# data[].attributes.body
Can I collect data from my students without creating biases? Will my students identity be protected when collecting data?

# data[].attributes.body
What strategies can I implement in the classroom to give my students confidence with applications?

# data[].attributes.body
Students struggle with reading comprehension in a mathematics classroom.

# data[].attributes.headline
1. Word Problem Solving in Contemporary Math Education: A Plea for Reading Comprehension Skills Training

# data[].attributes.subject
1. Word Problem Solving in Contemporary Math Education: A Plea for Reading Comprehension Skills Training

# data[].attributes.body
In this article, a study was shown analyzing 80 high and low achieving groups of students to find correlation between reading comprehension and their success for word problems in a mathematics course.  Boonen, Koning, Jolles and Schoot (2016) write, “effectively solving a mathematical word problem is assumed to depend not only on students’ ability to perform the required mathematical operations, but also on the extent to which they are able to accurately understand the text of the word problem”.  After reading through this article, the authors explain that they assessed students’ mental representation skills and reading comprehension skills (2016).  This is important to analyze as the teacher needs to understand if students are memorizing concepts or if they are able to apply previously learned knowledge to applications.

# data[].attributes.attachment_link.readable_text
Abstract

Successfully solving mathematical word problems requires both mental representation skills and reading comprehension skills. In Realistic Math Education (RME), however, students primarily learn to apply the first of these skills (i.e., representational skills) in the context of word problem solving. Given this, it seems legitimate to assume that students from a RME curriculum experience difficulties when asked to solve semantically complex word problems. We investigated this assumption under 80 sixth grade students who were classified as successful and less successful word problem solvers based on a standardized mathematics test. To this end, students completed word problems that ask for both mental representation skills and reading comprehension skills. The results showed that even successful word problem solvers had a low performance on semantically complex word problems, despite adequate performance on semantically less complex word problems. Based on this study, we concluded that reading comprehension skills should be given a (more) prominent role during word problem solving instruction in RME.

Keywords: word problem solving, mental representation skills, reading comprehension skills, Realistic Math Education, consistency effect

Introduction

In the last decades, mathematical word problem solving has gained much attention from both researchers and educational practitioners ( Campbell, 1992; Hegarty et al., 1995; Hajer, 1996; Depaepe et al., 2010; Hickendorff, 2011, 2013; Moreno et al., 2011; Boonen et al., 2013; Swanson et al., 2013). Mathematical word problems refer to mathematical exercises that present relevant information on a problem as text, rather than in the form of mathematical notation ( Rasmussen and King, 2000; Timmermans et al., 2007). Hence, effectively solving a mathematical word problem is assumed to depend not only on students' ability to perform the required mathematical operations, but also on the extent to which they are able to accurately understand the text of the word problem ( Lewis and Mayer, 1987; Hegarty et al., 1995; Van der Schoot et al., 2009; Jitendra and Star, 2012). Both of these aspects are related in such a way that developing a deeper understanding of the text of the word problem serves as a crucial step before the correct mathematical computations can be performed. Hence, a key challenge for word problem solvers is to get an adequate understanding of the problem statement ( Lee et al., 2009; Thevenot, 2010; Boonen et al., 2013).

Two individual skills are relevant in this regard. First, an important factor contributing to a deeper understanding of the text of the word problem is the ability to construct a rich and coherent mental representation containing all (the relations between the) solution-relevant elements that are derived from the text base of the word problem ( De Corte et al., 1985; Hegarty et al., 1995; Pape, 2003). That is, word problem solvers have to use a problem-model strategy in which they translate the problem statement into a qualitative mental representation of the problem situation hidden in the text ( Pape, 2003; Van der Schoot et al., 2009). This mental representation subsequently allows them to make a solution plan and execute the required mathematical operations. Although successful word problem solvers appear to employ such a problem-model strategy by drawing on their mental representation skills, less successful problem solvers often adopt an impulsive, superficial direct translation strategy, in which they only focus on selecting the presented numbers that, in turn, form the basis for their mathematical calculations ( Verschaffel et al., 1992; Hegarty et al., 1995).

The second important individual skill in word problem solving success substantiated by research evidence is the influence of a student's reading comprehension abilities ( Pape, 2003; Van der Schoot et al., 2009; Boonen et al., 2013). It has been suggested that reading comprehension abilities are especially helpful in dealing with semantic-linguistic word problem characteristics such as the sequence of the known elements in the text of the word problem, the degree to which the semantic relations between the given and unknown quantities of the problem are made explicit, and the relevance of the information in the text of the word problem ( De Corte et al., 1985, 1990; Verschaffel et al., 1992; Marzocchi et al., 2002).

Moreover, reading comprehension skills appear to be more important in overcoming such textual complexities than being able to use one's mental representation skills ( De Corte et al., 1985, 1990). This might explain why the use of a problem-model strategy is not sufficient in all circumstances. That is, word problems containing semantically complex features require both accurate mental representation skills and reading comprehension skills, whereas for word problems with a lower semantic-linguistic complexity, well-developed mental representational skills might be sufficient.

These findings suggest that, to teach students how to effectively solve mathematical word problems, mental representation skills and reading comprehension skills should both be part of the mathematics education program. Particularly, paying attention to semantic-linguistic features of word problems is relevant to help students improve their word problem solving success, as word problems become semantically more complex as students progress in their educational career, for example, when they make the transition to secondary education. Word problems offered in secondary school subjects like geometry, physics and biology, include more verbal information and generally contain more complex semantic-linguistic text features ( Silver and Cai, 1996; Helwig et al., 1999).

