Inverse Relationship of Addition and Subtraction
In my fourth year of teaching second grade, I noticed a pattern with students I will focus on the relationship between addition and subtraction in word problems. Free worksheets, online interactive activities and other resources to support children learning the relationship between addition and subtraction. progressing in Year 1 to: Understand the operation of addition, and of subtraction (as 'take away', 'difference', and. 'how many more to make') and use the.
Different types of WPs are presented for various student groups, in different schools or different age groups. For example, Swanson et al. The mathematical WPs they used were: However, the available literature has already shown that different categories of WPs may lead to different solution strategies and different error types.
For instance, different semantic problem types result in different errors Vicente et al. Obviously, different scientific studies reporting results for different student or age groups cannot be easily compared to one another when they use different WP types; it cannot be determined whether differences should be attributed to group or study manipulation or differences in the used stimulus material.
The Relationship between Addition and Subtraction in Problem Solving
In the following, we outline the major distinctions discussed in the literature. Besides the difficulty level, WPs have been categorized with regard to various other attributes. Based on standard algebra text books, Mayer categorized WPs according to their frequency. For instance, the change problem —where there is a start, a change, and a result state —can be subdivided into three subcategories depending on which state is the unknown.
The mathematical content of WPs can also serve as a basis for categorization. Algebra WPs typically require translation into a mathematical formula, whereas arithmetic WPs are solvable with simple arithmetic or even mental calculation. In contrast to arithmetic WPs, algebraic reasoning WPs share the same numerals and signs Powell and Fuchs, and the manipulation of those numbers and signals differs based on the question or expected outcome Kieran, However, the distinction is not that straightforward, as in some cases both methods can be applied.
For instance, in a study by Van Dooren et al. However, solution approaches are not easily dissociable between arithmetic and algebraic problems. If a WP is intended to be solved with an equation, in some cases a simple arithmetic approach is enough Gasco et al. Under some circumstances, it is even easier to solve WPs via alternative arithmetic strategies than by deriving algebraic equations. US children perform better on a story problem if it is in a money context and the numbers involve multiples of 25 Koedinger and Nathan, While the distinction between algebra and arithmetic WPs is important for investigation and evaluation, in this review we concentrate mainly on arithmetic WPs.
Standardized phrases and the idea that every problem is solvable are other important attributes of many, but not all WPs. Textbooks generally suggest implicitly that every WP is solvable and that every numerical information is relevant Pape, They usually provide standardized phrases and keywords that are highly correlated with correct solutions Hinsley et al. There are so-called non-standard WPs Jimenez and Verschaffel, which can be non-solvable WPs or if they are solvable some have multiple solutions and may contain irrelevant data.
In the recent literature, non-standard WPs are getting more and more attention Yeap et al.
Children give a high level of incorrect answers to non-standard WPS because these seem to contradict their mathematics-related beliefs learned in the classroom. Reusser presented 97 first and second graders with the following sentence: How old is the captain?
The rationale behind such studies is that always-solvable textbook problems with standardized phrases and including only relevant numerical information are hardly ecologically valid. Real-life WPs are not standardized, contain irrelevant information, and a solution may not always exist. The above subcategories, which essentially characterize specific sets of WP properties, have a direct impact on human performance in WP. For space limitations, we cannot discuss the impact of all subcategories in detail, but we illustrate their impact on performance and strategies with two examples: For instance, Change problems [cf.
Then he gave five marbles to Tom. How many marbles does Joe have now? Then Tom gave him some more marbles. Now Joe has eight marbles. How many marbles did Tom give him?
For Change 3 almost all the children used an addition strategy e. In sum, the subcategories introduced in this section influence both performance and the choice of solution strategies. Indeed, solution strategies have systematically been in the focus of WP research and addressed the following questions: Why do they make different errors and at which level of the solution process they do so?
Which kind of semantic representation do they create of the WP? Which skills are necessary for the solution process? The first theories on WP solution processes Kintsch and Greeno, have drawn on the text comprehension theories of Mayer and Van Dijk and Kintsch When solving problems, the solver first integrates the textual information into an appropriate situation model or a mental representation of the situation being described in the problem, which then forms the basis for a solution strategy.
