Singledigit, three addend sums of the type a + b + c offer a rich opportunity to directly observe the range of strategies that different participants may use because they afford the possibility of measuring a partial sum (i.e., a + b or a + c or b + c). For example, while computing the sum 9 + 7 + 1, do participants go in order by first adding 9 + 7 and then adding 1, or do they incur the cost of going out of order by adding 9 + 1 in order to obtain the partial sum of 10, which makes the subsequent addition of 7 less effortful? Informed by findings in simple and complex arithmetic, we investigated the problem types and participant characteristics that can predict out of order switching behavior in such threeaddend sums. To test our hypotheses, we tasked participants, first in an online study, and then in an inperson study to complete 120 singledigit, three addend problems. We found that participants switched the order of addition to prioritize efficiency gains in contexts in which the partial sum addends were small or equal to each other, or when doing so led to a partial sum of 10, or led to a partial sum that is equal to the third remaining integer. Response latency data confirmed that participants were deriving efficiencies in the manner we expected. Related to individual differences, our findings showed that participants with higher levels of math education were most likely to seek efficiency benefits whenever they were on offer.
Imagine being asked to perform the following addition involving three singledigit numbers: 9 + 7 + 1. How might you proceed? One approach is to advance from left to right and add 9 + 7 to get a partial sum of 16, and then to add 1 to get 17 as the final answer. Another approach is to notice that the first and third addend (9 + 1) total to the partial sum of 10 which simplifies the subsequent addition of 7 to get the final answer of 17. The latter approach relies upon the greater ease of adding any number to 10 and produces a more efficient overall process – even though it involves the initial cognitive cost of summing the addends out of order (the efficiency gains are even more tangible in sums of the type 97 + 82 + 3). In this work, we seek to investigate the mathematical strategies people use to solve such three addend sums.
A large prior literature (see
Three addend sums of the type a + b + c offer a rich opportunity to directly observe the range of strategies that different participants may use because such sums afford the possibility of measuring a partial sum (i.e., a + b or a + c or b + c). This partial sum, which participants are required to explicitly provide before entering the final sum, provides an objective indication of the particular strategy used by a participant – without relying on selfreports – which is an important goal for the psychologists researching mathematical cognition (
Strategies for three addend sums may, for example, include a preference to process addends in order in order to reduce the cognitive load associated with going out of order. Alternatively, participants may elect to switch orders to favor partial sums involving smaller numbers (e.g., for 9 + 2 + 3, compute the partial sum 2 + 3), partial sums that sum to 10 (e.g., for 6 + 2 + 8, compute the partial sum 2 + 8), or partial sums that feature the same digit (e.g., for 6 + 5 + 6, compute the partial sum 6 + 6). They therefore provide an attractive platform to investigate the particular types of strategies used by participants and the contexts in which they use them.
To get purchase on these issues, we began with an examination of prior findings related to multiplestrategy use in simple and complex arithmetic. We then sought to apply learnings from these contexts to develop hypotheses related to three addend addition. In particular we sought to examine two central questions: First, for what types of three addend problems do participants switch addend order? Second, does participant mathematical experience predict the contexts in which addend switching is more or less likely?