The Netherlands, like many other countries, currently places great emphasis on the teaching of word problem solving in contemporary mathematics education ( Ruijssenaars et al., 2004; Elia et al., 2009). The teaching of mathematics in the Netherlands takes place within the context of a domain-specific instructional approach, called Realistic Mathematics Education (RME, Van den Heuvel-Panhuizen, 2003), where the process of mathematical word problem solving plays an important role ( Van den Boer, 2003; Barnes, 2005; Prenger, 2005; Van den Heuvel-Panhuizen, 2005; Hickendorff, 2011). Studies investigating the educational practice of RME show that the teaching of mental representation skills receives much attention in word problem solving instruction ( Van den Heuvel-Panhuizen, 2003; Van Dijk et al., 2003; Elia et al., 2009). However, reading comprehension skills enabling students to become sensitive to semantic-linguistic complexities in a word problem appear to be trained fewer and less explicitly in the instructional practice of RME, in spite of its proven importance in previous studies (e.g., De Corte et al., 1985, 1990; Hegarty et al., 1992). This is presumably because teachers may underestimate or are not aware of the importance of reading comprehension skills for solving word problems ( Hajer, 1996; Van Eerde, 2009). Thus, the current approach to teaching word problem solving appears to emphasize the development of mental representation skills, but seems to pay less attention to the role of reading comprehension skills. In this respect, the way in which word problem solving is taught in the RME curriculum does not seem to be aligned with what is currently known from research about the factors involved in effective word problem solving.

Based on the above analysis of the RME curriculum it seems legitimate to assume that students attending such a curriculum may be at a disadvantage when semantic-linguistic characteristics of a word problem have to be taken into account. That is, students from an RME curriculum are likely to experience difficulties when ask to solve mathematical word problems with a high semantic-linguistic complexity. To test this assumption, we compared students' performance on word problems obtained while following the RME curriculum to their performances on an independent word problem solving task. First, we classified students as successful or less successful word problem solvers with the help of a mathematics test that is part of the RME curriculum, viz., the CITO Mathematics test. This test can be considered a method-specific (i.e., RME-specific) mathematics test of students' word problem solving performance, as it builds upon the currently used instructional method for word problem solving. Hence, this test reflects the skills that students learn in the RME classroom, in order to solve word problems ( Doorman et al., 2007; Hickendorff, 2011). Second, we examined students' performance on an independent word problem solving test, which contained either word problems that they could solve by only using their mental representation skills, or word problems that required them to also rely on their reading comprehension skills for handling semantic-linguistic complexities in the word problems. This procedure provides an advantage over prior studies of, among others, Hegarty et al. (1995), Pape (2003), and Van der Schoot et al. (2009), which typically used the main dependent variable of the study (i.e., problem solving success) as an outcome measure as well as a means to classify students into successful and less successful word problem solvers. The classification used in the present study, on the other hand, is based on an external, well-established measure of mathematical word problem solving, which is independent of the main dependent variable of the study (i.e., word problem solving success). This allowed us to make more meaningful group comparisons.

As previously mentioned, a key aspect that differentiates successful from less successful word problem solvers concerns their ability to construct an accurate mental representation of the problem text. Previous studies have shown that asking students to solve compare problems, especially inconsistent compare problems (see Example 1), is a suitable method for investigating whether or not they have effectively constructed an accurate mental representation of the problem statement (e.g., Pape, 2003; Van der Schoot et al., 2009).

[Example 1 - inconsistent word problem]

At the grocery store, a bottle of olive oil costs 7 euro.

That is 2 euro more than at the supermarket.

If you need to buy seven bottles of olive oil, how much will it cost at the supermarket?

[Example 2 - consistent word problem]

At the grocery store, a bottle of olive oil costs 7 euro.

At the supermarket, a bottle of olive oil costs 2 euro more than at the grocery store.

If you need to buy 7 bottles of olive oil, how much will you pay at the supermarket?

In inconsistent word problems like the one presented in Example 1, the translation process requires the identification of the pronominal reference 'that is' as the indicator of the relation between the value of the first variable ('the price of a bottle of olive oil at the grocery store') to the second ('the price of a bottle of olive oil at the supermarket'). This identification is necessary to become cognizant of the fact that, in an inconsistent compare problem, the relational term 'more than' refers to a subtraction operation rather than to an addition operation. So, inconsistent word problems create greater cognitive complexity than consistent word problems (see Example 2), requiring students to ignore the well-established association between more with increases and addition, and less with decreases and subtraction ( Schumacher and Fuchs, 2012). Empirical evidence corroborates this interpretation by showing that word problem solvers make more (reversal) errors on inconsistent than on consistent word problems (i.e., consistency effect, Lewis and Mayer, 1987; Pape, 2003; Van der Schoot et al., 2009). Especially students who fail to build an accurate mental representation of the problem statement, and thus immediately start calculating with the given numbers and relational term, seem to be less successful on inconsistent word problems ( Hegarty et al., 1995).

In the present study, we expected neither successful nor less successful problem solvers to experience difficulties with solving consistent compare word problems. However, we did assume that successful word problem solvers in the RME curriculum would experience less difficulties with correctly solving inconsistent compare problems as a result of their reliance on mental representation skills (acquired during word problem solving instruction in RME), than less successful problem solvers who employ a more superficial problem solving approach ( Verschaffel et al., 1992; Van der Schoot et al., 2009).

It is important to keep in mind that this only holds for consistent and inconsistent compare problems with a low semantic complexity; that is, problems that only tap into students' ability to construct an accurate mental representation. If the semantic complexity of compare problems increases, we expected that even students classified as successful word problem solvers (according to our classification based on the RME instruction) may come to experience difficulties with correctly solving inconsistent compare problems. In this case, correctly solving a word problem requires students to use both mental representational skills and reading comprehension skills, while word problem solving instruction in RME (presumably) has provided students only with considerable training in the first of these two skills.

A relatively well-studied and accepted way to increase the semantic complexity of (inconsistent) compare problems is to manipulate the relational term ( Lewis and Mayer, 1987; Van der Schoot et al., 2009). According to the lexical marking principle ( Clark, 1969), it is more difficult to process marked relations terms (such as 'less' in the antonym pair 'more-less,' 'narrow' in 'wide-narrow' or 'short' in 'tall-short') than unmarked relational terms (e.g., more, wide, tall). Consistent with this, research has shown that students find it easier to convert the unmarked relational term 'more than' into a subtraction operation than the marked relational term 'less than' into an addition operation ( Clark, 1969; Lewis and Mayer, 1987; Kintsch, 1998; Pape, 2003; Van der Schoot et al., 2009). In the present study, we therefore refer to word problems containing a marked relational term ('more than') as semantically more complex word problems, whereas word problems with an unmarked relational term ('less than') are referred to as semantically less complex word problems (see Examples 3 and 4 for examples of marked and unmarked word problems respectively). Importantly, the difficulties experienced with solving marked inconsistent word problems lie in the fact that these problems draw on students' use of their mental representation skills as well as on their reading comprehension skills. Accordingly, the influence of reading comprehension skills on word problem solving can only be studied for students who mentally represent the problem statement accurately, that is, the group of successful problem solvers in our study. So, although our group of successful word problem solvers may draw upon their mental representation skills, the insufficient attention to reading comprehension skills in the educational practice of RME is likely to cause them to experience difficulties with correctly solving (semantically complex) marked inconsistent word problems.