This approach was further applied by Thevenot and Oakhill, ; Jimenez and Verschaffel, ; Kingsdorf and Krawec, An important foundation of those approaches is that solving WPs is not a simple translation of problem sentences into equations Paige and Simon, Often both WPs and the corresponding numerical problems are done without language translation Schley and Fujita, Several researchers have focused on abstraction as a reductive process involved in the translation process in the WPs.
Successfully solving WPs has been argued to require at least three distinct processes Nesher and Teubal, Typically only the latter process is assumed to be shared with common arithmetic tasks. Flexible mental computation according to Greeno involves recognition of equivalence among objects that are decomposed and recombined in different ways.
Number sense refers to an intuitive feeling for numbers and their various uses and interpretations; an appreciation for various levels of accuracy when figuring; the ability to detect arithmetical errors, and a common-sense approach to using numbers Above all, number sense is characterized by a desire to make sense of numerical situations pp.
Sense-making is emphasized in all aspects of mathematical learning and instruction. The classroom climate is conducive to sensemaking. Mathematics is viewed as the shared learning of an intellectual practice.
This is more than simply the acquisition of skills and information. Children learn how to make and defend mathematical conjectures, how to reason mathematically and what it means to solve a problem. Mental Computation Mental computation according to Trafton refers to nonstandard algorithms for computing exact answers.
It is also referred to as the process of calculating an exact arithmetic result without the aid of an external computational or recording aid. Hope, ; Reys, It is recognized as both important and useful in everyday living as well as valuable in promoting and monitoring higher-level mathematical thinking Reys et al. A National Statement on Mathematics for Australian Schools Australian Education Council and the Curriculum Corporation, was released in recommending substantial change in emphasis among mental, written and calculator methods of computation and between approximate and exact solutions.
A major objective is to redirect the computational curriculum in schools to reflect a balance in the emphasis on methods of solution. Before the Statement, the curriculum was divided as: According to Boulware mental arithmetic has its origin during the second quarter of the nineteenth century. The idea of building a broader foundation of meaning and understanding in arithmetic gave rise to Mental Arithmetic as it was known in the middle of the nineteenth century with Warren Colburn considered as pioneer in the field of mental arithmetic.
Before his time, arithmetic had reached a point of extreme abstraction according to Boulware. The second half of the century witnessed the decline in interest and understanding of the purpose of mental arithmetic.
With the coming of more writing paper, cheap pencils, with the rise of industry and its accompanying needs for persons skilled in computation, the practical or computational phase of arithmetic took on importance around the turn of the century. The emphasis in arithmetic at that time was the teaching of isolated facts, followed by drill upon these facts.
High among the purposes stated for the study of arithmetic many authors of the time placed speed, memory and accuracy by mechanical rules. There was an emphasis in arithmetic on drill for perfection and automatic response at the expense of meaning and understanding.
Ina dissertation by Boulware is representative of the quest for the development of "meaning" in mental computation stirred by Brownwellwho urged that meaning and seeing sense in what is being learned should be the central focus of arithmetic instruction. Boulware's conception of mental computation is as follows: Mental arithmetic deals with number as a unified, consistent system, and not as an aggregate of unrelated facts.
It proceeds to the analysis of number combinations by processes of meaningful experiences with concrete numbers, reflective thinking in number situations, seeing relationships, and discovery of new facts as an outgrowth of known facts pp. Inin an article by Sister Josefina there seems to begin interest in mental computation and in the NCTM yearbook on computational skills there appears an article by Trafton where the need for including proficiency with estimation and mental arithmetic as goals for the study of computation is presented.
A good number of studies and articles about mental computation appeared in the period of the s e. With the increase of studies in cognitive skills and number sense e. Simon, ; Resnick, ; Silver, ; Schoenfeld, ; Greeno, ; Sowder, and more recent studies mentioned in this chapter, mental computation is suggested to be related to number sense, needed for computational estimation skills and considered a higher order thinking skill.
In a study by Reys, Reys and Hope they argued that the low mental computation performance reported in this study most likely reflected students' lack of opportunity to use mental techniques they constructed based on their own mathematical knowledge.
The study of Reys, Reys, Nohda and Emori assessed attitude and computational preferences and mental computation performance of Japanese students in grades 2, 4, 6, and 8. A wide range of performance on mental computation was found with respect to all types of numbers and operations at each grade level. The mode of presentation visual or oral was found to significantly affect performance levels, with visual items generally producing higher performance.
