Developing Students’ Algebraic Thinking in Earlier Grades, Lessons from China and Singapore

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Developing Students’ Algebraic Thinking in Earlier Grades: Lessons from China and Singapore

Jinfa Cai, Swee Fong Ng, and John C. Moyer

Abstract In this chapter, we discuss how algebraic concepts and representations are developed and introduced in the Chinese and Singaporean elementary curricula. We particularly focus on the lessons to be learned from the Chinese and Sin- gaporean practice of fostering early algebra learning, such as the one- problem- multiple-solutions approach in China and pictorial equations approach in Singapore.

Using the lessons learned from Chinese and Singaporean curricula, we discuss four issues related to the development of algebraic thinking in earlier grades: (1) To what extent should we expect students in early grades to think algebraically? (2) What level of formalism should we expect of students in the early grades? (3) How can

Authors’ note: The preparation of this paper has been supported by grants from the National Science Foundation (ESI-0114768; ESI-0454739). Any opinions expressed herein are those of the authors and do not necessarily represent the views of the National Science Foundation. This chapter is a revised version of an article published in ZDM (Cai, J., Lew, H. C., Morris, A., Moyer, J. C., Ng, S. F. & Schmittau, J. (2005). The Development of Students’ Algebraic thinking in Earlier Grades: A Cross-Cultural Comparative Perspective. Zentralblatt fuer Didaktik der Mathematik (International Journal on Mathematics Education), 37(1), 5–15, DOI

10.1007/BF02655892), where we summarized findings from analyses of Chinese, Russian, South Korean, Singaporean, and U.S. elementary curricula. In this chapter, we only focused on the Chinese and Singaporean curricula. We are grateful for the productive collaboration with Drs.

H. C. Lew, A. Morris, & J. Schmittau.

J. Cai (

⁾

Department of Mathematical Sciences, University of Delaware, Newark, USA e-mail:[email protected]

S.F. Ng

National Institute of Education, Nanyang Technological University, Singapore, Singapore e-mail:[email protected]

J.C. Moyer

Department of Mathematics, Statistics and Computer Science, Marquette University, Milwaukee, USA

e-mail:[email protected]

J. Cai, E. Knuth (eds.), Early Algebraization, Advances in Mathematics Education, DOI10.1007/978-3-642-17735-4_3, © Springer-Verlag Berlin Heidelberg 2011

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we help students make a smooth transition from arithmetic to algebraic thinking?

and (4) Are authentic applications necessary for students in early grades?

Introduction

Algebra has been characterized as the most important “gatekeeper” in mathematics.

It is widely accepted that to achieve the goal of “algebra for all,” students in elementary school should have experiences that prepare them for the more formal study of algebra in the later grades (National Council of Teachers of Mathematics [NCTM]

2000). However, curriculum developers, educational researchers, and policy makers are just beginning to explore the kinds of mathematical experiences elementary students need to prepare them for the formal study of algebra at the later grades (Britt and Irwin, this volume; Carpenter et al.2003; Carraher and Schliemann2007;

Kaput1999; Mathematical Sciences Education Board1998; NCTM2000; Schifter 1999; Stacey et al.2004).

For example, there is evidence that U.S. students are ill-prepared for the study of algebra (Silver and Kenney2001). One of the challenges teachers in the United States face is the lack of a coherent K-8 curriculum that can provide students with algebraic experiences that are both early and rich (Schmidt et al.1996). Customarily, algebra has not been treated explicitly in the school curriculum until the traditional algebra course offered in middle school or high school (NCTM2000). Moreover, according to a rigorous academic analysis by the American Association for the Ad- vancement of Sciences (AAAS2000), the majority of textbooks used for algebra in the United States have serious weaknesses. In addition, there is evidence that most elementary school teachers in the United States are not adequately prepared to integrate algebraic reasoning into their instructional practices (e.g., van Dooren et al.

2002). By comparison, it is plausible that the preparation of Chinese and Singa- porean elementary school teachers benefits from their own elementary school education, in which the formal study of algebra begins much earlier than in the United States.

Most U.S. students do not start the formal study of algebra until eighth or ninth grade, and many of them experience difficulties making the transition from arithmetic to algebra because they have little or no prior experience with the subject (Silver and Kenney2001). The unsatisfactory findings from national and international assessments (e.g., NAEP, TIMSS) indicate a need to develop U.S. students’

algebraic thinking in the early grades. In several countries (e.g., China and Singa- pore), students begin the formal study of algebra much earlier. However, it is likely that, across these settings, curriculum developers, educational researchers, and policy makers faced challenges similar to those of their counterparts in the United States and other countries as they attempted to develop early and appropriate algebraic experiences for younger children (Cai and Knuth2005; Carpenter et al.2003;

Kieran2004; Stacey et al.2004).