[Example 3 - marked word problem]

At the grocery store, a bottle of olive oil costs 7 euro.

At the supermarket, a bottle of olive oil costs 2 euro less than at the grocery store.

If you need to buy seven bottles of olive oil, how much will you pay at the supermarket?

[Example 4 - unmarked word problem]

At the grocery store, a bottle of olive oil costs 7 euro.

That is 2 euro less than at the supermarket.

If you need to buy seven bottles of olive oil, how much will it cost at the supermarket?

According to several researchers, the extent to which successful word problem solvers might be able to overcome difficulties with correctly solving marked inconsistent word problems is related to their reading comprehension skills (e.g., Lee et al., 2004; Van der Schoot et al., 2009). Translating a marked relational term like 'less than' into an addition operation is found to be closely associated with general measures of reading comprehension ( Lee et al., 2004; Van der Schoot et al., 2009). This suggests that reading comprehension skills, together with mental representation skills, might be necessary to deal with semantically complex word problems. The present study therefore also takes into account students' general reading comprehension ability.

In sum, the present study aimed to test the following hypotheses:

We hypothesized that, as a result of difficulties with constructing a coherent mental representation of word problems, less successful word problem solvers in the RME curriculum would make more errors on both unmarked and marked inconsistent word problems than on unmarked and marked consistent word problems.

We hypothesized that, as a result of paying insufficient attention to reading comprehension skills in the teaching of word problem solving, successful word problem solvers in the RME curriculum would experience difficulties with solving semantically complex, marked inconsistent word problems, but not with solving semantically less complex, unmarked, inconsistent word problems.

We hypothesized that, as a result of the alleged relation between reading comprehension ability and the ability to overcome the semantic-linguistic complexities of a word problem, a positive relation for successful problem solvers exists between reading comprehension ability and the number of correctly solved marked inconsistent word problems.

Materials and Methods

Selection of Participants

Data from 80 Dutch sixth-grade students (42 boys, 38 girls) from eight elementary schools in the Netherlands were collected. These students had a mean age of 11.72 years ( SD = 0.40). They were almost equally divided in two groups (by means of the median split method) on the basis of their score on the CITO (Institute for Educational Measurement) Mathematics test (2008). This selection procedure resulted in a group of less successful word problem solvers ( N = 41) and a group of successful word problems solvers ( N = 39). The CITO Mathematics test is a nationwide standardized test that reflects the way in which word problem solving is instructed in Realistic Mathematics Education. The test contains elements like mental arithmetic (addition, subtraction, multiplication, and division), complex applications (problems involving multiple operations) and measurement and geometry (knowledge of measurement situations), all of which are offered as mathematical word problems. The internal consistency of this test was high (Cronbach's α = 0.95, Janssen et al., 2010).

Parents provided written informed consent based on printed information about the purpose of the study. This study was carried out in accordance with the ethical procedures of the Vrije Universiteit Amsterdam.

Instruments and Procedure

The two measurement instruments that were used in this study were administrated to the students by three trained independent research assistants in a session of approximately 45 min.

Data Analysis

A 2 × 2 × 2 analysis of variance (ANOVA) was conducted with Consistency (consistent vs. inconsistent) and Markedness (unmarked vs. marked) as within-subject factors and Group (less successful vs. successful word problem solvers) as the between-subject factor. Follow-up tests were performed using paired sample t-tests. The partial eta-squared () was calculated as a measure of effect size ( Pierce et al., 2004). According to Pierce et al. (2004), values of 0.02, 0.13, and 0.26 represent small, medium, and large effect sizes respectively.

In the present study, the role of reading comprehension in the four word problem types was examined by calculating the product-moment correlations (Pearson's r) between reading comprehension and the difference score between the unmarked inconsistent and consistent word problem types, and the correlation between reading comprehension and the difference score between the marked inconsistent and consistent word problem types. These difference scores reflect the differences in performance between the consistent and inconsistent word problem types, and can be taken as a measure of the extent to which students are able to construct a mental representation of the described problem situation. The lower the difference score, the less word problem solvers suffer from the inconsistency. The correlations were first calculated for the less successful and successful word problem solvers together, and then, to test the third hypothesis, for each of these groups separately.

Our approach deviates from, but provides an important advantage over, the study by Van der Schoot et al. (2009), who added reading comprehension as a covariate in the repeated measures ANOVA. That is, the results obtained by Van der Schoot et al. (2009) could provide only limited insight into the exact locus of the covariate's effect, as it was not known which group (less successful or successful word problem solvers) or in which word problem type (consistent unmarked/marked or inconsistent unmarked/marked) reading comprehension played a role. Moreover, it turns out that the repeated measures ANCOVA does change the main effects of the repeated measures compared to assessing the main effects via a simple repeated measures ANOVA (see Thomas et al., 2009). So, the approach used in the present study enabled us to obtain more specific insight into the precise role of reading comprehension in word problem solving. In all analyses an alpha of 0.05 was used to test the significance of the results.

Results

The overall means ( M) and standard deviations ( SD) for the main factors in this study, as well as their intercorrelations, are displayed in . As can be seen, there was a significant main effect of Consistency [ F(1,78) = 23.84, p = 0.00, = 0.23], indicating that consistent word problems were completed more accurately than inconsistent word problems (i.e., consistency effect). There was no significant main effect of Markedness [ F(1,78) = 2.64, p = 0.11], suggesting that overall not more errors were made on marked than on unmarked word problems. The main effect of Group was also not significant [(1,78) = 1.15, p = 0.29)], indicating that overall successful problem solvers did not show a higher problem solving performance than less successful problem solvers.

