem as “zero minus nine” but said the answer is nine, “Because zero doesn’t add nothing. It’s just the—it stayed the other number.” Another second grader (4.C04), read the problem with both operational signs, “Zero plus take away nine,” but wrote 9, and explained, “Because you can’t add anything if you have zero. That’s too crazy.”
Overall, fifth graders who started off with strategies that indicated a whole number knowledgestructure (i.e., absolute value, L – S/S – L, both signs, and potentially Neg = 0), sometimes used strategies that indicated knowledge of integers at the knowledgepiece level. In fact, five of the fifth graders who first used the
Although fifth graders overall used the
So it’s basically just adding regular nine plus two, but since there’s the negative symbol in front of the nine, you know it won’t be a positive number, because it has a negative for the first one. So it’s negative eleven because of the negative nine.
Students’ (particularly fifth graders’) increased use of the
The
I took this [negative sign] out and did a minus, because you can’t do – I don’t think you can do…negative nine
On the next problem 3 + −3, the same student used a similar process and said, “I took out the plus, and then I did – I put the subtraction sign over the plus, and then three minus three equals zero.” Although
Students who started at a negative number and either counted left or right demonstrated more advanced knowledge of integer order. However, many of these students continued to rely on wholenumber value or operations knowledgepieces. For example, some students counted left (i.e., counted down in linear values) because they interpreted larger numbers as those with larger absolute values and addition as getting larger; therefore, they incorrectly counted for −9 + 2 but correctly counted for −4 + −3. For example, one fifth grader (4.Y04) explained for −9 + 2, “Since you have negative nine and you're adding – you're adding two to that. So you're going to keep – you're going to stay in the negatives…I counted just negative ten, then negative eleven.” Likewise, some students counted right (i.e., counted up in linear values) because they interpreted larger numbers as those with larger linear values and addition as counting up in the sequence; therefore, they incorrectly counted for −1 + −7 and correctly counted for −9 + 2. Incorrect use of the
Both fifth and second graders who started off using a strategy that indicated negative number knowledge (i.e., subtract make negative, add make negative, identity, count right, count left, zero pair) sometimes used wholenumber strategies. For example, some students who initially started at a negative number and counted right (or up in linear value from a negative) on −9 + 2 largely used strategies that would result in a correct answer, even if the strategy itself drew on a wholenumber strategy. For example, a fifth grader (4.X13) explained solving −9 + 2, “I would do negative nine and take it up two to negative seven, because that would take it closer to the positives.” When we asked, “And how did you know that you have to go up?” She responded, “Because it’s the – addition.” Such a response requires that students use the unary meaning of the minus sign and draws on the linear value of numbers (as opposed to absolute values). On the next problem, when solving 3 + −3, the same fifth grader reasoned, “Three plus negative three would be zero since positive going against the negative, and it’s the same number – would take it down to zero,” using language suggestive of starting at positive three and taking away three. Therefore, although some students might similarly use a subtraction strategy 3 – 3 to solve 3 + −3 but have no knowledge of negative numbers, this fifth grader did have knowledge that influenced his decision to solve it this way.
Exploring the relations between the strategy use and the placement of negative signs in the problems illustrates that fifth graders’ interpreting and using of the negative sign was less dependent on where the negative sign was in a problem and more relevant to the magnitude of the integers to which they were attached. In particular, fifth graders preferred the
In some cases, to use the
Some second and fifth graders recognized the inverse property, that a number and its inverse would make a zero pair. One insightful but uncommon strategy, only demonstrated by the fifth graders, was the
Second and fifth graders’ reading and use of the negative signs across the integer addition problems paints a complex picture of how their knowledgepieces and knowledgestructure of whole numbers and integer addition interacted, as reflected in their solution strategies.
