Using the "Identifying a Pattern" Strategy to Solve Mathematical Word Problems of Proportional Quantities at Grade 5 – Vietnam

Ratios and proportions that describe relationships between quantities are the foundation for students to understand and develop many concepts and skills in mathemat ics. Therefore, they play an important role in students’ learn ing Mathematics. This paper will present the results of the comparison of mathematical word problems relating to proportional quantities in Mathematics textbooks of Vietnam and USA from the perspective of pattern identification. It also indicates that the design and implementation of some activities to teach the word problems in which the analysis to identify patterns is a necessary strategy. The study results revealed that using “identifying a pattern” problems is a useful tool to promote problem solving competency for elementary school students in Vietnam.


Introduction
In a study about students' difficult ies in solving mathematical word problems fro m their teachers' perspectives, Seifi et al. (2012) showed that students in elementary grades had a weak foundation because their teachers taught them inappropriate strategies. In order to overcome students' difficult ies, most of the teachers suggested helping students in teaching them to look for a pattern. New Jersey Mathematics Curriculu m Framework (1997) emphasizes that every Mathemat ics teacher needs to assist students in recognizing, generalizing, and using patterns that exist in numbers, in shapes, and in the world around them.
It can be seen that identifying patterns in solving problems is considered as a helpful strategy. Patterns often appear in Primary Mathematics textbooks of America and Singapore with a variety of act ivities, such as identifying rules, extending patterns and solving word problems with patterns. However, in Vietnamese primary textbooks, the authors have not seen solving problems , where looking for a pattern is a strategy.
According to the trend of international integration, Vietnamese education is imp lementing a curriculu m, which transforms content-based instruction to competency-based education. Therefore, enhancing problem solving competency for primary students becomes an essential element in teaching and learning goals of "General Education Program: Mathematics" in Vietnam. In order to solve problems effect ively, elementary students need to be provided with d iscovery tools, including the strategy of looking for a pattern.
In this article, fro m analy zing using "identifying a pattern" in solving word problems relating to p roportional quantities in textbooks and implementing an experiment with Grade 5 students, the authors believe that using patterns could bring more fruitfu l for students in solving mathematical word problems.

Problems of Identifying a Pattern
A mathematical pattern can be described as any predictable regularity, usually involving nu merical, spatial or logical relationships. In every pattern, the various elements are organized in some regular fashion. Once a pattern is established, it is easy to predict what happens next because a pattern can be extended or continued after it 106 Using the "Identifying a Pattern" Strategy to Solve M athematical Word Problems of Proportional Quantities at Grade 5 -Vietnam has been identified (Mulligan, 2009). According to Warren, "The power of mathematics lies in relations and transformations which rise to patterns and generalizations. Abstracting patterns are the basics of structural knowledge and the goal of mathematics learn ing" (cited by Mulligan, 2009). Because the patterns can be found everywhere -in nature, nu mbers, and in shape, the strategy of looking for a pattern is one of the most frequently used strategies in solving mathematics problems.
Pattern-based thinking, using patterns to analyze and solve problems, is a powerful tool for doing mathematics in primary schools level. It is suitable for describing relationships and making the foundation for fu rther work on with algebraic functions in higher grades. In grade 5-6 levels, the key components of pattern-based thinking are exploring, analyzing, and generalizing patterns, viewing rules and input/output situations as functions (New Jersey Mathematics Curriculu m Framework, 1997, p.345). In elementary schools, the strategy of looking fo r a pattern is an extension of Drawing a Tab le and Creating an Organized List. These are two strategies used in solving mathematics problems. Loc (2016) proposed the following model of identify ing a pattern (see Figure 1).  (Loc, 2016) In the problem of "identifying a pattern", students need to find a rule fro m the connection of the data with the given informat ion. Once the pattern has been identified, students can predict what will happen next to find the solution.
To solve problems based on identifying a pattern, students have two indispensable requirements, including (1) determining quantities in the problem and the relationship between them, (2) using visual representations, words, symbols or numbers to describe the rule.
When students are faced with non-routine problems that have no standard method to solve, they usually give up easily because they do not know how to get started. The ability to discover and analyze patterns becomes a critical tool to help them be oriented. At this t ime, the problem of "identifying a pattern" is a part of the process of solving non-routine problems.
The process of solving a problem using the "identify ing a pattern" strategy can be performed as follows:  List the given information and identify the required information.  Make an organized list or create a table.  Determine the rule through analyzing data (the process of "identifying a pattern").  Use the pattern to find the missing information and direct to the correct solution to the problem.