In the context of simple arithmetic in general, and onedigit addition in particular, it is well understood that people use a range of different strategies. For example, a common strategy for simple onedigit addition is the retrieval of the answer from longterm memory (
Importantly, participant strategy may be influenced by a variety of factors related to the characteristics of problem at hand as well as factors related to the participant’s profile. Factors related to the problem include operand order (e.g., participants may use a different strategy for 6 + 7 than they do for 7 + 6;
Factors related to the participant’s profile include a participant’s mathematical experience and knowledge (
A growing body of evidence (see
In particular, common strategies for multidigit addition may include direct retrieval (e.g., recalling that 12 + 47 = 59), rounding the first and/or second operand up or down (e.g., 12 + 47 may rounded as 10 + 50 + 2 – 3), or columnar retrieval (2 + 4 and 2 + 7). Prior work has identified at least nine strategies that are used by children and adults (
As in the case of simple arithmetic, participant strategy in complex arithmetic is known to depend on the characteristics of the problem at hand. Once again, a key contributor to strategy choice is known to be problem difficulty. In general, problem difficulty may vary with the size of the operands – the larger the operands the more difficult the problem (
Participant strategy in complex arithmetic is also known to be affected by age and basic arithmetic fluency (
As discussed above, findings from both simple and complex arithmetic suggest that participants deploy a wide range of strategies that depend on problem characteristics as well as participant profiles. In this work, we sought to understand the range of strategies that people use for solving three addend sums. To achieve this goal, we presented participants with sums of different characteristics and examined if and when they added sums out of order. We reasoned that adding sums out of order involves increased cognitive effort relative to the more automatized and less effortful left to right processing of addends. It would therefore be undertaken only when doing so reduced processing difficulty.
Examining partial sums may not reveal the full breadth of all strategies used in three addend sums: for example, asking for a partial sum precludes information about decomposition strategies. Nevertheless, examining the pattern of preferred partial sums does provide an objective measure for a broad range of strategies in three addend sums.
Findings related to the impact of particular participant profiles in strategy selection suggests that participants with greater mathematical education tend to use a broader range of more efficient strategies. We therefore focused on measuring the impact of mathematical education on strategy choice. This single factor is not the only individual difference variable that may impact strategy choice; Nevertheless, we reasoned that it was likely to be an impactful variable – particularly in the context of the present studies that featured adult participants.
Our central guiding assumption in developing hypotheses in the context of three addend sums was that in general, participants would seek less difficult computations that are associated with decreased cognitive effort and postpone or avoid more difficult computations that are associated with increased cognitive effort. Psychologists have long believed that humans (and other animals) select actions to minimize effort (
Computations that rely on declarative or procedural knowledge are likely to be less cognitively effortful and therefore less difficult. Declarative knowledge is information that can be retrieved from memory without hesitation (
What addend pairs are likely to feature low difficulty based on declarative and procedural knowledge? We propose that lowdifficulty pairs due declarative knowledge include those in which both addends are small (≤ 5, e.g., 2 + 3), as well as those in which both addends are equal (e.g., 8 + 8). The ‘small number’ proposal is supported by the problem size effect according to which reaction times to problems of the type ‘a + b = ?’ increase with the size of the problem’s addends (
We next illustrate how information about lowdifficulty partial sums can be used to predict three addend addition strategies. For example, in the sum 3 + 9 + 2, we hypothesize that people are likely to first add 3 + 2 (a low difficulty partial sum per the ‘small number’ proposal) to obtain the partial sum of 5 and then add 9 to obtain 14. Similarly, in the sum 8 + 7 + 8, we hypothesize that people are likely to first add 8 + 8 (a low difficulty sum per the ‘equal addends’ proposal) to obtain the partial sum of 16, and then add 7 to obtain 23. Related to the ‘sum to 10 proposal’, for example, in the sum 9 + 7 + 1, we hypothesize that people are likely to first add 9 + 1 – since doing so yields a partial sum of 10 which then affords a lowdifficulty addition with 7 to obtain 17. Finally, we also hypothesize that people may proactively seek to perform sums to make the partial sum equal to the third addend. For example, in the sum 7 + 5 + 2, we hypothesize that people are likely to add 5 + 2 to obtain a partial sum of 7, which then provides access to the lowdifficulty 7 + 7 (per the ‘equal addends’ proposal). These considerations lead us to the first of six hypotheses tested in the present work:
Next, we consider the consequences of people’s tendency to postpone or avoid cognitive effort. Due to the problem size effect (
The cognitive effortbased reasoning also suggests that three addend sums in which there are no switching advantages and do not feature two large numbers in the first two slots should be less likely to be completed out of order than other sums. This is because the cognitive effort of adding out of order will not be mitigated by other gains related to reduceddifficulty. For example, consider the sum 6 + 3 + 8. In such cases, related to Hypothesis 1, there are no small number pairs, equal numbers, partials that add to 10, or partials that add to a third number. Further, related to Hypothesis 2, the first two numbers are not large (and therefore not avoided). Here, there is no apparent benefit of going out of order but doing so would involve the cognitive load related to keeping outoforder numbers in working memory. These considerations lead to the third of six hypotheses tested in the present work.