Table 1

Regarding the interacting effects between Consistency and Markedness, the analysis revealed a significant interaction [ F(1,78) = 7.64, p = 0.01, = 0.09] showing that overall the consistency effect was present for marked word problems but absent for unmarked word problems. Of more interest, in light of our hypotheses, is that, as expected, the Consistency × Markedness interaction differed for less successful and successful word problem solvers. This was evidenced by a significant three-way interaction between Consistency, Markedness, and Group [ F(1,78) = 4.32, p = 0.03, = 0.05]. In , word problem solving performance is presented as a function of consistency (consistent vs. inconsistent) and markedness (marked vs. unmarked) for less successful problem solvers (), and for successful problem solvers (), respectively.

As shown in , the main effect of Consistency [ F(1,38) = 8.16, p = 0.01, = 0.18] indicates that less successful word problem solvers showed the consistency effect. Given the non-significant Consistency × Markedness interaction [ F(1,38) = 0.25, p = 0.62], the consistency effect was present for both marked and unmarked word problems. No significant main effect of Markedness was found [ F(1,38) = 0.12, p = 0.74]. So, less successful word problem solvers performed significantly lower on both the unmarked and marked inconsistent word problem types, compared to the consistent unmarked and marked word problem types [ t(38) = 1.86, p = 0.04; t(38) = 2.57, p = 0.01 respectively].

As can be seen in , the group of successful problem solvers resembled the less successful problem solvers in that there was a main effect of Consistency [ F(1,40) = 16.29, p = 0.00, = 0.29], but no significant main effect of Markedness [ F(1,40) = 0.27, p = 0.61]. However, in contrast to the group of less successful problem solvers, the consistency effect in the group of successful problem solvers was present for marked but absent for unmarked word problems [Consistency × Markedness interaction: F(1,40) = 17.44, p = 0.00, = 0.30]. This indicates that successful word problem solvers performed significantly lower on marked inconsistent compared to marked consistent word problems [ t(40) = 5.07, p = 0.00], whereas performance on unmarked consistent and unmarked inconsistent word problem types did not differ significantly [ t(40) = 1.52, p = 0.13].

In sum, these findings show that less successful word problem solvers demonstrated the consistency effect on both semantic-linguistically simple (i.e., unmarked) and complex (i.e., marked) word problems, whereas successful word problem solvers only demonstrated the consistency effect when the word problem text contained complex semantic-linguistic features (i.e., marked).

Regarding the role of reading comprehension skills in word problem solving the following findings were obtained. Overall, there was a significant correlation between reading comprehension and mathematics scores obtained from the curriculum-specific RME test ( r = 0.59, p = 0.00). This suggests that students with higher reading comprehension scores also showed higher scores on the RME mathematics test. To obtain more detailed insight into the role of reading comprehension skills in solving marked and unmarked word problems, reading comprehension scores were correlated with the difference scores (inconsistent - consistent) computed for the marked and unmarked word problem types. Results showed that reading comprehension was significantly correlated with the difference score for unmarked word problems ( r = 0.19, p = 0.04) and had a marginally significant correlation with the difference score for marked word problems ( r = 0.17, p = 0.06). This suggests that overall reading comprehension abilities are relevant to solving both marked and unmarked word problems.

When looking at the successful and less successful problem solvers separately, the results showed, similar to the overall findings, that reading comprehension was significantly correlated with the scores on the RME-specific mathematics test for both successful ( r = 0.48, p = 0.00) and less successful problem solvers ( r = 0.64, p = 0.00). So, for successful and less successful problem solvers higher reading comprehension abilities were associated with higher RME mathematics scores. Furthermore, successful word problem solvers ( M = 46.42, SD = 2.66) scored significantly higher on the standardized reading comprehension test than less successful word problem solvers ( M = 35.02, SD = 1.27) [ t(53.32) = 3.87, p = 0.00].

More specific analyses focusing on the hypothesized relation between reading comprehension skills and solving marked inconsistent word problems revealed the following pattern of findings. In line with our expectations, the results of the correlational analyses between reading comprehension and the difference scores for marked and unmarked word problems showed that only in the group of successful word problem solvers the difference score for the marked word problem type was significantly related to reading comprehension ( r = -0.40, p = 0.01). Importantly, reading comprehension was not correlated with the successful word problem solvers' difference scores for unmarked word problems ( r = -0.27, p = 0.10). Furthermore, in the group of less successful word problem solvers, reading comprehension was also not correlated with the difference scores computed for either unmarked ( r = -0.04, p = 0.76) or marked word problems ( r = -0.04, p = 0.83).

So, only in the group of successful word problem solvers, a higher reading comprehension score was associated with a smaller difference score. That is, the vulnerability for the consistency effect on marked word problems was lower for students who have higher reading comprehension abilities. This suggests that students with higher reading comprehension abilities appear to suffer less from being primed to an inconsistent arithmetic operation (i.e., being directed toward a subtraction operation by 'less than' while addition is required) in solving marked inconsistent word problems.

Discussion

This study was motivated by the observation that contemporary RME primarily teaches students to use their mental representation skills, and focuses much less on using reading comprehension skills, to solve mathematical word problems. Against this background, we set out to investigate the assumption that students from an RME curriculum experience difficulties when having to solve mathematical word problems that are semantic-linguistically complex. We therefore designed a study in which we not only manipulated the extent to which mental representation skills were required, but also varied the semantic complexity of the word problems by using a marked (i.e., high semantic complexity) or unmarked (i.e., low semantic complexity) relational term in the word problem text. Moreover, we classified students as successful and less successful word problem solvers on the basis of their performance on an independent and well-established RME-specific mathematics test.

Using this classification procedure, it was hypothesized that less successful word problem solvers would experience difficulties with correctly solving inconsistent word problems irrespective of their semantic complexity (Hypothesis 1). This hypothesis was confirmed by our analyses, which showed that less successful word problem solvers performed poorly on both marked and unmarked inconsistent word problems. Successful word problem solvers, on the other hand, were able to effectively solve inconsistent word problems that had a low semantic complexity. So, these findings show that the RME-based classification in successful and less successful problem solvers was also reflected in our experimental word problem solving task.