A Comparison between Textbooks of Vietnam and America on Word Problems Relating to Proportional Quantities from the Perspective of Pattern Identification
In this part, the authors present results of the comparison between Go Math Grade 5 & Grade 6 (A merican textbooks) and Toan 5 (Vietnamese textbook) about problems of directly proportional quantities (name in Vietnamese).
In American mathemat ics curriculu m, fro m Kindergarten to h igh school, looking for a pattern is considered one of the strategies that are needed to equip students in solving mathematical problems in general and word problems in part icular. The level of the crit ical components of pattern-based thinking is identified to be more and mo re co mp licated at the higher grades. Students develop higher-level thinking throughout their work with patterns in many types of problems. According to NCTM (2000), in grades 5-6, students express understanding of patterns, relations, and functions as follows: (1) Describe, extend, and make generalizations about geometric and numeric patterns; (2) Represent and analyze patterns and functional relationships using words, tables, and graphs.
In Go Math Grade 5, the numerical pattern is part of the standards of computational and algebraic thinking. Identifying a pattern is one of the strategies used to solve word problems of proportional quantities. The example is illustrated in the figure below (see Figure 2). In the above problem, to assist students in finding answers, a table o f nu merical data representing relationships is proposed. In the table, there are 3 given values and other values adding to make 3 regular increasing sequences. There are two rules: (1) the number of extra lives is increased by 3, and the number of gold coins is increased by 6 after a level; (2) the number of gold coins is twice as great as the number of extra lives in any level. The rules are described by words and symbols.
It can be seen that the common way to identify rules in a pattern according to Go Math Grade 5 is finding the difference between two consecutive numbers and finding out whether the numbers have been mult iplied or d ivided by any given number. At the grade level, the concept of proportions or ratios does not appear in describ ing patterns.
Go Math Grade 6 continues to solving this type of problem with describing the pattern relating to equivalent ratios (see Figure 3). The problem asks to find the amount of gas used to travel 48 miles if using 2 gallons can travel 12 miles. In the table, students have to find many numbers of a sequence whose rule is hidden. The numbers in colu mns 2 and 3 are intended to assist students in exploring the pattern faster. By addition to 2, students can find the missing number of gas used, but the answer is based on equivalent ratios In general, according to Go Math, the way to assist students in describing patterns is using a data table to represent terms of quantities. Fro m the given values, students implement regular additions to consecutive terms of each quantity by the same number. Based on this rule, the relationship of mult iple or ratio between t wo quantities is found out.
In order to solve the problem of proportional quantities, while Go Math focuses on using a pattern, Toan 5 presents two methods of verbally solving as follows:  Method of rate mentions a rate of t wo values of the same quantity. For examp le, because 3 hours is three times as much as 1 hour, if travelling 4 kilo meters is in 1 hour then travelling 12 kilo meters is in 3 hours (in this case, 3 is a factor).  Method of reducing to a unit aims to find the value of 1 unit. For examp le, if we travel 12 kilo meters in 3 hours then in 1 hour (1 unit of t ime) we travel 4 kilometres, so in 5 hours we travel 20 kilometres.
When the value of 1 unit of a certain quantity is decimal, the method of reducing to the unit is not used because the problem o f proportional quantities is learned earlier than the concept of decimal nu mbers. Besides, if the value of the quantity is not mu ltiple of other quantity, students do not solve problems by the method of rate.
Toan 5 also illustrates two quantities that are directly proportional in the following data table in Figure 4.

