Grade 3 students’ mathematization through modeling: Situation models and solution models with...

Gms

Aa

b

a

A

KMEWRPM

bHScmS3s

1

1

caIe

0d

Journal of Mathematical Behavior 31 (2012) 15– 28

Contents lists available at ScienceDirect

The Journal of Mathematical Behavior

journa l h o me pag e: ww w.elsev ier .com/ locate / jmathb

rade 3 students’ mathematization through modeling: Situationodels and solution models with mutli-digit subtraction problem

olving

ki Murataa,∗, Sailaja Kattubadib

Stanford University School of Education, 520 Galvez Mall, Stanford, CA 94305, United StatesAashayein Foundation, 3056 25th street, Port Arthur, TX 77642, United States

r t i c l e i n f o

vailable online 26 October 2011

eywords:athematics education

lementary educationhole number

epresentationsroblem solvingathematization

a b s t r a c t

In considering mathematics problem solving as a model-eliciting activity (Lesh & Doerr,2003; Lesh & Harel, 2003; Lesh & Zawojewski, 2008), it is important to know what stu-dents are modeling for the problems: situations or solutions. This study investigated Grade3 students’ mathematization process by examining how they modeled different types ofmulti-digit subtraction situation problems. Students’ modeling processes differed from oneproblem type to another due to their prior experiences and the complexity of the prob-lems. This study showed that students make their own distinctions between solution andsituation models in their mathematization process. Mathematics curricula and teachingshould consider these distinctions to carefully facilitate different model development ofand support student understanding of a content topic.

© 2011 Elsevier Inc. All rights reserved.

Problem solving has been considered important in U.S. mathematics curricula and teaching. However, we have longeen debating and disagreeing about the best way to teach it to our students (e.g., see Charles, 1987; Franke & Casey, 1997;iebert et al., 1996; Kilpatrick, Martin, & Shifter, 2003; National Council of Teachers of Mathematics, 1989, 2000; Pape, 2004;choenfeld, 1992). If problem solving is to serve as a central mathematical experience rather than as word problems thatome at the end of textbook chapters, there is a strong need to carefully examine how students interpret problem situations,odel them, and connect them to appropriate mathematical solutions. Common Core State Standards (Council of Chief

tate School Officers, 2010) also emphasizes on modeling in student problem solving processes. By examining how Grade students modeled three different types of subtraction situation problems (Separate, Compare, and Part-Part-Whole), thetudy aims to understand the mathematization processes as students come to see mathematics in different situations.

. Perspectives

.1. Problem solving

Problem solving is at the heart of mathematics. Over the past decades, many educators have debated how to make it more
entral to students’ mathematical experiences (e.g., see Charles, 1987; Franke & Casey, 1997; Hiebert et al., 1996; Kilpatrick etl., 2003; Lesh & Zawojewski, 2008; National Council of Teachers of Mathematics, 1989, 2000; Pape, 2004; Schoenfeld, 1992).n his comprehensive literature review, Schoenfeld (1992) summarizes the history of problem solving in U.S. mathematicsducation, how it was considered and treated in the past decades, and describes different views, approaches, and definitions
∗ Corresponding author. Tel.: +1 650 723 2832; fax: +1 650 723 7578.E-mail addresses: [email protected] (A. Murata), [email protected] (S. Kattubadi).

732-3123/$ – see front matter © 2011 Elsevier Inc. All rights reserved.oi:10.1016/j.jmathb.2011.07.004

dx.doi.org/10.1016/j.jmathb.2011.07.004

http://www.sciencedirect.com/science/journal/07323123

http://www.elsevier.com/locate/jmathb

mailto:[email protected]

mailto:[email protected]

dx.doi.org/10.1016/j.jmathb.2011.07.004

16 A. Murata, S. Kattubadi / Journal of Mathematical Behavior 31 (2012) 15– 28

that guided the turbulent journey. Philosophical and pedagogical differences, however well intentioned, made our attentionswing between extremes (from the back-to-basics to the purely exploratory teaching approaches), and students have sufferedfrom the lack of consistency (Boaler, 2008). Problem solving today remains one of the central foci of the mathematicscurriculum without a shared view of what it is (Anderson, 2002; Hiebert et al., 1996; Schoenfeld, 1992).