Hypothesis  Illustrative Example  Hypothesized Partial Sum  Rationale 

1a (small)  5+9+2  5+2  Small Numbers added first 
1b (equal)  8+5+8  8+8  Equal numbers added first 
1c (10s)  9+7+1  9+1  Partial Sum to 10 
1d (3^{rd} num)  9+5+4  5+4  Partial Sum equals 3^{rd} Number 
2 (large)  8+9+3  8+3 or 9+3  Larger Sums Avoided 
3 (control)  6+3+8  6+3  Increased cognitive effort of going out of order; No associated benefits 
Are some people more sensitive to reduceddifficulty related benefits increased cognitive effort related costs? We next examine three additional hypotheses related to individual differences in three addend sums.
How might adults vary in their use of three addend strategies based on their mathematical education? One possibility is that upon attaining adulthood, most adults have encountered so many instances of singledigit addition that they are at the performance ceiling for all such pairs, and there are no meaningful differences in levels of difficulty across adults. Another possibility – and the one we develop in the present work – is that differences in difficulty persist into adulthood and remain salient enough to result in meaningful differences in addition strategies for sums involving three addends.
In this work, we used the number of mathematics courses in high school and beyond as a proxy for the level of automaticity related to number facts. We reasoned that a participant who had completed courses in, for example, algebra, precalculus, and calculus, had – due to repetition – had developed a greater facility than a participant who had not completed these courses.
Consider a group with relatively higher levels of math education. We hypothesize many of the scenarios described in Hypothesis 1 are most likely to be salient for this ‘high math education’ group. They are thus more likely (relative to a group with low math education) to go out of order in contexts in which they can maximize their opportunity to tackle addend pairs that offer them decreased difficulty. In particular, this group is more likely to go out of order to add small addends (Hypothesis 1a), equal addends (Hypothesis 1b), or complete partial sums that requires adding a number to 10 (Hypothesis 1c), or complete partial sums that require summing two equal digits (Hypothesis 1d). These considerations lead to the fourth of six hypotheses tested in the present work.
Conversely, the relative difficulty of summing two large numbers (c.f. Hypothesis 2) is likely to be higher for ‘low math education’ participants. Our fifth hypothesis is that these participants are more likely to pursue out of order sums to avoid adding two larger digits.
Finally, consider the case of numbers that did not offer any switching advantages and do not feature two large numbers in the first two slots (c.f. Hypothesis 3). We propose that low math education participants are more likely to switch for such sums than high math education participants. This is because lower levels of knowledge about number facts may often cause individuals in the low math group to be more likely to rely on backup math strategies such as verbal counting (
Hypothesis  Illustrative Example  Participants More Likely to Go Out of Order  Rationale 

4a (small)  5 + 9 + 2  Higher Math Education  Reduce Difficulty 
4b(equal)  8 + 5 + 8  Higher Math Education  Reduce Difficulty 
4c (10s)  9 + 7 + 1  Higher Math Education  Reduce Difficulty 
4d (3^{rd} num)  9 + 5 + 4  Higher Math Education  Reduce Difficulty 
5 (large)  8 + 9 + 3  Lower Math Education  Avoid Difficult Partials 
6 (control)  6 + 3 + 8  Lower Math Education  No switching advantages. Backup Strategies Used 
We tested the above hypotheses in two studies described below. In Study 1, we asked a sample of Internet workers of varying math education levels to solve 120 threeaddend sums with different problem characteristics. This study allowed us to test each of the six hypotheses discussed above. We will note that at the outset of this study, it was not clear to us that adults – many of whom would have encountered thousands of instances of singledigit addition – would meaningfully vary in experienced difficulty levels related to additions and whether such variances would lead to detectable differences in computation strategies. In Study 2 we reused the same problem sets from Study 1 in a laboratory setting. This afforded the opportunity to confirm that participants infact used the strategies that we inferred from their partial sums in Study 1. In Study 2, we also measured response latencies for each question type.
