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Word Problems in Mathematics Education

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• Lieven Verschaffel 2 ,
• Fien Depaepe 2 &
• Wim Van Dooren 2  

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Definition and Function of Word Problems

Word problems are typically defined as verbal descriptions of problem situations wherein one or more questions are raised the answer to which can be obtained by the application of mathematical operations to numerical data available in the problem statement (Verschaffel et al. 2000 ). As such they differ both from bare sums presented in written (e.g., 4 + 5 = ?; 5x + 2 = 22) or oral form (e.g., How much is 40 divided by 5?; What is the mean of the numbers 12, 17, 17, 18?), as well as from quantitative problems encountered in real life (e.g., Which type of loan should we take? Can I drive home from here without filling the tank?).

Importantly, the term “word problem” does not necessarily imply that every task that meets the above definition represents a true problem , in the cognitive-psychological sense of the word, for a given student, i.e., a task for which no routine method of solution is available and which therefore requires the activation of...

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Blum W, Niss M (1991) Applied mathematical problem solving, modeling, applications, and links to other subjects: state, trends and issues in mathematics instruction. Educ Stud Math 22:37–68

Article   Google Scholar  

Carpenter TP, Moser JM (1984) The acquisition of addition and subtraction concepts in grades one through three. J Res Math Educ 15:179–202

De Corte E, Greer B, Verschaffel L (1996) Mathematics teaching and learning. In: Berliner D, Calfee R (eds) Handbook of educational psychology. Macmillan, New York. Chapter 16

Google Scholar  

Fuson KC (1992) Research on whole number addition and subtraction. In: Grouws DA (ed) Handbook of research on mathematics teaching and learning. Macmillan, New York, pp 243–275

Goldin GA, McClintock E (eds) (1984) Task variables in mathematical problem solving. Franklin, Philadelphia

Lave J (1992) Word problems: a microcosm of theories of learning. In: Light P, Butterworth G (eds) Context and cognition: ways of learning and knowing. Harvester Wheatsheaf, New York, pp 74–92

Lesh R, Doerr HM (eds) (2003) Beyond constructivism. Models and modeling perspectives on mathematical problem solving, learning and teaching. Erlbaum, Mahwah

Schoenfeld AH (1991) On mathematics as sense-making: an informal attack on the unfortunate divorce of formal and informal mathematics. In: Voss JF, Perkins DN, Segal JW (eds) Informal reasoning and education. Erlbaum, Hillsdale, pp 311–343

Schoenfeld AH (1992) Learning to think mathematically. Problem solving, metacognition and sense-making in mathematics. In: Grouws DA (ed) Handbook of research on mathematics teaching and learning. Macmillan, New York, pp 334–370

Swetz F (2009) Culture and the development of mathematics: a historical perspective. In: Greer B, Mukhupadhyay S, Powell AB, Nelson-Barber S (eds) Culturally responsive mathematics education. Taylor and Francis, Routledge, pp 11–42

Thorndike E (1922) The psychology of arithmetic. Macmillan, New York

Book   Google Scholar  

Verschaffel L, Greer B, De Corte E (2000) Making sense of word problems. Swets & Zeitlinger, Lisse

Verschaffel L, Greer B, De Corte E (2007) Whole number concepts and operations. In: Lester FK (ed) Second handbook of research on mathematics teaching and learning. Information Age, Greenwich, pp 557–628

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Lieven Verschaffel, Fien Depaepe & Wim Van Dooren

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Verschaffel, L., Depaepe, F., Van Dooren, W. (2020). Word Problems in Mathematics Education. In: Lerman, S. (eds) Encyclopedia of Mathematics Education. Springer, Cham. https://doi.org/10.1007/978-3-030-15789-0_163

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CONCEPTUAL ANALYSIS article

Developing problem-solving expertise for word problems.

• School of Education, University of New England, Armidale, NSW, Australia

Studying worked examples impose relatively low cognitive load because learners’ attention is directed to learn the schema, which is embedded in the worked examples. That schema encompasses both conceptual knowledge and procedural knowledge. It is well-documented that worked examples are effective in facilitating the acquisition of problem-solving skills. However, the use of worked examples to develop problem-solving expertise is less known. Typically, experts demonstrate an efficient way to solve problems that is quicker, faster, and having fewer solution steps. We reviewed five studies to validate the benefit of worked examples to develop problem-solving expertise for word problems. Overall, a diagram portrays the problem structure, coupled with either study worked examples or complete multiple example–problem pairs, facilitates the formation of an equation to solve words problems efficiently. Hence, an in-depth understanding of conceptual knowledge (i.e., problem structure) might contribute to superior performance of procedural knowledge manifested in the reduced solution steps.

Introduction

Do we need to teach students how to develop expertise in mathematical problem solving? If so, how can mathematics educators accomplish such a goal? To explore how experts solve linear equations, Star and Newton (2009) interviewed eight experts in the domain of algebra (e.g., mathematics teachers). Regarding a linear equation, such as 7( n  + 13) = 42, one expert viewed the division by 7 on both sides as the first step would give “a nice, clean answer.” Another expert commented on an inefficient way of using distribution to remove the bracket as a first step—“Distributing, I would have had to deal with fractions and finding common denominators and things would not have been as nice” (p. 6). The findings indicate that the solution produced by experts is typically fewer steps, faster, and quicker.

Within the framework of cognitive load theory, the objective of the present paper was to review prior studies ( Ngu et al., 2009 , 2014 , 2016 , 2018 ; Ngu and Yeung, 2013 ) to examine the impact of worked examples upon the development of problem-solving expertise for word problems. More specifically, we examined students’ solution strategies to determine evidence of problem-solving expertise. Across five studies, we attributed students’ ability to use a single equation to solve a category of word problems with fewer solution steps as the demonstration of problem-solving expertise. Furthermore, evidence of problem-solving expertise also includes an adaption of the solution procedure of similar problems to solve transfer problems that differ slightly from similar problems in terms of problem structure. We will begin by discussing differential performance between experts and novices in different domains.

Differential Performance Between Experts and Novices

The seminal work of De Groot (1965) uncovered differential knowledge base of chess configurations between expert and novice chess players. When presented with a specific chess configuration, the expert chess players relied on schemas that contained thousands of chess configurations to guide the best move. In contrast, novices lacked appropriate schemas related to chess configurations to guide the best move. The findings suggest that expertise resides in having schemas, which contain domain-specific knowledge for a domain ( Tricot and Sweller, 2014 ).

According to Chi et al. (1981) , experts categorized physics word problems in accord with a specific principle (e.g., conservation of energy). Novices, on the other hand, categorized physics word problems based on surface features (e.g., inclined planes). In mathematics domain, experts were capable of categorizing a group of word problems based on the underlying principle or shared problem structure ( Silver, 1979 ). Clearly, the findings by Chi et al. (1981) and Silver (1979) suggest that the presence of schemas differentiates problem-solvers’ levels of knowledge and therefore levels of problem-solving expertise.

More recent research has shifted emphasis on the acquisition of both conceptual and procedural knowledge as evidence of mathematical proficiency ( Rittle-Johnson et al., 2001 ; Baroody et al., 2007 ). The conceptual knowledge refers to knowledge of the underlying principle that connects interrelated mathematical concepts for a specific topic, whereas procedural knowledge refers to the application of a sequential actions to obtain the solution ( Rittle-Johnson et al., 2001 ). Hiebert and Lefevre (1986) suggested that competence in conceptual knowledge assists in the execution of procedural knowledge. A review by Bethany Rittle-Johnson et al. (2015) indicated bidirectional relationship between conceptual knowledge and procedural knowledge. The gaining of conceptual knowledge facilitates the gaining of procedural knowledge and vice versa. Accordingly, we would expect an expert in mathematics domain to possess a schema that would reflect competence in both conceptual knowledge and procedural knowledge specific to a topic.

Blessing and Anderson (1996) examined how learners skipped steps after having acquired algebraic rules to solve problems. Apparently, once novice learners became expert learners, they could recognize a specific pattern that allowed them to skip intermediate steps mentally and create fewer solution steps. An advantage of step skipping performance is that it permits expert learners to solve problems more easily, quickly and efficiently. Likewise, the experts in geometry proof problems could infer from the diagram the whole statement schema related to a geometrical shape (e.g., congruent triangles-shared side; Koedinger and Anderson, 1990 ). Then, they used a minimal number of identical angles and sides to proof that the two triangles are congruent. In short, the step skipping performance exhibited by experts reflects the presence of a schema pertaining to conceptual knowledge and procedural knowledge. Presumably, that schema allows the experts to use conceptual knowledge to refine the solutions steps, resulting in fewer solution steps.

Mayer (1985) suggested that the presence of schematic knowledge is critical to success in solving word problems. Drawing on their schematic knowledge, the problem-solvers could select values and variable from the problem text, and integrate these in an equation for solution. In other words, the ability to identify structural elements (values, variable) from problem text and express these in an equation reflects the availability of a schema for a category of problems. For example, one can use a t -test to solve a category of statistic word problems that share a similar problem structure ( Quilici and Mayer, 1996 ). Indeed, differential ability to construct a mathematics-specific equation to solve word problems is a critical factor that differentiates successful and unsuccessful problem-solvers ( Hegarty et al., 1995 ). Successful problem-solvers were more likely than unsuccessful problem-solvers to construct a mathematics-specific equation for generating a solution. Hence, the ability to match a problem with a known solution path, and use a single equation to solve word problems is regarded as the demonstration of problem-solving expertise ( Blessing and Ross, 1996 ). While past research has identified superior performance of successful problem-solvers on word problems, addressing the issue of developing problem-solving expertise for word problems is less clear.

Research has examined how novices become experts across diverse domains ( Ericsson, 2006 ). Apparently, a substantial length of 10 years of practice is required to develop expertise across a range of domains, such as music, sports, and so on. Ericsson (2006) has recommended the use of deliberate practice to gain expertise in a domain. Specifically, the deliberate practice activities target a learner’s weakness of a particular aspect of the subject matter. For example, the deliberate practice activities requiring students to calculate the area of geometrical shapes (which was identified as a weak area of students) had improved their performance on geometry problems ( Pachman et al., 2013 ). Apart from the study by Pachman et al. (2013) , there is limited research investigating the development of problem-solving expertise for word problems. We argue that learning with the aid of worked examples can address such an issue. Because the use of worked examples to enhance mathematics learning is one of the cognitive load effects, we will discuss the theoretical rationale of cognitive load theory and instructional design in the next section.

Cognitive Load Theory

Cognitive load theory is an instructional theory that has influenced the design of instructions across diverse domains ( Sweller et al., 2011 ). The NSW Department of Education has advocated teachers to examine evidence-based research generated by cognitive load theory (e.g., worked examples) to improve instructional practices in different disciplines, and one of which is mathematics education ( NSW Education: Centre for Education Statistics and Evaluation, 2017 ). This study will review evidence-based research to support the use of worked examples to enhance problem-solving skills for word problems across mathematics and chemistry curriculum.

Cognitive load theory emphasizes the alignment between human cognitive architecture and instructional design to facilitate learning. The human cognitive architecture has a long-term memory that provides a huge storage for knowledge structure in the form of schemas. Most of the schemas are obtained from the long-term memory of other people. Early work by Miller (1956) indicated that it also has a limited working memory that can process seven elements at any given time, but more recent research indicates that it can process about four elements ( Cowan, 2001 ). Furthermore, information readily disappears without being rehearsed ( Peterson and Peterson, 1959 ). Once the information has been processed successfully in the working memory, it will be stored in the long-term memory in the form of schemas.

Cognitive load theory distinguishes three types of cognitive load (intrinsic, extraneous, germane). The intrinsic cognitive load is imposed by the complexity of materials that in turn, is governed by the level of element interactivity that a task contains. The level of element interactivity is determined by the interaction between elements, which must be processed simultaneously to allow understanding to occur. An element refers to anything that requires to be learned (e.g., a number, symbol, and a procedure; Chen et al., 2017 ). The intrinsic cognitive load depends on the complexity of the materials, and learners’ expertise level. The intrinsic cognitive load imposed on working memory increases as the level of element interactivity of the task increases. However, once novices gain expertise, they can “chunk” multiple interactive elements into a schema and store this in the long-term memory. Because we can process the schema retrieved from the long-term memory as a single entity in the working memory, it reduces the intrinsic cognitive load imposed on the working memory. Hence, the limitation of the working memory occurs when processing novel information, but not schemas from the long-term memory. From the perspective of expertise development, an automated schema ( Cooper and Sweller, 1987 ) will free up working memory to allow problem-solvers to deal with aspects of the transfer problem that are unfamiliar.

The extraneous cognitive load is imposed by inappropriate instructional designs that are ineffective for learning. Hence, extraneous cognitive load should always be eliminated by instructional designers. For example, in the domain of geometry problems, splitting learners’ attention between the diagram and the solution steps causes a split-attention effect, imposing extraneous cognitive load that impairs learning ( Tarmizi and Sweller, 1988 ). We can eliminate the split-attention effect by placing individual solution steps at relevant positions in the diagram.

The germane cognitive load is evoked by appropriate instructional designs that are effective for learning. More recent development of cognitive load theory suggests that germane cognitive load does not impose an independent cognitive load. Rather, it is part of the intrinsic cognitive load given that a learner invests germane cognitive load to understand the intrinsic nature of the task ( Sweller, 2010 ). The design of the variability practice increases germane cognitive load, but it benefits learning ( Paas and Van Merriënboer, 1994 ; Likourezos et al., 2019 ). For example, under the variability practice condition, learners are expected to invest germane cognitive load to identify a shared problem structure across a category of problems that differs in problem contexts. As will be discussed later, the provision of a diagram that depicts conceptual knowledge (problem structure) of percentage problems increases germane cognitive load and thus it contributes toward learning ( Ngu et al., 2014 , 2018 ).

The three types of cognitive load (intrinsic, extraneous and germane) have implication for designing effective instructions. To optimize the acquisition of problem-solving skills, we need to minimize extraneous cognitive load, optimize germane cognitive load and to ensure that the intrinsic cognitive load of the material is appropriate for learners. One such effective instructional method is the use of worked examples, which has been demonstrated across multiple studies ( Sweller et al., 2011 ).

The Worked Example Effect

One of the most widely researched cognitive load effects is the worked example effect. The worked example effect occurs when studying worked examples resulted in better learning outcomes and imposes lower cognitive load than solving the same problems particularly for novices in a domain ( Sweller et al., 2011 ; Renkl, 2014 ). A worked example provides detailed solution steps to solve a problem. The solution steps of a worked example encompass a schema required to solve a category of problems. Such a schema is regarded as domain-specific knowledge for a category of problems that share a similar problem structure ( Tricot and Sweller, 2014 ).

According to cognitive load theory, studying worked examples allow learners to focus on an understanding of the relation between problem states and problem-solving operators (e.g., algebra rules; Sweller and Cooper, 1985 ). Accordingly, studying worked examples impose low cognitive load and thus facilitates schema acquisition. In other words, studying worked examples represents an efficient way to overcome the limitation of working memory resources. In contrast, problem-solving approach imposes extraneous cognitive load because cognitive resources are used to search for a solution path, which interferes with the acquisition of schema. As highlighted in a review by Gog et al. (2019) , practice problem-solving compels learners to search for a solution procedure, which not only consumes a lot of cognitive resources but also time ( McLaren et al., 2016 ) and thus is not an efficient way to acquire schema. In essence, the worked example effect relies on the “borrowing and reorganizing principle” of information processing ( Sweller, 2010 ; Chen et al., 2015 ). It makes senses to borrow the schemas from the long-term memory of experts in a domain instead of using cognitive resources to search for a solution path as in the case of problem-solving approach.

Since the inception of cognitive load theory more than three decades ago, empirical studies that support the worked example effect across different domains are overwhelming ( Sweller et al., 2011 ). Early work on the worked example effect was found in learning algebra transformation problems ( Sweller and Cooper, 1985 ). Other studies in relation to the worked example effect are found in statistics ( Paas, 1992 ), physics ( van Gog et al., 2008 ), chemistry ( Ngu et al., 2009 ), and geometry ( Tarmizi and Sweller, 1988 ; Bokosmaty et al., 2015 ). Recently, researchers has extended the worked example effect to learn how to write Chinese characters in which each character consists of various components ( Lu et al., 2020 ). Presenting isolated component of each Chinese character in a variable format was more helpful for novice learners than the blocked format. The variable format allows novice learners to practice variable components of a Chinese character consecutively instead of a uniform component as in the case of a blocked format. In light of a volume of worked examples research, one may wonder how did researchers implement the worked example effect?

Implementation of Worked Examples

Research has found that students may merely look at worked examples rather than paying attention to the worked-out solution steps of worked examples ( Reed et al., 1985 ). Without paying attention to the solution steps, it is unlikely that learners can abstract a schema that is embedded in worked examples, and then use this to solve similar problems. Renkl and his colleagues have advocated the incorporation of prompts or self-explanation ( Renkl, 1999 ; Schworm and Renkl, 2006 ) to help learners focus on the underlying concepts embedded within worked examples. In order to prevent students from studying worked examples superficially, Sweller and Cooper (1985) required students to study a worked example paired with a problem. They reasoned that students would be more motivated to study a worked example, if they knew that they needed to solve a similar problem after studying the worked example. This pioneer work of studying a worked example paired with a problem becomes a blue print for effective implementation of the worked example effect ( Sweller et al., 2011 ). As attested by the findings of Trafton and Reiser (1993) , requiring learners to study a block of six worked examples, followed by solving a block of six problems was less effective than “study-one and solve-one” strategy. Indeed, more recent research has compared study examples only, example-problem pairs, problem–example pairs and problem-solving ( van Gog et al., 2011 ). Unsurprisingly, study examples only and complete example–problem pairs were better than either problem–example pairs or problem-solving only in terms of investing less effort in the acquisition phase and achieve better learning outcomes. The authors argued that novice learners, in particular, may not be able to diagnose their own errors in the solution. Therefore, problem–example pairs condition in which novice learners solved a problem paired with a worked example (which can act as feedback) may not be helpful. Of the five studies that we will discuss, one study uses study examples only and the other four studies use example–problem pairs. Importantly, across five studies, both studying worked examples only and completing example–problem pairs serves as direct instruction to facilitate the development of problem-solving expertise. In relation to word problems, it is important to know the types of knowledge required to solve word problem and the role of a diagram to enhance learning of word problems. We will discuss both of these in the next section.

Importance of a Diagram

According to Mayer (1982) , five types of knowledge are needed for solving word problems: linguistic, factual, schematic, strategic and algorithmic. Knowledge of the linguistic, factual and/or numerical components enables problem-solvers to translate and understand problem situation. The schematic knowledge enables problem-solvers to classify a problem with respect to a category of problems and the manner in which the problem can be solved. The strategic and algorithmic knowledge enables problem-solvers to plan a solution procedure for solving word problems.