Schroeder and Lester (1989) explain that there are three different approaches to teaching problem solving: (1) Teaching forproblem solving, in which content is taught first and then used in problem solving later, (2) Teaching about problem solving,in which heuristic strategies are taught for solving particular problems, and (3) Teaching through problem solving, in whichcontent is taught through non-routine problems. Considering how students need different experiences in developing fluencywith any mathematical processes, all three approaches can play roles in curriculum and teaching. The traditional instructionalapproach in the United States, however, has focused primarily on “teaching for problem solving,” where problem solving hasa focused end, and the solution strategy is treated in an algorithmic way to reach the correct answer (Hiebert et al., 1996;Schoenfeld, 1992). This is vastly different from the way problem solving is defined as a part of the vision of mathematicsclassrooms presented by the National Council of Teachers of Mathematics (NCTM, 2000):

Problem solving means engaging in a task for which the solution method is not known in advance. In order to find a solution,students must draw on their knowledge, and through this process, they will often develop new mathematical understanding.Solving problems is not only a goal of learning mathematics but also a major means of doing so . . . students should acquireways of thinking, habits of persistence, and curiosity, and confidence in unfamiliar situations that will serve them welloutside the mathematics classroom (p. 52).

Well-designed problems can invite students to explore their world mathematically, make sense of how it works, and helpbuild new mathematical understanding in the process. Problem situations are critical as they are designed to elicit students’existing knowledge and to connect it with the new learning (Hershkowitz, Schwarz, and Dreyfus, 2001; Nunokawa, 2005).

1.2. Modeling and problem solving: mathematizing processes

Lesh and colleagues discuss problem solving as a model-eliciting activity (Lesh & Doerr, 2003; Lesh & Harel, 2003; Lesh& Zawojewski, 2008), and argue that unlike traditional word problems requiring students to produce short answers toquestions about pre-mathematized situations, students make symbolic descriptions of situations (models) with problems,which are critical components providing conceptual tools to the process. This may involve development of a series of modelswhere students’ ideas are tested and revised accordingly as they come to focus more on structural and systemic aspects ofthe problems. This is also a process of mathematization. Mathematization occurs when one considers a real-life situation,generates a mathematical problem out of it, and uses mathematics to solve the problems (Freudenthal, 1978, 1991). Afterreflecting and constructing a model of the situation mathematically, the problem can be solved using mathematical tools andprocesses (e.g., numbers and formulas). The mathematizing process can be highly personal and individual at the beginning,as different students are likely to see different aspects of the situation as important (mathematically or not); however, overtime, students can come to focus on common mathematical objects in a problem.

In learning to solve problems with a new mathematical content, students may gradually gain competence over differentproblems that constitute the content topic. Nunokawa (2005) explains that students typically categorize problems andindicate the most appropriate approaches for that class, thus for any given content topic, a number of models may benecessary for successful problem solving. Different models are developed separately and locally, and they then graduallyextend to a larger class of problems as they become more sharable, transportable, and usable. Thus, developing specificmodels for different problem types may be the fist step for problem-solving competence.

Different models may also assist students’ problem solving processes differently. Researchers previously discussed thedifferences between models of problem solving (focusing on a modeled situation) and models for problem solving (focusingon quantitative relations in the situation; Gravemeijer, 1994, 1999; Klein, Breshuizen, & Treffers, 1998). As students reviseand work with models, they typically move from using models referentially to situations to using them more abstractly.That is, a problem situation often includes details that may not necessarily be mathematically important, and the initialrepresentation (model) of the situation can include information that is more situational than mathematical. Latter modelscan focus more on quantitative information that is essential for solving the problem.