In Study 1 we asked participants to complete three addend sums by first calculating a partial sum. Our goal was to test each of the six hypotheses described above.
We recruited 159 adult participants (median age range = 3540) through Amazon
Participants were walked through four example problems in which they were to add three integers in varying orders. For both the training phase and main task, participants were instructed to report both the partial sum of the first pair of integers they chose to add and the total sum of all three integers. For example, a potential partial sum of the problem: 9 + 7 + 1 is 10 and the total sum is 17. Participants were instructed to record their partial sum as a numeric value in a designated labeled text box. They were then instructed to record the total sum of all three integers in a separate but adjacently labeled text box. An example of a problem trial can be seen in
Participants were permitted to use pen and paper and asked to complete – in order – the 120 problems in one sitting without taking a break. All participants were given the same 120 problems, but the order of their presentation was randomized for each participant. Responses for the final sum of each problem were verified to be accurate in realtime. A notification would alert the participant if an incorrect total sum had been entered and they were prevented from continuing until they provided the correct answer.
The 120 questions belonged to 6 groups, each consisting of 20 questions. A summary of the types of questions in each group can be seen in
Group  Description  Hypothesis  Integers Adding to Specific Partial Sum 


First and Third  Second and Third  
A  Equal numbers  1b  4+ 

B  Partial sum to 10  1c  9+ 

C  Partial sum equals 3^{rd} addend  1d  8+ 

D  Partial sum to 12  Exploratory  5+ 

E  Partial sum to a multiple of 5  Exploratory  7+ 

F  Control  3  6+3+8  2+9+4 
Group A (equal numbers) contained two equal integers of the same value in each problem. Group B (partial sum to 10) contained two integers that created a partial sum of 10. Group C (partial sum to 3^{rd} addend) consisted of problems including two integers, that when added, would equal the value of the third remaining integer. Group D (partial sum to 12) was the first of two exploratory groups where two integers would create a partial sum to 12. We included this group since a duodecimal or base 12 system commonly occurs in daily mathematical interactions with clocks, calendars, and measurements (imperial feet and inches). Group E (partial sum to multiple of 5), the second of the two exploratory groups contained two integers that created a partial sum to an integer multiple of 5 (excluding the value 10). We included this partial sum as an exploratory group because counting by 5’s is often taught to children counting on their hands and feet. The final group was termed a control group as it consisted of randomly generated numbers under the condition that the numbers chosen did not fit the partial sum characteristics of the five previously mentioned groups.
The 120 problems were generated by software that was constrained by the requirements of each condition. For each group and each problem, the program initially randomly produced 3 ordered singledigit numbers between 1 through 9 (inclusively) for a problem. The software then checked that the problem met the partial sum requirements of the group it belonged to, did not simultaneously satisfy the requirements of a different group, and had not already been included in the problem list.
If any of the requirements were not met, the three numbers were discarded, and the program generated a new random set of 3 integers. This process was iterated until 20 problems had been generated for each respective group. For each group, 10 sets of questions were generated with the first and third integers of the problem adding up to the specific partial sum of its respective group, and 10 sets of questions were generated with the second and third integers of the problem adding up to the specific partial sum of its respective group.
After they completed all onehundredtwenty problems, participants were asked to report their prior experience with mathematics (see
Overall, Hypotheses 1 – 3 were fully supported. Hypothesis 4 was partially supported, Hypothesis 5 was not supported, and Hypothesis 6 was fully supported. We examine each result in turn.
