Research on the understanding of the inverse relation between addition and subtraction has concentrated, almost entirely, on the procedural question of the “short-cut strategy”.  This research deals with the extent to which children are able to solve a+b-b problems rapidly and correctly without having to calculate. However, the underlying development which leads to this insight has been rather neglected. We shall report studies which show that a crucial change in children’s understanding of the inverse relation takes place in the pre-school years. At this time many children progress from understanding the inversion of identity (if you add some stuff to an object and then take the same stuff away, you restore the status quo) to understanding the inversion of quantity (if you add 3 items to a set, and then take 3 different items away from it, the number of items in the set is as it was in the first place). We shall also show that children’s understanding of inversion at this time is not restricted to exact cancellation: they are as likely to judge correctly that a+b-c=>a when b>c as that a+b-b=a.  During the pre-school period there are large individual variations among children in their success with inversion problems, and our evidence suggests that these persist into the school years. We shall argue that there is a link between the strength of children’s underlying understanding of the inverse addition-subtraction relation and their ability to adopt “short-cut” inversion procedures

Research on the understanding of the inverse relation between addition and subtraction has concentrated, almost entirely, on the procedural question of the “short-cut strategy”.  This research deals with the extent to which children are able to solve a+b-b problems rapidly and correctly without having to calculate. However, the underlying development which leads to this insight has been rather neglected. We shall report studies which show that a crucial change in children’s understanding of the inverse relation takes place in the pre-school years. At this time many children progress from understanding the inversion of identity (if you add some stuff to an object and then take the same stuff away, you restore the status quo) to understanding the inversion of quantity (if you add 3 items to a set, and then take 3 different items away from it, the number of items in the set is as it was in the first place). We shall also show that children’s understanding of inversion at this time is not restricted to exact cancellation: they are as likely to judge correctly that a+b-c=>a when b>c as that a+b-b=a.  During the pre-school period there are large individual variations among children in their success with inversion problems, and our evidence suggests that these persist into the school years. We shall argue that there is a link between the strength of children’s underlying understanding of the inverse addition-subtraction relation and their ability to adopt “short-cut” inversion procedures

Research on the understanding of the inverse relation between addition and subtraction has concentrated, almost entirely, on the procedural question of the “short-cut strategy”.  This research deals with the extent to which children are able to solve a+b-b problems rapidly and correctly without having to calculate. However, the underlying development which leads to this insight has been rather neglected. We shall report studies which show that a crucial change in children’s understanding of the inverse relation takes place in the pre-school years. At this time many children progress from understanding the inversion of identity (if you add some stuff to an object and then take the same stuff away, you restore the status quo) to understanding the inversion of quantity (if you add 3 items to a set, and then take 3 different items away from it, the number of items in the set is as it was in the first place). We shall also show that children’s understanding of inversion at this time is not restricted to exact cancellation: they are as likely to judge correctly that a+b-c=>a when b>c as that a+b-b=a.  During the pre-school period there are large individual variations among children in their success with inversion problems, and our evidence suggests that these persist into the school years. We shall argue that there is a link between the strength of children’s underlying understanding of the inverse addition-subtraction relation and their ability to adopt “short-cut” inversion procedures

This paper deals with an aspect of the inverse relation between addition and subtraction that has received little research attention so far, namely the extent to which people are able to solve adaptively subtraction problems of the type a-b=. by means of indirect addition strategies (“how much do I have to add to b to get at a?”). After a review of the relevant math educational literature wherein a strong plea is made for the teaching and learning of this indirect addition strategy for solving small-difference subtraction problems like 21-18=. or 2012-1988=., we will report three recent studies done at our centre. In a first study adults were asked to solve three-digit subtractions with a small difference between the integers (e.g., 812-786=.) using the choice/no-choice method. Many adults were found to spontaneously apply the indirect addition strategy and to use it in a rather efficient and adaptive way. In a second study, 2^nd to 4^th graders solved small-difference subtractions up to 100 in two conditions: in the first condition, they could use their preferential strategy on each item; in the second condition, they were instructed to report at least one alternative strategy for solving each item. Generally speaking, children did not spontaneously apply indirect addition, but also could not generate it as an alternative solution method when explicitly asked for an alternative method. In a third study we compared the strategic performance on subtractions with small differences up to 100 of 2^nd to 4^th graders from regular classes with that of children from a school wherein the clever use of indirect addition got ample instructional attention. Although the children from the latter school generated somewhat more indirect addition strategies, the number of indirect additions remained remarkably low. Finally, we discuss theoretical and educational implications of our work.