Fuson (1992) discusses the tension between representing problem situations and representing solution procedures, astwo modes that do not always support one another. Because students can lack problem-type schemata for situations (withunspecified quantities to assign quantities of the given problem), many consider numbers in the problem to be centralin interpreting the problem, to the extent that they only see numbers in the problem and manipulate them to find theanswer without considering the problem setting (Carpenter, Hubert, & Moser, 1983; Fuson & Willis, 1988; Ishida, 1998;Lesh & Harel, 2003; Thompson & Hendrickson, 1983; Willis and Fuson, 1988). The heavy focus on quick solutions can guidestudents’ attention away from the problem contexts, which are critical in solving the problems. In classrooms, the distinctionbetween representing problem situations and representing mathematical solutions is often not discussed, and students (and
teachers) often use one but not the other, successfully or unsuccessfully. Both teaching and research will benefit if these twoformats are clearly differentiated.
In mathematics teaching, the focus is often placed on how to teach particular content and procedures at each level ofcurriculum, and representations of mathematical concepts are put aside for the sake of instruction of the procedures. Duval

A. Murata, S. Kattubadi / Journal of Mathematical Behavior 31 (2012) 15– 28 17

(matctpt

CsRar(

1

atfi

o

1

iGllibraus

ap

Fig. 1. Conceptual framework.

2006) discusses how semiotic representations must necessarily be used for any mathematical activity that encompassesathematical concepts, because mathematical concepts can only be accessed through representations. Ng and Lee (2009)

lso discuss that while students may use the model method in an algorithmic way, the model is a problem-solving heuristichat requires students to reflect on how they could accurately represent the information in the problem. The researchersall this “the art” of representation, and this process needs to be taught to students initially, as it becomes a tool for studentso solve mathematics problems later. Nunokawa (2000) also discusses how heuristic strategies should be taught as ways torobe problem situations rather than as templates of solutions. As students learn the suitability of a model for a problemype with experiences, they come to intuit mathematical objects schematically and abstractly (Wittmann, 1981).

In order to support students’ problem solving, their modeling processes need to be understood and facilitated. Commonore State Standards (Council of Chief State School Officers, 2010) emphasizes the importance of modeling in problemolving across grade levels, and how different quantitatives, representations and tools may be used to reason mathematically.esearch has so far fallen short in capturing such a process, as it has not acknowledged the essential role representationsnd models play in problem solving (Cai & Lester, 2005). Models are systems that students develop and use, instead of mereeferences to look at, and the models become tools to predict, describe, and explain their subsequent mathematical thinkingLesh & Zawojewski, 2008).

.3. Conceptual framework

Taking the ideas from the literature reviewed above, we considered mathematization as shown in Fig. 1. In mathematizing real-life situation, students model the situation mathematically, and the process requires them to extract information fromhe situation. Initially, students may focus on information that is specific to the situation but not necessarily mathematicalor the solution process (model of the situation). With some experience, they will come to zoom in on important quantitativenformation, to be used for problem solving (model for the situation).

By examining this mathematization process through students’ models, we investigate how students come to make sensef quantitative information in problem situations. It will help us understand how to guide their problem solving experiences.

.4. Number-line representations

Number-line representations are central in the study described in this paper. Number line is a common representationn mathematics curricula. In the Netherlands, open number lines have been used widely since 1990 (Beishuizen, 1993;ravemeijer, 1994; Treffers & De Moor, 1990), in the framework of Realistic Mathematics Education (RME). Open number

ine means that the number line is not marked with units, but the user of the representation places the quantities on the blankine to flexibly fit the problem. Klein et al. (1998) summarize the benefits of the number-line representation as follows: (1) its suitable with informal solution procedures due to its linear character, (2) it supports raising the level of students’ activityy helping them develop more-sophisticated strategies, (3) it has a natural and transparent character to represent non-outine problems, and (4) it involves students for higher cognitive involvement. Gravemeijer (1999) explains how modelsre embedded in the problem situations in RME, and students are supported to reinvent more formal mathematics throughsing the models. While students may initially view numbers in the problems tied to identifiable objects and units in the
ituation, they come to recognize them as entities of their own by using and revising models.
In U.S. mathematics curricula, number-line representations are also commonly seen. In analyzing and contrasting U.S.nd Japanese mathematics curricula, Murata (2008) found number-line representations were used more for number-onlyroblems than for situational problems (more than 50% for number-only problems, compared to less than 10% for situation


Fig. 2. Subtraction situation problem types.Adapted from Carpenter et al. (1999).

problems). Thus, number-line representations are typically not used to model problem situations in the United States, butas a specific tool (algorithm) to aid numerical calculations.