The strategies used to do mental computation were limited, with most subjects using frequently a mental version of a learned algorithm. In a study by G. Thompson about the effect of systematic instruction in mental computation upon fourth grade students' arithmetic, problem-solving and computation ability, a significant difference favored the group taught mental computation, with girls improving more than boys. According to Markovits and Sowder it would seem reasonable that if children were encouraged to explore numbers and relations through discussions of their own and their peers' invented strategies for mental computation, their intuitive understanding of numbers and number relations would be used and strengthened.
Okamoto found that children's understanding of the whole number system seemed to be a good predictor of their performance on word problems. Cross-cultural research has identified a variety of mental computation strategies generated by students, e. Sribner points out that individuals develop invented procedures suited to the particular requirements of their particular occupations. In a study on individuals who are highly skilled in mental arithmetic Stevensforty-two different mental strategies were observed.
Efficient, inefficient and unique strategies were identified for each of five groups grade 8. Dowker describes in a study the strategies of 44 academic mathematicians on a set of computational estimation problems involving multiplication and division of a simple nature.
Computational estimation was defined as making reasonable guesses as to approximate answers to arithmetic problems, without or before actually doing the calculation. Observing people's estimation strategies, Dowker suggests, may provide information not only about estimation itself, but also about people's more general understanding of mathematical concepts and relationships.
From this perspective Dowker concludes that estimation is related to number sense.
Sowder who agrees with this position points out that computational estimation requires a certain facility with mental computation. In a study by Beishuizenhe investigated the extent to which an instructional approach in which students use of the hundreds board supported their acquisition of mental computation strategies. In the course of his analysis, he found it necessary to distinguish between two types of strategies for adding and subtracting quantities expressed as two digit numerals as follows: The study's findings also suggest that instruction involving the hundreds board can have a positive influence on a student's acquisition of N10 strategies.
Fuson and Briars and others have also identified these strategies. Hope points out that because most written computational algorithms seem to require a different type of reasoning than mental algorithms, an early emphasis on written algorithms may discourage the development of the ability to calculate mentally. Lee recommends that perhaps it is time to investigate changing our traditional algorithms for addition and subtraction to left-to-right procedures.
According to Reys et al.
Word problems: a review of linguistic and numerical factors contributing to their difficulty
Mental computation can be viewed from the behaviorist perspective as a basic skill that can be taught and practiced. But it can also be viewed from the constructivist view in which the process of inventing the strategy is as important as using it. In this way it can be considered a higher-order thinking skill Reys et al.
Additionsubtraction and teaching strategies The Curriculum and Evaluation Standards for School Mathematics NCTM recognizes that addition and subtraction computations remain an important part of the school mathematics curriculum and recommends that an emphasis be shifted to understanding of concepts. Siegler indicated how important it is for children to have at least one accurate method of computation.
In a study by Engelhardt and Usnick while no significant difference between second grade groups using or not using manipulatives was found, significant differences in the subtraction algorithm favored those taught addition with manipulatives.
Usnick and Brown found no significant differences in achievement between the traditional sequence for teaching double-digit addition, involving nonregrouping and then regrouping, and the alternative, in which regrouping was introduced before non-regrouping examples in second graders. Ohlsson, Ernst, and Rees used a computerized model to measure the relative difficulty of two different methods of subtraction, with either a conceptual or a procedural representation.
The results of the use of the model suggested that regrouping is more difficult to learn than an alternative augmented method, particularly in a conceptual representation, a result that contradicts current practice in American schools. Dominick's study with third grade students suggested that students' confusion with the borrowing algorithm centered around a misunderstanding of what was being traded. Evans found that groups taught with pictorial representations or by rote learned to borrow in significantly less time than did a group using concrete materials in grades 2 and 3.
Sutton and Urbatch recommended the use of base-ten blocks, beans and bean sticks or beans and bean cups to serve as manipulatives to use for trading games and with the "transition board". A modified version of the base ten board. They also emphasized that attempting to teach addition and subtraction without initially preparing the student with trading games could be counterproductive and result in lack of understanding due to lack of preparation.
In a study which analyzed individual children's learning of multidigit addition in small groups in the second grade, results suggested that rarely did a child spontaneously link the block trades with written regrouping Burghardt, Fuson and Briars and P. Thompson found that the base ten blocks could be a helpful support for children's thinking, but many children do not seem spontaneously to use their knowledge of blocks to monitor their written multidigit addition and subtraction.