However, on semantically complex word problems even the successful problem solvers experienced difficulties, as indicated by the large number of errors they made on marked inconsistent word problems (Hypothesis 2). More concretely, successful word problem solvers found it more difficult to translate a marked relational term ('less than') into an addition operation, than to translate an unmarked relational term ('more than') into a subtraction operation.

These findings once again support prior observations that (subtle) semantic-linguistic elements of a word problem, more specifically the marked relational term, influence word problem solving success ( Clark, 1969; Lewis and Mayer, 1987; Kintsch, 1998; Pape, 2003; Van der Schoot et al., 2009). Moreover, they are in line with empirical work consistently reporting processing problems with marked terms, which are suggested to be caused by the semantic representation of negative poles of antonym pairs (e.g., more than vs. less than) like 'less than' being more fixed and complex, and therefore less likely to be reversed, than that of the positive poles like 'more than' (e.g., Lewis and Mayer, 1987; for a detailed explanation of the underlying mechanism, see, e.g., Clark, 1969). For example, earlier studies have shown that students are less able to recall marked terms accurately in memory tasks ( Clark and Card, 1969), have slower naming responses for marked terms in naming tasks ( Schriefers, 1990), have slower solution times for problems with marked adjectives in reasoning problems ( French, 1979), and, the finding replicated in this study, experience problems with reversing a marked inconsistent word problem (e.g., Pape, 2003; Van der Schoot et al., 2009).

Importantly, our results reveal the interesting situation that students classified as successful word problem solvers in an RME curriculum are unsuccessful in solving semantically complex (inconsistent) word problems. The fact that successful problem solvers were able to solve inconsistent word problems with a lower semantic complexity suggests that this poor performance on semantically complex word problems is not due to shortcomings in their mental representation skills. Rather, it seems that successful problem solvers particularly have difficulties to effectively handle semantic-linguistic complexities in word problems. This suggests that students lack the reading comprehension skills required for identifying and translating a primed mathematical operation to the 'word problem appropriate' mathematical operation. In the case of marked inconsistent word problems, this means that even successful students find it difficult to convert 'less than' into an addition operation. Although it could be argued that this is likely the result of the relatively little attention to the development of reading comprehension skills in the context of mathematical word problem solving in RME (e.g., Elia et al., 2009), this speculative interpretation needs to be further substantiated in future research.

Building upon prior studies (e.g., Lee et al., 2004; Van der Schoot et al., 2009), another aim of this study was to investigate whether reading comprehension skills could help (successful) word problem solvers to overcome the semantically complex marked relational term in an inconsistent word problem. In line with our expectations, reading comprehension was positively related to the performance on marked (but not unmarked) inconsistent word problems for the group of successful word problem solvers, whereas for the less successful group no significant relations were found between reading comprehension and word problem solving (Hypothesis 3).

These results provide corroborating evidence that general reading comprehension skills play an important role in students' ability to correctly solve semantically complex word problems. Moreover, our findings represent an advance over prior work by more specifically delineating which types of word problems and for which students reading comprehension ability might have an effect. This study shows that reading comprehension skills are especially helpful when it comes to improving the performance on semantically complex word problems by successful word problem solvers (as classified by the RME mathematics test). More specifically, reading comprehension skills are relevant for word problem solving primarily in helping students to effectively translate complex (i.e., marked) relations terms encountered in inconsistent word problems to the correct mathematical operation (i.e., addition). From this, it is evident that reading comprehension skills provide a valuable addition to mental representation skills for word problem solving, and that simply relying on mental representation skills is not sufficient to correctly solve semantically complex word problems. This suggests that in addition to teaching students to use their mental representation skills to solve word problems, word problem solving instruction should have sufficient attention for developing and using reading comprehension skills related to identifying and dealing with semantic-linguistic features in the word problem statement.

It is important to start developing such skills early in elementary school, as word problems get semantically more complex when students progress in their educational career, for example when making the transition from elementary to secondary education ( Silver and Cai, 1996; Helwig et al., 1999). Particularly in instructional approaches focused on word problem solving that show an imbalance between the amount of instruction time being devoted to the teaching of mental representation skills and reading comprehension skills, such as in RME, it is important to make teachers aware of this unequal distribution. Encouraging them to pay more attention to reading comprehension skills and teaching students how to deal with semantic-linguistic characteristics in word problems would then provide a good starting point to work toward more equally balanced word problem solving instructions. Moreover, it is useful to make a distinction between learning to process more subtle semantic-linguistic text features (like a marked relation term) and dealing with more general semantic text complexities (like the relevance of the information in the word problem text, the explicitness of the described relations, and the sequence of the known elements in the word problem text).

These and other practical aspects of the results, such as finding the optimal balance between the amount of instruction in strategic mental representation skills and reading comprehension skills, remain to be addressed in future research. Presumably, currently effective intervention programs that focus on both strategic mental representation skills and reading comprehension skills, such as schema-based instruction (e.g., Jitendra et al., 2002, 2011), and the Solve It! instruction method ( Montague et al., 2000; Krawec et al., 2013), could provide a fruitful starting point in pursuing this challenge.

Author Contributions

All authors listed, have made substantial, direct and intellectual contribution to the work, and approved it for publication.

Conflict of Interest Statement

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.

References

Schumacher R. F., Fuchs L. S. (2012). Does understanding relational terminology mediate effects of intervention on compare word problem? [J. Exp. Child Psychol. 111 607-628. 10.1016/j.jecp.2011.12.001PMC free article]

Swanson H. L., Lussler C. M., Orosco M. J. (2013). Cognitive strategies, working memory, and growth in word problem solving in children with math difficulties. J. Learn. Disabil. XX, 1-20. 10.1177/0022219413498771

# data[].attributes.attachment_link.title
Word Problem Solving in Contemporary Math Education: A Plea for Reading Comprehension Skills Training

# data[].attributes.attachment_link.description
Successfully solving mathematical word problems requires both mental representation skills and reading comprehension skills. In Realistic Math Education (RME), however, students primarily learn to apply the first of these skills (i.e., representational skills) in the context of word problem solving.