1.5. Students’ conceptions of multi-digit numbers and operations

Numerous studies examined how children think of numbers and how their number sense develops in the early years(e.g., see Fuson, 1992). Researchers agree that there are three levels of conception of quantities. In considering three quan-tities involved in the subtraction number relationships (� − � = �), children first use Count-All-and-Take-Away strategy byconsidering all quantities as collections of individual units (e.g., for 7 − 3, count 7 unitarily, then count 3, then count therest from one). Over time, children will abbreviate the subtrahend to find the difference, to use Count-Down or Count-Upstrategy (e.g. for 7 − 3, they may count down three times from 7 to find the answer, 4). At the final level, they understandhow different numbers relate to one another, and find the difference using the knowledge (e.g., understand that 3 and 3make 6, thus 7 − 3 must be 4 because 7 is 1 more than 6). For multi-digit numbers, while children use the same knowledge,the complexity is added by the linguistic and symbolic representations of the numbers (Carpenter, Franke, Jacobs, Fennema,& Empson, 1998; Fuson et al., 1997; Fuson, 1992). Number names in European languages are non-transparent in terms ofplace value, and young students often experience number names as an unstructured linear string of words, compared tochildren who speak Chinese-based languages (e.g., Chinese, Japanese, Korean) with explicit ten-ness in counting words. Forexample, in English, 11, 12, and 13 are called “Eleven,” “Twelve,” and “Thirteen,” while in Japanese they are called “Ten-one,”“Ten-two, and “Ten-three.” Therefore, even very young Japanese children hear and experience the “ten” as they count intoteen numbers with explicit number structure in the words. If students lack solid understanding of the base ten place valuesystem, multi-digit operation can easily become a set of steps to follow to find answers, which often further misguide themto consider multi-digit numbers as collections of unrelated single-digit numbers.

Previous research shows two general methods students use for multi-digit operations (Carpenter et al., 1998; Fusonet al., 1997; Fuson, 1992; Gravemeijer, 1994; Selter, 1998). For the decomposition method,1 students split the numbers byplaces and operate them individually by places (e.g., for 58 − 21, decompose 58 into 50 and 8, 21 into 20 and 1, subtracttens 50 − 20 = 30, then ones 8 − 1 = 7, then add them together 30 + 7 = 37). For the sequential-chunking method,2 studentsstart with the minuend and make sequential moves by the chunks of the subtrahend (e.g., for 58 − 21, 58 − 20 = 38, then38 = 1 = 37). Students may also combine some parts of these two approaches to find answers. For standard algorithms, theyuse the decomposition method by places. The number-line representations are more suited with the sequential-chunkingmethod, which has a different benefit to support students’ number sense development (Selter, 1998).

1.6. Subtraction problem types

Researchers agree on different subtraction situational problem types that are commonly taught in classrooms (Carpenter,Carey, & Kouba, 1990; Carpenter, Fennema, Franke, Levi, & Empson, 1999; Fuson, 1992, 2003; Gustein & Romberg, 1995).There are three broader types of subtraction situations for which any of the three quantities that are involved in the problemcan be missing (� − � = �). Fig. 2 summarizes these problem types regarding subtraction.

Research also shows that in the typical U.S. classroom, Separate problems are the main type taught and considered as
subtraction problems (take-away problems), but not other types. While in some other countries, different problem types areused in instruction (Stigler, Fuson, Ham, & Kim, 1986). Common Core Sate Standards (Council of Chief State School Officers,2010) stress the importance of using different situation problems as students learn to solve problems. In the present study,
1 Researchers used a variety of terms to refer to this method such as Decompose-Tens-and-Ones method, Multiunit method, Hundreds-Tens-Unit method,and Split method.