In this chapter, we discuss how algebraic concepts and representations are developed and introduced in the Chinese and Singaporean elementary curricula. We

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particularly focus on the lessons to be learned from the Chinese and Singaporean practice of fostering early algebra learning. Knowledge of the curricula and instructional practices of different nations can increase educators’ and teachers’ abilities to meet the challenges associated with the development of students’ algebraic thinking.

We should state at the outset that it is not our intent in this paper to evaluate the curricula or the instructional practices used in China and Singapore. Instead, our focus is on studying and understanding how the curricula (and instruction) in the two countries contribute to the development of students’ algebraic thinking. We believe that the understanding we gain by taking this international perspective will increase our ability to address the issues related to the development of students’

algebraic thinking in elementary school. We need to support students’ development of algebraic thinking in the early grades and help them appreciate the usefulness of algebraic approaches in solving various problems. Therefore, we are especially interested in the way the two curricula prepare students to make smooth transitions from informal to formal algebraic thinking.

In the sections that follow, we first present the unique features of the two curricula we studied. Then we discuss the lessons we can learn from China and Singapore regarding the development of students’ algebraic thinking in earlier grades.

Features of the Chinese and Singaporean Curricula

We analyzed the national curricula of China and Singapore to understand how the authors intended to help students develop algebraic thinking in elementary school (Curriculum Planning & Development Division2000; Division of Elemen- tary Mathematics1999). We were interested in the algebraic concepts included in the curriculum and in how these concepts are developed and represented. In particular, we analyzed each curriculum along three dimensions: (1) goal specification, (2) content coverage, and (3) process coverage (Cai2004a). Results from the analysis can be found in Cai (2004b) and Ng (2004). The focus of this chapter is to identify the unique features of Chinese and Singaporean curricula and then to illustrate how mathematics teachers and researchers can use the international perspective to tackle difficult issues related to the development of algebraic thinking in early grades.

Algebra Emphases in the Chinese and Singaporean Curricula

The four goals of the algebra standard in the Principles and Standards for School Mathematics (NCTM2000) are: Goal 1—Understand patterns, relations, and functions; Goal 2—Represent and analyze mathematical situations and structures using algebraic symbols; Goal 3—Use mathematical models to represent and understand quantitative relationships; and Goal 4—Analyze change in various contexts. We employed our previously developed case studies of the Chinese and Singaporean curricula (Cai2004b; Ng2004) to compare the emphases of each curriculum with

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these four goals. Our criteria for determining whether a curriculum emphasizes the development of a certain NCTM goal are that the curriculum must explicitly state a similar learning goal or must include extensive instructional activities clearly intended to help students achieve the goal.

There is a consistency between the curricular emphases of both the Chinese and Singaporean curricula. They both explicitly state that their main goal in teaching algebraic concepts is to deepen students’ understanding of quantitative relationships.

Furthermore, both curricula include extensive use of algebraic models that are intended to help achieve the overriding goal of deepening students’ understanding of quantitative relationships. These two features indicate that both curricula emphasize the development of goal 3 of NCTM’s algebra standard.

In fact, the algebraic emphasis in both Chinese and Singaporean elementary school mathematics is consonant with all but the fourth goal of the NCTM algebra standard. In China, the fourth goal is not addressed fully until the concept of function is formally introduced in junior high school, although qualitative analysis of change is done in elementary school. In Singapore, the fourth goal is not developed until students are in secondary school.

The Chinese Curriculum

The overarching algebra-related goal in the Chinese elementary curriculum is for students to better represent and understand quantitative relationships, numerically and symbolically. The main focus is on equations and equation solving. Variables, equations, equation solving, and function sense permeate the curriculum in grades 1 to 4. Equations and equation solving are formally introduced in the first half of grade 5. Once equation solving is introduced, it is applied to the learning of mathematical topics, such as fractions, percents, statistics, and proportional reasoning, in the second half of grade 5 and grade 6.