# data[].attributes.headline
2. Recognition of Students' Difficulties in Solving Mathematical Word Problems from the Viewpoint of Teachers

# data[].attributes.subject
2. Recognition of Students' Difficulties in Solving Mathematical Word Problems from the Viewpoint of Teachers

# data[].attributes.body
This article describes strategies to help students understand what to do to help understand and interpret word problems.  Through their findings, they observed that students had difficulties with reading the text, being unfamiliar with the context and taking an unsuccessful approach to set up the problem (Haghverdi, 2012, p.2923).  The studies that were documented in this article focus on both the teacher and the students.  The research describes the strategies the teachers use to help introduce and represent the applications to students and also describes the students approach to the problems.  They discovered that if a student is able to visualize and set up a diagram to understand the problem it will help increase their success in their comprehension.

# data[].attributes.headline
3. The Influence of Visual Representations and Context on Mathematical Word Problem Solving

# data[].attributes.subject
3. The Influence of Visual Representations and Context on Mathematical Word Problem Solving

# data[].attributes.body
I can relate to this article very well as one of the main goals from the author was explaining that students learn best from using visual representations for applications.  In my classes, students have a very difficult team reading the word problem and understand what it is asking.  Cankoy and Ozder (2011) “focus on the effect of the presence of visual representations in word problem solving”.  It was interesting to read through their findings and from this, I can take strategies and techniques and start implementing them into my classroom so I can have my students become more accustomed to solving applications.

# data[].attributes.headline
4. The Examining Mathematical Word Problems Solving Ability under Efficient Representation Aspect

# data[].attributes.subject
4. The Examining Mathematical Word Problems Solving Ability under Efficient Representation Aspect

# data[].attributes.body
This article explains difficulties that arise with students approaching word problems and suggests strategies for reading word problems better.  A few efficiency strategies consist of “comprehension of relations, asking questions, induction and predication, determination of importance and integration and compilation” (Amiripour, Rostamy-Malkhalifeh, & Sajadi, 2013, p.4).  In this article, the authors hypothesize relationships between using efficient representation and efficient word problem solving ability.  Using the research from this article will help give me a better idea of how to implement changes in the classroom.

# data[].attributes.body
What teaching strategies and classroom techniques can a teacher use to help improve students reading comprehension in a mathematics classroom?

# data[].attributes.headline
5. Promoting mathematics learning through problem solving in the K8 classroom: improving student success and deepening learning

# data[].attributes.subject
5. Promoting mathematics learning through problem solving in the K8 classroom: improving student success and deepening learning

# data[].attributes.body
This article describes new routines that teachers can implement in the classroom to allow the students to feel more comfortable and confident in their learning through problem solving.  The authors explain that students need to make sense of concepts and relationships in a mathematical way.  Curran and Ellis (2017) write, “once students generate new ideas and insights from problem solving, they are cognitively ready for teacher-led instruction that will help them to organize, refine and better understand how this new learning connects to prior knowledge”.  What I found helpful from this article was reading how the authors describe the changes in the classroom and how technology plays a big role in learning.  Establishing a new routine will help students make connections for real life applications.

file:///C:/Users/mpisani/Downloads/Promoting_mathematics_learning.PDF

# data[].attributes.headline
6. Learning to Think Mathematically: Problem Solving, Metacognition, and Sense Making in Mathematics

# data[].attributes.subject
6. Learning to Think Mathematically: Problem Solving, Metacognition, and Sense Making in Mathematics

# data[].attributes.body
After reading through this article, I learned that it is important to have your students to start thinking mathematically at all times.  Since solving word problems in a mathematics classroom is a challenging concept for students to master, is important that the teacher establishes a culture where students are thinking mathematically so they can handle complex situations.  Schoenfeld (2017) states that mathematics is conceptualized as the science of patterns.  This article encourages teachers to change curriculum content and instructional style to have students seeks solutions, not just memorize procedures.

# data[].attributes.headline
8. Student learning and perceptions in a flipped linear algebra course

# data[].attributes.subject
8. Student learning and perceptions in a flipped linear algebra course

# data[].attributes.body
In this article, the authors explain the advantages of having a flipped algebra classroom verses direct instruction.  Grandgenett, Hodge, Love, and Swift (2013) state, “students in the flipped classroom environment had a more significant increase between the sequential exams compared to the students in the traditional lecture section.”  In my own classroom, I’ve noticed that the more I lecture and talk in front of the classroom, the easier it is that I lose my students focus.  By incorporating a flipped classroom where the students are more engaged and running the show, the students now take a bigger role in the instruction and have to pay closer attention to detail with the content.  Giving the students a bigger role gives them more confidence which enhances student performance.  This article is important for problem solving in class as a flipped classroom would be a good technique to try to see growth in performance for solving word problems.

# data[].attributes.body
In this article, they list “eight processes and proficiencies that enable students to cultivate expertise in math concepts and to develop problem-solving skills germane to their grade level” (n.d.).  This article emphasizes the importance of adjusting to the common core standards and how the teacher role changes significantly.  Students need to become the center of the classroom and lead discussions.  Students need to become more driven and the teachers take a step back to watch and observe.  With this change in the role of the teacher, it holds the students more accountable and they are able to think of their own rather than be fed the information.  One of the biggest weaknesses for students solving word problems in a mathematics classroom is that they are given the set up and given too much information.  Taking a step back will enforce students expand their thinking and apply what they know so they are more comfortable approaching applications.

# data[].attributes.attachment_link.readable_text
December 2012/January 2013 | Volume 70 | Number 4
Common Core: Now What?

Jessica Hiltabidel

The most significant difference is how my role as a teacher changes. To successfully implement Common Core standards, a classroom needs to become student driven and student led. The teacher becomes more of a facilitator, an instigator of thought, encouraging students to persevere through the process of finding a technique for solving new problems.

The second difference is that my checklist of skills is gone. There are seemingly only five Common Core units applicable to middle school math (Number Systems, Expressions and Equations, Functions, Geometry, and Statistics and Probability). Although these units mirror skills currently required of 8th graders, the approach taken to developing these skills is completely different. The standards move away from the large quantity of material students were previously exposed to at each grade level and instead emphasize the depth at which skills are developed. Words like formulize and reason replace solve and explain in instructions to students.