Hypothesis 1a was related to the greater propensity among all adults to switch the order of addition when either the first and third addend were small and the second addend was large (e.g., 4 + 9 + 3), or when the first addend was large and the second and third addend were small (e.g., 9 + 4 + 3). In either case, we expected such sums to be completed out of order at greater rates than the out of order rates in the control group. This was indeed the case. The switching rate for the control group was 38.9%. When the large addend was specified as a number greater than or equal to 8, and the small addends were specified as numbers ≤ 5, the overall switching rate was 52.27%. The difference in switching rates is significant χ^{2}(1, 9989) = 46.18,
We next confirmed Hypothesis 1b – 1d (i.e., higher switching rates for equal numbers, partial sums involving adding a number to 10, and partial sums involving two equal numbers). Hypothesis 2 related to the greater propensity among all adults to switch the order of addition when both the first and second addend were large (9 + 8 + 3) was confirmed. Further, Hypothesis 3 related to reduced switching for Control trials relative to all other trials was also confirmed. The results are shown in
Group  Switch Rate %  χ^{2} 

Control Group  –  
Small Numbers (H1a: 4 + 9 + 3)  52.27  124.12*** 
Equal Addends (H1b: 8 + 5 + 8)  49.06  66.58*** 
Sums to Ten (H1c: 9 + 7 + 1)  50.09  80.69*** 
Sums to Third Addend (H1d: 9 + 4 + 5)  49.06^{a}  66.58*** 
Large Addends (H2: 9 + 8 + 3)  45.80  16.37** 
Control < Sample (H3)  43.58^{b}  34.07*** 
^{a}Identical rate with H1b is coincidental. ^{b}Switch Rate for NonControl Addends.
*
Neither the exploratory ‘sum to 12’ group (in which an out of order partial sum added to 12), nor the exploratory ‘Sum to Multiple of 5’ group were associated with an increased probability of switching relative to control trials.
Hypothesis 46 involved examining the consequences of (predicted) differences in levels of math education across adults. We predicted that a higher number of years of mathematics education would result in an increased proclivity to go out of order to sum small addends, equal addends, create a partial sum of 10, and to create a partial sum equal to the third number (Hypothesis 4). As shown in
Group  Estimate  Standard Error 

Small Addends (H4a)  0.07**  0.02 
Equal Addends (H4b)  0.17***  0.03 
Sums to Ten (H4c 
0.01  0.03 
Sums to Third Addend (H4d 
0.00  0.03 
Large Addends (H5)  0.05  0.05 
Control (6 + 3 + 8)  0.09**  0.03 
*
We further predicted that lower mathematics education would result in an increased proclivity to switch the order of sums in which the first two addends were large and in control sums. As shown in
Overall Hypotheses 46 suggested that higher math education was related to switching to leverage efficient strategies (e.g., double sums were switched at greater rates; see
Finally, we were intrigued by why people with lower levels of math education were going out of order in Control sums – for which there no normative benefits of switching. We explored the possibility that in the absence of the ability to readily recall sums, people may tend to compute a partial sum by starting with the largest number and adding the smallest number to it. For example, in the 6 + 3 + 8 problem, they might start with 8 and add 3 to get a partial sum of 11. We did not find supporting evidence for this ‘add from largest’ strategy since partial sums of the type max + min occurred at the same (statistical) rate as other partial sums in the ‘low education’ group.
Study 1 was an important first step in examining how participants solve three singledigit addend problems. However, it did not provide response latency data that is crucial to establish the relative strategy benefits of each type of strategy. Study 2 addressed this limitation.
Study 2 had three goals: First, we sought to replicate a portion of the results in Study 1 using an inperson study. We reasoned that an inperson study could validate the pattern of results obtained via online methods. Second, we sought to examine whether response latency data, measurable in an inperson study, was consistent with our assumptions related to problem difficulty and cognitive effort. We expected that response latency times for ‘equal addends’, ‘sum to 10’, and ‘sum to third addend’ would be less than response latency times for the control group. Third, we sought to use verbal reports to obtain participant justifications for their chosen strategy.