In the present paper we shall report several studies dealing with an aspect of the inverse relation between addition and subtraction that has received little research attention so far, namely the extent to which people are able to solve adaptively direct subtraction problems of the type a-b=. by means of indirect addition strategies (“how much do I have to add to b to get at a?”).

After a brief review of the relevant (math educational) literature , wherein a strong plea is made in favour of the teaching and learning of this indirect addition strategy for solving small-difference subtraction problems like 21-18=. or 2012-1988=., we will report three recent studies done at our centre about this topic.

In a first study adults were asked to solve three-digit subtractions with a small difference between the integers (e.g., 812-786=.) using the choice/no-choice method. Many adults were found to spontaneously apply the indirect addition strategy and to use it in a rather efficient and adaptive way.

In a second study, 2^nd to 4^th graders solved small-difference subtractions up to 100 (like 41-39=.) in two conditions: in the first condition, they could use their preferential strategy on each item; in the second condition, they were instructed to report at least one alternative strategy for solving each item.

Generally speaking, children not only did not spontaneously apply indirect addition to solve small-difference subtractions, but also could not generate it as an alternative solution method when explicitly asked for an alternative method.

In a third study we compared the strategic performance on subtractions with small differences up to 100 of 2^nd to 4^th graders from regular classes with that of children from a school wherein the clever use of indirect addition got ample instructional attention. Although the children from the latter school generated somewhat more indirect addition strategies, the number of indirect additions remained remarkably low.

We will end the paper with some theoretical, methodological and instructional implications.

Conceptual knowledge gives humans a deep and abstract understanding of general relations in a domain while procedural knowledge enables them to quickly and efficiently solve problems. Cognitive learning theories provide contradicting predictions as to the causal relations between conceptual and procedural knowledge in the course of knowledge acquisition: These relations are assumed to be either none-existent, or bi-directional, or uni-directional. To date, the empirical evidence for each of these hypotheses is quite rare and weak. In two experimental studies, we tried (1) to derive treatments adequate for influencing students’ conceptual and procedural knowledge about decimal fractions independently of each other and (2) to investigate whether an experimentally induced increase in one knowledge kind will subsequently lead to an increase in the other knowledge kind. The treatment for improving conceptual knowledge was derived from theories of conceptual change, while the treatment for improving procedural knowledge was based on theories of skill acquisition. To evaluate the breadth of the treatment effects, four typical measures of each knowledge kind were used. The samples of the two studies comprised a total of about 170 fifth-graders. Contrary to our expectations, the two treatments failed to show an at least partly independent influence on conceptual and procedural knowledge. So the causal interrelations of both knowledge kinds could not be investigated. The findings were consistent over the two studies as well as over the eight knowledge measures and confirm previous findings obtained with structural equation models. In sum, the results show that the conceptual and the procedural knowledge about relatively simple mathematics problems are so closely intertwined that distinguishing between them is of limited practical value. However, the generalizability of these findings is unclear and should be the subject of further research. Implications for cognitive learning theories, for educational practice, and for future studies are discussed.

Conceptual knowledge gives humans a deep and abstract understanding of general relations in a domain while procedural knowledge enables them to quickly and efficiently solve problems. Cognitive learning theories provide contradicting predictions as to the causal relations between conceptual and procedural knowledge in the course of knowledge acquisition: These relations are assumed to be either none-existent, or bi-directional, or uni-directional. To date, the empirical evidence for each of these hypotheses is quite rare and weak. In two experimental studies, we tried (1) to derive treatments adequate for influencing students’ conceptual and procedural knowledge about decimal fractions independently of each other and (2) to investigate whether an experimentally induced increase in one knowledge kind will subsequently lead to an increase in the other knowledge kind. The treatment for improving conceptual knowledge was derived from theories of conceptual change, while the treatment for improving procedural knowledge was based on theories of skill acquisition. To evaluate the breadth of the treatment effects, four typical measures of each knowledge kind were used. The samples of the two studies comprised a total of about 170 fifth-graders. Contrary to our expectations, the two treatments failed to show an at least partly independent influence on conceptual and procedural knowledge. So the causal interrelations of both knowledge kinds could not be investigated. The findings were consistent over the two studies as well as over the eight knowledge measures and confirm previous findings obtained with structural equation models. In sum, the results show that the conceptual and the procedural knowledge about relatively simple mathematics problems are so closely intertwined that distinguishing between them is of limited practical value. However, the generalizability of these findings is unclear and should be the subject of further research. Implications for cognitive learning theories, for educational practice, and for future studies are discussed.