108
Using the "Identifying a Pattern" Strategy to Solve M athematical Word Problems of Proportional Quantities at Grade 5 -Vietnam In describing the relationship, the textbook does not explain the principle of regular addition by the same number to each term of quantity, nor does it mention the ratio or mult iple between time and distance. The remark orients students towards two methods of verbal solving discussed above.

Comments
Through studying Go Math (Grades 5-6) and Toan 5, we can see some differences in solving problems of proportional quantities as follows (see Table 1):  (1) The relationship between the quantities in a ratio is multiplication in nature (not addition); (2) Although one of the numbers in the ratio is not a factor (or mu lt iple) of the other number, a unit rate of quantity is always found; (3) Equivalent ratios are not necessary integral mu ltiples of another ratio.
In Toan 5, although there is only one table illustrating the proportional relat ionship, it clearly describes a numerical pattern. Most of the Vietnamese students are taught to solve problems of proportional quantities by either the method of rate or the method of reducing to a unit.
However, if the value of 1 unit is a decimal or there are no divisible relationships between the given values, both two these methods cannot be an option for students. Furthermore, it is not always clearly visible to recognize mu ltip le or div isible relationships between the given values if they are large nu mbers. As a result, students make errors in calculating, even give up solving problems.

The Research Questions
Fro m the comments above, three research questions are posed as follows 1) In Vietnam, for solving problems of p roportional quantities, how can the strategy of using the problem "identifying a pattern" be taught to fifth graders? 2) Are students interested in using the strategy of "identifying a pattern"? 3) Is using the strategy of "identifying a pattern" effective to proportional quantities problem?

Learning Activities
To answer the three research questions above, we designed a study consisting of three learning act ivities in order to teach solving word problems of direct ly proportional quantities at Grade 5 by using the problem of "identifying a pattern" (see Appendix)

Participants
Experimental subjects were 30 fifth graders in primary schools in Bac Lieu city. These students have not still learned the directly proportional quantities according to Math 5 curriculu m. The study was carried out in October 2019.

Teaching Activities
In our study, there are three major activit ies carried out in 2 periods (70 minutes).
Because Grade 5 students have not learned about patterns, the first activity aims to introduce patterns, describe their rule in words, and extend them. Students approach the concept of the pattern via working out the problems in three given tables. In table 1, students identify a pattern relating to finding the relationship of map distance to the real distance, wh ich was taught in Grade 4. Map distance and real distance have a directly proportional relationship. Table 2 is for t wo quantities that are not proportional, and students discover relat ionships between work t ime and the money earned (+1 in the terms of work time, + 2 in the terms of money earned). Table 3 requires students to recognize the proportional relationship between the length and the width of the T-shirt. Another task in Activity 1 is identifying the equivalent ratios of quantities in each table for students to understand the ratio clearly. Through these illustrations of patterns, it is expected that the ratio of two d irectly proportional quantities is recognized.
The second activity aims to instruct students in the process of solving word problems by creating tables and using the "identifying a pattern" strategy. The problem designed has three tasks in relation to the proportional relationship between the nu mber of books and the amount spent. In tasks (1) and (2), students can use the method of rate or the method of reducing to a unit to find answers. For task (3), referring to the table with some given numbers describing the rule in words, students find all the missing numbers, and then find the answer to the task (2).
The third activity is for students to practice themselves on solving two problems. In Problem 1, two known methods are no longer effective; question b directs students towards finding smaller values and the unit rate. Problem 2 is more d ifficu lt because students need to find out two quantities, create a table, and d iscover relat ionships between them to get the answer.
After Activity 3, the teacher asks students to propose the general process used in solving word problems by using the "identifying a pattern" strategy. Finally, students are interviewed by asking the two following questions: 1) Do you like solving word p roblems with using the strategy of "identifying a pattern"? Why or why not? 2) Are there any difficult ies you faced in solving word problems with using the "identifying a pattern" strategy?