The term “variable” is not formally defined in Chinese elementary school mathematics. However, in the teacher’s guide for the national curriculum, teachers are reminded that variables can represent many numbers simultaneously, that they have no place value, and that representations of variables can be selected arbitrarily. In Chinese elementary school mathematics, variable ideas are used in three different ways. First, they are used as place holders for unknowns in equation solving. In grades 1–2, for example, a question mark, a picture, a word, a blanket, or a box is used to represent the unknowns in equations. Second, variables are viewed as pattern generalizers or as representatives of a range of values. Specifically, words, rather than letters, are used to represent variables in order to help 3rd grade students understand the meanings of formulas. For example, after examining several specific examples, the formulas for the areas of rectangles and squares are represented as area of a rectangle=its length ×its width and area of a square=its side×its side. However, teachers are counseled to emphasize the generalizable nature of the formulas. That is, for any rectangle, its area can be found by multiplying its length

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and width. In grades 5 and 6, letters are used to represent formulas for finding areas of squares, triangles, rectangles, trapezoids, and circles. The third use of variables is to represent relationships, such as direct proportionality (y/x=k)and inverse proportionality (xy=k).

The function concept is not formally introduced in the Chinese elementary curriculum. However, function ideas permeate the curriculum so that students’ function sense can be informally developed. According to the curriculum guide, the perva- sive use of function ideas in various content areas not only fosters students’ learning of the content topics, but also provides a solid foundation for the future learning of advanced mathematical topics in middle and high schools. In the early grades, the Chinese curriculum provides students with many opportunities to develop function sense at a concrete, intuitive level. Function sense is first introduced in the context of comparing and operating with whole numbers in grade 1 using a one-to-one map- ping. In grade 6, multiple representations (pictures, diagrams, tables, graphs, and equations) are used to represent functional relationships between two quantities.

These functional relationships are embedded in the curricular treatments of circles, statistics, and proportional reasoning.

The Chinese elementary school curriculum is intended to develop at least three thinking habits in students. The first thinking habit is to examine quantitative relationships from different perspectives. Students are consistently encouraged, and provided with opportunities, to represent a quantitative relationship in different ways.

Throughout the Chinese elementary school curriculum, there are numerous examples and problems that require students to identify quantitative relationships and represent them in multiple ways (Cai2004b). For example, in the following problem from Grade 2, the quantitative relationship involves the amount of money paid to the cashier, the change, and the cost of two batteries: Xiao Qing purchased two batteries. She gave the cashier 6 Yuans and got 4 Yuans in change back. How much does each battery cost? The teacher’s reference book recommends that teachers allow students to represent the quantitative relationship in different ways, such as the following:

The amount of money paid to the cashier−the cost of the two batteries=the change.

The cost of the two batteries+the change=the amount of money paid to the cashier.

The amount of money paid to the cashier−the change=the cost of the two batteries.

The second thinking habit is to solve a problem using both arithmetic and algebraic approaches. Expectations that students solve problems arithmetically in multiple ways and also algebraically in multiple ways can clearly be seen in the Chinese curriculum, and such expectations are common in Chinese classrooms by teachers (Cai2004c). Furthermore, students are asked to make comparisons between arithmetic and algebraic ways of representing quantitative relationships. For example, in Grade 5, teachers and students can discuss and compare different ways to solve the following problem: Liming elementary school has funds to buy 12 basketballs at 24 Yuans each. Before buying the basketballs, they decided to spend 144 Yuans of the funds for some soccer balls. How many basketballs can they buy?

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Arithmetic solutions:

Solution 1: Begin by computing the original funding and subtract the money spent on soccer balls:(24×12−144)÷24=144÷24=6 basketballs.

Solution 2: Begin by computing the number of basketballs that can no longer be bought: 12−(144÷24)=6 basketballs.

Algebraic solutions:

Solution 3: Assume that the school can still buyx basketballs:(24×12−144)= 24x.

Therefore,x=6 basketballs.

Solution 4: Assume that the school can still buyxbasketballs: 24×12=24x+144.

Solution 5: Assume that the school can still buyxbasketballs. 12=(144÷24)+x.

In the Chinese curriculum, after equation solving is introduced, students have opportunities to use an equation-solving approach to solve application problems as they learn statistics, percents, fractions, and ratios and proportions (Cai2004b).

This arrangement is built into the curriculum to deepen the students’ understanding of quantitative relationships and to help students appreciate the equation-solving approach. For example, students in Grade 6 are encouraged to use four different methods to solve the following percent problem: A factory modified its production procedures. After that, the cost of making one product was 37.40 Yuans which is 15%

lower than the cost before the production procedures were modified. What was the cost of making the same product before the production procedures were modified?