Informed by the effectsizes in Study 1, we recruited 50 community volunteers (22 females) with a mean age of 24.5 years. Participants were asked to complete an identical set of problems to those participants in Study 1. However, Study 2 was conducted inperson. Unlike in Study 1, there were no meaningful differences in the prior math experience of the participant sample (all had completed at least one college level math course). We therefore exclusively focused on strategy differences due to problem characteristics – and not due to differences in math education.
The participant instructions and protocols for Study 2 were similar to those in Study 1 with two key differences. First, after entering a partial sum, participants were required to press the ‘tab’ key, which made an additional text field appear – in which they then entered their final response. This alteration allowed us to measure participant response latencies for partial sums as well as response latencies for final sums. Second, after completing 120 sums, participants were asked to provide verbal reports of their strategy selection process for eight problems in which they switched the addition order – two each from Group A (equal addends), Group B (Partial sum to 10), Groups C (Partial sum equals 3rd number), and Group F (Control). In these questions, participants were provided a report of the partial sum they obtained, and then asked to provide an explanation of why they picked their strategy via writing 12 sentences in an open response format.
An unusual feature of the study was that due to concerns related to the COVID19 pandemic, participants were asked to wear a mask and surgical gloves for the duration of the study. It is possible that this slightly increased the response latencies we recorded.
Overall, the results of Study 2 closely resembled the results of Study 1 – thereby providing an inperson replication of an online study. Further, patterns of response latencies supported assumptions related to problem difficulty and cognitive effort. Finally, participant explanations for their strategy choices were inline with hypothesized justifications for factors contributing to switching.
Replicating Study 1, the switching rates for ‘equal addends’, ‘sum to 10’, and ‘sum to third addend’ problem types were significantly higher than for control problems (
Group  Switch Rate %  χ^{2} 

Control Group  –  
Small Numbers (4+9+3)  53.35  58.53*** 
Equal Addends (8+5+8)  53.12  53.09*** 
Sums to Ten (9+7+1)  51.83  45.72*** 
Sums to Third Addend (9+4+5)  51.08  41.17*** 
Large Addends (9+8+3)  51.54  22.54*** 
Control < Sample  30.03  34.07*** 
*
We had hypothesized that the total response latencies of problems in the ‘equal addends’, ‘sum to 10’, and ‘sum to third addend’ groups would be less than that of the overall response latency of problem types in the Control group. This was found to be the case.
The response latency across all problems was 6.23 seconds with a standard deviation of 3.34 seconds. We removed 212 outliers (out of 6000 responses) with response latencies greater than 20 seconds (~ +4 standard deviations from the mean). The response latency for problems in the control group was 6.61 seconds; for problems in the ‘equal addends’ group it was 6.13 seconds,
We split the latency times into the pre and post partial sum intervals. The results are shown in
Notably, the lower average response latency of the ‘Equal Addends’ problem type was exclusively due to a reduced latency in the pre partial sum interval. This is consistent with our assumption of greater facility related to adding a number to itself. For example, in computing the sum 8 + 5 + 8, we would predict that 8 + 8 has low response latency – and that is what we observed.
Contrastingly, the lower average response latency of the ‘Sum to Ten’ problem type was exclusively due to a reduced latency in the post partial sum interval. This is consistent with our assumption of greater facility related to adding any number to ten. For example, in computing the sum 8 + 7 + 2, we would predict that 10 + 7 – a computation that occurs after the partial sum of 8 + 2 has been computed will have response latency – and that is what we observed. A similar pattern was seen in ‘Sum to Third’ problem type where the post partial sum interval was exclusively responsible for the reduced average response latency compared to the control group. This is consistent with our assumption of greater facility related to adding a partial sum to itself. For example, in adding 9 + 4 + 5, the efficiencies appear to occur after 4 + 5 have been added to obtain a partial sum of 9 – which can then quickly be added to itself.