Mayer suggested that the greatest hurdle for solving word problems is to represent word problems amenable for generating solutions. Prior studies have demonstrated the power of a diagram to represent word problems by displaying the relationship among quantitative values and variable in order to aid in the construction of a mathematical relationship for solutions ( Mayer and Gallini, 1990 ; Hegarty and Kozhevnikov, 1999 ; Ng and Lee, 2009 ; Schwonke et al., 2009 ; Zahner and Corter, 2010 ; Jitendra et al., 2011 ). However, the design of a diagram matters. As revealed by Hegarty and Kozhevnikov (1999) , schematic diagrams displaying spatial information that captures the problem structure facilitates higher solution success than the non-schematic diagrams (pictorial diagrams) that illustrate cover stories. In other words, a schematic diagram can assist a learner to translate abstract relationships within the problem text and make it concrete. Indeed, using a schema-based instruction incorporating a diagram, which shows mathematical relationship (values and variable) cited in the problem text has improved learning of word problems (e.g., Jitendra et al., 2011 ). Accordingly, of the five studies that we reviewed, four studies incorporate diagrams that seek to capture the problem structure of word problems. For example, the equation approach in Study 3 and Study 4 has a horizontal line in which a shorter length of this line represents a fraction of a percentage quantity. Furthermore, in our review, three out of the five studies involve cross-cultural mathematics education—we will discuss this in the next Section.

Cross-cultural Mathematics Education

Research has indicated that students from different cultural backgrounds (China vs. U.S) use different approaches (algebra vs. non-algebra) to solve word problems ( Cai, 2000 ). The Chinese students tended to use the algebra approach, which requires the formulation of an equation, and then solve for the unknown variable, x . On the other hand, U.S students preferred the use of non-algebra approach, which may rely on concrete visual representations (e.g., drawing a picture) to solve word problems. Regarding the content knowledge of solving linear equations, Ngu and Phan (2020) found that Australian pre-service teachers were inferior to Malaysian pre-service teachers. An analysis of primary mathematics education curriculum reveals that Asian countries (e.g., China, Korea, and Japan) have introduced the topic of linear equations in primary mathematics curriculum, but not Western countries ( Cai et al., 2005 ). Hence, an earlier exposure to algebra may have helped Asian students to build a stronger algebra foundation than their peers in Western countries. In the current review of five studies, we included cross-cultural studies to highlight differential performance between Australian students and Malaysian students especially in regard to the use of algebra in gaining problem-solving expertise for word problems.

Target Domain and Research Questions

Mathematics is a common thread across science, technology, engineering and mathematics (STEM) disciplines ( Fitzallen, 2015 ). We examined word problems that include within-domain word problems in mathematics curriculum as well as between-domain word problems in STEM curriculum. For Studies 1 and 2, we focused on between-domain word problems in a chemistry context ( Ngu et al., 2009 ; Ngu and Yeung, 2013 ). The target domain is the molarity chemistry word problems. For Studies 3, 4 and 5, we focused on within-domain word problems in mathematics context—the percentage change word problems ( Ngu et al., 2014 , 2016 , 2018 ). While each study involved a comparison of different instructional approaches, we focused on the algebra approach that requires the integration of relevant information in an equation to solve a category of word problems that shares a similar problem structure. Of the five studies, four have diagrams to represent word problems. Our main aim was to examine the effect of the algebra approach for acquiring problem-solving expertise. Moreover, we included participants from either Asia (Studies 1, 2 and 4) or Australia (Study 3) or both Asia and Australia (Study 5). The purpose was to examine differential development of problem-solving expertise for word problems across different cultural settings. Specifically, we addressed two research questions:

1. Are there a proportion of students who acquire expertise for solving word problems after studying worked examples?

2. Is there differential development of problem-solving expertise for word problems between Australian students and Malaysian students in regard to the equation approach (algebra approach)?

The Study 1 aimed to facilitate students’ learning of molarity problems, which is a type of word problems in a chemistry context ( Ngu et al., 2009 ). The Study 1 compared three computer-based formats for learning molarity problems: (a) static-solution format, (b) no-solution format, and (c) interactive-solution format ( Figure 1 ) The design of the three computer-based format was based on the hierarchical network problem representation proposed by Nathan et al. (1992) . The strength of the network problem representation depends on its ability to depict a hierarchical level of concepts (values, variable) and their relation without its irrelevant cover story. In essence, it allows the learner to visualize how the values, variable and their relation can be integrated in an equation, such as mass/RFM = MV/1000 for solution. It should be noted that the equation, such as mass/RFM = MV/1000, represents the problem structure of molarity problems. Thus, the conceptual knowledge of molarity problems was scaffolded by a hierarchical level of concepts (values, variable) and their relation, whereas its procedural knowledge was revealed in the solution steps.

Figure 1 . A worked example of a molarity problem: (A) static-solution format, (B) no-solution format, and (C) interactive-solution format. Source: Ngu et al. (2009) .

The three computer-based formats differ in the design of the solution steps. For the static-solution format, all three solution steps are placed at relevant positions in the diagram, which eliminates a split-attention effect ( Tarmizi and Sweller, 1988 ). For the no-solution format, no solution steps are provided. The cognitive load involved in deducing the solution steps would be low, given that the diagram explicitly displays the problem structure in which the solution steps are embedded. For the interactive-solution format, learners can learn from the diagram in three ways: (i) click relevant positions and have all three solution steps continuously—static-solution format, (ii) do not click the diagram—no-solution format, and (iii) click a relevant position to view one solution step at a time. The availability of different options to explore the diagram would expect to impose high cognitive load. Thus, it was hypothesized that the static-solution format would be better than the no-solution format, which in turn, would be better than the interactive-solution format.

We implemented a pre-test—intervention—post-test design. The pre-test shared identical content as the post-test and it provided a baseline score for students. The pre-test (or post-test) consisted of 5 similar problems and 4 transfer problems. The similar problems were isomorphic to the acquisition problems because both shared the same solution procedure (problem structure), whereas an adaptation of the solution procedure was required to solve transfer problems. The means for the pre-test ranged from 4 to 9% for the similar problems and 0 to 7% for the transfer problems. Forty-two Asian students aged about 15 years old from a secondary school participated in the study. A chemistry teacher introduced pre-requisite knowledge of molarity problems (e.g., molar mass and atomic mass) a week prior to the computer session. On the day of testing, all students completed a pre-test. Then, they were randomly assigned to the interactive-solution format (14 students), the static-solution format (14 students) and the no-solution format (14 students). Students across the three formats completed the computer session (25 min) in a laboratory where a computer was assigned to each student. First, they studied a brief instruction in regard to the use of the computer (e.g., use of icons and menu). Second, they studied an instruction sheet which provided the definition of molarity and two worked examples showing how to solve molarity problems. Third, students across the three formats studied (and not solved) eight molarity problems with the aid of a computer. According to Paas and Van Merriënboer (1994) , studying worked examples only may potentially eliminate the negative effect caused by students attending to incorrect solutions generated, which could interfere with learning. Lastly, all students undertook a post-test. Students might learn from studying the instruction sheet; however, we expected the dominant learning to occur when they studied multiple worked examples with the aid of a computer.

One way ANOVA performed on similar problems revealed a significant difference between the three formats, F (2, 39) = 4.06, p = 0.03. A follow-up Tukey test indicated that the difference was between no-solution format ( M = 0.36, SD = 0.41) and interactive-format ( M = 0.06, SD = 0.12) where p = 0.025. Again, one way ANOVA performed on transfer problems showed a significant difference among the three formats, F (2, 39) = 5.48, p = 0.00. A follow-up Tukey test revealed that the difference was between static-solution format ( M = 0.43, SD = 0.33) and interactive-solution format ( M = 0.07, SD = 0.15) where p = 0.006. Overall, the results indicated that the no-solution format or static-solution format outperformed the interactive-solution format for molarity problems across the similar problems and transfer problems.

We computed the Relative condition efficiency, E = P − M 2 where E = efficiency, P = performance and M = mental effort to examine the efficiency of the three instructional formats. The Relative condition efficiency attributes the performance outcomes to the cognitive load involved in processing instructional materials. One way ANOVA on E values was non-significant on similar problems, F (2, 39) = 2.08, p = 0.14, but it was significant on transfer problems at 10% level, F (2, 39) = 2.64, p = 0.08, indicating that the no-solution format was better than the static-solution format, which in turn was better than the interactive-solution format.

On examining students’ solution strategies, students in the interactive-solution format did not skip solution steps. However, six students (43%) from the static-solution format, and 8 students (57%) from the no-solution format skipped solution steps 1 and 2. They wrote mass/RFM = MV/1000 (first step), and substituted values and a variable ( M ) to solve the problem (second step; Table 1 ). The demonstration of a two-step strategy parallels prior studies of expertise development in problem-solving, whereby students could retrieve a single equation needed to solve molarity problems (e.g., Blessing and Ross, 1996 ). Presumably, the gaining of the conceptual knowledge had led to the execution of procedural knowledge efficiently, resulting in fewer solution steps.

Table 1 . The Study, type of word problems, evidence of expertise, percentage of students demonstrated expertise development for similar problems.

Once again, the objective of this Study 2 was to facilitate students’ learning of molarity chemistry problems ( Ngu and Yeung, 2013 ). As indicated in Figure 2 , the equation worked example consists of three equation steps. The emphasis is placed on the construction of an equation for solution (step 3). The design of the equation worked example does not represent the design of a typical worked example in which all solution steps to obtain a solution are provided. Instead, the equation worked example only portrays three key equation steps to solve molarity problems. It should be stressed that the three solution steps contain both conceptual knowledge and procedure knowledge for solving the molarity problems. The equation worked example was compared with the text editing condition. The text editing condition requires learners to scrutinize the problem text and indicate whether it contains missing, irrelevant or relevant information for solution. It places emphasis on identifying relevant information for solution; but, it falls short of addressing the procedural knowledge of solving the molarity problems.

Figure 2 . An equation worked example. Source: Ngu and Yeung (2013) .

The sample consisted of 22 Asian students aged about 17 years old from a secondary school. Students were randomly assigned to two groups: (i) text editing, and (ii) equation worked examples ( Table 1 ). On Day 1, the chemistry teacher introduced pre-requisite knowledge (e.g., atomic mass and molar mass) pertaining to molarity problem. On Day 2, we implemented the experimental procedure which consisted of an acquisition phase and a test phase. We did not include a pre-test because all test materials and procedures had to align with the school curriculum. The post-test was similar to Study 1—it comprised 5 similar problems and 4 transfer problems. During the learning phase (20 min), students studied an instruction sheet and completed seven example–problem pairs. The instruction sheet provided the definition of molarity and a worked example illustrating how to solve a molarity problem. Again, the dominant learning would occur during which students completed multiple example-problem pairs rather than studying the instruction sheet. Each pair consisted of an equation worked example and a similar problem ( Figure 2 ).

The equation worked examples group ( M = 0.42, SD = 0.34) outperformed the text editing group ( M = 0.16, SD = 0.25) on similar problems, t (20) = 2.00, p = 0.05. A significant difference between the equation worked example group ( M = 0.45, SD = 0.31) and the text editing group ( M = 0.14, SD = 0.26) was observed for transfer problems, t (20) = 2.60, p = 0.02. Thus, the results favored the equation worked examples group irrespective of similar problems or transfer problems.

Concerning the solution strategy, of those 11 students in the text editing group, only one demonstrated a two-step strategy (9%) typically shown by expert problem-solvers. However, six out of 11 students in the equation worked examples group demonstrated a two-step strategy (55%). Having acquired the schema for the molarity problem, students realized that they can skip intermediate steps 1 and 2. Hence, they retrieved the equation that integrated relevant information, mass/MM = MV/1000 (first step), and then substituted values and a variable to solve for M in step 3 (second step). Consistent with prior studies ( Koedinger and Anderson, 1990 ; Blessing and Anderson, 1996 ), such step skipping performance to generate a two-step strategy indicates expertise development for molarity problems ( Table 1 ). We attribute the development of expertise for molarity problems to the design of the equation worked examples that imposes low cognitive load. It is possible that the acquisition of conceptual knowledge of the underlying problem structure (mass/MM = MV/1000) facilitates the gaining of procedural knowledge, leading to the generation of reduced solution steps.

This Study 3 is related to learning how to solve percentage change problems, which is a type of word problems in everyday situations ( Ngu et al., 2014 ). Sixty 8th grade Australian students (mean age = 14) were randomly assigned to: (i) unitary approach, (ii) pictorial approach, and (iii) equation approach ( Table 1 ). Consider a percentage change problem used in the study, “ If your father wants to increase your weekly allowance of $20 by 5%, what is your new allowance? .” Central to the unitary approach is the unit percentage concept. This unitary approach consists of three solution steps: (i) 100%+ 5% = 105% (increase by 5%), (ii) $20÷100 = $0.2 (calculate 1%), and (iii) $0.2 × 105 = $21 (calculate 105%). Each solution step cannot be understood independent of other solution steps. The interaction between elements within each solution step and across the three steps would constitute a high level of element interactivity and thus intrinsic cognitive load. Moreover, to calculate the sub-goal of the unit percentage (1%), the learner needs to integrate information from two separate sources [100% in (i) and $20 in (ii)], which will cause a split-attention effect ( Lee and Kalyuga, 2011 ). Thus, the combined consequences of high level of element interactivity and extraneous cognitive load would render this approach ineffective.

The diagram of the pictorial approach aims at depicting the proportion concept—the alignment between quantity and percentage. A rectangular bar diagram is divided into 10 equal chunks and each chunk represents 10%. The alignment between quantity ($20) and percentage (100%) not only eliminates a split-attention effect ( Lee and Kalyuga, 2011 ) but it also acts as a point of reference to calculate a sub-goal (quantity) that corresponds to 1, 5, 10%, etc. The learners need to learn the solution steps with reference to the diagram. Similar to the unitary approach, the learners need to process the interaction between multiple elements within each solution step as well as between the three solution steps. The germane cognitive load is increased to deduce the proportion of 10%:$2 and 5%:$1, leading to the calculation of new allowance, 105%:$21. Nonetheless, the pictorial approach may not be better than the unitary approach. The diagram would impose cognitive load when the learner needs to deduce a quantity that corresponds to % other than a multiple of 10% (e.g., 17%).

For the equation approach, similar to the schema-based instruction proposed by Jitendra et al. (2011) , a diagram depicting a horizontal line is used to scaffold conceptual knowledge (problem structure) of percentage change problems. The germane cognitive load is increased to process the horizontal line, which aims at helping learners to translate the problem structure that consists of two components: (i) original allowance, and (ii) increased amount. In addition, the horizontal line also plays a crucial role in mapping the problem structure to an equation: New allowance = original allowance + increased amount. Within the topic of percentage problems, students would have learned percentage quantity prior to learning percentage change problems. Therefore, they were expected to process the increased amount, such as ($20 × 5%) as a single element. Overall, the processing of the equation approach entails the manipulation of two elements [$20 and ($20 × 5%)], which would impose low cognitive load. In other words, building on the prior knowledge of percentage quantity has lowered the intrinsic cognitive load of learning the percentage change problems. It was hypothesized that the equation approach would be better than the unitary and pictorial approach on learning how to solve percentage change problems.

We conducted a pre-test—intervention—post-test design. Again, the pre-test which had similar content as the post-test served as a baseline to examine subsequent learning gain. The post-test consisted of 10 similar problems and 3 transfer problems. The means for the pre-test ranged from 7 to 17% for the similar problems and 0 to 2% for the transfer problems. The learning phase required students to study an instruction sheet and complete six example–problem pairs that took 20 min. The instruction sheet provided the definition of percentage, the review of the percentage quantity and a worked example showing how to solve a percentage increased problem. Each example–problem pair consisted of a worked example and an isomorphic problem ( Figure 3 ). Once again, students might benefit from studying the instruction sheet, but learning was expected to occur predominately via the completion of multiple example–problem pairs.

Figure 3 . A worked example of the percentage change problem. Source: Ngu et al. (2014) .

Students who scored 80% or above in the pre-test and those who did not complete all test materials were excluded from the final data analysis. One-way ANOVA performed on similar problems showed significant difference between the three groups, F (2, 52) = 10.88, p = 0.01. Post-hoc Tukey test indicated significant differences between the unitary approach ( M = 0.57, SD = 0.43) and pictorial approach, ( M = 0.29, SD = 0.36) where p = 0.04; and also between the equation approach ( M = 0.85, SD = 0.25) and pictorial approach, ( M = 0.29, SD = 0.36) where p = 0.01; but non-significant difference between the unitary approach ( M = 0.57, SD = 0.43) and equation approach ( M = 0.85, SD = 0.25) was found.

Similarly, one-way ANOVA performed on transfer problems indicated significant difference between the three groups, F (2, 52) = 8.83, p = 0.01. Furthermore, Post-hoc Tukey test revealed significant differences between the equation approach ( M = 0.58, SD = 0.43) and unitary approach ( M = 0.19, SD = 0.36) where p = 0.01; and between the equation approach ( M = 0.58, SD = 0.43) and pictorial approach ( M = 0.11, SD = 0.27) where p = 0.01. Overall, the equation approach was better than the pictorial approach for the similar problems, and the other two approaches for the transfer problems.

In regard to the solution strategy, of those 17 students who received the equation approach, 14 could (82%) integrate relevant information in a single equation (first step) and calculate the answer (second step; Table 1 ). Once again, Study 3 shows the power of worked examples to assist students to develop expertise in which they not only identified the problem structure and used an equation to solve the problems, but also exhibited step skipping performance which is consistent with prior research ( Koedinger and Anderson, 1990 ; Blessing and Anderson, 1996 ). We attribute the generation of a two-step strategy ( Figure 4 ) as a result of the scaffold provided by the diagram that facilitates in-depth understanding of conceptual knowledge of the percentage change problems, and the solution steps that illustrate its procedure knowledge.

Figure 4 . A solution of a percentage increased problem that shows step skipping performance.

The target domain for Study 4 was the percentage change problems, which are similar to Study 3. Fifty-nine Asian students were randomly assigned to either the unitary approach or the equation approach. The design of the unitary approach and equation approach is the same as in Study 3. The research design involved a learning phase and a test phase. Again, we were unable to include a pre-test owing to the need to align the testing materials with the school curriculum. We focused on the equation approach ( Table 1 ). The equation approach was similar to the equation approach in Study 3.