Solution 1: If the pre-modification cost is viewed as the unit 1, then the current cost is 15% less than the cost before modification.

Letx=the cost before the modification.

x−15%x=37.4 (1−15%)x=37.4 85%x=37.4 x=44

Answer: The cost before the modification was 44 Yuans.

Solution 2: If the pre-modification cost is viewed as the unit, then 37.40 is 15% less, or 1−15%=85% of the cost before the modification. Therefore, the cost before the modification is 37.4÷85%=44.

Solution 3: Similarly, the figure below can be used to represent the problem if the pre-modification cost is viewed as the unit. Since the current cost is 15% less than the cost before the modification, the current cost is 85% of the previous cost. There- fore, the cost before the modification was 37.4÷85%=44.

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| | The cost before the modification

37.40 Yuans

| | . . . ....

The current cost 15%

Solution 4: Because the 85% also represents a ratio, the ratio of costs before and after the modification is 85 to 100. It must be the same as 37.40 Yuans tox, where xis the cost before the modification.

Hence, 37.4/x=85/100, x=44.

Undoubtedly, the approach of asking students to use and compare these four different types of solutions (algebraic, arithmetic, pictorial, and ratio) is based on the principle that considering multiple perspectives can foster a deep understanding of the relationship between quantities.

According to the Chinese curriculum, solving a problem using both an arithmetic approach and an algebraic approach helps students build arithmetic and algebraic ways of thinking about problem solving. At the elementary school level, Chinese students solve problems like the examples given in this section, all of which can be solved arithmetically. As might be expected, it is common at the beginning of their transition to algebraic problem solving for students to wonder why they need to learn an equation-solving approach. However, after a period of time using both approaches, students come to see the advantages of using equations to solve these types of problems. In recent years, several researchers have discussed the notion of “algebra in arithmetic” (e.g., Britt and Irwin, this volume; Russell et al., this volume). While Chinese school mathematics does not explicitly claim this notion, the use of both arithmetic and algebraic approaches to solve problems can help show students the algebra in arithmetic. Although this practice contrasts the arithmetic and algebraic approaches, it also helps soften the boundaries between them.

There are three objectives in teaching students to solve problems both arithmetically and algebraically: (1) to help students attain an in-depth understanding of quantitative relationships by representing them both arithmetically and algebraically; (2) to guide students to discover the similarities and differences between arithmetic and algebraic approaches, so they can understand the power of a more general, algebraic approach; and (3) to develop students’ thinking skills as well as flexibility in using appropriate approaches to solve problems. Post et al. (1988) in- dicated that “first-describing-and-then-calculating” is one of the key features that make algebra different from arithmetic. Comparisons between the arithmetic and algebraic approaches can highlight this unique feature.

The third thinking habit in Chinese school mathematics is to use inverse operations to solve equations. Starting in the first grade, subtraction is defined as the inverse of addition. Although the term “solve” is not used at grade 1, students learn

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to solve equations starting at grade 1 and they continue solving equations throughout the entire curriculum. For example, students in the first grade are guided to think about the following question: “If 1+( )=3, what is the number in ( )?” In order to find the number in ( ), the subtraction 3−1=2 is introduced. Throughout the first grade, students are consistently asked to solve similar problems. In the second grade, multiplication and division with whole numbers are introduced in Chinese elementary school mathematics. Division is first introduced using equal sharing. Division is also presented as the inverse of multiplication: “What multiplied by 2=8?” That is, “If( )×2=8, what is the number in ( )?”

In addition, the Chinese curriculum emphasizes generalizing from specific examples. By examining specific examples, students are guided to create generalized expressions. Students develop this habit of mind at a variety of points in the curriculum, but especially when formulas for finding perimeters, areas, and volumes are introduced, when operational laws are presented, or when the averaging algorithm is discussed. In particular, generalizing is intertwined in the three habits of minds mentioned above.

The Singaporean Curriculum

In Singapore, some algebraic concepts are formally introduced in elementary grade six (age 12+). At this level children are taught how to construct, simplify, and evaluate algebraic expressions in one variable. The notion of letters as variables is introduced at this level. The concept of equations and other structural aspects of algebra are developed in lower secondary years (age 13+onwards). However, the Singa- porean elementary mathematics curriculum provides a wide variety of experiences to help younger children develop algebraic thinking, and this development is made possible by using “model methods” or “pictorial equations” to analyze parts and wholes, generalize and specify, and do and undo.

