Inverse reference in subtraction performance: An analysis from arithmetic word problems

Studies of elementary calculation have shown that adults solve basic subtraction problems faster with problems presented in addition format (e.g., 6?+?_?=?13) than in standard subtraction format (e.g., 13 – 6?=?_). Therefore, it is considered that adults solve subtraction problems by reference to the inverse operation (e.g., for 13 – 6?=?7, “I know that 13 is 6?+?7”) because presenting the subtraction problem in addition format does not require the mental rearrangement of the problem elements into the addition format. In two experiments, we examine whether adults' use of addition to solve subtractions is modulated by the arrangement of minuend and subtrahend, regardless of format. To this end, we used arithmetic word problems since single-digit problems in subtraction format would not allow the subtrahend to appear before the minuend. In Experiment 1, subtractions were presented by arranging minuend and subtrahend according to previous research. In Experiment 2, operands were reversed. The overall results showed that participants benefited from word problems where the subtrahend appears before the minuend, including subtractions in standard subtraction format. These findings add to a growing body of literature that emphasizes the role of inverse reference in adults' performance on subtractions.     

Number words in young children's conceptual and procedural knowledge of addition, subtraction and inversion

Three studies addressed children's arithmetic. First, 50 3- to 5-year-olds judged physical demonstrations of addition, subtraction and inversion, with and without number words. Second, 20 3- to 4-year-olds made equivalence judgments of additions and subtractions. Third, 60 4- to 6-year-olds solved addition, subtraction and inversion problems that varied according to the inclusion of concrete referents and number words. The results indicate that number words play a different role in conceptual and procedural development. Children have strong addition and subtraction concepts before they can translate the physical effects of these operations into number words. However, using number words does not detract from their calculation procedures. Moreover, consistent with iterative relations between conceptual and procedural development, the results suggest that inversion acquisition depends on children's calculation procedures and that inversion understanding influences these procedures.     

Math Preference and Mastery Relationship in Middle School Students with Autism Spectrum Disorders

We conducted this study to determine the relationship between math preference and mastery for five middle school students with autism spectrum disorders. We randomly presented several math addition and subtraction problem formats to determine the students’ preferences. Results indicated that preference was idiosyncratic across students. In addition, preference was not related to mastery in some students. Results are discussed within a theoretical framework of matching law. Implications for practitioners are discussed.

Concept-procedure interactions in children's addition and subtraction

A 3-week problem-solving practice phase was used to investigate concept-procedure interactions in children’s addition and subtraction. A total of 72 7- and 8-year-olds completed a pretest and posttest in which their accuracy and procedures on randomly ordered problems were recorded along with their reports of using concept-based relations in problem solving and their conceptual explanations. The results revealed that conceptual sequencing of practice problems enhances children’s ability to extend their procedural learning to new unpracticed problems. They also showed that well-structured procedural practice leads to improvement in children’s ability to verbalize key concepts. Moreover, children’s conceptual advances were predicted by their initial procedural skills. These results support an iterative account of the development of basic concepts and key skills in children’s addition and subtraction.     

Preschoolers’ nonsymbolic arithmetic with large sets: Is addition more accurate than subtraction?

Adult and developing humans share with other animals analog magnitude representations of number that support nonsymbolic arithmetic with large sets. This experiment tested the hypothesis that such representations may be more accurate for addition than for subtraction in children as young as 3½ years of age. In these tasks, the experimenter hid two equal sets of cookies, visibly added to or subtracted from the sets, and then asked 3½-year-olds which set had more cookies. Initial set size was either large (7 or 9) or very large (18 or 30), and the final sets differed by either a high proportion (ratio of 1:2) or a low proportion (difference of 1 cookie). Children’s addition performance exceeded chance, as well as their subtraction performance, across set sizes and proportions, whereas subtraction performance did not exceed chance. Arithmetic performance was also independent of counting ability. Addition performance was remarkably accurate when ratios between outcomes were close to 1, in contrast to previous findings. Interpretations for the asymmetry between addition and subtraction are discussed with respect to the nature of representations for nonsymbolic arithmetic with large sets.     