The Fuson and Briars study suggested that frequent solving of multidigit addition or subtraction problem accompanied by children's thinking about the blocks and evaluating their written marks procedure, might be a powerful means to reduce the occasional trading errors made by children. The study also suggested that counting methods that use fingers, are not necessarily crutches that later interfere with more complex tasks.
Fuson and Fuson found that in all of the groups studied, children were accurate and fast at counting up for subtraction as at counting on for addition. This contrasts with the usual finding that subtraction is much more difficult than addition over the whole range of development of addition and subtraction solution strategies.
Sequence counting on and counting up according to Fuson and Fuson are abbreviated counting strategies in which the number words represent the addends and the sum. In both strategies the counting begins by saying the number word of the first addend. Thornton's study provides evidence that children who were given an opportunity to learn a counting up meaning for subtraction as well as counting down counting back from minuendpreferred the counting up meaning. In a series of studies by Bright, Harvey and Wheeler they defined an instructional game as a game for which a set of instructional objectives has been determined.
These instructional objectives may be cognitive or affective and are determined by the persons planning the instruction, before the game is played by the students who receive the instruction in it.
The results of the studies suggest that: Hestad found that the use of a card game was effective for third grade students in introducing new mathematical concepts and maintaining skills. In a study by Cobb the use of the hundreds board by second graders' in a classroom where instruction was broadly compatible with recent reform recommendations NCTM,was investigated. The role played by the use of the hundreds board over a week period in supporting the conceptual development of four second graders was studied.
Particular attention was given to the transition from counting on to counting by tens and ones. The hundreds board is a ten-by-ten grid from either 0 to 99 or 1 to The results indicated that the children's' use of the hundreds board did not support the construction of increasingly sophisticated concepts of ten.
However, children's use of the hundred board did appear to support their ability to reflect on their mathematical activity once they had made this conceptual advance. The utility of the hundreds table in teaching computation has been also recognized by Beishuzen ;Hope, Leutzinger, Reys and Reys ; Thornton, Jones and Neal and Van de Walle and Watkins Teachers ' Pedagogical Beliefs about Mathematics Teaching Learning and Assessment We can learn more about how invisible components in the teaching and learning situation can contribute to or detract from the quality of the mathematical learning that takes place by focusing on the culture according to Nickson It is important, he points out, in exploring the mathematics classroom from the perspective of the culture, it generates, to remember that we are concerned with the people in the setting and what they bring to it.
Nickson adds that we must increase our sensitivity to the importance of their hidden knowledge, beliefs, and values for mathematics education. One of the major shifts in thinking in relation to teaching and learning of mathematics in recent years has been with respect to the adoption of differing views about the nature of mathematics as a discipline.
The view of mathematics that has informed and historically transfixed most mathematics curriculum has been, according to Lakatos, one of considering that mathematics as consisting of "immutable truths and unquestionable certainty". Such a view does not take into account how mathematics changes and grows and is waiting to be discovered Nickson, Brown and Cooney note that the intensity of the teachers' beliefs is very important in the classroom culture. The traditional detachment of mathematics content from shared activity and experience, so that it remains at an abstract and formal level, constructs barriers around the subject, according to Nickson, that sets it apart from others areas of social behavior.
The message conveyed is that is has to be accepted unquestioningly and from which no deviation is permitted. The classroom culture will mirror this unquestioning acceptance. The visibility and acceptance of what is done or not done in mathematics are factors in stopping teachers from engaging in activities that they may instinctively feel are appropriate but might challenge the supposedly inviolable essence of mathematics as they themselves were taught.The Inverse Relationship of Addition and Subtraction
In investigating the relationship between what teachers believe about how children learn mathematics and how those teachers teach mathematics, A. Thompson points out that studies have examined the congruence between teachers' beliefs and their practice and findings have not been consistent. Researchers such as Grant and Shirk have reported a high degree of agreement between teachers' professed views of mathematics teaching and their instructional practice, where as others have reported sharp contrasts e.
Carter, ; Cooney, ; Shaw, ; Thompson, It has been argued Nickson, ; Ball, that bringing teachers into the arena of research activity can be an important step in increasing their understanding of research processes and results and their relation to classroom practice. Each mathematics classroom will vary according to the actors within it. The unique culture of each classroom is the product of what teachers bring to it in terms of knowledge, beliefs, and values, and how these affect the social interactions within that context.