Data analysis was based on 57 students who completed all test materials. The equation group ( M = 0.78, SD = 0.18) outperformed the unitary group ( M = 0.67, SD = 0.22) on similar problems, t (55) = 2.01, p = 0.05. The equation group ( M = 0.38, SD = 0.40) was marginally better than the unitary group ( M = 0.22, SD = 0.29) for the transfer problems, t (55) = 1.74, p = 0.09. The unitary approach imposed significantly higher mental effort than the equation approach, t (55) = 2.76, p = 0.008, r = 0.35 (a medium effect). In addition, using the relative condition efficiency, E = P − M 2 , the equation approach was significantly more efficient than the unitary approach, t (55) = 2.83, p = 0.006, r = 0.36 (a large effect).

On examining students’ solution strategies, 48% wrote a single equation, for example, 88 + (80 × 10%) (first step), and calculate the answer (second step). In addition, 44% wrote percentage quantity, for example, (80 × 10%; first step), and then added 80 to (80 × 10%) (second step). Once again, the success of the equation approach is clearly seen in students’ solution strategies. Hence, an in-depth understanding of the conceptual knowledge might contribute to the superior performance of procedural knowledge manifested in the reduced solution steps. Moreover, the robustness of the equation approach was confirmed in Asian context. It should be noted that 37% of Malaysian students in the unitary approach who did not have access to the equation approach used a modified version of the equation approach (two-step equation approach), such as (i) $250 × 12% = $30 (Step 1), and (ii) $250 + $30 = $280 (Step 2). Indirectly, this implies that Malaysian students may have stronger foundation than Australian students in Study 3 in regard to the use of algebra for solving word problems.

This Study 5 documented how to solve challenging percentage-change problems, which poses a challenge to students because the goal is to find the original quantity after a change of its original quantity ( Parker and Leinhardt, 1995 ; Ngu et al., 2018 ). An example of a challenging percentage-change problem is shown in Figure 4 : The sale price of an item including a 10% Good and Service Tax (GST) is $264. Find the price of the item excluding GST . The Australian sample (55 students, mean age = 16), and the Malaysian sample (75 students, mean age = 16) participated in the study. Students in each sample were randomly assigned to three groups (equation, unitary, unitary–equation approaches; Table 1 ).

As shown in Figure 5 once again, the equation approach is accompanied by a horizontal line depicting two components: (i) original price, and (ii) increased amount. However, a variable, such as x , is used to denote the original price, and the increased amount of the original price ( x  × 10%). Again, the germane cognitive load is increased to process the problem structure portrayed in the horizontal line, resulting in the generation of an equation: Sale price = original price + increased amount. Similar to the Study 3 or Study 4, the diagram aimed to uncover conceptual knowledge of the challenging percentage-change problems. The subsequent solution steps involve the substitution of values ($264, 10%), a variable ( x ) to form an equation, such as $264 = x + x × 10%, and solve for x .

Figure 5 . A worked example of a challenging percentage-change problem. Source: Ngu et al. (2018) .

The unitary approach shares similar design features as the unitary approach in Study 3 or Study 4. The main concept of this unitary approach is the unit percentage. The learners are required to calculate 1%, and then a multiple of 1% to obtain the answer. Basically, it comprises three solution steps: (i) 100% + 12% = 112% (markup 12%), (ii) $34 ÷112 = $0.3035 (calculate 1%), and (iii) $0.3035 × 100 = $30.35 (calculate 100% which is the original price). Similar to the Study 3 or Study 4, the need to integrate information from the first solution step (112%) and the second solution step ($34) will cause a split-attention effect. Thus, the use of working memory to deal with the split-attention effect as well as the high element interactivity arises from the interaction between multiple elements within and across the solution steps would hinder effective learning.

The main difference between the unitary–pictorial approach and the unitary approach is that the former has a diagram. The germane cognitive load is increased to process the diagram which depicts the proportion concept, aligning the quantity to its corresponding percentage. In particular, the alignment between $34 and 112% not only eliminates a split-attention, but also facilitates the calculation of a subgoal (1%). Hence, the unitary–pictorial would impose lower cognitive load than the unitary approach.

We used a pre-test—intervention—port-test design. The pre-test had the same number and type of questions as the post-test—it provided the baseline score to examine the impact of different instructional approaches upon learning to solve challenging percentage-change problems. The means pre-test score for Australian students ranged from 0–3%, and Malaysian students was 0%.

Again, the learning phase was similar to Study 3. Students were given 20 min to study an instruction sheet and complete six example–problem pairs. The instruction sheet provided two worked examples, one of which was percentage increased problem and the other was percentage decreased problem. For each example–problem pair, they studied a worked example ( Figure 5 ) and solve a similar problem. Once again, the main learning was expected to occur when students completed the example–problem pairs though they may also benefit from studying the instruction sheet.

Students who scored 80% or above in the pre-test were excluded from the final data analysis. In relation to the post-test, Malaysian students ( M = 0.36, SD = 0.23) outperformed Austrian students ( M = 0.16, SD = 0.27) for the for the equation approach in line with the hypothesis, t (37) = 2.52, p = 0.02. The Australian students and Malaysian students did not differ on the unitary approach, ( Ms = 0.23 versus 0.30), t (36) = 0.97, p = 0.34, nor on the unitary–pictorial approach, ( Ms = 0.47 versus 0.40), t (42) = 0.86, p = 0.40. In contrast to the hypothesis, Australian students did not outperform Malaysian students for the unitary–pictorial approach. Using pairwise comparisons, for Australian students, the unitary–pictorial group outperformed both the unitary group ( p = 0.02), and the equation group ( p = 0.00). In contrast, no differences were observed between the three groups for Malaysian students.

We analyzed the number of students who used respective solution strategies across the unitary approach, unitary–pictorial approach and equation approach for the post-test. A chi-square test indicated significant differences favoring Malaysian students for the equation approach, χ 2 (1, N  = 39) = 27.57, p  < 0.001, the unitary approach, χ 2 (1, N  = 38) = 5.43, p  = 0.02, and the unitary–pictorial approach, χ 2 (1, N  = 44) = 6.12, p  = 0.01. However, for the unitary–pictorial approach, Australian and Malaysian students demonstrated step skipping performance in that they generated two-step strategy. For example, consider a test item: A shirt has been discounted 60% and now costs $80. What did it cost originally? By discarding the first step “40% represents $80,” students wrote two steps: (i) 80 ÷ 40 = 2, and (ii) 2 × 100 = $200. A chi-square test indicated no difference between Australian students and Malaysian student on two-step strategy, χ 2 (1, N  = 44) = 0.86, p  = 0.36.

Regarding the equation approach, importantly, 14 out of 24 (58%) Malaysian students in the post-test skipped some aspects of the solution procedure for the same test item above. They integrated relevant information and expressed it in an equation: $80 =  x – 60% x (first step) and solve for x (second step; Figure 6 ). For Australian students, of those 5 students who provided accurate answers, none of them exhibited problem-solving expertise. In fact, many Australian students struggled with algebra (e.g., equation solving skills) and thus did not benefit from the equation approach. It is possible that Australian students could benefit from an alternative algebra approach that splits the solution procedure in two stages. In stage 1, the learner calculates % after a discount of 60%, which is 40%. In stage 2, the learner forms an equation, such as 40% x = $80, solve for x . This alternative approach may be easier because the variable ( x ) appears only once instead of twice ($80 =  x – 60% x ) in the equation ( Koedinger et al., 2008 ).

Figure 6 . A solution of a challenging percentage decreased problem that shows step skipping performance.

Taken together, the results from Study 4 and Study 5 suggests that Malaysian students were more competent in using the algebra approach for solving word problems. The Australian students were not inferior to Malaysian students in regard to the use of non-algebra approach (unitary–pictorial approach) for solving word problems. Consistent with the results obtained in Study 3 and Study 4, once students had gained expertise, they were capable of using a single equation to represent the problem structure for subsequent generation of a solution. Once again, this illustrates the power of worked examples to facilitate the acquisition of domain-specific knowledge related to challenging percentage-change problems, leading to the generation of reduced solution steps.

This paper is primarily concerned with the review of five prior studies that documented the power of worked examples to help students in developing expertise for word problems across Mathematics and Science curriculum. The review has provided an answer to the research questions. Across five studies, approximately 50% of students who were exposed to the worked examples approach has acquired problem-solving expertise for word problems. The implementation of studying worked examples only (Study 1), and example–problem pairs (Studies 2, 3, 4, and 5) imposes relatively low cognitive load, thus facilitating the acquisition of problem-solving expertise. As summarized in Table 1 , across five studies, students could generate an equation based on the information provided in the problem text (first step), and then solve the problem (second step). The demonstration of step skipping performance reflects the availability of a schema for a category of word problems, which encompasses conceptual knowledge and procedural knowledge. It appears that the presence of conceptual knowledge enhances the execution of procedural fluency in the form of fewer solution steps. In relation to cross-cultural comparison, Malaysian students outperformed Australian students for the equation approach (algebra approach) but not the unitary–pictorial approach (non-algebra approach). Presumably, such phenomenon is due to a stronger algebra foundation for Malaysian students.

Theoretical Considerations

Research has indicated the benefit of providing of a diagram to enhance learning of word problems in the domain of probability ( Schwonke et al., 2009 ). This is particularly the case when learners were informed that the function of a diagram aimed at bridging the relation between problem text and the equation. Thus, the provision of a diagram across Studies 1, 3, 4, and 5 that scaffolds the problem structure plays a critical role in developing problem-solving expertise. More specifically, in Study 1, studying the diagram that displayed a hierarchical order of the relation between individual structural elements had provided insights into the formation of an equation, leading to the generation of a two-step strategy. In Studies 3, 4 and 5, a horizontal line was divided into two different lengths that corresponded to two different quantities (e.g., an increased amount as a fraction of the whole amount), thus scaffolding the conceptual knowledge of percentage change problems. Such a display of visual information assisted students to formulate an equation to solve percentage problems efficiently. While Study 2 did not have a diagram, an emphasis of having three key solution steps that encompass both conceptual knowledge and procedural knowledge allowed students to infer that the third equation step was a critical step that contained the problem structure expressed in an equation for solution. On the other hand, without the aid of a diagram to scaffold the problem structure (such as the proportion concept in the unitary approach) across Study 3–5, learners struggled to acquire skills for solving word problems.

Previous research has focused on analogical learning to facilitate the acquisition of schema for word problems ( Reed, 1989 ). It highlights the mapping of concepts between two problems that share a similar problem structure. However, it falls short of providing instructional support to integrate relevant information and express this in an equation. In contrast, a worked example accompanied by a diagram is effective, because the diagram provides clues to organize problem structure from the problem text and express this in an equation. In addition, completing multiple example–problem pairs enables students to gain familiarity with a category of word problems that shares a similar problem structure and thus a similar solution procedure. Accordingly, students developed problem–solving expertise for word problems as a result of exposing to worked examples.

Practical Implication for Mathematics Education

The Australian educators are encouraged to consider the merit of cognitive load theory in designing instructions for effective learning ( NSW Education: Centre for Education Statistics and Evaluation, 2017 ). The current popular mathematics textbooks (e.g., Vincent et al., 2012 ) do not include the use of worked examples especially the implementation of example–problem pairs to enhance mathematics learning. Hence, it is timely to promote greater use of worked examples to develop problem-solving expertise especially for word problems that presents a challenge to students.

The use of worked examples to facilitate expertise development also depends on students’ prior knowledge ( Kalyuga et al., 2003 ). For example, students who are weak in basic algebra concepts, such as the meaning of variable, factorization, and equation solving skills, may not benefit from the equation approach for learning challenging percentage problems, let alone develop expertise for this type of problems. Thus, it is important to strengthen students’ prior knowledge of algebra before exposing them to the use of the algebra approach for learning word problems. Furthermore, having prior knowledge of equivalent fractions (e.g., 1/10 = 10/100) would have facilitated the processing of the proportion concept in a diagram (e.g., unitary–pictorial approach) with fewer elements ( Carlson et al., 2003 ).

Limitations and Future Directions

This review highlights the importance of using the algebra approach (i.e., the equation approach) to facilitate the acquisition of problem-solving expertise for word problems. However, there are other instructional approaches which could have achieved the same purpose. For example, in Study 5, the unitary–pictorial approach had also shown to be effective for challenging percentage-change problems. The diagram in the unitary–pictorial approach scaffolded the relation between quantity and percentage based on proportional reasoning. Consequently, students demonstrated step skipping performance, which was reflected in their solution strategies. Thus, future review should explore the design of different worked example formats that has the potential to facilitate the development of problem-solving expertise.

Early work by Cooper and Sweller (1987) indicates the importance of an extended practice time to enable students to automate the schema to solve not only similar problems but also transfer problems that require the adaptation of the solution procedure. Students across the five studies demonstrated step skipping performance despite a relatively short learning phase of about 25 min. Nonetheless, future research should provide a longer learning phase to enable students to develop not only expertise in solving similar problems, but also skills to solve transfer problems. In addition, a longer learning phase would also allow more students (e.g., more than 50%) to develop expertise for solving word problems.

Across the five case studies, the use of worked examples has facilitated the development of problem-solving expertise for a category of word problems sharing the same schema. Mayer (1982) distinguished standard problems (similar problems) and non-standard problems (transfer problems). Mayer regarded standard problems as a category of word problems that share the same schema. Unlike standard problems, non-standard problems do not share the same schema. In fact, an adaptation of the solution procedure for solving standard problems is required to solve non-standard problems. Overall, the strength of worked examples lie in its ability to assist learners to develop expertise for solving similar problems (standard problems), and to a lesser extent, transfer problems (non-standard problems).

Prior studies have uncovered the challenge for problem-solvers to interpret differences in syntax (word order) within specific type of word problems. Clement (1982) found that students tended to interpret the syntax of word problems in a static rather than relational manner. They tended to produce 6S = P for a statement “There are six times as many students as professors at this university.” In a related study, successful problem-solvers were capable of using relational key words (e.g., less) to form an equation accurately, whereas unsuccessful problem-solvers interpreted the syntax literally, leading to an incorrect equation ( Hegarty et al., 1995 ). In fact, the main issue here is failure for problem-solvers to interpret relational variables which stand for variables rather than objects. Strengthening learners’ prior knowledge of the concept of variable would alleviate learners’ working memory capacities for processing numerical and translation dimensions of word problems. Additional research is needed to verify this proposition.

Drawing on the review of the five studies, the main ideas and interpretations presented pointing to the efficacy of studying worked examples alone or completing multiple example–problem pairs to develop problem-solving expertise for word problems. A typical worked example provides a diagram that scaffolds conceptual knowledge (problem structure) and procedural knowledge (solution steps). There is evidence that an in-depth understanding of conceptual knowledge (i.e., problem structure) might contribute to superior procedural fluency manifested in the reduced solution steps. Thus, we urge mathematics educators to consider the incorporation of worked examples in mathematics classroom to assist students in gaining problem-solving expertise for word problems.

Author Contributions

BN and HP contributed equally to the articulation, conceptualization, and write-up of this manuscript. All authors contributed to the article and approved the submitted version.

Conflict of Interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher’s Note

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Baroody, A. J., Feil, Y., and Johnson, A. R. (2007). An alternative reconceptualization of procedural and conceptual knowledge. J. Res. Math. Educ. 38:115. doi: 10.2307/30034952

CrossRef Full Text | Google Scholar

Blessing, S. B., and Anderson, J. R. (1996). How people learn to skip steps. J. Exp. Psychol. Learn. Mem. Cogn. 22, 576–598. doi: 10.1037/0278-7393.22.3.576

Blessing, S. B., and Ross, B. H. (1996). Content effects in problem categorization and problem solving. J. Exp. Psychol. Learn. Mem. Cogn. 22, 792–810. doi: 10.1037/0278-7393.22.3.792

Bokosmaty, S., Sweller, J., and Kalyuga, S. (2015). Learning geometry problem solving by studying worked examples:Effects of learner guidance and expertise. Am. Educ. Res. J. 52, 307–333. doi: 10.3102/0002831214549450

Cai, J. (2000). Mathematical thinking involved in U.S. and Chinese students’ solving of process-constrained and process-open problems. Math. Think. Learn. 2, 309–340. doi: 10.1207/s15327833mtl0204_4

Cai, J., Lew, H. C., Morris, A., Moyer, J. C., Ng, S. F., and Schmittau, J. (2005). The development of students’ algebraic thinking in earlier grades: A cross-cultural comparative perspective. Int. J. Math. Educ. 37, 5–15. doi: 10.1007/BF02655892

Carlson, R., Chandler, P., and Sweller, J. (2003). Learning and understanding science instructional material. J. Educ. Psychol. 95, 629–640. doi: 10.1037/0022-0663.95.3.629

Chen, O., Kalyuga, S., and Sweller, J. (2015). The worked example effect, the generation effect, and element interactivity. J. Educ. Psychol. 107, 689–704. doi: 10.1037/edu0000018

Chen, O., Kalyuga, S., and Sweller, J. (2017). The expertise reversal effect is a variant of the more general element interactivity effect. Educ. Psychol. Rev. 29, 393–405. doi: 10.1007/s10648-016-9359-1

Chi, M. T. H., Feltovich, P. J., and Glaser, R. (1981). Categorization and representation of physics problems by experts and novices. Cogn. Sci. 5, 121–152. doi: 10.1207/s15516709cog0502_2

Clement, J. (1982). Algebra word problem solutions: thought processes underlying a common misconception. J. Res. Math. Educ. 13, 16–30. doi: 10.2307/748434

Cooper, G., and Sweller, J. (1987). Effects of schema acquisition and rule automation on mathematical problem-solving transfer. J. Educ. Psychol. 79, 347–362. doi: 10.1037/0022-0663.79.4.347

Cowan, N. (2001). The magical number 4 in short-term memory: a reconsideration of mental storage capacity. Behav. Brain Sci. 24:87. doi: 10.1017/S0140525X01003922

De Groot, A. (1965). Thought and Choice in Chess . The Hague: Mouton.

Google Scholar

Ericsson, K. A. (2006). “The influence of experience and deliberate practice on the development of superior expert performance,” in The Cambridge Handbook of Expertise and Expert Performance . eds. K. A. Ericsson, N. Charness, P. J. Feltovich, and R. R. Hoffman (New York, NY: Cambridge University Press), 683–704.

Fitzallen, N. (2015). “STEM education: what does mathematics have to offer?” in Mathematics Education in the Margins: Proceedings of the 38th Annual Conference of the Mathematics Education Research Group of Australasia . eds. M. Marshman, V. Geiger, and A. Bennison (Sunshine Coast: MERGA), 237–244.