2 This method has been called Begin-with-One-Number method, Multiunit Sequential methods, Stepwise method, and Curtailed-Sequential countingmethod.

ap

2

m

123

3

3

Wbrdc

stM3msTt

tcot

abfitpra

t1c

3

maod


ll three types of problems were used for students to solve, although students had different familiarity levels with theseroblem types.

. Research questions

This study investigated Grade 3 students’ mathematization process by examining how they modeled different types ofulti-digit subtraction situational problems. We asked the following research questions:

. How do Grade 3 students model different subtraction problem situations?

. How do their models change over time?

. What do these changes indicate about students’ mathematization processes?

. Methods

.1. Participants and settings

Twenty-one Grade 3 students participated in this study. They attended a Grade K – 5 public elementary school in theestern United States. The city where the school is located has mixed-income SES, with 24% of the students at the school

elonging to socio-economically disadvantaged households, and 32% of the students are ELL (English Language Learners)eceiving special instruction at the school.3 All twenty-one students were from one classroom, and they reflected the schoolemographics. The classroom teacher, Andrea,4 had taught for 2 years at this grade level prior to the year this study wasonducted.

Andrea participated in a professional development program, lesson study, in which the instructional unit taught for thistudy had been planned (Murata, Bofferding, Pothen, Taylor, & Wischnia, in press). Lesson study is a collaboration-basedeacher professional development that originated in Japan more than 100 years ago (Lewis & Tsuchida, 1998; Lewis, Perry, &

urata, 2006; Murata & Takahashi, 2002; Murata, 2010, 2011). Three teachers who taught different grade levels (Grades 2,, and 4) worked together in this lesson study, and all three planned and taught related but different instructional units onulti-digit subtraction at the end of the collaboration. Although lesson study typically seen outside of Japan is small-scale,

chool-based, and single-grade focused, it can come in many different formats, sizes, and structures in Japan (Murata &akahashi, 2002). The group of teachers in the current study designed their own lesson study, that included all three of themeaching separately in own classrooms, being observed by the other teachers, and receiving feedback.

The first author of this paper participated in the lesson study as a facilitator to focus the teachers’ work and guide theirhinking. She set the agenda and summarized their work at the beginning and the end of each meeting. She also providedontent support by answering teachers’ questions about the mathematics topics and asked questions to probe their thinkingf the topic further. When appropriate, she also provided research literature and curricular resources for teachers to use ashey planned lessons.

When the teachers initially met to identify a content topic for their collaboration, they wanted a topic that spannedcross their grades. They chose multi-digit subtraction since they all found the topic challenging to teach and wanted toetter support their students’ learning. The teachers only had general ideas of what aspects of multi-digit subtraction toocus at the beginning of lesson study. As the meetings progressed in the school year, however, they became increasinglynterested in the uses of representations in their instruction. The teachers considered this approach to be an effective wayo connect student thinking more concretely than focusing only on numbers and symbols. Grade 4 teacher brought a unitlan from a resource curriculum (Pearson, 2008) whose objective was to represent student thinking using number-lineepresentations. The teachers liked the material, and they started with the existing unit and changed the problem situationsnd numbers slightly for their own instructional units for individual classrooms.

This paper discusses student learning in the Grade 3 classroom for several reasons. Andrea was the least experiencedeacher (with two years of teaching experience) of the three who participated in the lesson study (the other teachers had3 and 14 years of experience, respectively), and she was most open and willing to learn. She was also open to data beingollected during the entire instructional unit.

.2. Data collection

During the four-lesson instruction of the unit spanned over two weeks, students solved problems and shared their solutionethods with the whole class. There were two main problems students solved as a whole class (one part-part-whole problem

nd one compare problem, described later), and for the rest of the unit, they solved seven multi-digit situational problemsn their own as independent seatwork. These problems were printed individually on separate worksheets and comprisedifferent types of problems. Fig. 3 lists the problems.