Solving arithmetic and algebra word problems is a key component at every level of the Singapore elementary mathematics curriculum (Curriculum Planning & De- velopment Division [CPDD]1999,2000). In 1983, the Singapore Ministry of Ed- ucation officially introduced into the elementary mathematics curriculum a heuristic involving diagram- or model-drawing. This heuristic was intended as a tool for solving arithmetic, as well as algebraic, word problems involving whole numbers, fractions, ratios and percents (Kho1987). It was believed that if students were provided with the means to visualize a word problem—be it a simple arithmetic word problem or an algebra word problem—the structural underpinning of the problem would be made overt. Once children understood the structure of the problem, they were more likely to solve it (Kho1987).

In the earlier grades, pictures of real objects are initially used to model problem situations, but then the pictures are replaced by the more abstract rectangles. For example, actual pictures of cars, and then rectangles, are used to solve the following problem in second grade: Ali has 8 toy cars and David has 6 toy cars. How many

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Fig. 1 Pictorial equation

solving Raju and Samy shared $410 between them. Raju re-

ceived $100 more than Samy. How much money did Samy receive?

2 units=$410 - $100

=$310 1 unit=$155 Samy received $ 155

toy cars do they have altogether? This pictorial approach, which becomes increas- ingly more complex as the grade level progresses is introduced in the first grade’s teacher’s guide. As students advance through the primary years, the model method is used to solve algebra problems involving unknowns, the part-whole concept and proportional reasoning. In each case, the rectangles allow students to treat unknowns as if they are knowns. This is because the unknowns are represented by unit rectangles that can be treated as if they are knowns, even though the unit represents an unknown number of objects.

Figure1 is an example from Grade 5. Samy’s rectangle or unit is the genera- tor of all the relationships presented in the problem. Raju’s rectangle is dependent upon Samy’s, with Raju’s share represented by a unit identical to Samy’s plus another rectangle representing the relational portion of $100 more. Using the model drawing, a pictorial equation representing the problem is formed, and if the letterx replaces Samy’s unit, then the algebraic equationx+x+$100=$410 is produced.

The use of the rectangle as a unit representing the unknown provides a pictorial link to the more abstract idea of letters representing unknowns. The entire structure of the model can be described as a pictorial equation.

In summary, children solve word problems using the “model method” to construct pictorial equations that represent all the information in word problems as a cohesive whole, rather than as distinct parts. To solve for the unknown, children undo the operations that are implied in the pictorial equation. This approach helps further enhance their knowledge of the properties of the four operations. The intent of using the “model method” described above is to provide a smooth transition from working with unknowns in less abstract form to the more abstract use of letters in formal algebra in secondary school.

Besides the use of pictorial equations to analyze part-whole relationships, the second big idea in the Singaporean curriculum is developed by exploring the structures in patterns. Students are provided with both numeric and geometric pattern recognition activities. Such activities require students to specify and then generalize the rule they construct to continue the pattern they see. The inclusion of many such activities in the curricular materials suggests that the Singapore primary mathematics syllabus places great importance on the thinking processes—generalizing and specifying.

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Fig. 2 Inverse operations in the second grade Singaporean curriculum

Furthermore children are provided a variety of activities that foster the development of other algebraic thinking habits—doing and undoing, building rules to represent functions, and also abstracting from computation (Driscoll1999). For example, the process of “doing-undoing” is emphasized in the Singaporean teacher’s guide, specifically when teachers first introduce the four operations. In particular, in the unit “Addition and Subtraction,” notes in the second grade teacher’s guide suggests that teachers highlight the relationship between addition and subtraction as well as multiplication and division, as evidenced by the example in Fig.2(TG2A, 1995, pp. 23–24).

The habit of mind, building rules to represent functions, as well as the notion of letters as variables, is developed using a functional approach. Through this approach, children engage in activities that help them develop the pointwise notion of function by looking for the relations exhibited in sets of ordered pairs representing the input and output of problem situations (e.g.(1,2), (2,4), (3,6), . . . , (x,2x)).

Tasks move from simple numerical activities where children are engaged in doing and undoing to using letters to generalize the operation that produces the output from a given input. As children perform these tasks, they address the question,

“What’s the rule for this pattern?” This notion is first developed through supple- mentary activities in the second grade teacher’s guide (Ng2004) designed to help students memorize the multiplication facts and solve word problems (see Fig.3).