Encouraging students to share and compare solution methods is a key component of reform efforts in mathematics in many countries, but experimental studies that more conclusively demonstrate the benefits of sharing and comparing ideas for student learning are largely absent. In a series of studies, we experimentally evaluated a potentially pivotal component of this instructional approach that is supported by basic research in cognitive science: the value of students comparing multiple solution methods. Our investigations focused on 10-12 year olds studying computational estimation (e.g., mentally estimating the product of 23 * 57) and 13-14 year olds studying algebra linear equation solving (e.g., solving equations such as 3(x + 1) = 12). In all studies, students learned the mathematical content in one of two conditions (assigned randomly): 1) comparing and contrasting alternative solution methods (i.e. contrasting cases), where two worked examples were presented on the same page, accompanied by two reflection questions that asked students to compare and contrast the two worked examples; or 2) reflecting on the same solution methods one at a time, where the same worked examples were presented on two separate pages, with a single reflection question focusing on only one worked example on each page. In both conditions, students worked with a partner in their regular mathematics classrooms to study and explain worked examples. Our results indicate that students in the contrasting cases group were more accurate in their performance on procedural knowledge items (including transfer items), showed greater procedural flexibility, and also showed comparable gains in conceptual knowledge. In particular, comparison seemed to facilitate attention to and adoption of non-conventional methods. These findings suggest potential mechanisms behind the benefits of comparing contrasting solutions and ways to support effective comparison in the classroom. Overall, it seems to pay to compare.

Encouraging students to share and compare solution methods is a key component of reform efforts in mathematics in many countries, but experimental studies that more conclusively demonstrate the benefits of sharing and comparing ideas for student learning are largely absent. In a series of studies, we experimentally evaluated a potentially pivotal component of this instructional approach that is supported by basic research in cognitive science: the value of students comparing multiple solution methods. Our investigations focused on 10-12 year olds studying computational estimation (e.g., mentally estimating the product of 23 * 57) and 13-14 year olds studying algebra linear equation solving (e.g., solving equations such as 3(x + 1) = 12). In all studies, students learned the mathematical content in one of two conditions (assigned randomly): 1) comparing and contrasting alternative solution methods (i.e. contrasting cases), where two worked examples were presented on the same page, accompanied by two reflection questions that asked students to compare and contrast the two worked examples; or 2) reflecting on the same solution methods one at a time, where the same worked examples were presented on two separate pages, with a single reflection question focusing on only one worked example on each page. In both conditions, students worked with a partner in their regular mathematics classrooms to study and explain worked examples. Our results indicate that students in the contrasting cases group were more accurate in their performance on procedural knowledge items (including transfer items), showed greater procedural flexibility, and also showed comparable gains in conceptual knowledge. In particular, comparison seemed to facilitate attention to and adoption of non-conventional methods. These findings suggest potential mechanisms behind the benefits of comparing contrasting solutions and ways to support effective comparison in the classroom. OverEncouraging students to share and compare solution methods is a key component of reform efforts in mathematics in many countries, but experimental studies that more conclusively demonstrate the benefits of sharing and comparing ideas for student learning are largely absent. In a series of studies, we experimentally evaluated a potentially pivotal component of this instructional approach that is supported by basic research in cognitive science: the value of students comparing multiple solution methods. Our investigations focused on 10-12 year olds studying computational estimation (e.g., mentally estimating the product of 23 * 57) and 13-14 year olds studying algebra linear equation solving (e.g., solving equations such as 3(x + 1) = 12). In all studies, students learned the mathematical content in one of two conditions (assigned randomly): 1) comparing and contrasting alternative solution methods (i.e. contrasting cases), where two worked examples were presented on the same page, accompanied by two reflection questions that asked students to compare and contrast the two worked examples; or 2) reflecting on the same solution methods one at a time, where the same worked examples were presented on two separate pages, with a single reflection question focusing on only one worked example on each page. In both conditions, students worked with a partner in their regular mathematics classrooms to study and explain worked examples. Our results indicate that students in the contrasting cases group were more accurate in their performance on procedural knowledge items (including transfer items), showed greater procedural flexibility, and also showed comparable gains in conceptual knowledge. In particular, comparison seemed to facilitate attention to and adoption of non-conventional methods. These findings suggest potential mechanisms behind the benefits of comparing contrasting solutions and ways to support effective comparison in the classroom. Overall, it seems to pay to compare.all, it seems to pay to compare.