3 District data.4 All the names appearing in this paper are pseudonyms.


Fig. 3. Independent worksheet problems5.

Andrea created these problems in consultation with curricular resource materials. All the worksheets were collected forthe study. As discussed in the section above, the problem set included different types of problems (Separate, Part-Part-Whole,and Compare).

This study was a part of a larger study in which additional data were collected. These include videotapes of lesson studymeetings and classroom lessons (transcribed later), copies of curricular materials teachers used to plan lessons, written lessonplans, student work (in addition to the worksheets described above), and researchers’ observation notes. For this study, weused student work as a primary data source. We also used lesson materials (lesson plans) and videotapes (transcripts) of thelessons as secondary data sources, as will be explained later.

3.3. Data analysis

All worksheets were carefully examined, and all students’ work was summarized and described in terms of how theymodeled and solved the problems. We were first interested in knowing the different solution methods students used. Thuswe coded their work exclusively for this purpose. We cross-coded a few pieces of student work at a time, and compared thecoding results until we reached 80% agreement. Subsequently, we discussed why we disagreed and reached 100% agreement.We then focused more on the models (representations) students used in their solution processes. There were two broadercategories of models: number-line models and number-only models. We coded the models as number-line models whenstudents used open number lines and plotted numbers in the problems and then found answers by finding the difference,either by using a standard algorithm, or decomposing the numbers by places and subtracting. We coded the models number-only models when students used numbers only (without any drawings) to find the answer. The second author coded allstudent work, and the first author reviewed the entire coding results to attain a reasonable degree of agreement. Thenumber-line representations were further coded as either solution model (showing only mathematical quantities neededto solve the problems) or situation model (presenting additional information to depict the situation). Fig. 4 summarizes theexamples of these models.

For solution models, students typically marked two quantities (289 and 477) on the number line and find the distancebetween the points by chunking the distance. For situation models, students started with the zero point, considered thefirst distance traveled (289 miles) from the origin, and found the second distance to the destination. We considered this asa situation model because including the zero made the model inclusive of all three quantities in the problem (� − � = �).
In representations, the inclusion of zero helped model whole problem situations as the representations now appropriatelyshowed all three quantities involved to clarify the relationships among them (this will be explained in detail below). For thiscoding, we coded all the student work separately, and the results were compared. The agreement was 100%. The reliability
5 In the original problems, actual students’ names were used. Thus, all names were changed for this paper.


FO

ws

ftowsir(s

aCf

adiAACt

4

4

ppm

ig. 4. Examples of models used by students for Problem 3 (Heather and Vicki are on their way to Legoland which is 477 miles away from San Francisco.n the first day, they traveled 289 miles. How many more miles did they have left to travel?).

as high because the data were straightforward and the distinction was clear. We also noted if students used addition orubtraction in using different models.

We constructed a large spreadsheet to record methods and models used by individual students to see how they changedrom one problem to next, and to examine the patterns of change over time. We noted the orders and the relations betweenhe models students used for each problem type: we wanted to see if students chose one primary model to solve all problems,r if they selectively used different models for different problem types. We also noted the accuracy of the student answers,hich could be affected by computational as well as conceptual errors. The accuracy rates were high: for Problems 1–7, they

cored 100%, 78%, 100%, 67%, 100%, 100%, 100%, respectively. The high rates resulted because the subtraction content taughtn the unit was not difficult for the students. As explained earlier, the lesson study’s goal for the unit was to use commonepresentations to connect different student thinking of problems, thus Andrea purposefully chose manageable quantitiesthree digit numbers) so that students would not be challenged in the calculation process and could focus on representingituations.

After all the worksheets were analyzed, we compiled the results by problems and by models in separate tables. We alsonalyzed students’ model choices according to different problem types by grouping problems: Separate (Problems 1 and 5),ompare (Problems 2, 4, and 6), and Part-Part-Whole (Problems 3 and 7), to determine possible patterns of student choicesor each problem type.