The functional approach provides students in Grade 3 with experiences extending sequentially written number patterns, and with tasks that require them to look for number patterns presented in tables. Here students devise the rule linking numbers in one row/column with numbers in another row/column, thus continuing the informal introduction to pointwise functions begun in second grade. In the Singapore mathematics curriculum, these activities are generally presented to students within a context that is familiar to them.

The complexity of the doing and undoing process increases in Grade 3. For example, the teacher’s guide (Ng2004) challenges students to determine the input number from the output after two given operations. Add 30 to a number. Then sub- tract 35 from the sum. The answer is 27. What is the original number?

At Grade 6, a developmental approach to the informal introduction of functions (pictorial to symbolic) is used to introduce non-recursive rules and the concept of letter as variable.

Lessons from Chinese and Singaporean School Mathematics

What lessons can we learn from Chinese and Singaporean curricula about developing students’ algebraic thinking in earlier grades? In the sections that follow we

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Fig. 3 Informal introduction to functions in the second grade Singaporean curriculum

present insights from our curricular analyses. In particular, we address the following issues: the reason curricula should expect students in early grades to think algebraically, the level of formalism and generalization expected of students, the nature of support for helping students make a smooth transition from arithmetic to algebraic thinking, and the role authentic applications play in fostering algebraic thinking.

Why Should Curricula Expect Students in Early Grades to Think Algebraically?

We realize that the question of whether we should expect students in early grades to think algebraically is not an issue these days. Based on recent research on learning, there are many obvious and widely accepted reasons for maintaining this expecta- tion. However, we raise the question of why in order to offer a less obvious reason for developing algebraic ideas in the earlier grades, namely that resistance to algebra would be reduced if we could remove the misconception that arithmetic and algebra are disjointed subjects.

Although one can make an eloquent argument in favor of studying algebra at the secondary level (e.g., Usiskin1995), in reality, the need to learn algebraic ideas currently is not as universally accepted as the need to learn arithmetic, history, or writing (Usiskin1995). Even those who have taken an algebra course and have done well can live productive lives without ever using it. Therefore, many middle and high school students are not motivated to learn algebra. We believe that resistance to algebra can be more effectively addressed by helping students form algebraic habits of thinking starting at elementary school. If students and teachers routinely spent the first five or six years of elementary school simultaneously developing arithmetic and algebraic thinking (with differing emphases on both at different stages of learning), arithmetic and algebra would come to be viewed as being inextricably interconnected. We believe an important outcome would be that the study of algebra in secondary school would become a natural and non-threatening extension of the mathematics of the elementary school curriculum.

Although it is widely accepted that we should expect students in early grades to think algebraically, the real question is how can we prepare students in earlier grades

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Fig. 4 Fifth grade

Singaporean student solution Furniture Problem: At a sale, Mrs. Tan spent $530 on a table, a chair and an iron. The chair cost $60 more than the iron. The table cost $80 more than the chair. How much did the chair cost?

to think algebraically? “Although there is some agreement that algebra has a place in the elementary school curriculum, the research basis needed for integrating algebra into the early mathematics curriculum is still emerging, little known, and far from consolidated” (Carraher and Schliemann2007, p. 671). Our analyses indicate that the Chinese and Singaporean curricula could be useful references for those wishing to help elementary students develop a stronger sense of the connections between arithmetic and algebra. Specifically, the Chinese and Singaporean curricula provide concrete examples of promising ways to integrate arithmetic and algebraic ideas in the earlier grades.

Are Young Children Capable of Thinking Algebraically?

Our research clearly shows that elementary Chinese and Singaporean students are capable of using algebraic approaches to solve problems (Cai2003,2004b,2004c;

Ng and Lee2005). For example, when 151 5^th grade Singaporean students were asked to solve the “Furniture Problem,” shown in Fig.4, nearly a half of them cor- rectly used the pictorial equation approach to solve the problem.

In comparative studies involving Chinese and U.S. 6^thgrade students, we found that U.S. 6^thgrade students tended to use concrete problem-solving strategies, while Chinese students tended to use generalized problem-solving strategies involving letter symbols as generalized representatives of ranges of values (Cai and Hwang 2002).