The first problem we presented, −9 + 2, closely aligns with 9 + 2, and would not be likely to prime a whole number subtraction feature pattern because there was no number before the negative sign. Therefore, our results are not surprising that almost half of the second graders did not read or use the negative sign on this problem, similar to how it would not be surprising if in a book they read soufflé as “sue ful,” having no prior experience to suggest that the line over the “e” is anything but an errant mark. On the other hand, background knowledge of negative numbers helped the fifth graders identify the negative signs, much like having heard soufflé pronounced and knowing that it is a food could help students realize that the accent is important. Therefore, consistent with prior studies, in our study, students’ existing knowledge played a role in recognizing the key problem features and applying a set of operations on them in their solution strategies (e.g.,
Students’ first use of the negative sign was most likely to occur on the first problem. In terms of entrenchment, this result is surprising because the minus sign’s position in −9 + 2 does not match a whole number knowledgepiece and could easily be ignored; we would have expected more students to use the negative sign as the problems progressed (after they saw the negative sign more and in locations that align more with whole number feature patterns and their CCSN). Instead, the results suggest that they were attuned to the perceptual difference between this problem and the previously given whole number addition problem. Second graders’ willingness to accept that the subtraction problem 9 – 2 might be miswritten as −9 + 2 suggests that they might benefit from instruction contrasting these and other problems that share similar features but are not equivalent (
Fifth graders frequently interpreted the problems as addition problems and then qualified that the answer would be negative because the problem had a negative. Their focus on adding first highlights that their addition problem features and notions of absolute value in their CCSN were more entrenched, preventing them from interpreting numbers’ values in terms of their linear order (i.e., −7 is two more than −9) and supporting their thinking about numbers in relation to their magnitudes (i.e., −11’s magnitude is two more than −9’s). Such students might benefit from a focus on contrasting addition as getting more in terms of linear value versus absolute value and representing their thinking.
Those who did not acknowledge the negative sign on −9 + 2, were most likely to use it on the second problem 3 + −3, which closely matches a feature pattern for whole number subtraction. The location of negative sign inbetween two numbers for 3 + −3 could allow students who did not know about negative numbers to use the negative sign as a subtraction sign. One possible explanation to why second graders were mostly inclined to interpret the negative sign as a subtraction operation across problems may be due to interpreting the plus symbol or addition operation as “and” or as an indication to do the next step. Therefore, when second graders read 3 + −3 as three plus minus three, they combined three with the next step of taking away three as seen with 4.G02 (see section L – S/S – L). Although second graders who used the
Contrary to the previously discussed results, students often stopped using the negative sign on the additive inverse set: 3 + −3 or −8 + 8. According to
Students’ use of the negative sign in their solution strategies differed depending on their knowledgestructure and interpretation of knowledgepieces. The most common strategy of
Exploring strategy use over the series of problems and in relation to the problem structures indicated that fifth graders were more inclined to use multiple strategies compared to second graders. Previous studies explained strategy variability as a predictor of conceptual change, as entrenchment and strategy variability are negatively correlated. However, less variability does not mean deeper entrenchment (e.g.,
Even though the structure of each problem involves the location of the negative sign(s), students’ strategy use was also largely dependent on the negative numbers’ absolute values. For instance, when negative number’s absolute value was less than or the same as the positive number (e.g., 7 + −3, −1 + 8, 3 + −3, and −8 + 8), regardless of whether the negative number was first or second, more fifth graders applied the
However, the counting down and crossing zero to get to negative numbers was not common for 1 + −3 or 4 + −6. Instead, it is likely that for some fifth graders, their use of
Another interesting insight into fifth graders’ strategy was the use of
In our study, we explored students’ problem interpretations and solution strategies within integer addition problems. Our results indicate that entrenchment of knowledgepieces and knowledgestructures play a role in students’ encounters with negative integer addition problems. By expanding prior work on entrenchment to integers and evaluating students’ strategies in terms of knowledgepieces and knowledgestructure, we highlight that previously entrenched patterns can be useful in unlikely ways, suggesting that instruction could better leverage students’ entrenched patterns in relation to more sophisticated strategies and new problem types. Much of students’ difficulties with integers lies in the conflicts they experience with their prior learning (largely in school) being limited to wholenumber knowledgepieces and a wholenumber knowledgestructure. Further research should continue to explore how introducing integer knowledgepieces at the beginning of schooling could influence the development of integer knowledgestructures. Although students might take longer to make sense of both positive and negativerelated knowledgepieces, they may ultimately develop an integer knowledgestructure sooner and with fewer conflicts (as opposed to developing a wholenumber knowledgestructure, breaking it apart to revise the knowledgepieces, and putting it back together).