Lesson materials (e.g., lesson plans) and video transcripts of all lessons were reviewed to see how the lessons supportednd affected students’ modeling of problem situations. Andrea’s lesson plans indicated the parts of the lessons that wereesigned to facilitate student discussion of problem situation modeling. Thus we paid particular attention to those sections

n the video transcripts and our observation field notes. The chart was created to contrast the different models students andndrea shared in classroom discussions, and the descriptions of these models were added. Special attention was paid whenndrea and the students were making (or failing to make) connections among different models and problem situations.ompared to the analysis of student work, the analysis of other data sources were secondary to the purpose of this paper;hey were used to add contextual information to the primary results of student work.

. Results and discussion

.1. Student modeling of the seven problems

Table 1 summarizes the solution methods students used to solve the seven problems.Although student methods for Separate problems stayed consistent (Problems 1 and 5), they generally used more decom-

osition methods than the sequential-chunking methods over time for Compare (Problems 2, 4, and 6) and Part-Part-Whole
roblems (Problems 3 and 7). This also reflects the fact that students used more number-line models at the beginning, butore number-only algorithms later on for the latter two problem types.Tables 2–4 summarize the models students used as they solved the seven problems, separated by different problem types.


Table 1Frequency of student methods by problems.

Problem 1 Problem 2 Problem 3 Problem 4 Problem 5 Problem 6 Problem 7Separate Compare P-P-W Compare Separate Compare P-P-W

Sequential-chunking methods 33% 61% 54% 33% 33% 39% 25%Decomposition methods 67% 39% 43% 66% 67% 50% 75%Mix method (hybrid of the two) 0 0 3% 0 0 11% 0

Table 2Frequency of models used for Separate Problems.

Problem 1 Problem 5

Number-line model 33% 33%Using addition (0) (0)Using subtraction (33%) (33%)

Number-only model 67% 67%Using addition (0) (0)Using subtraction (67%) (67%)

Table 3Frequency of models used for Part-Part-Whole Problems.

Problem 3 Problem 7

Number-line model 54% 25%Using addition (42%) (0)Using subtraction (12%) (25%)

Number-only model 43% 75%Using addition (0) (0)Using subtraction (43%) (75%)

Table 4Frequency of models used for Compare Problems.

Problem 2 Problem 4 Problem 6

Number-line model 61% 33% 39%Using addition (61%) (33%) (22%)Using subtraction (0) (0) (17%)

Number-only model 39% 66% 61%Using addition (22%) (22%) (11%)Using subtraction (17%) (44%) (50%)

Student models for Separate problems (Problems 1 and 5) stayed consistent over time. This may be because studentshad a plenty of experience with this type of problem previously, and were set with their model choice as they solved theseproblems. On the other hand, students were less familiar with Compare (Problems 2, 4, and 6) and Part-Part-Whole problems(Problems 3 and 7), thus they used number-line models initially, and then switched to number-only models later.

They also used more addition-related models first for Part-Part-Whole and Compare problems, then moved on to usemore subtraction-related models. Table 5 summarizes this shift.

For Part-Part-Whole problems, 42% of students used the addition model for Problem 3, while 100% used the subtractionmodel for Problem 7. For Compare problems, 83% of students started out with addition models, and 67% used the subtractionmodel for Problem 6. This shift is reasonable because these problems were presented as additive situations (Fig. 3), thusstudents considered them as addition problems. Students’ earlier model uses for these problem types indicate that theychose their models to closely relate to the problem situations.

For each problem type, students came to use subtraction with a number-only model more exclusively over time. Thismay indicate that students gradually generalized all problem situations to understand them as subtraction situations, and
also more abstractly. For Part-Part-Whole problems, 75% of students used the number-only subtraction model for Problem7, and for Compare problems, 50% of students used it for Problem 6. It appears that once they made the connection amongdifferent problem types, they used the common model generally for all problems.
Table 5Frequency of addition or subtraction approaches by problems.


Using addition 0 83% 42% 55% 0 33% 0Using subtraction 100% 17% 55% 44% 100% 67% 100%


4

Ac

ttsAmse

dttcfa

m

cgA

JABA

HFc

Fig. 5. Examples of student models for Problem 1.