Gog, T. V., Rummel, N., and Renkl, A. (2019). “Learning how to solve problems by studying examples,” in The Cambridge Handbook of Cognition and Education . eds. Dunlosky J. and Rawson K. A. (New York, NY, US: Cambridge University Press), 183–208.

Hegarty, M., and Kozhevnikov, M. (1999). Types of visual-spatial representations and mathematical problem solving. J. Educ. Psychol. 91, 684–689. doi: 10.1037/0022-0663.91.4.684

Hegarty, M., Mayer, R. E., and Monk, C. A. (1995). Comprehension of arithmetic word problems: a comparison of successful and unsuccessful problem solvers. J. Educ. Psychol. 87, 18–32. doi: 10.1037/0022-0663.87.1.18

Hiebert, J., and Lefevre, P. (1986). “Conceptual and procedural knowledge in mathematics: An introductory analysis,” in Conceptual and Procedural Knowledge: The Case of Mathematics . ed. J. Hiebert (Hillsdale, NJ: Erlbaum), 1–27.

Jitendra, A. K., Star, J. R., Rodriguez, M., Lindell, M., and Someki, F. (2011). Improving students’ proportional thinking using schema-based instruction. Learn. Instr. 21, 731–745. doi: 10.1016/j.learninstruc.2011.04.002

Kalyuga, S., Ayres, P., Chandler, P., and Sweller, J. (2003). The expertise reversal effect. Educ. Psychol. 38, 23–31. doi: 10.1207/s15326985ep3801_4

Koedinger, K. R., Alibali, M. W., and Nathan, M. J. (2008). Trade-offs between grounded and abstract representations: evidence From algebra problem solving. Cogn. Sci. 32, 366–397. doi: 10.1080/03640210701863933

PubMed Abstract | CrossRef Full Text | Google Scholar

Koedinger, K. R., and Anderson, J. R. (1990). Abstract planning and perceptual chunks: elements of expertise in geometry. Cogn. Sci. 14, 511–550. doi: 10.1207/s15516709cog1404_2

Lee, C., and Kalyuga, S. (2011). Effectiveness of on-screen pinyin in learning Chinese: an expertise reversal for multimedia redundancy effect. Comput. Hum. Behav. 27, 11–15. doi: 10.1016/j.chb.2010.05.005

Likourezos, V., Kalyuga, S., and Sweller, J. (2019). The variability effect: when instructional variability is advantageous. Educ. Psychol. Rev. 31, 479–497. doi: 10.1007/s10648-019-09462-8

Lu, J., Kalyuga, S., and Sweller, J. (2020). Altering element interactivity and variability in example-practice sequences to enhance learning to write Chinese characters. Appl. Cogn. Psychol. 34, 837–843. doi: 10.1002/acp.3668

Mayer, R. E. (1982). Memory for algebra story problems. J. Educ. Psychol. 74, 199–216. doi: 10.1037/0022-0663.74.2.199

Mayer, R. E. (1985). “Mathematical ability,” in Human Abilities: An Information-Processing Approach . ed. R. J. Sternberg (New York, NY: Freeman), 127–150.

Mayer, R. E., and Gallini, J. K. (1990). When is an illustration worth ten thousand words? J. Educ. Psychol. 82, 715–726. doi: 10.1037/0022-0663.82.4.715

McLaren, B. M., van Gog, T., Ganoe, C., Karabinos, M., and Yaron, D. (2016). The efficiency of worked examples compared to erroneous examples, tutored problem solving, and problem solving in computer-based learning environments. Comput. Hum. Behav. 55, 87–99. doi: 10.1016/j.chb.2015.08.038

Miller, G. A. (1956). The magical number seven, plus or minus two: some limits on our capacity for processing information. Psychol. Rev. 63, 81–97. doi: 10.1037/h0043158

Nathan, M. J., Kintsch, W., and Young, E. (1992). A theory of algebra-word-problem comprehension and its implications for the design of learning environments. Cogn. Instr. 9, 329–389. doi: 10.1207/s1532690xci0904_2

Ng, S. F., and Lee, K. (2009). The model method: Singapore children’s tool for representing and solving algebraic word problems. J. Res. Math. Educ. 40, 282–313. doi: 10.5951/jresematheduc.40.3.0282

Ngu, B. H., Mit, E., Shahbodin, F., and Tuovinen, J. (2009). Chemistry problem solving instruction: a comparison of three computer-based formats for learning from hierarchical network problem representations. Instr. Sci. 37, 21–42. doi: 10.1007/s11251-008-9072-7

Ngu, B. H., and Phan, H. P. (2020). “An examination of pre-service teachers’ content knowledge on linear equations: A cross-cultural study,” in Progress in Education. Vol. 64. ed. R. V. Nata (New York, NY: Nova Science Publishers, Inc.), 1–34.

Ngu, B. H., Phan, H. P., Hong, K. S., and Usop, H. (2016). Reducing intrinsic cognitive load in percentage change problems: The equation approach. Learn. Individ. Differ. 51, 81–90. doi: 10.1016/j.lindif.2016.08.029

Ngu, B. H., and Yeung, A. S. (2013). Algebra word problem solving approaches in a chemistry context: equation worked examples versus text editing. J. Math. Behav. 32, 197–208. doi: 10.1016/j.jmathb.2013.02.003

Ngu, B. H., Yeung, A. S., Phan, H. P., Hong, K. S., and Usop, H. (2018). Learning to solve challenging percentage-change problems: a cross-cultural study from a cognitive load perspective. J. Exp. Educ. 86, 362–385. doi: 10.1080/00220973.2017.1347774

Ngu, B., Yeung, A., and Tobias, S. (2014). Cognitive load in percentage change problems: unitary, pictorial, and equation approaches to instruction. Instr. Sci. 42, 685–713. doi: 10.1007/s11251-014-9309-6

NSW Education: Centre for Education Statistics and Evaluation (2017). Cognitive load theory: Research that teachers really need to understand. Sydney, Australia. Available at:  http://www.cese.nsw.gov.au/images/stories/PDF/cognitive_load_theory_report_AA1.pdf (Accessed March 27, 2022).

Paas, F. G. W. C. (1992). Training strategies for attaining transfer of problem-solving skill in statistics: a cognitive-load approach. J. Educ. Psychol. 84, 429–434. doi: 10.1037/0022-0663.84.4.429

Paas, F., and Van Merriënboer, J. J. G. (1994). Variability of worked examples and transfer of geometrical problem-solving skills: a cognitive-load approach. J. Educ. Psychol. 86, 122–133. doi: 10.1037/0022-0663.86.1.122

Pachman, M., Sweller, J., and Kalyuga, S. (2013). Levels of knowledge and deliberate practice. J. Exp. Psychol. Appl. 19, 108–119. doi: 10.1037/a0032149

Parker, M., and Leinhardt, G. (1995). Percent: A privileged proportion. Rev. Educ. Res. 65, 421–481. doi: 10.3102/00346543065004421

Peterson, L., and Peterson, M. J. (1959). Short-term retention of individual verbal items. J. Exp. Psychol. 58, 193–198. doi: 10.1037/h0049234

Quilici, J. L., and Mayer, R. E. (1996). Role of examples in how students learn to categorize statistics word problems. J. Educ. Psychol. 88, 144–161. doi: 10.1037/0022-0663.88.1.144

Reed, S. K. (1989). Constraints on the abstraction of solutions. J. Educ. Psychol. 81, 532–540. doi: 10.1037/0022-0663.81.4.532

Reed, S. K., Dempster, A., and Ettinger, M. (1985). Usefulness of analogous solutions for solving algebra word problems. J. Exp. Psychol. Learn. Mem. Cogn. 11, 106–125. doi: 10.1037/0278-7393.11.1.106

Renkl, A. (1999). Learning mathematics from worked-out examples: analyzing and fostering self-explanations. Eur. J. Psychol. Educ. 14, 477–488. doi: 10.1007/bf03172974

Renkl, A. (2014). Toward an instructionally oriented theory of example-based learning. Cogn. Sci. 38, 1–37. doi: 10.1111/cogs.12086

Rittle-Johnson, B., Schneider, M., and Star, J. R. (2015). Not a one-way street: bidirectional relations between procedural and conceptual knowledge of mathematics. Educ. Psychol. Rev. 27, 587–597. doi: 10.1007/s10648-015-9302-x

Rittle-Johnson, B., Siegler, R. S., and Alibali, M. W. (2001). Developing conceptual understanding and procedural skill in mathematics: an iterative process. J. Educ. Psychol. 93, 346–362. doi: 10.1037/0022-0663.93.2.346

Schwonke, R., Berthold, K., and Renkl, A. (2009). How multiple external representations are used and how they can be made more useful. Appl. Cogn. Psychol. 23, 1227–1243. doi: 10.1002/acp.1526

Schworm, S., and Renkl, A. (2006). Computer-supported example-based learning: when instructional explanations reduce self-explanations. Comput. Educ. 46, 426–445. doi: 10.1016/j.compedu.2004.08.011

Silver, E. A. (1979). Student perceptions of relatedness among mathematical verbal problems. J. Res. Math. Educ. 10, 195–210. doi: 10.2307/748807

Star, J., and Newton, K. (2009). The nature and development of experts’ strategy flexibility for solving equations. ZDM 41, 557–567. doi: 10.1007/s11858-009-0185-5

Sweller, J. (2010). Element interactivity and intrinsic, extraneous, and germane cognitive load. Educ. Psychol. Rev. 22, 123–138. doi: 10.1007/s10648-010-9128-5

Sweller, J., Ayres, P., and Kalyuga, S. (2011). Cognitive Load Theory . New York: Springer.

Sweller, J., and Cooper, G. A. (1985). The use of worked examples as a substitute for problem solving in learning algebra. Cogn. Instr. 2, 59–89. doi: 10.1207/s1532690xci0201_3

Tarmizi, R. A., and Sweller, J. (1988). Guidance during mathematical problem solving. J. Educ. Psychol. 80, 424–436. doi: 10.1037/0022-0663.80.4.424

Trafton, J. G., and Reiser, B. J. (1993). The contributions of studying examples and solving problems to skill acquisition. In Proceedings of the 15th Annual Conference of the Cognitive Science Society . eds. M. Polson; Hillsdale: Lawrence Erlbaum; (1017–1022).

Tricot, A., and Sweller, J. (2014). Domain-specific knowledge and why teaching generic skills does not work. Educ. Psychol. Rev. 26, 265–283. doi: 10.1007/s10648-013-9243-1

van Gog, T., Kester, L., and Paas, F. (2011). Effects of worked examples, example-problem, and problem-example pairs on novices’ learning. Contemp. Educ. Psychol. 36, 212–218. doi: 10.1016/j.cedpsych.2010.10.004

van Gog, T., Paas, F., and van Merriënboer, J. J. G. (2008). Effects of studying sequences of process-oriented and product-oriented worked examples on troubleshooting transfer efficiency. Learn. Instr. 18, 211–222. doi: 10.1016/j.learninstruc.2007.03.003

Vincent, J., Price, B., Caruso, N., Romeril, G., and Tynan, D. (2012). Maths World 9 Australian Curriculum Edition . South Yarra, VIC: Macmillan.

Zahner, D., and Corter, J. E. (2010). The process of probability problem solving: use of external visual representations. Math. Think. Learn. 12, 177–204. doi: 10.1080/10986061003654240

Keywords: problem-solving, expertise, worked examples, word problems, cognitive load

Citation: Ngu BH and Phan HP (2022) Developing Problem-Solving Expertise for Word Problems. Front. Psychol . 13:725280. doi: 10.3389/fpsyg.2022.725280

Received: 15 June 2021; Accepted: 21 March 2022; Published: 03 May 2022.

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Copyright © 2022 Ngu and Phan. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY) . The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Bing Hiong Ngu, [email protected]

Disclaimer: All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article or claim that may be made by its manufacturer is not guaranteed or endorsed by the publisher.

A Math Word Problem Framework That Fosters Conceptual Thinking

This strategy for selecting and teaching word problems guides students to develop their understanding of math concepts.

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Word problems in mathematics are a powerful tool for helping students make sense of and reason with mathematical concepts. Many students, however, struggle with word problems because of the various cognitive demands. As districtwide STEAM professional development specialists, we’ve spent a lot of time focusing on supporting our colleagues and students to ensure their success with word problems. We found that selecting the right word problems, as well as focusing on conceptual understanding rather than procedural knowledge, provides our students with real growth.

As our thinking evolved, we began to instill a routine that supports teaching students to solve with grit by putting them in the driver’s seat of the thinking. Below you’ll find the routine that we’ve found successful in helping students overcome the challenges of solving word problems.

Not all word problems are created equal

Prior to any instruction, we always consider the quality of the task for teaching and learning. In our process, we use word problems as the path to mathematics instruction. When selecting the mathematical tasks for students, we always consider the following questions:

• Does the task align with the learning goals and standards?
• Will the task engage and challenge students at an appropriate level, providing both a sense of accomplishment and further opportunities for growth?
• Is the task open or closed? Open tasks provide multiple pathways to foster a deeper understanding of mathematical concepts and skills. Closed tasks can still provide a deep understanding of mathematical concepts and skills if the task requires a high level of cognitive demand. 
• Does the task encourage critical thinking and problem-solving skills?
• Will the task allow students to see the relevance of mathematics to real-world situations?
• Does the task promote creativity and encourage students to make connections between mathematical concepts and other areas of their lives?

If we can answer yes to as many of these questions as possible, we can be assured that our tasks are rich. There are further insights for rich math tasks on NRICH and sample tasks on Illustrative Mathematics and K-5 Math Teaching Resources .

Developing conceptual understanding

Once we’ve selected the rich math tasks, developing conceptual understanding becomes our instructional focus. We present students with Numberless Word Problems and simultaneously use a word problem framework to focus on analysis of the text and to build conceptual understanding, rather than just memorization of formulas and procedures. 

• First we remove all of the numbers and have students read the problem focusing on who or what the problem is about; they visualize and connect the scenario to their lives and experiences. 
• Next we have our students rewrite the question as a statement to ensure that they understand the questions.
• Then we have our students read the problem again and have them think analytically. They ask themselves these questions: Are there parts? Is there a whole? Are things joining or separating? Is there a comparison? 
• Once that’s completed, we reveal the numbers in the problem. We have the students read the problem again to determine if they have enough information to develop a model and translate it into an equation that can be solved.
• After they’ve solved their equation, we have students compare it against their model to check their answer.  

Collaboration and workspace are key to building the thinking

To build the thinking necessary in the math classroom , we have students work in visibly random collaborative groups (random groups of three for grades 3 through 12, random groups of two for grades 1 and 2). With random groupings, we’ve found that students don’t enter their groups with predetermined roles, and all students contribute to the thinking.

For reluctant learners, we make sure these students serve as the scribe within the group documenting each member’s contribution. We also make sure to use nonpermanent vertical workspaces (whiteboards, windows [using dry-erase markers], large adhesive-backed chart paper, etc.). The vertical workspace provides accessibility for our diverse learners and promotes problem-solving because our students break down complex problems into smaller, manageable steps. The vertical workspaces also provide a visually appealing and organized way for our students to show their work.  We’ve witnessed how these workspaces help hold their attention and improve their focus on the task at hand.

Facilitate and provide feedback to move the thinking along

As students grapple with the task, the teacher floats among the collaborative groups, facilitates conversations, and gives the students feedback. Students are encouraged to look at the work of other groups or to provide a second strategy or model to support their thinking. Students take ownership and make sense of the problem, attempt solutions, and try to support their thinking with models, equations, charts, graphs, words, etc. They work through the problem collaboratively, justifying their work in their small group. In essence, they’re constructing their knowledge and preparing to share their work with the rest of the class. 

Word problems are a powerful tool for teaching math concepts to students. They offer a practical and relatable approach to problem-solving, enabling students to understand the relevance of math in real-life situations. Through word problems, students learn to apply mathematical principles and logical reasoning to solve complex problems. 

Moreover, word problems also enhance critical thinking, analytical skills, and decision-making abilities. Incorporating word problems into math lessons is an effective way to make math engaging, meaningful, and applicable to everyday life.

Why Word Problems Are Such a Struggle for Students—And What Teachers Can Do

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Give Cindy Cliche a math word problem, and she can tell you exactly where most students are going to trip up.

Cliche, the district math coordinator in the Murfreesboro City school district in Tennessee, has spent decades teaching elementary schoolers how to tackle their first word problems and now coaches teachers in how to do the same. Kids’ struggles, for the most part, haven’t changed, she said.

Take this problem, which students might work on in 1st grade: There are some bunnies on the grass. Three bunnies hop over, and then there are five total. How many bunnies were there to begin with?

The problem is asking about a change: What’s the starting, unknown quantity of bunnies, if adding 3 to that quantity equals 5? In other words, x + 3 = 5 . But most 1st graders don’t make that connection right away, Cliche said. Instead, they see the numbers 3 and 5, and they add them.

“Nine times out of 10 they’re going to say, ‘eight,’” Cliche said. “They’re number pluckers. They take this number and this number and they add them together or they take them apart.”

This is one of the biggest challenges in word problem-solving, educators and researchers agree—getting students to understand that the written story on the page represents a math story, and that the math story can be translated into an equation.

Making this connection is a key part of early mathematical sense-making. It helps students begin to understand that math isn’t just about numbers on a page, but a way of representing relationships in the world. And it’s one of the ways that kids learn to unite conceptual understanding of problems with the procedures they will need to solve them.

“When students struggle [with word problems], it tends to be everything else they have to do to get to the calculation,” said Brian Bushart, a 4th grade teacher in the West Irondequoit schools in Rochester, N.Y.

There are evidence-backed strategies that teachers can use to help students make these connections, researchers say.

These approaches teach students how to understand “math language,” how to devise a plan of attack for a problem, and how to recognize different problem types. And though they provide students tools and explicit strategies, these techniques are designed to support kids’ sense-making, not circumvent it, said Lynn Fuchs, a research professor in the department of special education at Vanderbilt University.

The goal, she said, is “understanding the full narrative of what’s being presented.”

How word problems are used in early grades

Story problems serve a few different purposes in early grades, said Nicole McNeil, a professor of psychology at the University of Notre Dame who studies students’ cognitive development in math.

They can help connect children’s preexisting knowledge to the math they’re learning in class—"activating that knowledge kids have in their everyday life, and then showing, how do mathematicians represent that?” McNeil said.

Cliche likes to use word problems in this way to introduce the concept of dividing by fractions.

“We’ll tell the kids, ‘I have three sandwiches here and I need to divide them in half so that everyone will get a piece,’” she said. “‘How many people can I feed?’”

After students solve the problem, Cliche introduces the operation that students could use to divide by fractions—marrying this conceptual understanding with the procedure that students would use going forward.