Finally, written participant explanations of why they switched the order of addends, coded by an experimenter who was blind to the condition, were almost always of three main types: Either participants reported that they did not know their reasons for switching, or they reported an idiosyncratic reason (e.g., “I have always liked the Number 6.”), or they identified the expected justification for the particular problem type (e.g., “I wanted the numbers to sum to ten.”). The explanation breakdown for each problem type was as follows (the % may not sum to 100 because of a minority of responses being classified into an ‘other’ category): ‘Equal Addends’ – Don’t know 4%, Idiosyncratic 15%, Expected 72%; ‘Sum to 10’ – Don’t know 4%, Idiosyncratic 5%, Expected 84%; ‘Sum to Third’ – Don’t know 31%, Idiosyncratic 12%, Expected 55%; ‘Control’ – Don’t know 41%, Idiosyncratic 48%.
Prior research has demonstrated that computations involving simple and complex arithmetic feature multiple strategies whose deployment depends on factors related to the characteristics of the problem and the profile of the participant. In this work we made progress towards understanding how these factors influence the computation of singledigit, threeaddend sums. These sums offer the advantage that they often require the computation of a partial sum prior to the computation of the final answer, and the particular partial sum chosen is often indicative of the underlying strategy implemented by the participant. Thus, examining mathematical strategies in the context of threeaddend sums have the potential to reduce reliance on selfreport
We hypothesized that participants would switch the order of addition to prioritize addends that were small (5 + 9 + 3), or equal (8 + 5 + 8) or when doing so led to a partial sum of 10 (9 + 7 + 1), or a partial sum that was equal to the remaining third addend (9 + 4 + 5). Each of these hypotheses was confirmed. We also hypothesized that participants would prefer going out of order if doing so would avoid or postpone adding sums with large addends (8 + 9 + 3) since they required greater effort, sooner. Our results supported this hypothesis. We also predicted that people would be less likely to go out of order when doing so did not offer efficiency gains. Consistent with this hypothesis, participants’ switching rates for problems in the control group was lower than for problems in other groups. These results were confirmed both via an online and an inperson study. Response latency data in the inperson study data demonstrated that the efficiency gains in these problems were predictably either before the partial sum (e.g., when equal addends were present) or were after the partial sum (e.g., when a partial sum of 10 was computed).
Related to individual differences, consistent with our hypotheses, we found that participants with greater mathematical experience were more likely (compared to participants with lesser mathematical experience) to go out of order to exploit efficiencies related to adding relatively small numbers and adding equal numbers. Further, as hypothesized, individuals with lesser mathematical experience were more likely (compared to participants with greater mathematical experience) to go out of order in problems in the control group.
Notably, our results did not support our hypotheses that people with greater mathematical background were more likely (relative to people with lesser mathematical experience) to approach problems in which a partial sum led to 10 or in which a partial sum led to a number equal to the remaining third addend – even though across all participants people did prefer to switch in such sums. This is consistent with prior observation that indicates that computational skill and problemsolving may be separable aspects of mathematical cognition (
Our results also did not support the hypothesis that people with greater mathematical background were less likely (relative to people with lesser mathematical experience) to avoid problems that featured two large addends. At first appraisal, the lack of an interaction for such problems suggests that there is no meaningful difference in the automaticity levels of adding large addends for participants with greater or lesser mathematical experience. An alternative reappraisal is that it is not the absolute level of automaticity that determines choice but the relative level of automaticity. The partial sum of 9 + 8 may indeed have greater absolute levels of automaticity for people with greater levels of mathematical experience than for people with lower mathematical experience, but the relative difference in automaticity levels between the different participants may be relatively constant. It is, therefore, possible that all participants switched away from large addends at equivalent rates because relative to other available partial sums, these addends offered relatively lower automaticity. We acknowledge that this explanation is post hoc and requires rigorous empirical testing.
Our results did support our hypothesis that participants with lower levels of mathematical experience are more likely to switch away from control problems in which there were no normative automaticity advantages. We had predicated this hypothesis on the assumption that participants with lower automaticity may use the min addition strategy (
We close this work with the observation that while the study of three addend sums is firmly rooted in the mathematical cognition literature, it has the potential to open a new domain for research into valuebased decision making (
The authors have no funding to report.
The authors have declared that no competing interests exist.
The authors have no additional (i.e., nonfinancial) support to report.