**Effect of Schema Use in Solving Word Problems: Emphasis on Linguistic Difficulties **

**By Namirah Fatmanissa (Sampoerna University)**

A word problem is a problem put in the context of daily life, differentiating it from other types of problem.

The context in word problems demands students to read, understand, and use their mathematical understanding.

A word problem is not only written in the form of a mathematical expression but also is combined with daily language or picture. It needs complex solving steps i.e. reading, understanding, transforming into a mathematical model, processing mathematical model, interpreting the result to the context, and check the result.

This will need students to make more effort compared to non-word problems.

Students’ ability in solving math word problems has been considered low, especially caused by difficulties in understanding information provided. Schema can be a helpful tool for students to face such difficulties. The aim of this article is to discuss the effectiveness of schema in resolving difficulties based on linguistic aspects constructing of word problems (multiple representation systems, vocabulary, and grammar, and syntax).

In Indonesia, several studies showed students' low performance in solving word problems. Moreover, a survey conducted by the Center for Development and Empowerment of Mathematics Teachers and Educational staff (PPPPTK) stated that more than 50% of Indonesian teachers complained about students' difficulties in solving word problems. Adding to that, word problems were often found in National Examination.

Responding to this issue, several studies had been conducted in Indonesia. Some studies focused on students’ error analysis in solving word problems. Interestingly, these studies found that linguistic difficulties; such as the ability to understand vocabulary, translate daily vocabulary to mathematical expressions, and understand symbols; were predominantly found. These studies did not specifically discuss the linguistic difficulties although they were often found. Other studies focused on experiments in using particular strategies to help students improve their performance to solve word problems. For example, the use of Polya’s steps, think-talk-write, and problem-posing.

To better understand the linguistic difficulties of the word problems, it is important to understand the linguistic aspects constructing them. Word problem is constructed by three main aspects i.e. multiple representation system, vocabulary, and grammar and syntax. These aspects become the character of word problems from a language perspective. Difficulty due to multiple representation systems refers to the difficulty to make meaning among the three systems: daily language, symbols, and visual representation. Difficulty due to vocabulary refers to the difficulty understanding the meaning of vocabulary despite its location in the sentence. Difficulty due to grammar and syntax is the difficulty to make meaning of the whole sentence because of the structure of words in it or how the sentence is constructed.

Linguistic difficulty in solving word problems was not only found in basic mathematics topics, but also in high school topics like derivatives. This study found that students faced difficulties during word problem solving of derivatives in diverse cases involving transforming meaning among visual representation, symbol, and language.

In accordance with that, it is imperative to investigate specific strategies but focusing on how to help students facing linguistic difficulties in solving word problems. Schema-based strategy instruction is known to benefit students in exercising their ability to understand and apply the linguistic aspects of word problems. Schema is a framework, diagram, or plan that is used to organize information. Schema can be in the form of a diagram to organize the information given in word problems visually or in the form of parts and mathematical expressions. The aim of this research is to analyze the effectiveness of schema in resolving difficulties based on linguistic aspects constructing word problems, especially in the derivative topics.

Data was collected through a mixed-method study with an embedded quasi-experimental model to test the effectiveness of schema use. The quantitative study was conducted using a pretest-posttest comparison group design, while the qualitative analysis was done to gain information on how students solve word problems in both groups. Groups being compared are the treatment group that was taught to use schema and the control group that was not.

Participants were 69 students of 11th grade; 33 belonged to the treatment group, while the rest belonged to the control group. The homogeneity of both groups was tested through a t-test for comparing means of pretest score obtained before the treatment ( 𝑝 = 0.775 ). Besides, to support whether the socio-linguistic backgrounds of students in both groups were similar, a survey of students’ daily languages and ethnicity was conducted and showed no difference in both groups' backgrounds.

The treatment class was taught to use schema to solve word problems, but the control class was not. Both classes were given pretest before the teaching periods and posttest after it. The topic for both classes was derivative. The teaching design for both classes was different in several ways. The teaching design for the treatment group was in line with the schema process based on Powell (Powell, 2011) and Boonen et al, while the control class was taught to solve word problems in derivative by reading, making a plan, determining a solution, and interpreting without being guided to utilize schema in that process.

Pretest and posttest were given in the form of six essay questions. The test items were constructed with the goal to check whether the aspects of linguistic difficulties appeared. The test items readability and clarity were consulted by two mathematics education experts. The student’s work was analyzed by first giving code(s) of an aspect of linguistic difficulties which appeared. The codes were “MRS” (for multiple representation system), “V” (for vocabulary), and “GS” (for grammar and syntax). The codes were recorded, counted, and students’ works were classified based on it. After being classified, distinct cases of difficulty in each code were further explored through interviews.

The effect of schema teaching was analyzed by checking whether the teaching of using schema helped students to face their linguistic difficulties based on each aspect (i.e. multiple representation system, vocabulary, and grammar and syntax). This was done by comparing the number of difficulties between the treatment and control group found in the posttest. Given that the pre-treatment equivalency test showed no significant difference between the two groups (𝑝 = 0.775), showing the same ‘starting point’ between groups, analyzing this through posttest was adequate. The quantitative analysis was done by having statistics tests and calculating effect size. The data being used as the number of difficulties found in each aspect on pretest and posttest.

Between groups, there was a significant difference of difficulties based on each aspect. Treatment group showed significantly less difficulties in multiple representation system (𝑝 = 0.007), vocabulary ( 𝑝 = 0.048) , and grammar and syntax (𝑝 = 0.000) aspect. This showed that schema teaching was helpful for students in facing linguistic difficulties during word problem-solving. The effect sizes for all aspects showed moderate (for MRS and vocabulary aspect) and strong effect (for grammar and syntax aspect). It can be inferred that the schema teaching helped students on facing linguistic difficulties, especially due to the complexity of grammar and syntax of the word problem.

In general, the schema is proven to be effective in helping students solve word problems. Schema teaching is also effective in resolving mathematical linguistic difficulties during word problem-solving in all aspects of linguistic difficulties. Adding to that, students being taught schema showed more attention and application upon the whole sentence or information provided by the problem and were more self-directed in determining solving steps towards problem goal.

This study does not claim that schema teaching will benefit all types of students on all mathematics topics due to its participants' selection and mathematics topic limitation. Responding to that, it is hoped that in the future, studies on word problems, linguistic difficulties, and schema teaching will be more expanded, for example by investigating more mathematics topics or different types of students.

**Namirah Fatmanissa (Sampoerna University)**

Namirah Fatmanissa is an experienced mathematic educator. She is currently a mathematics lecturer at Faculty of Education, Sampoerna University. Namirah has taught Problem Solving and Project-Based Learning, Calculus and Analytical Geometry 2, and Real Analysis. Namirah graduated from Sampoerna School of Education (SSE) with a degree in Mathematics Education. She earned a Master of Education from Universitas Pendidikan Indonesia in 2018. Namirah’s field of interest is mostly on contextual teaching in Mathematics Education.

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