Children's profiles of addition and subtraction understanding

The current research explored children's ability to recognize and explain different concepts both with and without reference to physical objects so as to provide insight into the development of children's addition and subtraction understanding. In Study 1, 72 7- to 9-year-olds judged and explained a puppet's activities involving three conceptual relations: (a) a+b=c, b+a=c; (b) a-b=c, a-c=b; and (c) a+b=c, c-b=a. In Study 2, the self-reports and problem-solving accuracy of 60 5- to 7-year-olds were recorded for three-term inverse problems (i.e., a+b-b=?), pairs of complementary addition and subtraction problems (i.e., a+b=c, c-b=?), and unrelated addition and subtraction problems (e.g., 3-2). Both studies highlighted individual differences in the concepts that children understand and the role of concrete referents in their understanding. These differences were related to using efficient procedures to solve unrelated addition and subtraction problems in Study 2. The results suggest that a key advance in children's conceptual understanding is incorporating subtractive relations into their mental representations of how parts are added to form a whole.     

Better elementary number processing in higher skill arithmetic problem solvers: Evidence from the encoding step

This paper addresses the relationship between basic numerical processes and higher level numerical abilities in normal achieving adults. In the first experiment we inferred the elementary numerical abilities of university students from the time they needed to encode numerical information involved in complex additions and subtractions. We interpreted the shorter encoding times in good arithmetic problem solvers as revealing clearer or more accessible representations of numbers. The second experiment shows that these results cannot be due to the fact that lower skilled individuals experience more maths anxiety or put more cognitive efforts into calculations than do higher skilled individuals. Moreover, the third experiment involving non-numerical information supports the hypothesis that these interindividual differences are specific to number processing. The possible causal relationships between basic and higher level numerical abilities are discussed.     

Strategies in subtraction problem solving in children

The aim of this study was to investigate the strategies used by third graders in solving the 81 elementary subtractions that are the inverses of the one-digit additions with addends from 1 to 9 recently studied by Barrouillet and Lépine. Although the pattern of relationship between individual differences in working memory, on the one hand, and strategy choices and response times, on the other, was the same in both operations, subtraction and addition differed in two important ways. First, the strategy of direct retrieval was less frequent in subtraction than in addition and was even less frequent in subtraction solving than the recourse to the corresponding additive fact. Second, contrary to addition, the retrieval of subtractive answers is confined to some peculiar problems involving 1 as the subtrahend or the remainder. The implications of these findings for developmental theories of mental arithmetic are discussed.     

Children’s understanding of addition and subtraction concepts

After the onset of formal schooling, little is known about the development of children’s understanding of the arithmetic concepts of inversion and associativity. On problems of the form a + b − b (e.g., 3 + 26 − 26), if children understand the inversion concept (i.e., that addition and subtraction are inverse operations), then no calculations are needed to solve the problem. On problems of the form a + b − c (e.g., 3 + 27 − 23), if children understand the associativity concept (i.e., that the addition and subtraction can be solved in any order), then the second part of the problem can be solved first. Children in Grades 2, 3, and 4 solved both types of problems and then were given a demonstration of how to apply both concepts. Approval of each concept and preference of a conceptual approach versus an algorithmic approach were measured. Few grade differences were found on either task. Conceptual understanding was greater for inversion than for associativity on both tasks. Clusters of participants in all grades showed that some had strong understanding of both concepts, some had strong understanding of the inversion concept only, and others had weak understanding of both concepts. The findings highlight the lack of developmental increases and the large individual differences in conceptual understanding on two arithmetic concepts during the early school years.     

Effects of menstrual cycle phase on aggression measured in the laboratory

This study investigated the effects of menstrual cycle phase on aggression in two groups of women, which differed in the severity of their self-reported perimenstrual symptoms. A low- and a high-symptom group were recruited using the Menstrual Distress Questionnaire (MDQ) to define the groups. Twenty-two subjects (11 low and 11 high symptom) participated across one menstrual cycle: during the premenstrual, menstrual, midfollicular, and ovulatory phases. The Point Subtraction Aggression Paradigm was used to assess aggression on each day of participation. There were three main findings; a) rates of aggressive responding did not vary across phases of the menstrual cycle; b) the high-symptom group emitted higher rates of aggressive responding across the menstrual cycle than did the low-symptom group; and c) rates of aggressive responding correlated with the MDQ's behavioral and psychological scales and not the somatic scales. These findings indicate that the menstrual cycle phase does not differentially affect this laboratory measure of aggression. The differences found between the two symptom groups parallel a few reports indicating that women who differ in retrospectively reported mood and behavioral changes related to their menstrual cycle also differ on a number of other psychometric measures. Aggr. Behav. 24:9–26, 1998. © 1998 Wiley-Liss, Inc.