This difference is most apparent in the Common Core's Standards of Mathematical Practice. Unlike Maryland's current curriculum, these Standards of Mathematical Practice (or practice standards) emphasize the "how" when it comes to teaching, not simply the "what." They list eight processes and proficiencies that enable students to cultivate expertise in math concepts and to develop problem-solving skills germane to their grade level (see " The Standards of Mathematical Practice.")

Productive Struggle

During my professional development on implementing the Common Core standards, I examined how engaging students in productive struggle helps them get comfortable with the practice standards. I realized what I needed to add to my classroom practice to ensure that all eight practice standards were brought to life. To allow students the opportunity to persevere, use tools appropriately, and so on, I needed to release control of my classroom. I needed to let students struggle-independently and collaboratively-using inquiry as the primary instructional tool.

The established lesson practice within Baltimore City Public Schools is "I do, we do, you do." Teachers introduce a skill; students then complete examples as a class, with the teacher guiding them through each step; then students demonstrate their learning through independent practice.

Inquiry-based lessons, in contrast, begin with "you do" and give students the opportunity to figure things out for themselves. They rely on the first of the eight practice standards: Make sense of problems and persevere in solving them. The theory is that if a student works through the steps on his or her own, that student has a higher likelihood of retaining the information than if he or she were simply given steps to follow.

In theory, this approach sounds wonderful. But how can we expect students operating below grade level to succeed if left to their own devices? Given the wide diversity of economic background and proficiency levels in my class, how were my students going to learn to make sense of problems and persevere in solving them?

These questions ran through my mind during my training. Fortunately, I learned approaches that strengthened my ability to teach through inquiry, including three powerful takeaways:

Teachers must engage students with new concepts-physically and in other learning modalities.

Effective questioning can spur inquiry.

It's possible to teach math effectively without using directives or modeling.

Teaching with All Modalities

If you physically engage a student while stimulating his or her brain, you create the best chance for that kid to actively participate in inquiry and retain learning. Kinesthetic movement is one of the four modalities (along with visual, auditory, and tactile) that educator Neil Fleming has identified through which people process new information and experiences.

At the beginning of math classes, I do quick warm-ups involving movement, such as "Shake Down, Count Down." Students start at a given number-say, eight-and shake their right hand eight times, then their left hand, their right foot, and their left foot. Once you've worked out all four limbs, shake each limb seven times, then six, and down to one. In addition to introducing movement, the silliness helps build a classroom culture in which students aren't afraid to poke fun at one another or make mistakes.

I try to keep this inquiry-promoting atmosphere going as I introduce concepts. For example, in a lesson on unit rates, my students first count how many jumping jacks they can do in three minutes, then calculate how many they could do in one minute. I have them browse store circulars to find prices for items, calculate the unit rate of each item they identify, and complete a worksheet showing those calculations.

We then discuss the items and prices students chose and how they calculated the unit rate of various items with the information they had. Class discussion not only uses auditory skills, but also enables students to engage in several of the standards for math practice, such as critiquing others' reasoning.

Questioning Effectively

As an instigator of thought, I try to ask the right question-one that enables unsure teenagers to progress to the next level of learning. I've learned to use "hook questions" to engage students' curiosity and encourage inquiry in math.

A hook question has no one right answer. It doesn't even have to be something to which you know the answer. A good hook question encourages students to become investigators and to seek the answer from outside resources-textbooks, the Internet, even their communities. This open-endedness might also lead students to use several of the Common Core's practice standards, such as, make sense of problems (Standard 1) or look for and make use of structure (Standard 7).

For example, as students learn how to solve inequalities, we tell them that if you multiply or divide by a negative number, you have to flip the sign of the inequality. As I explained this rule, one of my students asked, "Why?" As I thought about it, I realized I wasn't sure. I could show this rule with an example, but I couldn't explain why it existed.

I made discovering the answer to this question students' homework for the night and challenged them to find the answer before me. Students were allowed to seek the answer from any source. They loved the challenge. There was an entire thread about this homework on Facebook as students asked friends in high school to help explain the rule. Even a math teacher from another state chimed in.

I'm confident that most of the students involved will never again forget to flip the inequality symbol. If they do, they can go back to their archived Facebook messages and rediscover why they need to-because when you multiply or divide by a negative number, you change the value into its opposite (a negative becomes a positive and vice versa). To accommodate this change in sign, you must also change the inequality symbol.

My students who struggle with math take one look at a word problem and immediately raise their hand and ask for help. Instead of asking what they need help with, I now ask them what they already know. For example, in a problem involving a system of equations, most of my students know there are four methods of solving: elimination, substitution, graphing, and guess and check. Most students can usually identify these methods, so I ask, "Which method can you use?" Then I allow the student to guess and check their way through the problem or try any of the other methods. At least this gives them a way to plunge in.

A more advanced student may solve the same problem by immediately choosing the simplest method. I'd ask this learner, "How can you answer this problem using one or two methods that differ from the one you already used?"

Teaching a Required Skill Through Inquiry

Let's look at how I taught one skill my students must master with my old approach and then with my Common Core-friendly approach. In Maryland, 8th grade students must be able to write inequalities to represent relationships, using a variable; appropriate relational symbols (>, ≥, <, ≤, and =); no more than three operational symbols; and rational numbers from - 1,000 to 1,000. The skill is taught as part of a unit involving writing, solving, and graphing solutions to equations and inequalities.

In the past, I've taught this unit according to the district's curriculum guide, which acknowledges the importance of real-world connections. For example, the problem the district's curriculum guide used to introduce students to writing inequalities deals with wanting to see a PG-13 movie. Students come up with a list of ages that would be allowed to attend, then think of real-world situations and questions for which multiple answers could be correct (such as, "At what ages are people allowed to drive cars?"). The difference between these practices and inquiry-based practices comes in the next step.

In my district's curriculum, teachers next guide students through an activity in which they explore an inequality word problem, such as,

To receive a discount at the Half-a-Dozen Flags Amusement Park, a group of visitors must purchase a minimum of $600 worth of full-day and half-day tickets. Twenty-six tourists in a group purchased full-day tickets at $15 each. Write and solve an inequality to calculate how many more tourists in this group would need to each buy a $9.50 half-day ticket so this group qualifies for the discount.

The teacher directs students to take three steps:

1. identify the different events,

2. label them with a variable,
and

3. fill in the values for each event.