But word problems can also be used in the opposite direction, to see if students can apply their understanding of equations they’ve learned to real-world situations, McNeil said.

And there’s another, practical reason that teachers practice word problems: They’re ubiquitous in curriculum and they’re frequently tested.

There are lots of different kinds of problems that kids could work on in math classes, said Tamisha Thompson, a STEAM (for science, technology, engineering, the arts, and math) instructional coach in the Millbury public schools in Massachusetts, and a doctoral student in learning sciences at Worcester Polytechnic Institute.

Many story problems have one right answer, but there are also problems that could have multiple answers—or ones that aren’t solvable. Spending more time with a broader diversity of problems could encourage more creative mathematical thinking, Thompson said. “But we’re really driven by standardized tests,” she said. “And standardized tests typically have one right answer.”

In general, between 30 percent and 50 percent of standardized-test items in math feature these kinds of story problems, said Sarah Powell, an associate professor in the department of special education at the University of Texas at Austin.

“Until things change, and until we write better and different tests, if you want students to show their math knowledge, they have to show that through word problem-solving,” Powell said.

Why students struggle with word problems

Sometimes, students struggle with word problems because they don’t know where to start.

Just reading the problem can be the first hurdle. If early-elementary schoolers don’t have the reading skills to decode the words, or if they don’t know some of the vocabulary, they’ll struggle, said McNeil.

That can result in students scoring low on these portions of standardized tests, even if they understand the underlying math concepts—something McNeil considers to be a design flaw. “You’re trying to assess math, not reading twice,” she said.

Then, there’s math-specific vocabulary. What do words like “fewer than,” or “the rest,” mean in math language, and how do they prompt different actions depending on their placement in a problem?

Even if students can read the problem, they may struggle to figure out what it’s asking them to do, said Powell. They need to identify relevant information and ignore irrelevant information—including data that may be presented in charts or graphs. Then, they have to choose an operation to use to solve the problem.

Only once students have gone through all these steps do they actually perform a calculation.

Teaching kids how to work through all these setup steps takes time. But it’s time that a lot of schools don’t take, said Cliche, who has also worked previously as a state math trainer for Tennessee. Word problems aren’t often the focus of instruction—rather, they’re seen as a final exercise in transfer after a lot of practice with algorithms, she said.

A second problem: Many schools teach shortcut strategies for deciphering word problems that aren’t effective, Powell said.

Word problem “key words” charts abound on lesson-sharing sites like Teachers Pay Teachers . These graphic organizers are designed to remind students which math words signal different operations. When you see the word “more,” for example, that means add the numbers in the problem.

Talking with students about the meaning of math vocabulary is useful, said Powell. But using specific words as cues to add or subtract is a flawed strategy, Powell said, because “there is no single word that means an operation.” The word “more” might mean that the numbers need to be added together—or it might mean something else in context. Some problems have no key words at all.

In a 2022 paper , Powell and her colleagues analyzed more than 200 word problems from Partnership for Assessment of Readiness for College and Careers (PARCC) and Smarter Balanced math tests in elementary and middle school grades. Those tests are given by states for federal accountability purposes.

They found that using the key words strategy would lead students to choose the right operation to solve the problem less than half the time for single-step problems and less than 10 percent of the time for multistep problems.

Evidence-based strategies for helping struggling students

So if key words aren’t an effective strategy to support students who struggle, what is?

One evidence-based approach is called schema-based instruction . This approach categorizes problems into different types, depending on the math event portrayed, said Fuchs, who has studied schema-based instruction for more than two decades.

But unlike key words, schemas don’t tell students what operations to use. Instead, they help students form a mental model of a math event. They still need to read the problem, understand how that story maps onto their mental model, and figure out what information is missing, Fuchs said.

One type of schema, for example, is a “total” or “combine” problem, in which two quantities together make a total: “Jose has five apples. Carlos has two apples. How many apples do they have together?” In this case, students would need to add to get the answer.

But this is also a total problem: “Together, Jose and Carlos have seven apples. If Jose has five apples, how many apples does Carlos have?”

Here, adding the two numbers in the problem would bring students to the wrong answer. They need to understand that seven is the total, five is one part of the total, and there is another, unknown part—and then solve from there.

To introduce schemas, Vanderbilt’s Fuchs said, “we start with a child and the teacher representing the mathematical event in a concrete way.”

Take a “difference” problem, which compares a larger quantity and a smaller quantity for a difference. To demonstrate this, an early-elementary teacher might show the difference in height between two students or the difference in length of two posters in the room.

Eventually, the teacher would introduce other ways of representing this “difference” event, like drawing one smaller and one larger rectangle on a piece of paper. Then, Fuchs said, the teacher would explain the “difference” event with a number sentence—the formula for calculating difference—to connect the conceptual understanding with the procedure. Students would then learn a solution strategy for the schema.

Children can then use their understanding of these different problem types to solve new problems, Fuchs said.

There are other strategies for word-problem-solving, too.

• Attack strategies . Several studies have found that giving students a consistent set of steps they can use to approach every problem has positive effects. These attack strategies are different from schemas because they can be used with any problem type, offering more general guidance like reminders to read the problem and pull out relevant information.
• Embedded vocabulary. A 2021 study from Fuchs and her colleagues found that math-specific vocabulary instruction helped students get better at word problem-solving. These vocabulary lessons were embedded into schema instruction, and they focused on words that had a specific meaning in a math context—teaching kids the difference between “more than” and “then there were more,” for example.
• ‘Numberless’ problems . Some educators have also developed their own strategies. One of these is what’s called “numberless” word problems. A numberless problem has the same structure as a regular story problem but with the quantities strategically removed. An initial statement might say, for example, “Kevin found some bird feathers in the park. On his way home, he lost some of the feathers.”

With numberless problems, instead of jumping to the calculation, “the conversation is the goal,” said Bushart, the 4th grade teacher from New York, who has created a website bank of numberless problems that teachers can use .

The teacher talks with students about the change the story shows and what numbers might be reasonable—and not reasonable. The process is a form of scaffolding, Bushart said: a way to get students thinking conceptually about problems from the start.

Balancing structure and challenge

These approaches all rely on explicit teaching to give students tools that can help them succeed with problems they’re likely to see often in class or on tests.

But many math educators also use word problems that move beyond these common structures, in an attempt to engage students in creative problem-solving. Figuring out how much structure to provide—and how much challenge—can be a delicate balance.

These kinds of problems often require that students integrate real-life knowledge, and challenge them to “think beyond straightforward applications of mathematical situations,” said McNeil of Notre Dame.

There may be an extra number in the problem that kids don’t have to use. Or the problem might pose a question that would lead students to a nonsensical answer if they just used their procedural knowledge. For example: 65 students are going on a field trip. If each bus can hold 10 students, how many buses are needed?

Students might do the calculation and answer this question with 6.5, but that number doesn’t make sense, said McNeil—you can’t have half a bus.

In a 2021 study , McNeil and her colleague Patrick Kirkland rewrote some of these challenging questions in a way that encouraged students to think more deeply about the problems. They found that middle school students who worked on these experimental problems were more likely than their peers to engage in deep mathematical thinking. But, they were also less likely to get the problems correct than their peers who did standard word problems.

Other research, with young children, has found that teaching students how to transfer their knowledge can help them work through novel problems.

When students are given only problems that are all structured the same way, even minor changes to that format can prevent them from recognizing problem schemas, said Fuchs.

“What we found in our line of work is that if you change the way the word problem reads, in only very minor ways, they no longer recognize that, this is a ‘change’ problem, or a ‘difference’ problem,” she said, referencing different problem schemas.

In the early 2000s, she and her colleagues tested interventions to help students transfer their knowledge to more complex, at times open-ended problems. They found that when children were taught about the notion of transfer, shown examples of different forms of the same problem type, and encouraged to find examples in their own lives, they performed better on novel, multistep problems than their peers who had only received schema instruction.

The results are an example of how explicit instruction can lay the groundwork for students to be successful with more open-ended problem-solving, Fuchs said.

Exactly how to sequence this learning—when to lean into structure and when to release students into challenge—is an open question, McNeil said.

“We need more researchers focused on what are the best structures? What order should things go in? What is the appropriate scope and sequence for word problems?” she said. “We don’t have that information yet.”

A version of this article appeared in the May 10, 2023 edition of Education Week as Why Word Problems Are Such a Struggle for Students—And What Teachers Can Do

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Solving Word Problems using Schemas: A Review of the Literature

Sarah r. powell.

Vanderbilt University

Solving word problems is a difficult task for students at-risk for or with learning disabilities (LD). One instructional approach that has emerged as a valid method for helping students at-risk for or with LD to become more proficient at word-problem solving is using schemas. A schema is a framework for solving a problem. With a schema, students are taught to recognize problems as falling within word-problem types and to apply a problem solution method that matches that problem type. This review highlights two schema approaches for 2 nd - and 3 rd -grade students at-risk for or with LD: schema-based instruction and schema-broadening instruction. A total of 12 schema studies were reviewed and synthesized. Both types of schema approaches enhanced the word-problem skill of students at-risk for or with LD. Based on the review, suggestions are provided for incorporating word-problem instruction using schemas.

Since Pólya (1945) introduced four steps for solving word problems (understand the question, devise a plan, carry out the plan, and look back and check), teachers have been encouraged to provide more systematic instruction on problem solving in mathematics. Word-problem instruction has become vital for students. High-stakes standardized tests like the National Assessment of Educational Progress (NAEP; National Assessment Governing Board, 2009 ) place heavy emphasis on mathematics word problems and national educational organizations like the National Council of Teachers of Mathematics ( NCTM; 2000 ) heavily value the teaching of problem solving across grades K through 12. Many researchers have investigated methods for teaching problem solving to general-education students ( Marshall, 1995 ; Schoenfeld, 1992 ; Shavelson, Webb, Stasz, & McArthur, 1988 ) and, in more recent years, to students with learning disabilities (LD) (e.g., Case, Harris, & Graham, 1992 ; Mastropieri, Scruggs, & Shiah, 1997 ; Miller & Mercer, 1993 ).

Over the last two decades, a sizeable literature has begun to accumulate with an emphasis on helping students develop schemas to solve word problems in mathematics (e.g., Fuchs, Fuchs, Finelli, Courey, & Hamlett, 2004 ; Fuchs, Seethaler, et al., 2008 ; Griffin & Jitendra, 2009 ; Jitendra & Hoff, 1996 ; Willis & Fuson, 1988 ). A schema is a framework, outline, or plan for solving a problem ( Marshall, 1995 ). In mathematics, students can use schemas to organize information from a word problem in ways that represent the underlying structure of a problem type. Pictures or diagrams, as well as number sentences or equations, can be used to represent schemas.

Often, word problems can be differentiated into types of problems. The problem type is determined by what is happening in the word-problem narrative. For example, students may be given the following information: There are 7 blue birds and 4 red birds sitting on a tree . If, however, students are asked, How many birds are on the tree? , the problem type is combining or totaling (the birds). If students are asked, Five blue birds flew away, how many blue birds are left sitting in the tree? , the problem type is finding the change (in the number of blue birds). Combining or totaling is different from finding a change in that the examples represent two distinct problem types. Two distinct schemas can be used to solve the problems. Once students determine the problem type, they can apply a schema (i.e., diagram, equation, or plan) to assist in solving the word problem. In the elementary grades, most word problems can be sorted into only a few types ( Riley & Greeno, 1988 ). If students know a schema for each type, and understand how to sort problems into the problem types and apply the solution method for each schema, then students should be able to solve most word problems ( Cooper & Sweller, 1987 ).

The first purpose of the present paper was to review and synthesize the literature on schemas within word-problem instruction to determine (a) what schemas were taught to elementary students at-risk for or with LD, (b) how these schemas were taught, and (c) what effects were associated with solving word problems using schemas. The second purpose was to provide suggestions for classroom teachers on how to teach students to use schemas to solve word problems.

Word-Problem Difficulty

Students at-risk for or with LD often struggle with word-problem solving ( Parmar, Cawley, & Frazita, 1996 ). For example, Wilson and Sindelar (1991) worked with second- through fifth-graders with LD. On a test of addition and subtraction word problems, these students performed significantly below third-grade students without LD. At second- and fourth-grade, Englert, Culatta, and Horn (1987) tested 24 students with LD on 16 addition word problems. When compared to grade-level peers, the students with LD demonstrated significantly lower accuracy on word-problem solutions. More recently, Jordan and Hanich (2000) administered 14 addition and subtraction word problems to 20 second-grade students at-risk for or with mathematics LD and 29 second-grade students without LD. Students at-risk for or with LD answered fewer word problems correct than students without LD and employed less efficient strategies. These findings were corroborated with a larger group of second-grade ( Hanich, Jordan, Kaplan, & Dick, 2001 ) and third-grade students ( Jordan & Montani, 1997 ). Moving beyond simple word problems, Fuchs and Fuchs (2002) administered 10 word problems comprising four types (i.e., shopping list, buying bags, half, and pictograph) and 10 multi-step word problems with tables and graphs to fourth-grade students. The performance of 40 students with LD was compared to normative data collected from typical fourth-grade students. On both word problem sets, students with LD scored significantly lower than students in the normative group with effect sizes (ESs) ranging from 0.49 to 1.10 favoring the normative group.

Word problems may pose a challenge for students at-risk for or with LD because numerous steps and skills are necessary to solve a word problem ( Parmar et al., 1996 ). Additionally, students may struggle with comprehension of the text of the word problem ( Cummins, Kintsch, Reusser, & Weimer, 1988 ). Many students with LD struggle with mathematics and reading difficulty; therefore, embedding mathematics within a linguistic context may challenge students who also have reading deficits ( Fuchs, Fuchs, Stuebing, et al., 2008 ). To solve a word problem, students must use the text to identify missing information, derive a plan for solving for the missing information, and perform a calculation to find the missing information. Even when complex calculations are not required, students with LD struggle with problem solving compared to their average-performing peers ( Pellegrino & Goldman, 1987 ).

Instruction for Students with LD

To help students at-risk for or with LD become more efficient and accurate word-problem solvers, explicit word-problem instruction may be warranted ( Parmar et al., 1996 ). For example, Kroesbergen, Van Luit, and Maas (2004) randomly assigned 265 8- to 11-year-old students with LD or behavior disorders to receive constructivist multiplication instruction, explicit multiplication instruction, or regular classroom instruction (control). Students in the constructivist and explicit conditions received 30 lessons over 4 to 5 months. Intervention focused on multiplication automaticity and problem solving. In the constructivist condition, students were encouraged to discuss different approaches to solving a multiplication problem and then determine whether they could use one of these approaches to solve the problem. In the explicit condition, students were told how a problem should be solved and were provided examples of good problem-solving strategies. Students were always told which strategies to use. In the control condition, students followed the school's regular mathematics curriculum which included instruction on multiplication. At posttest, explicit instruction students significantly outperformed constructivist and control students on a computation multiplication measure and a measure of multiplication word problems. Kroesbergen et al. concluded that explicit or direct mathematics instruction, but not discovery or constructivist learning, may benefit lower-performing students.

Several explicit approaches exist for teaching students at-risk for or with LD to solve word problems ( Jitendra & Xin, 1997 ). These include diagramming or drawing the word problem ( van Garderen, 2007 ); identifying key words in a word problem and solving the problem based on the key word; utilizing computer-assisted instruction with explicit step-by-step work ( Mastropieri et al., 1997 ); using a mnemonic device to guide word-problem solving ( Miller & Mercer, 1993 ); learning metacognitive strategies to monitor word problem-solving progress ( Case et al., 1992 ); and using a checklist of steps to solve word problems along with monitoring work with metacognitive strategies ( Montague, Warger, & Morgan, 2000 ). An additional approach to teaching word-problem solving to students at-risk for or with LD, which has been developed over the last 20 years, is using schemas to solve word problems (e.g., Fuchs, Fuchs, Finelli, et al., 2004 ; Jitendra & Hoff, 1996 ). Word-problem instruction using schemas differs from typical word-problem instruction (e.g., key words, checklist of steps) because students first identify a word problem as belonging to a problem type and then use a specific problem-type schema to solve the problem. In conventional word-problem instruction, students may organize word-problem information or follow a mnemonic device to work step-by-step through the problem; however, students are not taught to determine a problem type and solve word problems according to a problem-type schema. In this paper, the research conducted to evaluate using schemas in word-problem solving was reviewed and synthesized.

Literature Search

The studies selected for this literature review met four criteria. First, the implemented treatments incorporated explicit instruction on solving a word problem though a schema. Second, studies needed to include, but not necessarily be limited to, students at-risk for or with LD. Third, study participants comprised students in second or third grade. These were the target grades because this is often when identification of students with LD occurs ( Fletcher, Lyon, Fuchs, & Barnes, 2006 ) and when written word-problem solving is a major focus of the curriculum as opposed to less formal, oral problems presented in kindergarten and first grade. Fourth, studies needed to be published in a peer-reviewed journal. I conducted searches in electronic databases including ERIC, PsycInfo, and ProQuest using the following terms: schema, word problem, story problem , and problem solving . Then, I read the titles and abstracts of articles to identify studies that fit the four criteria resulting in 12 word-problem solving studies. In all 12 studies, instruction focused on addition and subtraction word problems, which are the two operations most commonly found in second- and third-grade instruction and on standardized tests ( Hudson & Miller, 2006 ).

Overview of Included Studies

Each schema study reviewed in this paper is outlined in Table 1 . Study publication dates ranged from 1996 to 2009. Across studies, almost 4000 students were included. Of these students, 411 were at-risk for LD and 173 were identified as receiving special education services. Jitendra and colleagues tended to work with students with LD whereas Fuchs and colleagues generally worked with students at-risk for LD. The researchers utilized a variety of experimental designs: single subject (1), group teaching without assignment to treatment conditions (1), student random assignment (4), matched pairs random assignment (2), and classroom random assignment (5). Instruction occurred during school hours in individual settings (3), in small groups (2), and in large groups (8). Assessments for determining instructional effects were experimenter-designed in all 12 studies. Two of the studies included standardized assessments as well as experimenter-designed measures in the testing battery. In ten of the studies, more than one assessment was administered.