The daily experiences of students in mathematics classes of teachers with positive attitudes were found to be substantially different from those of students in classrooms of teachers with negative attitudes in a study by Karp Overall, teachers with negative attitudes toward mathematics employed methods that fostered dependency and provided instruction which was based on rules and memorization, relied on an algorithmic presentation, concentrated on correct answers and neglected cognitive thought processes and mathematical reasoning, whereas teachers with positive attitudes were found to encourage student initiative and independence.
Swetman found no significant relationship between teachers' mathematics anxiety and students' attitude toward mathematics in grades 3 to 6. Attitude toward mathematics however, became more negative as grade increased in teachers and students. Teacher influence on student achievement At the time of a study by Good and Grouws,comparatively few studies had included observational measures that detail how the teacher functions as an independent variable in order to influence student achievement.
Teacher effectiveness as operationally defined in their study appeared to be associated strongly with the following clusters: Brophy's study found that most investigative efforts had focused on curricular content and students' learning without careful consideration of teachers' instructional practices.
Loef found that more successful teachers in grade 1 represented differences among addition and subtraction problems on the basis of the action in the problem and the location of the unknown, and they organized their knowledge on the basis of the level of the children's understanding of the problem in context.
Hiebert and Carpenter note that it seems evident that procedures and concepts should not be taught as isolated bits of information, but it is less clear what connections are most important or what kind of instruction is most effective for promoting these connections.
In mathematics, Barr found that seven out of nine fourth-grade teachers used their textbooks by moving lesson by lesson through the book. In contrast, Freeman and Porter and Stodolsky found most mathematics teachers to be selective in their use of textbook lessons, problem sets, and topics, although topics not included in the texts were only occasionally added to the instructional program.
Sosniak and Stodolsky found in a study of four fourth-grade teachers that the influence of textbooks on teachers' thinking and on instruction was somewhat less than the literature indicates.
Their results suggest that patterns of textbook use and thinking about these materials were not necessarily consistent across subjects even for a single teacher, and that the conditions of elementary teachers' work encouraged selective and variable use of textbook materials.
In a study by Stigler, Fuson, Ham, and Myongan analysis is made of addition and subtraction word problems in American and Soviet elementary mathematics textbooks. The data suggests that American children entering first grade can solve the simple kinds of addition and subtraction word problems on which American texts spend so much time.
Another study on text books is one by Ashcraft and Christy in which they study the frequency of arithmetic facts in elementary texts. The study tabulated the frequency with which simple addition and multiplication facts occur in elementary school arithmetic texts for grades The results indicated a "small-facts bias" in both addition and multiplication.
The small facts bias in the presentation of basic arithmetic, at least to the degree observed, probably works against a basic pedagogical goal, mastery of simple facts. It may also provide a partial explanation of the widely reported problem size or problem difficulty effect, that children's and adults' responses to large basic facts are both slower and more error prone than their solutions to smaller facts. In a study by Porter elementary school mathematics is used as a context for considering what could be learned from careful descriptions of classroom content.
Teachers log and interviews show that large numbers of mathematics topics are taught for exposure with no expectation of student mastery: Porter argues that "ultimately teachers must decide what is best for their students and within the limits of their own knowledge, time and energy.
Also influencing teachers' behaviors are teachers' attitudes and beliefs about teaching and mathematics. Bush in a study about factors related to changes in elementary student's anxiety found that mathematics anxiety tended to decrease as teachers in grades spent more time in small group instruction, had more years of experience, and took more post-bachelor's mathematics courses.
According to a study by Tangrettifindings indicated that the elementary teachers that participated in the study were not adequately prepared to meet NCTM expectations. Their teaching focus was found to be an algorithmic approach with emphasis on numeration and computation. Lack of confidence in content areas beyond arithmetic were reported as contributing to the lack of preparedness of elementary teachers to implement innovative curriculum.
Wood, Cobb and Yackel report that after participating in a study, changes occurred in a teacher's second grade beliefs about the nature of mathematics from rules and procedures to meaningful activityabout learning from passivity to interacting and about teaching from transmitting information to guiding students' development of knowledge.
A similar result was reported in a study by Zilliox In-service elementary school teachers felt they were teaching more and better mathematics lessons, were more comfortable with student use of hands-on materials and with managing small groups, and had a different sense of student capabilities and different expectations for student behavior after participating in the study.