We have some of the anonymous data available through the Purdue University Research Repository: PURR (for access see
We presented a portion of this work at the American Educational Research Association conference in 2019. This work has not been previously published.
This research was supported by a National Science Foundation CAREER award DRL1350281.
This research was approved by Purdue University’s Institutional Review Board.
Strategies  Integer Addition Problems 


−9 + 2  3 + −3  −1 + −7  −8 + 8  4 + −6  0 + −9  7 + −3  −1 + 8  1 + −3  −4 + −3  −2 + 3  
Absolute value^{a}  9%  5%  1%  8%  7%  4%  10%  9%  8%  3%  7% 
Identity  0%  0%  0%  0%  0%  61%  0%  0%  0%  0%  0% 
L – S^{b}  1%  34%  5%  28%  7%  1%  24%  20%  4%  2%  18% 
S – L  1%  0%  0%  0%  0%  0%  0%  0%  0%  0%  0% 
Neg = 0  2%  1%  0%  1%  2%  1%  2%  1%  2%  0%  1% 
Subtract make negative  22%  2%  3%  0%  18%  2%  7%  2%  13%  4%  2% 
Add make negative  28%  28%  68%  31%  30%  18%  31%  32%  31%  65%  33% 
Both signs  3%  7%  3%  4%  4%  4%  5%  4%  5%  5%  4% 
Count right  24%  15%  5%  14%  19%  1%  15%  19%  20%  3%  23% 
Count left  8%  5%  11%  3%  4%  4%  1%  5%  4%  15%  5% 
Zero pair  0%  1%  0%  2%  0%  0%  1%  3%  2%  0%  4% 
^{a}We have included this strategy to account for the fifth graders who ignored the negative sign in some of the problems.
^{b}This code applies to solving 3 + −3 as 3 – 3 or −8 + 8 as 8 – 8.
Strategies  Integer Addition Problems 


−9 + 2  3 + −3  −1 + −7  −8 + 8  4 + −6  0 + −9  7 + −3  −1 + 8  1 + −3  −4 + −3  −2 + 3  
Absolute value^{a}  44%  47%  44%  56%  49%  58%  51%  55%  53%  56%  53% 
Identity  0%  0%  0%  0%  0%  9%  0%  0%  0%  0%  0% 
L – S^{b}  24%  27%  22%  11%  13%  9%  27%  22%  9%  11%  20% 
S – L  0%  0%  2%  0%  11%  5%  0%  0%  9%  0%  2% 
Neg = 0  9%  4%  9%  15%  5%  7%  4%  11%  7%  11%  11% 
Subtract make negative  2%  0%  2%  0%  2%  0%  0%  0%  0%  2%  0% 
Add make negative  5%  5%  11%  5%  5%  5%  5%  5%  7%  9%  7% 
Both signs  5%  11%  4%  2%  7%  5%  7%  4%  7%  7%  4% 
Count right  5%  4%  0%  2%  4%  0%  0%  0%  4%  0%  0% 
Count left  4%  0%  4%  2%  2%  0%  2%  2%  2%  2%  2% 
Zero pair  0%  0%  0%  4%  0%  0%  0%  0%  0%  0%  0% 
^{a}We have included this strategy to account for the fifth graders who ignored the negative sign in some of the problems.
^{b}This code applies to solving 3 + −3 as 3 – 3 or −8 + 8 as 8 – 8.
We use the term
Some teachers likely mentioned negatives as part of their instruction but as far as we know they did not have dedicated lessons on negative numbers and associated operations with them. A few fifthgrade students had prior exposure from a previous study when they were in earlier grades and they may have also had informal experiences that we are not aware of.
The authors have declared that no competing interests exist.
We presented a portion of this work at the American Educational Research Association conference in 2019. This work has not been previously published.
The authors have no additional (i.e., nonfinancial) support to report.