Two Approaches to Schema Instruction

In this section, two approaches to schema instruction are discussed. The first, referred to as schema-based instruction , teaches students to use schematic diagrams to solve addition and subtraction word problems ( Jitendra, Griffin, Deatline-Buchman, & Sczesniak, 2007 ; Jitendra & Hoff, 1996 ). The student reads a word problem, selects a schema diagram into which the word problem fits, and uses the structure of the diagram to solve the problem. In more recent studies, students are taught to use a mathematical equation (i.e., 4 + ? = 7), after filling in a schematic diagram, to solve the problem ( Griffin & Jitendra, 2009 ). The work by Jitendra and colleagues uses schema-based instruction. By contrast, Fuchs et al. (2003) uses a second approach to schema instruction, schema-broadening instruction . Schema-broadening instruction is similar to schema-based instruction in that students read the word problem and select a schema (from the taught schema) to solve word problems. Schema-broadening instruction differs from schema-based instruction because students are taught to transfer their knowledge of problem types to recognize problems with novel features (e.g., different format, additional question, irrelevant information, unfamiliar vocabulary, or information presented in charts, graphs, or pictures) as belonging to a problem type for which they know a solution. As with Jitendra and colleagues, Fuchs and colleagues also teach students to set up and solve mathematical equations (e.g., X – 3 = 7) representing the structure of problem types ( Fuchs et al., 2009 ).

In terms of schema instruction, the schema-based instruction of Jitendra and colleagues differs from the schema-broadening instruction of Fuchs and colleagues in one primary way. With schema-broadening (but not schema-based) instruction, students receive explicit instruction on transfer to novel problems. The schemas that Jitendra and colleagues used rely on diagrams for organizing word-problem work. (See Figure 1 for an example.) Fuchs and colleagues, in contrast, teach students to organize word-problem information in sections or in mathematical equations. (See Figures 2 and ​ and3 3 for examples.)

Schema-Based Instruction

To understand how schema-based instruction may benefit students with LD, Jitendra and Hoff (1996) worked with three third- and fourth-grade students with LD. During 13 to 16 days of intervention, students learned to recognize defining features of addition and subtraction word-problem types, classify problems in terms of problem types, map the word problem information onto the schema's diagram, and use the diagram to solve the problem. Jitendra and Hoff taught three schemas: change, group, and compare. All three students demonstrated positive growth as the study progressed and maintained skills 2 to 3 weeks after the final intervention session, with only a slight decline in scores. Through this multiple-baseline, single-subject design, Jitendra and Hoff demonstrated the possible benefit of using schemas for teaching word-problem solving to students with LD.

Working with a larger number of students, Jitendra et al. (1998) recruited 34 second- to fifth-grade students who performed below the 60 th percentile on a word-problem measure. Students were randomly assigned to receive small-group schema instruction or small-group traditional instruction during 17 to 20 sessions. Schema instruction focused on change, group, and compare problems. Students learned how to identify the schema for a word problem and to use a schema diagram to organize the problem's information. The traditional instruction followed a basal mathematics program focused on general mathematics skills and was implemented to control for tutoring time. At posttest, students participating in schema tutoring outperformed students in the traditional tutoring on experimenter-designed measures of word problems. A delayed posttest, administered one week after tutoring commenced, continued to favor schema students over traditional students. Jitendra et al. also recruited 24 average-performing third graders to serve as a normative sample. At posttest, the schema-tutoring students performed comparably to students in the normative sample, whereas traditional-tutoring students did not. These results favoring schema instruction led Jitendra et al. to conclude that word-problem instruction using schemas is more advantageous to students at-risk for LD than traditional word-problem instruction.

In the next phase of this research program, Jitendra moved from small-group schema intervention to whole-class schema-based instruction. Jitendra, Griffin, Deatline-Buchman, et al. (2007) provided schema-based instruction similar to Jitendra and Hoff (1996) with students receiving instruction on using schematic diagrams to solve change, combine or group, and compare problems. Students were taught to fill word-problem information into a problem type's corresponding schematic diagram and then generate a mathematical equation (i.e., a number sentence with missing information) to help solve the problem. A question mark was used to mark the missing information (i.e., ? + 5 = 10). Across three classrooms, 38 lower-performing third-grade students, 9 of whom were identified with LD, received schema-based instruction. Instruction lasted 15 weeks with three 30-min sessions per week. On two experimenter-designed word-problem posttests, students in the three classrooms demonstrated improvement from pretest although the improvement was not significant. Jitendra, Griffin, Deatline-Buchman, et al. concluded that lower-performing students and students with LD need and benefit from explicit word problem-solving instruction focused around schemas. With the absence of control classrooms for comparison purposes or significant growth from pre- to posttest, Jitendra, Griffin, Deatline-Buchman, et al. indicated, but did not verify, that schema instruction may be beneficial for students at-risk for or with LD.

Comparing schema-based instruction to another word-problem solving approach, Jitendra, Griffin, Haria, et al. (2007) randomly assigned 88 third-grade students to two conditions: schema-based instruction and general-strategy instruction. Four of the 88 participants were identified with LD. Schema-based instruction focused on the change, combine, and compare problem types as in Jitendra, Griffin, Deatline-Buchman, et al., (2007) whereas students receiving general-strategy instruction were taught four steps to solve a word problem (i.e., read and understand, plan, solve, and check) along with four strategies to assist in solving a word problem (i.e., use manipulatives, act it out or draw a diagram, write a number sentence, and use information from a graph). Similar to Jitendra, Griffin, Deatline-Buchman, et al., students receiving schema instruction learned to identify the schema of a word problem, fill word-problem information into a schematic diagram, and then generate an equation to help solve the word problem. Students used different schematic diagrams for each of the three problem types, and the use of schematic diagrams was faded toward the end of instruction on each problem type. Many students, however, continued to draw schematic diagrams independently. After all three problem types were introduced, tutors taught the students to solve two-step problems that combined two schemas. All students received 41 lessons, each lasting approximately 25 min. From pre- to posttest, students in the schema-based condition outperformed students in the general-strategy condition on an experimenter-designed word-problem measure, with an ES of 0.52. The same measure, administered six weeks after posttest, again showed students in the schema-based condition outperforming general-strategy condition students (ES = 0.69). The number of students with LD was small ( n = 4), so results for students with disabilities were not presented by Jitendra, Griffin, Deatline-Buchman, et al. separate from the main analysis. Therefore, conclusions about the benefit of schema-based instruction for students with LD could not be inferred.

Interestingly, Griffin and Jitendra (2009) also compared schema-based instruction to general-strategy instruction with third-grade students but did not replicate the results from Jitendra, Griffin, Haria, et al. (2007) . Students from three classrooms ( n = 60; 5 with LD) were matched based on performance on a standardized mathematics test and then the pairs were randomly assigned to schema-based or general-strategy instruction. Schema-based and general-strategy instruction were similar to that provided in Jitendra, Griffin, Haria, et al., except that instruction was provided in 20 lessons lasting 100 min each. Schema instruction included completing schematic diagrams and generating equations. The final four lessons comprised instruction on two-step problems where tutors taught students to solve problems using two schemas. On an experimenter-designed word-problem measure, there were no significant differences between the two groups at posttest or at 12-week maintenance (even though both groups demonstrated growth from pretest to posttest to maintenance). On a measure of word-problem solving fluency administered three times throughout instruction, there were significant differences favoring schema-based instruction at the beginning of treatment. These effects, however, faded over the course of the study: At posttest, schema-based and general-strategy groups performed similarly. Griffin and Jitendra attributed the inconsistency of this finding to the fact that instruction was provided in 100-min sessions and once a week rather than shorter sessions occurring several times a week.

Jitendra and colleagues' program of research on schema-based instruction is impressive and demonstrates that students at-risk for or with LD may benefit from explicit schema instruction. These researchers taught students to use three schemas (i.e., change, combine or group, and compare) on different types of word problems with two operations (i.e., addition and subtraction). Even though the specific nature of schema-based instruction varied in small ways from study to study, the majority of students benefitted from learning about different schema and applying the schema to solve word problems. Across studies, two instructional design features were consistently incorporated within schema-based instruction. First, interventions were of long duration (13 to 45 lessons), and second, explicit instruction focused on recognizing a problem's schema, using a diagram based on the schema, and solving the problem. The research by Jitendra and colleagues offers a solid foundation for future schema-based investigations and provides strategies that teachers can use to enhance the performance of their students with LD on word problems.

Schema-Broadening Instruction

As in Jitendra and colleagues' schema-based instruction, schema-broadening instruction relies on schemas for conceptualizing word problems. Some of Fuchs and colleagues' schema-broadening instruction comprises problem types (i.e., shopping list, half, buying bags, pictograph) that are notably different from the problem types used by Jitendra and colleagues. Other schema-broadening problem types of Fuchs and colleagues (i.e., total, difference, and change) are similar to the combine, compare, and change problem types of Jitendra and colleagues. Schema-broadening instruction includes a focus on transfer features to help students expand their conceptualization of the schema. Thus, schema-broadening instruction helps students recognize a novel problem (with unfamiliar problem features such as different format, additional question, irrelevant information, unfamiliar vocabulary, or information presented in charts, graphs, or pictures) as belonging to the schema for which they know a problem solution strategy.

To pinpoint the effects of explicit transfer instruction within schema-broadening instruction, Fuchs et al. (2003) randomly assigned 24 third-grade classrooms ( n = 375) to four conditions: problem-solution instruction, partial-problem-solution-with-transfer instruction (to control for instructional time), full problem-solution-with-transfer-instruction, or control, business-as-usual instruction with a 6-lesson introductory general-problem solving unit that all 24 classrooms received. Students receiving special education services ( n = 23) were distributed across the four conditions. After this introductory unit, problem-solution instruction was presented over the next 20 lessons, in which students were explicitly taught to understand and recognize four schema (i.e., shopping list, half, buying bags, and pictograph) and to apply rules for solving problems for each schema. Students in the partial-problem-solution-plus-transfer condition received only 10 solution lessons but also received 10 transfer lessons. The transfer lessons included explicit instruction on the meaning of transfer and instruction to broaden schema to address problems with different formats, unfamiliar vocabulary, additional questions, and broader problem-solving contexts. Students in the full-problem-solution-plus-transfer condition received all 20 solution lessons and all 10 transfer lessons. In terms of classroom performance from pre- to posttest, students in the problem-solution, partial-problem-solution-with-transfer, and full-problem-solution-with-transfer classrooms outperformed control classrooms on an experimenter-designed immediate-transfer measure (ESs = 2.61, 2.15, and 1.82, respectively). On a far-transfer measure, students who received the partial- or full-solution-plus-transfer-instruction significantly outperformed control classrooms. Additionally, classrooms that received the full-solution-plus-transfer instruction improved more than classrooms that received the solution instruction alone. For students with disabilities, however, the results were not as promising. In the partial-problem-solution condition, 60-80% of the students were unresponsive to treatment. Students in the problem-solution and full-problem-solution-with-transfer conditions demonstrated greater levels of response. This study, as well as a similar study conducted with 24 classrooms of 366 students by Fuchs, Fuchs, Prentice, et al. (2004) , demonstrated the added value of schema instruction with an explicit focus on transfer schemas. Interestingly, in Fuchs, Fuchs, Prentice, et al. students in special education demonstrated significant gains over control students with ESs of 0.87 to 1.96.

To further extend this research program on schema-broadening instruction, Fuchs, Fuchs, Finelli, et al. (2004) randomly assigned 24 classrooms, with 351 students, to three conditions: schema-broadening instruction that addressed three transfer features, schema-broadening instruction that addressed six transfer features, and business-as-usual control. Twenty-nine students received special education services. All classrooms received six sessions about generic word problem-solving steps. Schema-broadening classrooms also received 28 lessons focused on the four schemas taught in Fuchs et al. (2003) . The schema-broadening instruction condition addressed three transfer features (i.e., different format, different question, or different vocabulary). The six-feature schema-broadening instruction condition addressed different format, different question, different vocabulary, irrelevant information, combined problem types, and mixing of transfer features. On experimenter-designed measures with the shortest transfer distance (unfamiliar problems but without novel features), students participating in both schema-broadening instruction conditions performed comparably but significantly better than control (ESs = 3.69 and 3.72, respectively). On measures assessing word problems with medium transfer distance (i.e., different format, question, or vocabulary transfer features), again there were no significant differences between the two schema-broadening instruction conditions, which outperformed the control group (ESs = 1.98 and 2.71, respectively). However, on the measure assessing the greatest transfer distance (i.e., involving all six transfer features), students in the schema-broadening instruction condition that incorporated all six transfer features demonstrated a significant advantage with an ES of 2.71 over control students and an ES of 0.72 over students in the narrower schema-broadening instruction treatment. Students with disabilities demonstrated similar gains to those of students without disabilities. Fuchs, Fuchs, Finelli, et al. demonstrated that students benefit from explicit schema-broadening instruction focused on a wide variety of transfer features.

In an expansion of Fuchs, Fuchs, Finelli, et al. (2004) , Fuchs and colleagues tested how real-life problem-solving skills might provide added benefit to schema-broadening instruction ( Fuchs et al., 2006 ). From 30 classrooms, 445 third-grade students (34 of whom received special education services) were randomly assigned by classroom to schema-broadening instruction, schema-broadening and real-life instruction, or business-as-usual control. All 30 classrooms received six 40-min sessions on general problem-solving strategies. Both schema-broadening treatments received an additional 30 sessions on four problem types. Additionally, schema-broadening plus real-life instruction classrooms received explicit instruction via video on real-life problem solving skills (i.e., review the problem, determine extra steps necessary for solving the problem, find important information without number, figure out important information not provided within the problem, reread, and ignore irrelevant information). On experimenter-designed measures of immediate and medium word-problem transfer, both schema-broadening treatments outperformed control classrooms with ESs ranging from 3.59 to 6.84. On a far transfer task, the added benefit of explicit real-life problem solving emerged on an open-ended question about what the student could buy. Students could use information from a pictograph, a price chart, or their own experiences to answer the question. On this question, the schema-broadening plus real-life students outperformed schema-broadening students (ES = 1.83). In this way, Fuchs et al. (2006) demonstrated how the combination of schema-broadening and real-life problem-solving instruction is beneficial for solving word problems. Results for students with disabilities, however, were not disaggregated from the entire sample, thus it was unclear if these students performed in a similar manner.

To investigate the effect of schema-broadening instruction for students at-risk for LD, Fuchs, Fuchs, Craddock, et al. (2008) randomly assigned 119 classrooms to receive schema-broadening instruction or to participate in a business-as-usual control group. Then, within each whole-class condition, 243 students at-risk for or with LD were randomly assigned to receive small-group schema-broadening tutoring or to remain in their whole-class condition without tutoring. In this way, 28 students received business-as-usual whole-class instruction and no schema-broadening tutoring, 51 students received whole-class schema-broadening instruction but no schema-broadening tutoring, 56 students received business-as-usual whole-class instruction with schema-broadening tutoring, and 108 students received whole-class schema-broadening instruction plus schema-broadening tutoring. The schema-broadening instruction at the classroom level provided explicit instruction on solving the four problem types (i.e., shopping list, half, buying bags, and pictograph) over 16 weeks. Tutoring occurred 3 times a week for 13 weeks following completion of three weeks of whole-class instruction. Tutoring sessions lasted 20 to 30 min in small groups of two to four students. For students who received whole-class schema-broadening instruction, tutored students outperformed students who did not receive tutoring on experimenter-designed measures (ES = 1.13). In a similar way, for students in business-as-usual classrooms, tutored students outperformed students who did not receive tutoring (ES = 1.34). Importantly, students who received two tiers of schema-broadening instruction (whole class and small-group tutoring) significantly outperformed students who received schema-broadening tutoring without whole-class schema-broadening instruction. This finding suggests that the combination of whole-class instruction and small-group tutoring provided the best outcome for students struggling with word problems. Whole-class instruction was beneficial alone as was small-group tutoring; however, the combination proved better than one or the other.

Two other studies in the Fuchs's program of research ( Fuchs et al., 2009 ; Fuchs, Seethaler, et al., 2008 ) rely on schema-broadening instruction but with problem types (i.e., change, total, and difference) that parallel those used by Jitendra and colleagues (i.e., change, combine, and compare). In these tutoring studies (conducted on a one-to-one basis), however, students were also explicitly taught to set up and solve mathematical equations that represent the underlying schema of the word problems similar to Griffin and Jitendra (2009) , Jitendra, Griffin, Deatline-Buchman, et al. (2007) , and Jitendra, Griffin, Haria, et al. (2009). In a pilot study, Fuchs, Seethaler, et al. (2008) randomly assigned 35 third-grade students at-risk for or with LD to two conditions: schema-broadening instruction tutoring with mathematical equations or no-tutoring control. All students performed below the 26 th percentile on global math and reading tests. Students in the schema-broadening condition received individual instruction over 12 weeks with sessions conducted 3 times a week, 30 min per session. Instruction focused on the three problem types with three transfer features (irrelevant information, important information embedded within charts, graphs, or pictures, and double-digit numbers). First, students learned to understand and identify the three schemas (i.e., problem types), to set up an equation to represent each schema (i.e., 3 + X = 9), and to solve equations. Then, explicit instruction to broaden schema to the three transfer features occurred. Students receiving schema-broadening tutoring demonstrated significantly better growth than control students on an experimenter-designed test of word problems (ES = 1.80) and on a test of word problems designed by a research team not affiliated with the study (ES = 0.69). On a standardized test of problem solving, however, there were no significant differences.

Expanding the pilot study to focus on the effects of treatment as a function of difficulty subtype (i.e., students at-risk for or with mathematics LD alone versus students at-risk for or with mathematics and reading LD) and controlling for tutoring time with a contrasting math tutoring condition, Fuchs et al. (2009) randomly assigned 133 third-grade students, blocking by difficulty subtype and by site (i.e., Nashville vs. Houston) to three conditions: number combinations tutoring, schema-broadening word-problem tutoring, or no-tutoring control. Students in the two tutoring conditions received individual tutoring on word problems or on number combinations 3 times a week for 15 weeks, each time for 20 to 30 min. Word-problem tutoring relied on schema-broadening instruction with mathematical equations similar to Fuchs, Seethaler, et al. (2008) . Growth from pre- to posttest on an experimenter-designed word-problem measure, including problems that required transfer, indicated that students in word-problem tutoring significantly outperformed students in number-combinations tutoring and in the control group (ESs = 0.83 and 0.79, respectively). On a standardized test of problem solving, students in word-problem tutoring significantly outperformed students in the control group (ES = 0.28). Additionally, difficulty subtype did not moderate the effect of schema-broadening instruction with equations. That is, students at-risk for or with mathematics and reading LD and students at-risk for mathematics without reading LD responded comparably well to the treatments.

Expanding beyond whole-class, schema-broadening instruction to incorporate mathematical equations, the research conducted by Fuchs et al. (2009) and Fuchs, Seethaler, et al. (2008) revealed how students at-risk for or with LD may benefit from tutoring that combines schema-broadening instruction with instruction on setting up and solving addition and subtraction mathematical equations. Because students did not receive concurrent whole-class instruction and individual word-problem tutoring as in Fuchs, Fuchs, Craddock, et al. (2008) , future research may investigate the added value of such a combination with schema-broadening plus mathematical equations instruction provided at the whole-class and small-group or individual tutoring levels.