In the inquiry-based instruction that undergirds the Common Core State Standards, a teacher would instead ask open-ended questions, as a facilitator might.

In my newly designed lesson on inequalities, I gave pairs of students a "deconstructed" question. This meant I rewrote questions found in a math textbook to include the bare minimum of information and allowed students to determine their own procedures. One textbook question read, "To go to Disney World costs $50 for each adult ticket and $30 for each child's tickets. X tickets are bought of each. If the total spent is more than $800, how many tickets were bought?"

I deconstructed this problem so it would push students to demonstrate Practice Standard 1 (make sense of the problem and persevere in solving it) without frustrating them with extra words and exact figures right off. (This adjustment especially helped my English language learners and students with individualized education programs). I gave them this question: "You're taking a trip to Disney World. How many tickets do you buy?"

Initially, students clamored about not having enough information to move forward. Once they were given the instruction of solving the problem any way they wanted as long as they could explain the process they followed to get their answer, they were on their way. Some students were practical: They bought enough tickets for their family and friends. Others bought enough tickets for the Baltimore Ravens football team or all members of their favorite boy band.

After students had presented their ideas and we discussed them as a class, I asked, "How much money would you spend to take all these people to Disney World?"

This time, students were energized and focused on the process of discovery. One group used the Internet to figure out the actual cost of a ticket. Another said they found a coupon for a 100 percent discount so didn't have to pay anything! By the time we were finished, all my students could identify what key information they would need to solve a word problem like this and what simple operations they could use. When I presented them with the original textbook question, they could make use of the structure they'd discovered and see this complicated question as a set of simple steps.

1. Multiply the cost of an adult ticket by how many adults went.

2. Multiple the cost of a child ticket by how many children went.

3. Add for your total.

4. Make sure this sum is larger than $800.

Investigating Results

As I continued changing my teaching approach throughout the 2011-12 school year, I wondered if this transition from knowledge provider to instigator of thought was benefitting students. I decided to investigate by analyzing test results. My school district requires students to take an assessment in February that mimics the end-of-year Maryland State Assessment and covers all the math skills 8th grade students are accountable for.

Overall, the performance of my students on this benchmark assessment was poor. Only 17 percent scored in the proficient or advanced range, meaning their score was at 60 percent or above. Of the remaining students, only 20 percent scored in the 50 percent range. These results seem to suggest that my inquiry-based instruction approach aligned to the Common Core's demands wasn't working.

However, an item analysis of each question told a different story. By this February benchmark, I'd only taught four inquiry-based lessons. On average, students scored better on three of the four test items that assessed skills I'd taught through inquiry than they did on items assessing skills I'd taught traditionally. For example, 75 percent of my students could recall my inquiry-based lesson on the skill of writing inequalities and selected the correct answer.

In the next few years, math teachers will discover that their roles must change if they are to help students meet these Standards for Mathematical Practice. After spending a year exploring these standards, I have three general suggestions for how to start this shift.

First, take time to read and understand the standards for mathematical practice. Identify which ones you already emphasize in your classroom and analyze how you must change your classroom so that you can strengthen students' skill with the remaining standards.

Second, join a professional learning community, which will provide you with the support necessary to make difficult transitions, commiserate about failures, and celebrate successes. If there are no opportunities through your district, take advantage of national conferences or webinars on the Common Core standards available through many associations.

Finally, realize that you have the power to create change for your students. So much of education reform is done to teachers and students. This is a change that can be done by teachers. By equipping yourself with the tools to engage, challenge, and enrich students' learning experiences, you can transform schools into laboratories in which students learn to successfully investigate, discover, and solve problems.

1. Make sense of problems and persevere in solving them.

2. Reason abstractly and quantitatively.

3. Construct viable arguments and critique the reasoning of others.

4. Model with mathematics.

5. Use appropriate tools strategically.

6. Attend to precision.

7. Look for and make use of structure.

8. Look for and express regularity in repeated reasoning.

Endnotes

Fleming, N. (1995). I'm different; not dumb: Modes of presentation (V.A.R.K.) in the tertiary classroom. Proceedings of the 1995 Annual Conference of the Higher Education and Research Development Society of Australasia, 18, 308-313.

Fried, R. L. (2001). The passionate teacher: A practical guide. Boston: Beacon Press.

Jessica Hiltabidel teaches 8th grade math at John Ruhrah Elementary Middle School in Baltimore.

# data[].attributes.attachment_link.description
December 2012/January 2013 | Volume 70 | Number 4 Common Core: Now What? Jessica Hiltabidel The most significant difference is how my role as a teacher changes. To successfully implement Common Core standards, a classroom needs to become student driven and student led.

# data[].attributes.headline
9. What teachers say about student difficulties solving mathematical world problems in grades 2-5

# data[].attributes.subject
9. What teachers say about student difficulties solving mathematical world problems in grades 2-5

# data[].attributes.body
This article describes results from a study observing students and their success rate with understanding and solving word problems.  In this study, results show that students who had the most success work the problem independently and are able to identity key words (Brunn, Lopez-Mohler, Pearce, & Skinner, 2013).  At the same time, this article also shows the influence state assessment has on the instruction for mathematical word problems.  I can relate to this topic as I feel that our Algebra 1 course is starting to change to cater to the PARCC exam and the wording of problems given on that exam.  Word problems are assigned with a new approach and students have to be able to analyze and internalize the information on their own and then relate it to real life application.

# data[].attributes.headline
7. Determining Attitudes and Anxiety of Levels of Students in Need of Protection Towards Mathematics Course

# data[].attributes.subject
7. Determining Attitudes and Anxiety of Levels of Students in Need of Protection Towards Mathematics Course

# data[].attributes.body
This article talks about the anxiety that students face in a mathematics classroom and how their lack of self-esteem brings a negative attitude to these students which interferes with their success.  In my classroom, anxiety and low self-esteem are two of the main characteristics of a lot of the students that I have.  Confidence is a key factor in the success of students and their understanding of course content.  From the data collected in the research, studies in this article show that some factors to anxiety in the classroom are gender, ethnicity, and achievement level.  This is relatable to my classes and I hope I can improve instruction in my classroom to help eliminate anxieties for my students.