For example, in solving the Odd Number Pattern Problem shown in Fig.5, the U.S. and Chinese students had almost identical success rates (70%) when they were asked to find the number of guests who entered on the 10^thring. However, the success rate for Chinese students (43%) was higher than that of the U.S. students (24%) when they were asked to find the ring number on which 99 guests would enter the party (χ2(1, N =253)=10.23,p < .01). This appears to be due to the fact that more Chinese than U.S. students used abstract strategies to answer the question. In- deed, fully 65% of Chinese students choosing an appropriate strategy for Question C used an abstract strategy, compared to only 11% for the U.S. sample. In contrast, the majority (75%) of U.S. students chose concrete strategies, compared to 29% of

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Sally is having a party.

The first time the doorbell rings, 1 guest enters.

The second time the doorbell rings, 3 guests enter.

The third time the doorbell rings, 5 guests enter.

The fourth time the doorbell rings, 7 guests enter.

The guests keep arriving in the same way. On the next ring a group enters that has 2 more persons than the group that entered on the previous ring.

A. How many guests will enter on the 10th ring? Explain or show how you found your answer.

B. Write a rule or describe in words how to find the number of guests that entered on each ring.

C. 99 guests entered on one of the rings. What ring was it? Explain or show how you found your answer.

Fig. 5 Odd number pattern problem

the Chinese students. Abstract strategies (e.g. solve fornif 99=2n−1) are more efficient than concrete strategies (e.g., repeatedly adding 2 until 99 is reached or making an exhaustive table or list) to answer the third question, which involves “undoing” (i.e., finding the ring number when the number of entering guests is known).

This Odd Number Pattern Problem was administered along with other tasks to some 4^th, 5^th, and 6^thSingaporean students (Cai2003). It was found that 12% of the 4^th graders, 16% of the 5^thgraders, and 37% of the 6^thgraders used abstract strategies.

The findings from these studies about Chinese and Singaporean students’ use of abstract strategies to solve problems like the Odd Number Pattern suggest that young children are capable of thinking algebraically. The findings also suggest that the Chinese and Singaporean approaches are beneficial for helping elementary students develop algebraic thinking.

How Can We Help Students to Think Arithmetically and Algebraically?

According to Kieran (2004), in the transition from arithmetic to algebra, students need to make many adjustments in the way they think, even those students who are quite proficient in arithmetic. Kieran particularly suggested the following five types of adjustments in developing an algebraic way of thinking: (1) Focus on relationships and not merely on the calculation of a numerical answer, (2) Focus on inverses of operations, not merely on the operations themselves, and on the related idea of doing/undoing, (3) Focus on both representing and solving a problem rather than on merely solving it, (4) Focus on both numbers and letters, rather than on numbers alone, (5) Refocus on the meaning of the equal sign. Helping students make a smooth transition from arithmetic to algebraic thinking is a common goal in the two Asian curricula. There are at least three ideas in the Chinese and Singaporean curricula that help students make the adjustments needed to develop algebraic ways of thinking.

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The first is related to the use of inverse operations to solve equations. For example, in Chinese elementary schools, addition and subtraction are introduced simultaneously at the first grade, and subtraction is introduced as the inverse operation of addition (Cai2004a,2004b, 2004c). The idea of equation and equation solving permeates the introduction of both subtraction and division. Similar approaches are taken in the Singaporean curricula (see Fig.2). There is no doubt that this inverse operation approach to subtraction and division can help students make two of the adjustments suggested by Kieran: (1) Focus on relationships and not merely on the calculation of a numerical answer, and (2) Focus on inverses of operations, not merely on the operations themselves, and on the related idea of doing/undoing.

The second idea is the use of pictorial equation solving in the Singaporean curriculum. Pictorial equation solving clearly can help students to focus on both rep- resenting and solving a problem rather than on merely solving it, as suggested by Kieran. Before the calculation of the numerical answer can begin, the model drawing has to make clear the relationships between the different objects. The pictorial equation solving approach also focuses on both numbers and letters, rather than on numbers alone, as well as on both representing and solving a problem rather than on merely solving it. Although rectangles, rather than letters, are used to represent variables in the pictorial equation solving approach, the model drawings show how children must learn to be flexible in their use of the rectangles, just as they do in later years when they use letters to represent variables. In a given question, some rectangles can be used to represent unknown values and others can be used to represent known parameters.

The third idea is that solving problems is done using both arithmetic and algebraic approaches in Chinese curriculum. This idea, incorporating the Chinese practice of using both arithmetic and algebraic approaches to solve problems, can help students to focus on both numbers and letters, rather than on numbers alone.