Results
In Activity 1, students can quickly recognize rules and fin ish finding missing numbers. For Table 1, students discovered 3 types of pattern: Answer 1: "The real d istance is always twice as long as the map distance" (17/30 students).
Answer 2: "When the map distance is increased by 3, the real distance is increased by 6" (10/30 students).
Answer 3: "The measure of map distance and real distance is increased by the same factor" (3/30 students).
Similarly, in Table 2, all students describe the rule of "the amount of earned money is increased by 2 after the number of time work is increased by 1". To find the amount of earned money for 10 hours, 10/30 students implemented mult iplying the amount of earned money for 5 hours by 2 because they base on "10 is a double of 5". Students' wrong answers indicate that they have not understood deeply about proportional quantities. The others who found the amount of earned money for 6, 7, 8, 9 hours before 10 hours have a true answer.
Students describe the rule in Table 3 as fo llo ws: "the length of each size is increased by 3 while the width is increased by 2". It is clearly shown that students did not pay attention to the ratio of the length to the width. In Activity 2, students prioritize the method of reducing to a unit because they are familiar with this method in solving problems in p revious grades. The first solving step is finding the price of a book wh ich is 7000 VND, then mu ltip lying the number of books by 7000 to find the amount of money buying 24 books, buying 9 books. So me students have difficu lty in finding the nu mber of books corresponding to 126,000 VND due to the division of 126,000 by 7,000. There were no students who chose "24 is twice 12" to find the amount of money for purchasing 24 books by multip lying the amount of money purchasing for 12 books by 2. It means that the rate of the numbers of books is not chosen the first time. However, after working with the table in task 3, students get the answer by finding the amount of money for buying 9 books based on 3 books, 24 books based on 12 books. Although the rule in the table described as "the amount of spent money is always equal to the number of notebooks multip lied by 7000" or "the number o f money is 7000 t imes the nu mber o f books", students found missing numbers based on the mu ltip licat ion relationship between terms of the same quantity.
In Problem 1 of Activ ity 3, based on "every 1500 people will increase by 30 people", students easily find out "every 500 people will increase by 10 people" and "every 50 people will increase by 1 person". Fro m that, they fill in the table of numbers according to the rule of increasing and recognize "every 5000 people will increase by 100 people". Students also discovered that the ratio of increasing population in the town is 1 out of 50.
In Problem 2 (Activity 3), students have difficu lty in determining the variables and are confused about how to present the solution. After creating the table, some students find the answer but do not know how to present the solving process.
To sum up, almost students perform requirements well in Activities 1 and 2 like finding out the missing numbers and describing patterns. However, there are still some errors in words and meanings, such as "difference between the number of notebooks is 7,000 VND" and "when the amount of time work increases, the amount of money also increases". Students have some d ifficu lties in deciding variables and presenting the solving process.

Discussions and Conclusions
After doing three learning activities with patterns, students find out the general process to solve problems by using pattern identification. It can be concluded that using patterns helps students realize the constant ratio relationship of two proportional quantities that the previous methods do not. Moreover, extending patterns overcome 110 Using the "Identifying a Pattern" Strategy to Solve M athematical Word Problems of Proportional Quantities at Grade 5 -Vietnam some limitations on finding multiples as well as factors.
According to the results of the interview, 30/30 students stated that solving problems by using the strategy of "identifying a pattern" is more co mfortable than the method of reducing to a unit and present fewer errors in performing numerical operations because the numbers in the table have clear rules. They agree that pattern identification is an effective strategy for solving word problems. However, students also state that it is difficult to use words to describe patterns and present the solving process in non-routine problems. According to the common presentation in elementary school, each solution for a problem must always include the solution sentences and formula of numerical operations.
The positive results above indicate that Vietnamese students should be approached using the method of "identifying a pattern" in solving word problems as an effective strategy. Besides, it is essential to introduce both patterns of proportional and non-proportional quantities to encourage students to identify many types of patterns.
In Vietnam primary mathemat ical syllabus, before grade 5, students are acquainted with the strategy of "identifying a pattern" relating to numbers, shapes, forming tables of addition and mult iplication, but this is not exploited in solving word problems yet. Fro m the experimental and survey results, we think that the teaching process designed above is relatively appropriate.
In conclusion, to develop students' problem solving competency through problem solving, it is very necessary to provide them strategies and tools, including the method of "identifying a pattern".