A Framework for Teaching Word Problems in the Primary Grades

Across the two lines of work applying schema theory to word-problem solving, Jitendra and colleagues and Fuchs and colleagues provide evidence that students, including those at-risk for or with LD, may benefit from this explicit approach to word-problem instruction at the classroom and tutoring levels. In the schema-based instruction of Jitendra and colleagues, students learned to use schematic diagrams to solve word problems. In Jitendra's more recent research, students also learned to set up and solve an equation to find the word-problem answer after filling in a schematic diagram. The schema-broadening instruction of Fuchs and colleagues incorporated explicit schema instruction about word problem transfer features so students could learn how to recognize novel problems as belonging to the schemas they learned, but without reliance on schematic diagrams. Additionally, in the schema-broadening instruction of Fuchs et al. (2009) and Fuchs, Seethaler, et al. (2008) , students learned to use mathematical equations to represent the structure of a word problem.

Limitations

Before proceeding, it is important to discuss a few limitations across the two lines of schema work. First, and perhaps most importantly, many of the measures used to determine treatment effects were designed by the experimenters conducting the research. Some measures included word problems almost identical to those presented to students during instruction, perhaps raising questions about the generalizability of the word-problem instruction. When standardized tests of problem solving were administered, effects were either not significant or not as large as on the experimenter-designed measures. Second, a few of the studies, especially those by Fuchs and colleagues, included students at-risk for LD and not necessarily students with identified LD. Some of these students at-risk for LD received special education services; most did not. Within these studies for students at-risk for LD, the results for students with disabilities were not disaggregated from the primary sample. The same holds true for some of the other studies conducted by both Fuchs and colleagues and Jitendra and colleagues ( Fuchs et al., 2006 ; Fuchs et al., 2009 ; Fuchs, Fuchs, Craddock, et al., 2008 ; Fuchs, Seethaler, et al., 2008 ; Griffin & Jitendra, 2009 ). Therefore, it remains unclear if the interventions benefit students at-risk for LD, students with LD, or both.

Implications for Practice

Other schema investigations (e.g., Jitendra, DiPipi, & Perron-Jones, 2002 ; Jitendra, Hoff, & Beck, 1999 ; Xin, Jitendra, & Deatline-Buchman, 2005 ; Xin & Zhang, 2009 ) suggest using schema to solve word problems in the intermediate grades. That aside, the focus of the present literature review was on the primary grades, where the literature provides the basis for conceptualizing a framework for teaching word problems to students at-risk for or with LD that comprises the following features. First, instruction should be explicit. Across the two lines of schema work, schema were introduced in an explicit manner, and teachers or tutors often modeled or provided worked examples of word problems using each schema. It is not surprising that students at-risk for or with LD benefitted from explicit instruction, given that other mathematics researchers not focused on schema instruction (e.g., Kroesbergen et al., 2004 ; Mercer, Jordan, & Miller, 1996 ) have demonstrated the benefits of explicit instruction for students at-risk for or with LD. Across all the schema studies at the primary grades, students learned one word-problem schema at a time and had adequate practice (i.e., for days or weeks) on the schema before learning another schema.

Next, word-problem instruction should be organized. Students with LD profited from organizing word problems via schemas and having an explicit method for conceptualizing their solutions for each schema. This solution method could be a schematic diagram ( Jitendra et al., 1998 ), a mathematical equation ( Fuchs et al., 2009 ; Griffin & Jitendra, 2009 ), or a way of organizing information ( Fuchs, Fuchs, Finelli, et al., 2004 ). Because methods that work for one student may not work for another, it is important that teachers familiarize themselves with various explicit methods for helping students learn schema approaches to word problems so they can best help their students.

A schematic diagram may help some students organize their word-problem work, as in Figure 1 . With a schematic diagram, students fill in the relevant numbers from the word problem. The area of the schematic diagram that represents the question of the word problem (i.e., the word-problem solution) is left blank or filled in with a question mark. Students then learn how to solve for the blank space in the diagram to solve the word problem by calculating the answer. For example, the following is a typical elementary-school word problem: A classroom has 15 students. If 6 of the students are boys, how many students are girls? Using the schemas employed by Jitendra et al. (1998) , this word problem falls under the “group” schema because there is a larger set (i.e., the classroom) with smaller sets (i.e., boys and girls) within the larger set. The larger set (15) and one of the smaller sets (6) are defined within the text of the word problem. The other smaller set is the missing information needed to answer the word-problem question. After a student selects the word-problem schema (i.e., group), they fill in a schematic diagram. The schematic diagrams assist in organizing the word-problem information in pictorial fashion, which, as Jitendra and colleagues have demonstrated, may be beneficial for students at-risk for or with LD.

Another approach that may assist students in organizing word-problem work is to decide on a problem's schema and then use a mathematical equation to represent the underlying structure of the schema. Working with the word problem just discussed: A classroom has 15 students. If 6 of the students are boys, how many students are girls? , Fuchs et al. (2009) categorized this problem as falling within a “total” schema. In a “total” schema, parts are put together for a total. Instead of using a schematic diagram, students are taught a mathematical equation (i.e., P1 + P2 = T) that represents the two parts (i.e., P1, P2) put together for a total (i.e., T). After students decide the word problem's schema, they write the mathematical equation to help organize their word-problem work. (See Figure 2 for a worked example.) Students fill in the relevant numbers from the word problem and write an X for the missing part of the equation. Students then solve for X (i.e., solve the word-problem question). As demonstrated by Fuchs and Jitendra, using an equation to represent the schema is also an effective approach for strengthening the word-problem skill of students at-risk for or with LD.

For more difficult word problems, Fuchs and colleagues (e.g., Fuchs, Fuchs, Finelli, et al., 2004 ) have demonstrated how students can organize their word-problem work without a schematic diagram or mathematical equation, while relying on their knowledge of the problem schema. The word problem, Maya wants to buy 2 bags of pencils for $3 each, 4 notebooks for $2 each, and 6 folders for $1 each. How much will Maya spend? , would fall under the “shopping list” schema because multiple items of various prices are purchased. Instead of using a schematic diagram or equation for a “shopping list” problem, students draw vertical lines on their paper to organize their work. (See Figure 3 for a worked example.) Students calculate one part or step of the shopping list problem (i.e., pencils, notebooks, folders) in each section and calculate the overall cost of items in the right-most section. The vertical lines assist students in organizing the word-problem information and their work, but drawing the lines is not a necessity. That is, students could use their knowledge of schemas to solve the word problem with or without the lines.

Several other dimensions of a word-problem teaching framework using schema theory also emerge across the two lines of schema. These include practice in sorting word problems into schema, many instructional sessions, and multiple settings (i.e., whole class, small group, individual) for schema instruction to occur. With regard to sorting word problems into schema, both Jitendra and colleagues and Fuchs and colleagues made a point of mixing word-problem types so students had to differentiate what problems belonged to which schema. Some of this practice was explicit via flash cards ( Fuchs, Seethaler, et al., 2008 ). Some of the schema identification practice was embedded within the lesson, whereby teachers and tutors presented students with word problems, students had to decide which schema represented the word problem, and then use the structure of the schema to solve the word problem.

Across schema studies, the number of instructional sessions varied from 13 to 45 sessions. For all studies but one, students were taught or tutored multiple times each week, and students demonstrated significant gains from the schema instruction. Only the results of Griffin and Jitendra (2009) proved disappointing, with the absence of significant differences between the schema-based instruction and a comparison group. The authors attributed this lack of significance to the fact that instruction was provided once a week instead of multiple times each week. The significant results from the other 11 studies highlighted in this review suggest that instruction should be of sufficient duration (i.e., weeks and months, not days) and occur multiple times each week.

Finally, in the body of work reviewed in this paper, schema instruction occurred in whole-class, small-group tutoring, and individual tutoring settings. Students at-risk for or with LD benefitted from the schema instruction in all three of these settings. One study ( Fuchs, Fuchs, Craddock, et al., 2008 ) isolated the effects of conducted schema instruction provided in whole-class arrangement versus small-group tutoring settings. They concluded that the combination of both whole-class teaching and small-group tutoring may optimally enhance outcomes for students at risk for LD and that tutoring was essential for promoting strong outcomes. Teachers, therefore, should be mindful that whole-class instruction may not be enough for students with or at risk for LD, and additional tutoring (i.e., Tier 2 or 3 within a Response to Intervention framework) may be necessary to improve the word-problem outcomes of students with LD.

The linking features of these two schema approaches require students to (a) read a word problem, (b) recognize the underlying structure of the word problem as belonging to a specific schema, and (c) solve the word problem using a solution method that represents a schema. Whether students use schematic diagrams, mathematical equations, or another method to help them apply their knowledge of the word-problem schema, the research conducted by Jitendra and colleagues and Fuchs and colleagues demonstrates that students at-risk for or with LD may benefit from explicit word-problem instruction that incorporates schemas.

Acknowledgments

This research was supported by Award Number R01HD059179 from the Eunice Kennedy Shriver National Institute of Child Health & Human Development to Vanderbilt University. The content is solely the responsibility of the authors and does not necessarily represent the official views of the Eunice Kennedy Shriver National Institute Of Child Health & Human Development or the National Institutes of Health.

• Case LP, Harris KR, Graham S. Improving the mathematical problem-solving skills of students with learning disabilities: Self-regulated strategy development. Journal of Special Education. 1992; 26 :1–19. [ Google Scholar ]
• Cooper G, Sweller J. Effects of schema acquisition and rule automation on mathematical problem-solving transfer. Journal of Educational Psychology. 1987; 79 :347–362. [ Google Scholar ]
• Cummins DD, Kintsch W, Reusser K, Weimer R. The role of understanding in word problems. Cognitive Psychology. 1988; 20 :405–438. [ Google Scholar ]
• Englert CS, Culatta BE, Horn DG. Influence of irrelevant information in addition word problems on problem solving. Learning Disability Quarterly. 1987; 10 :29–36. [ Google Scholar ]
• Fletcher JM, Lyon GR, Fuchs LS, Barnes MA. Learning disabilities: From identification to intervention. New York, NY: Guilford Press; 2006. [ Google Scholar ]
• Fuchs LS, Fuchs D. Mathematical problem-solving profiles of students with mathematics disabilities with and without comorbid reading disabilities. Journal of Learning Disabilities. 2002; 35 :563–573. [ PubMed ] [ Google Scholar ]
• Fuchs LS, Fuchs D, Craddock C, Hollenbeck KN, Hamlett CL, Schatschneider C. Effects of small-group tutoring with and without validated classroom instruction on at-risk students' math problem solving: Are two tiers of preventions better than one? Journal of Educational Psychology. 2008; 100 :491–509. [ PMC free article ] [ PubMed ] [ Google Scholar ]
• Fuchs LS, Fuchs D, Finelli R, Courey SJ, Hamlett CL. Expanding schema-based transfer instruction to help third graders solve real-life mathematical problems. American Educational Research Journal. 2004; 41 :419–445. [ Google Scholar ]
• Fuchs LS, Fuchs D, Finelli R, Courey SJ, Hamlett CL, Sones EM, Hope SK. Teaching third graders about real-life mathematical problem solving: A randomized control trial. The Elementary School Journal. 2006; 106 :293–311. [ Google Scholar ]
• Fuchs LS, Fuchs D, Prentice K, Burch M, Hamlett CL, Owen R, et al.Jancek D. Explicitly teaching for transfer: Effects on third-grade students' mathematical problem solving. Journal of Educational Psychology. 2003; 95 :293–305. [ Google Scholar ]
• Fuchs LS, Fuchs D, Prentice K, Hamlett CL, Finelli R, Courey SJ. Enhancing mathematical problem solving among third-grade students with schema-based instruction. Journal of Educational Psychology. 2004; 96 :635–647. [ Google Scholar ]
• Fuchs LS, Fuchs D, Stuebing K, Fletcher J, Hamlett CL, Lambert W. Problem-solving and computational skill: Are they shared or distinct aspects of mathematical cognition? Journal of Educational Psychology. 2008; 100 :30–47. [ PMC free article ] [ PubMed ] [ Google Scholar ]
• Fuchs LS, Powell SR, Seethaler PM, Cirino PT, Fletcher JM, Fuchs D, et al.Zumeta RO. Remediating number combination and word problem deficits among students with mathematics difficulties: A randomized control trial. Journal of Educational Psychology. 2009; 101 :561–576. [ PMC free article ] [ PubMed ] [ Google Scholar ]
• Fuchs LS, Seethaler PS, Powell SR, Fuchs D, Hamlett CL, Fletcher JM. Effects of preventative tutoring on the mathematical problem solving of third-grade students with math and reading difficulties. Exceptional Children. 2008; 74 :155–173. [ PMC free article ] [ PubMed ] [ Google Scholar ]
• Griffin CC, Jitendra AK. Word problem-solving instruction in inclusive third-grade classrooms. The Journal of Educational Research. 2009; 102 :187–201. [ Google Scholar ]
• Hanich LB, Jordan NC, Kaplan D, Dick J. Performance across different areas of mathematical cognition in children with learning difficulties. Journal of Educational Psychology. 2001; 93 :615–626. [ Google Scholar ]
• Hudson P, Miller SP. Designing and implementing mathematics instruction for students with diverse learning needs. Boston, MA: Allyn & Bacon; 2006. [ Google Scholar ]
• Jitendra A, DiPipi CM, Perron-Jones N. An exploratory study of schema-based word-problem-solving instruction for middle school students with learning disabilities: An emphasis on conceptual and procedural understanding. The Journal of Special Education. 2002; 36 :23–38. [ Google Scholar ]
• Jitendra AK, Griffin CC, Deatline-Buchman A, Sczesniak E. Mathematical problem solving in third-grade classrooms. Journal of Educational Research. 2007; 100 :283–302. [ Google Scholar ]
• Jitendra AK, Griffin CC, Haria P, Leh J, Adams A, Kaduvettoor A. A comparison of single and multiple strategy instruction on third-grade students' mathematical problem solving. Journal of Educational Psychology. 2007; 99 :115–127. [ Google Scholar ]
• Jitendra AK, Griffin CC, McGoey K, Gardill MC, Bhat P, Riley T. Effects of mathematical word problem solving by students at risk or with mild disabilities. The Journal of Educational Research. 1998; 91 :345–355. [ Google Scholar ]
• Jitendra AK, Hoff K. The effects of schema-based instruction on mathematical word-problem-solving performance of students with learning disabilities. Journal of Learning Disabilities. 1996; 29 :422–431. [ PubMed ] [ Google Scholar ]
• Jitendra AK, Hoff K, Beck MM. Teaching middle school students with learning disabilities to solve word problems using a schema-based approach. Remedial and Special Education. 1999; 20 :50–64. [ Google Scholar ]
• Jitendra A, Xin YP. Mathematical word problem-solving instruction for students with mild disabilities and students at risk for math failure: A research synthesis. The Journal of Special Education. 1997; 30 :412–438. [ Google Scholar ]
• Jordan NC, Hanich LB. Mathematical thinking in second-grade children with different forms of LD. Journal of Learning Disabilities. 2000; 33 :567–578. [ PubMed ] [ Google Scholar ]
• Jordan NC, Montani TO. Cognitive arithmetic and problem solving: A comparison of children with specific and general mathematics difficulties. Journal of Learning Disabilities. 1997; 30 :624–634. [ PubMed ] [ Google Scholar ]
• Kroesbergen EH, Van Luit JEH, Maas CTM. Effectiveness of explicit and constructivist mathematics instruction for low-achieving students in the Netherlands. The Elementary School Journal. 2004; 104 :233–251. [ Google Scholar ]
• Marshall SP. Schemas in Problem Solving. New York, NY: Cambridge University Press; 1995. [ Google Scholar ]
• Mastropieri MA, Scruggs TE, Shiah R. Can computers teach problem-solving strategies to students with mild mental retardation? Remedial and Special Education. 1997; 18 :157–165. [ Google Scholar ]
• Mercer CD, Jordan L, Miller SP. Constructivistic math instruction for diverse learners. Learning Disabilities Research & Practice. 1996; 11 :147–156. [ Google Scholar ]
• Miller SP, Mercer CD. Mnemonics: Enhancing the math performance of students with learning difficulties. Intervention in School and Clinic. 1993; 29 :78–82. [ Google Scholar ]
• Montague M, Warger C, Morgan TH. Solve it! Strategy instruction to improve mathematical problem solving. Learning Disabilities Research & Practice. 2000; 15 :110–116. [ Google Scholar ]
• National Assessment Governing Board . National assessment of educational progress. Washington, DC: Institute of Education Sciences; 2009. [ Google Scholar ]
• National Council of Teachers of Mathematics . Principles and standards for school mathematics. Reston, VA: Author; 2000. [ Google Scholar ]
• Parmar RS, Cawley JF, Frazita RR. Word problem-solving by students with and without mild disabilities. Exceptional Children. 1996; 62 :415–429. [ Google Scholar ]
• Pellegrino JW, Goldman SR. Information processing and elementary mathematics. Journal of Learning Disabilities. 1987; 20 :23–32. 57. [ PubMed ] [ Google Scholar ]
• Pólya G. How to solve it. Princeton, NJ: Princeton University Press; 1945. [ Google Scholar ]
• Riley MS, Greeno JG. Developmental analysis of understanding language about quantities and of solving problems. Cognition and Instruction. 1988; 5 :49–101. [ Google Scholar ]
• Schoenfeld AH. Learning to think mathematically: Problem solving, metacognition, and sense-making in mathematics. In: Grouws D, editor. Handbook for research on mathematics teaching and learning. New York, NY: MacMillan; 1992. pp. 334–370. [ Google Scholar ]
• Shavelson RJ, Webb NM, Stasz C, McArthur D. Teaching mathematical problem solving: Insights from teachers and tutors. In: Charles RI, Silver EA, editors. The teaching and assessing of mathematical problem solving. Reston, VA: The National council of Teachers of Mathematics, Inc.; 1988. pp. 203–231. [ Google Scholar ]
• van Garderen D. Teaching students with LD to use diagrams to solve mathematical word problems. Journal of Learning Disabilities. 2007; 40 :540–553. [ PubMed ] [ Google Scholar ]
• Willis GB, Fuson KC. Teaching children to use schematic drawings to solve addition and subtraction word problems. Journal of Educational Psychology. 1988; 80 :192–201. [ Google Scholar ]
• Wilson CL, Sindelar PT. Direct instruction in math word problems: Students with learning disabilities. Exceptional Children. 1991; 57 :512–519. [ PubMed ] [ Google Scholar ]
• Xin YP, Jitendra AK, Deatline-Buchman A. Effects of mathematical word problem-solving instruction on middle-school students with learning problems. The Journal of Special Education. 2005; 39 :181–192. [ Google Scholar ]
• Xin YP, Zhang D. Exploring a conceptual model-based approach to teaching situated word problems. The Journal of Educational Research. 2009; 102 :427–441. [ Google Scholar ]

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14 Effective Ways to Help Your Students Conquer Math Word Problems

If a train leaving Minneapolis is traveling at 87 miles an hour…

Word problems can be tricky for a lot of students, but they’re incredibly important to master. After all, in the real world, most math is in the form of word problems. “If one gallon of paint covers 400 square feet, and my wall measures 34 feet by 8 feet, how many gallons do I need?” “This sweater costs $135, but it’s on sale for 35% off. So how much is that?” Here are the best teacher-tested ideas for helping kids get a handle on these problems.