Through the comparisons of the approaches without using letters (arithmetic approach) and with letters (algebraic approach), students are able to see the role of letters in the algebraic approaches. Incorporating the Chinese practice of using both arithmetic and algebraic approaches to solve problems can also focus on relations and not merely on the calculation of numeric answers, as well as focus on both representing and solving a problem rather than on merely solving it. In fact, the two different approaches can be viewed as different ways to represent quantitative relationships.

Are Authentic Applications Necessary for Students in Early Grades?

Some researchers and educators believe that the learning of algebraic ideas should always be anchored in real-world situations that the students are familiar with (e.g., Bell1996). Some U.S. Standards-based curricula reflect this view (e.g., Senk and Thompson2003). These curricula engage U.S. students in mathematical problems

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embedded in authentic contexts. The applied problem solving activities require U.S.

students to explore contextualized problems in depth, construct strategies and approaches based on their understanding of mathematical relationships, utilize a variety of tools (e.g., manipulatives, computers, calculators), and communicate their mathematical reasoning through drawing, writing, and talking.

Others believe algebra does not have to be learned using real-world situations because the essence of algebra is not applied (Kieran1992; Usiskin1995). Rather, at its core, algebraic knowledge is an understanding of mathematical structures and relationships. So the work of algebra should be to abstract properties of operations and structures, and the goal should be to learn the abstract structures themselves, rather than to learn how the structures can be used to describe the real world.

While applications are important in both the Chinese and Singaporean curricula, the contexts of application problems are not truly authentic. In the case of the Chinese curriculum, for example, the main focus is on equations and the process of equation solving itself, rather than on the use of applications to provide insight into the equation solving process. As a result, the development of equation and equation solving ideas in the Chinese elementary mathematics curriculum is done in three interrelated stages: (1) the intuitive stage, (2) the introduction stage, and (3) the application stage. After Chinese students have been formally introduced to equations and equation solving, there are opportunities to use an equation-solving approach to solve application problems as they learn statistics, percents, fractions, and ratios and proportions (Cai2004b). This arrangement is desirable in order to deepen the students’ understanding of quantitative relationships and to help students appreciate the equation-solving approach.

Conclusion

This study analyzes and compares how algebraic concepts and representations are introduced and developed throughout the Chinese and Singaporean curricula. It provides an international perspective to the question of the kinds of algebraic experiences elementary school students should have. In particular, the study identi- fies unique features of each curriculum. Both curricula state that their main goal in teaching algebraic concepts is to deepen students’ understanding of quantitative relationships, but the emphases and approaches to helping students deepen their understanding of quantitative relationships differ.

The Chinese elementary school curriculum emphasizes the examination of quantitative relationships from various perspectives. Students are consistently encouraged and provided with opportunities to represent quantitative relationships both arithmetically and algebraically. Furthermore, students are asked to make comparisons between arithmetic and algebraic ways of representing a quantitative relationship.

In Singapore, students are provided ample opportunities to make generalizations through number pattern activities. Equations are not introduced symbolically; instead, they are introduced through pictures. Such “pictorial equations” are used ex- tensively to represent quantitative relationships. The “pictorial equations” not only

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provide a tool for students to solve mathematical problems, but they also provide a means for developing students’ algebraic ideas.

In this chapter, we addressed four questions related to the development of algebraic thinking in earlier grades: (1) Why should curricula expect students in early grades to think algebraically? (2) Are young children capable of thinking algebraically? (3) How can we help students make a smooth transition from arithmetic to algebraic thinking? (4) Are authentic applications necessary for students in early grades? We found that both Chinese and Singaporean students in elementary school are expected to think algebraically. In fact, students in earlier grades are capable of thinking algebraically to solve problems. The earlier emphasis on algebraic ideas may indeed help students develop arithmetic and algebraic ways of thinking about problems.

Regarding the use of authentic applications in elementary school, both Chinese and Singaporean curricula include many application problems, but the contexts are not necessarily authentic. The Chinese elementary curriculum uses formal algebraic symbolism, but the Singaporean elementary curriculum does not.

It is important to indicate that any curriculum has a complex relationship to what actually occurs in classrooms. In this paper, the focus of our discussion has been on the intended treatment of algebraic ideas in the Chinese and Singaporean curricula.

Nevertheless, the Chinese and Singaporean curricula may serve as concrete examples of what can be implemented when we try to address the issues related to the development of students’ algebraic thinking in earlier grades.

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