1. Solve word problems regularly

This might be the most important tip of all. Word problems should be part of everyday math practice, especially for older kids. Whenever possible, use word problems every time you teach a new math skill. Even better: give students a daily word problem to solve so they’ll get comfortable with the process.

Learn more: Teaching With Jennifer Findlay

2. Teach problem-solving routines

There are a LOT of strategies out there for teaching kids how to solve word problems (keep reading to see some terrific examples). The important thing to remember is that what works for one student may not work for another. So introduce a basic routine like Plan-Solve-Check that every kid can use every time. You can expand on the Plan and Solve steps in a variety of ways, but this basic 3-step process ensures kids slow down and take their time.

Learn more: Word Problems Made Easy

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3. Visualize or model the problem

Encourage students to think of word problems as an actual story or scenario. Try acting the problem out if possible, and draw pictures, diagrams, or models. Learn more about this method and get free printable templates at the link.

Learn more: Math Geek Mama

4. Make sure they identify the actual question

Educator Robert Kaplinsky asked 32 eighth grade students to answer this nonsensical word problem. Only 25% of them realized they didn’t have the right information to answer the actual question; the other 75% gave a variety of numerical answers that involved adding, subtracting, or dividing the two numbers. That tells us kids really need to be trained to identify the actual question being asked before they proceed. 

Learn more: Robert Kaplinsky

5. Remove the numbers

It seems counterintuitive … math without numbers? But this word problem strategy really forces kids to slow down and examine the problem itself, without focusing on numbers at first. If the numbers were removed from the sheep/shepherd problem above, students would have no choice but to slow down and read more carefully, rather than plowing ahead without thinking. 

Learn more: Where the Magic Happens Teaching

6. Try the CUBES method

This is a tried-and-true method for teaching word problems, and it’s really effective for kids who are prone to working too fast and missing details. By taking the time to circle, box, and underline important information, students are more likely to find the correct answer to the question actually being asked.

Learn more: Teaching With a Mountain View

7. Show word problems the LOVE

Here’s another fun acronym for tackling word problems: LOVE. Using this method, kids Label numbers and other key info, then explain Our thinking by writing the equation as a sentence. They use Visuals or models to help plan and list any and all Equations they’ll use. 

8. Consider teaching word problem key words

This is one of those methods that some teachers love and others hate. Those who like it feel it offers kids a simple tool for making sense of words and how they relate to math. Others feel it’s outdated, and prefer to teach word problems using context and situations instead (see below). You might just consider this one more trick to keep in your toolbox for students who need it.

Learn more: Book Units Teacher

9. Determine the operation for the situation

Instead of (or in addition to) key words, have kids really analyze the situation presented to determine the right operation(s) to use. Some key words, like “total,” can be pretty vague. It’s worth taking the time to dig deeper into what the problem is really asking. Get a free printable chart and learn how to use this method at the link.

Learn more: Solving Word Problems With Jennifer Findlay

10. Differentiate word problems to build skills

Sometimes students get so distracted by numbers that look big or scary that they give up right off the bat. For those cases, try working your way up to the skill at hand. For instance, instead of jumping right to subtracting 4 digit numbers, make the numbers smaller to start. Each successive problem can be a little more difficult, but kids will see they can use the same method regardless of the numbers themselves.

Learn more: Differentiating Math 

11. Ensure they can justify their answers

One of the quickest ways to find mistakes is to look closely at your answer and ensure it makes sense. If students can explain how they came to their conclusion, they’re much more likely to get the answer right. That’s why teachers have been asking students to “show their work” for decades now.

Learn more: Madly Learning

12. Write the answer in a sentence

When you think about it, this one makes so much sense. Word problems are presented in complete sentences, so the answers should be too. This helps students make certain they’re actually answering the question being asked… part of justifying their answer.

Learn more: Multi-Step Word Problems

13. Add rigor to your word problems

A smart way to help kids conquer word problems is to, well… give them better problems to conquer. A rich math word problem is accessible and feels real to students, like something that matters. It should allow for different ways to solve it and be open for discussion. A series of problems should be varied, using different operations and situations when possible, and even include multiple steps. Visit both of the links below for excellent tips on adding rigor to your math word problems.

Learn more: The Routty Math Teacher and Alyssa Teaches

14. Use a problem-solving rounds activity.

Put all those word problem strategies and skills together with this whole-class activity. Start by reading the problem as a group and sharing important information. Then, have students work with a partner to plan how they’ll solve it. In round three, kids use those plans to solve the problem individually. Finally, they share their answer and methods with their partner and the class. Be sure to recognize and respect all problem-solving strategies that lead to the correct answer.

Learn more: Teacher Trap

Like these word problem tips and tricks? Learn more about Why It’s Important to Honor All Math Strategies .

Plus, 60+ Awesome Websites For Teaching and Learning Math .

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• How to Define a Research Problem | Ideas & Examples

How to Define a Research Problem | Ideas & Examples

Published on November 2, 2022 by Shona McCombes and Tegan George. Revised on May 31, 2023.

A research problem is a specific issue or gap in existing knowledge that you aim to address in your research. You may choose to look for practical problems aimed at contributing to change, or theoretical problems aimed at expanding knowledge.

Some research will do both of these things, but usually the research problem focuses on one or the other. The type of research problem you choose depends on your broad topic of interest and the type of research you think will fit best.

This article helps you identify and refine a research problem. When writing your research proposal or introduction , formulate it as a problem statement and/or research questions .

Table of contents

Why is the research problem important, step 1: identify a broad problem area, step 2: learn more about the problem, other interesting articles, frequently asked questions about research problems.

Having an interesting topic isn’t a strong enough basis for academic research. Without a well-defined research problem, you are likely to end up with an unfocused and unmanageable project.

You might end up repeating what other people have already said, trying to say too much, or doing research without a clear purpose and justification. You need a clear problem in order to do research that contributes new and relevant insights.

Whether you’re planning your thesis , starting a research paper , or writing a research proposal , the research problem is the first step towards knowing exactly what you’ll do and why.

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As you read about your topic, look for under-explored aspects or areas of concern, conflict, or controversy. Your goal is to find a gap that your research project can fill.

Practical research problems

If you are doing practical research, you can identify a problem by reading reports, following up on previous research, or talking to people who work in the relevant field or organization. You might look for:

• Issues with performance or efficiency
• Processes that could be improved
• Areas of concern among practitioners
• Difficulties faced by specific groups of people

Examples of practical research problems

Voter turnout in New England has been decreasing, in contrast to the rest of the country.

The HR department of a local chain of restaurants has a high staff turnover rate.

A non-profit organization faces a funding gap that means some of its programs will have to be cut.

Theoretical research problems

If you are doing theoretical research, you can identify a research problem by reading existing research, theory, and debates on your topic to find a gap in what is currently known about it. You might look for:

• A phenomenon or context that has not been closely studied
• A contradiction between two or more perspectives
• A situation or relationship that is not well understood
• A troubling question that has yet to be resolved

Examples of theoretical research problems

The effects of long-term Vitamin D deficiency on cardiovascular health are not well understood.

The relationship between gender, race, and income inequality has yet to be closely studied in the context of the millennial gig economy.

Historians of Scottish nationalism disagree about the role of the British Empire in the development of Scotland’s national identity.

Next, you have to find out what is already known about the problem, and pinpoint the exact aspect that your research will address.

Context and background

• Who does the problem affect?
• Is it a newly-discovered problem, or a well-established one?
• What research has already been done?
• What, if any, solutions have been proposed?
• What are the current debates about the problem? What is missing from these debates?

Specificity and relevance

• What particular place, time, and/or group of people will you focus on?
• What aspects will you not be able to tackle?
• What will the consequences be if the problem is not resolved?

Example of a specific research problem

A local non-profit organization focused on alleviating food insecurity has always fundraised from its existing support base. It lacks understanding of how best to target potential new donors. To be able to continue its work, the organization requires research into more effective fundraising strategies.

Once you have narrowed down your research problem, the next step is to formulate a problem statement , as well as your research questions or hypotheses .

If you want to know more about the research process , methodology , research bias , or statistics , make sure to check out some of our other articles with explanations and examples.

Methodology

• Sampling methods
• Simple random sampling
• Stratified sampling
• Cluster sampling
• Likert scales
• Reproducibility

 Statistics

• Null hypothesis
• Statistical power
• Probability distribution
• Effect size
• Poisson distribution

Research bias

• Optimism bias
• Cognitive bias
• Implicit bias
• Hawthorne effect
• Anchoring bias
• Explicit bias

All research questions should be:

• Focused on a single problem or issue
• Researchable using primary and/or secondary sources
• Feasible to answer within the timeframe and practical constraints
• Specific enough to answer thoroughly
• Complex enough to develop the answer over the space of a paper or thesis
• Relevant to your field of study and/or society more broadly

Research questions anchor your whole project, so it’s important to spend some time refining them.

In general, they should be:

• Focused and researchable
• Answerable using credible sources
• Complex and arguable
• Feasible and specific
• Relevant and original

Your research objectives indicate how you’ll try to address your research problem and should be specific:

A research aim is a broad statement indicating the general purpose of your research project. It should appear in your introduction at the end of your problem statement , before your research objectives.

Research objectives are more specific than your research aim. They indicate the specific ways you’ll address the overarching aim.

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Division Facts and Word Problems

Join us on a mathematical journey as we explore division facts and problem-solving strategies. This lesson is designed to help students strengthen their understanding of division, practice division facts, and apply their knowledge to solve word problems. Through engaging activities and real-world scenarios, students will develop confidence in their division skills and enhance their problem-solving abilities. Get ready to dive into the world of division!

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Difficulties Encountered In Mathematical Word Problem Solving Of The Grade Six Learners

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New research offers solutions for cybersecurity in hospitals

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In May, a major cyberattack disabled clinical operations for nearly a month at Ascension, a health care provider that includes 140 hospitals across the U.S. Investigators tracked the problem to malicious ransomware that had infected an employee's computer.

Health care systems offer juicy targets for cybercrime because of the valuable personal, financial, and health data they hold. A 2023 survey of health information technology and IT security professionals reported that 88% of their organizations had experienced an average of 40 attacks during the previous year.

One key vulnerability has been the increasing complexity of their IT systems, says Hüseyin Tanriverdi, associate professor of information, risk, and operations management at Texas McCombs. It's a result of decades of mergers and acquisitions forming larger and larger multihospital systems.

After a merger, they don't necessarily standardize their technology and care processes. The health system ends up having a lot of complexity, with different IT systems, very different care processes and disparate governance structures." Hüseyin Tanriverdi, associate professor of information, risk, and operations management at Texas McCombs

But complexity could also offer a solution to such problems, he finds in new research. With co-authors Juhee Kwon of City University of Hong Kong and Ghiyoung Im of the University of Louisville, he says that a "good kind of complexity" can improve communication among different systems, care processes, and governance structures, better protecting them against cyber incidents.

Complex vs. Complicated

Using data from 445 multihospital groups spanning 2009 to 2017, the team looked at the oft-repeated notion that complexity is the enemy of security.

They distinguished between two similar-sounding IT concepts that are key to the problem.

• Complicatedness  is a large number of elements in a system that interconnect and share information in structured ways.
• Complexity  occurs when a large number of elements interconnect and share information in unstructured ways -; as when integrating systems after mergers and acquisitions.

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Because complicated systems have structures, Tanriverdi says, it's difficult but feasible to predict and control what they'll do. That's not feasible for complex systems, with their unstructured connections.

Tanriverdi found that as health care systems got more complex, they became more vulnerable. The most complex systems -; with the largest varieties of health service referrals from one hospital to another -; were 29% more likely to be breached than average.

The problem, he says, is that such systems offer more data transfer points for hackers to attack, and more opportunities for human users to make security errors.

He found similar vulnerabilities with other forms of complexity, including:

• Many different types of medical services handling health data.
• Decentralizing strategic decisions to member hospitals instead of making them at the corporate center.

Setting data standards

The researchers also proposed a solution: building enterprise-wide data governance platforms, such as centralized data warehouses, to manage data sharing among diverse systems. Such platforms would convert dissimilar data types into common ones, structure data flows, and standardize security configurations.

"They would transform a complex system into a complicated system," he says. By simplifying the system, they would further lower its level of complication.

He tested the cybersecurity effects of creating such platforms. The result, he found, was that in the most complicated system, they would reduce breaches up to 47%.

Centralizing data governance reduces avenues for hackers to get in, Tanriverdi says. "With fewer access points and simplified and hardened cybersecurity controls, unauthorized parties are less likely to gain unauthorized access to patient data."

He recommends supplementing technical controls with stronger human ones, as well: training users in cybersecurity practices and better regulating who has access to various parts of the system.

Tanriverdi acknowledges a paradox in his approach. Investing in a new layer of technology may introduce more IT complexity at first. But in the long run, it's a good type of complexity that tames the existing -; and more hazardous -; kinds of complexity.

"Practitioners should embrace IT complexity, as long as it gives structure to information flows that were previously ad hoc," he says. "Technology reduces cybersecurity risks if it is organized and governed well."

University of Texas at Austin

Tanriverdi, H., et al. (2024). Taming Complexity in Cybersecurity of Multihospital Systems: The Role of Enterprise-wide Data Analytics Platforms.  MIS Quarterly . doi.org/10.25300/misq/2024/17752 .

Posted in: Device / Technology News | Healthcare News

Tags: Health Care , Healthcare , Hospital , Research , Technology

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5 facts about student loans

Americans owe about $1.6 trillion in student loans as of June 2024 – 42% more than what they owed a decade earlier. The increase has come as greater shares of young U.S. adults  go to college and as the cost of higher education increases.

Here are five facts about student loans in America based on a Pew Research Center analysis of data from several sources, including the Federal Reserve Board’s 2023 Survey of Household Economics and Decisionmaking .

Pew Research Center conducted this analysis to understand how many Americans have outstanding student loan debt and how this debt is associated with their economic well-being.

In this analysis, adults with student loan debt include those whose student loans are temporarily on hold or in forbearance. The analysis does not include debt incurred through credit cards or other types of loans used for education.

The analysis is mostly based on the Federal Reserve’s 2023 Survey of Household and Economic Decisionmaking (SHED). Conducted annually since 2013, the SHED measures U.S. adults overall financial well-being and difficulties meeting expenses. It also regularly includes a battery of questions on debts incurred for education, education decisions, and an assessment of the value of higher education. The 2023 SHED had 11,400 respondents, weighted to be representative of the U.S. adult civilian noninstitutionalized population.

One-in-four U.S. adults under 40 have student loan debt. This share drops to 14% among those ages 40 to 49 and to just 4% among those 50 and older.

Of course, not all Americans attend or graduate from college, so student loan debt is more common among the subset of people who have done so. Among adults under 40 who have at least a four-year college degree, for example, 36% have outstanding student loan debt.

Age differences reflect, in part, the fact that older adults have had more time to repay their loans. Still, other research has found that young adults are also more likely now than in the past to take out loans to pay for their education. In the 2018-2019 academic year, 28% of undergraduate students took out federal student loans. That’s up from 23% in 2001-2002, according to data from College Board – a nonprofit organization perhaps best known for its standardized admissions tests (like the SAT) that also documents trends in higher education.

The amount of student loan debt that Americans owe varies widely by their education level. Overall, the median borrower with outstanding student debt owed between $20,000 and $24,999 in 2023.

• Among borrowers who attended some college but don’t have a bachelor’s degree, the median owed was between $10,000 and $14,999 in 2023.
• The typical bachelor’s degree holder who borrowed owed between $20,000 and $24,999.
• Among borrowers with a postgraduate degree the median owed was between $40,000 and $49,999.

Looking at the same data another way, a quarter of borrowers without a bachelor’s degree owed at least $25,000 in 2023. About half of borrowers with a bachelor’s degree (49%) and an even higher share of those with a postgraduate education (71%) owed at least that much.

Adults with a postgraduate degree are especially likely to have a large amount of student loan debt. About a quarter of these advanced degree holders who borrowed (26%) owed $100,000 or more in 2023, compared with 9% of all borrowers. Overall, only 1% of all U.S. adults owed at least $100,000.

Young college graduates with student loans are more likely than those without this kind of debt to say they struggle financially. A quarter of college graduates ages 25 to 39 with loans say they are either finding it difficult to get by financially or are just getting by , compared with 9% of those without loans. And while only 29% of young college graduates with outstanding student loans say they are living comfortably, 53% of those without loans say the same.

Young college graduates with student loans still tend to have higher household incomes than their counterparts who haven’t completed college. For many young adults, student loans are a way to make an otherwise unattainable education a reality. Although these students have to borrow money to attend college, the investment might make sense if it leads to higher earnings later in life.

College graduates ages 25 to 39 who have student loan debt have higher household incomes than non-college graduates in the same age group (regardless of student loan status). But their household incomes are lower than those of young college graduates who don’t have student loan debt.

Around half of young college graduates with student loans (48%) have household incomes of at least $100,000. That compares with just 14% of non-college graduates. But among college graduates without student loan debt, 64% have household incomes of $100,000 or more.

Household income includes an individual’s income and the income of any spouse or partner living with them. So these differences may at least partly reflect the fact that college graduates are more likely to be married.

Young college graduates with student loan debt are more likely than those without debt to say their education wasn’t worth the cost . About a third (35%) of those ages 25 to 39 who have at least a bachelor’s degree and outstanding student loan debt say the benefits of their degree weren’t worth the lifetime financial costs. By comparison, 16% of young college graduates without outstanding student loans say the same.

Note: This is an update of a post originally published Aug. 13, 2019.

• Age & Generations
• Higher Education
• Student Loans

Richard Fry is a senior researcher focusing on economics and education at Pew Research Center .

Anthony Cilluffo is a former research analyst who focused on social and demographic trends at Pew Research Center .

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