.2. Classroom instruction and different models used by students

The instructional unit used two main situational subtraction problems for the whole-class discussion:

Problem A: Elizabeth’s family visited their grandmother who lives 534 miles away. On the first day they drove 319 miles.How much more did they have to drive on the second day?

Problem B: Ms. Andrea and Ms. Susan each drove from Coastal Town to other cities. Ms. Susan drove 117 miles to the capitolof California, Sacramento. Ms. Andrea drove 383 miles to Disneyland. How many more miles did Ms. Andrea drive?

ndrea created these problems as she thought that students would make connections between the ‘traveling’ problemontext and linear number-line representations.

In the first lesson of the unit, as students discussed what was happening in the situation in Problem A, Andrea helpedhem physically role played the situation by walking across the length of the classroom, identifying the points of departure,he hotel where the family stayed the first night, and the grandmother’s house. While most children’s initial drawings of theituation included elaborated and detailed pictures of family members, cars, buildings and scenery along the travel path,ndrea summarized the student discussion by showing the problem situation using a number-line as a solution model, byarking minuend and subtrahend on a line and finding the distance between them by chunking it. Following the summary,

tudents independently solved Problem 1 from Fig. 3. Many used the number line as a solution model, to follow the teacher’sxample (Fig. 5).

Andrea did not present subtraction as a topic of study for the unit, but merely gave the problem context and had studentsraw/model/discuss what they saw important in the context. Thus some students used addition to find the remainingravel distance, which was more contextually meaningful given the context (the family was going to travel more to get tohe destination), while the other students used subtraction. Thus the variations among their approaches were seen in theirhoice of additive or subtractive features to find the distance, and/or the different chunks they used in the process. In general,or Part-Part-Whole problems (Problems 3 and 7) and Separate problems (Problems 1 and 5), this approach was directlypplicable to the problem situations; however, it was not for Compare problems (Problems 2, 4, & 6) (Fig. 6).

As students worked on Problem B (in whole class), Andrea was confused that there could be more than one number-lineodel to show the problem. The students showed their thinking, as a situation model and solution model:Andrea was puzzled by the difference between the two models and wanted to clarify that in the following lesson. After

onsulting with her more experienced colleagues in the lesson study group, she chose to contrast these representations anduided student discussion at the start of the next lesson,ndrea: . . . we had a problem where I went to Disneyland, Ms. Susan went to Sacramento, and we wanted to know how much further I

drove than Ms. Susan. What type of problem did you figure out this was?ose: Comparing two numbers.ndrea: Comparing. What shows us that this is a comparing problem, a comparison problem?en: ‘Cause it’s how much more did you drive?ndrea: . . . how much more did I drive, and there’s a picture of that Mei helped us with. She drew two number lines (Example 6.2), to

show that I drove more. And we want to figure out this much (identify where the difference is shown in the representation). Andthen, after the lesson, we talked about how most people did their number line as like a close-up view of this little piece, here(Example 6.1). ‘Cause almost everybody did their number line from a 117 to 383. And so, few people did from 0 to 383. So whatmost people were drawing was this little piece up here (She shows the difference in Example 6.2 again). OK?

aving more than one representation for a problem pushed students and Andrea to think more carefully about the problem.ollowing this example, students discussed different problems and different models that accompanied each of the problems,ompared these models, and made connections between different models focusing on the problem situations.


Pro blem B: For spring bre ak, Ms. Andrea and Ms. Susan eac h dr ove fro m Co asta l Tow n toother cities. Ms. Susan drov e 117 miles to the ca pitol of California, Sacrame nto. M s.Andrea drove 383 miles to Disneyland. How man y mor e mile s did Ms. Andrea drive?Student Re prese ntatio n 6.1: Us ing repr esen tation as a gener al me thod ( Comp are problem )

+3 +8 0 +100 +8 3

117 12 0 20 0 300 38 3

Student Re prese ntatio n 6.2: Using re presentat ion t o sh ow the probl em situ ation (Comp are p robl em)

Ms Andrea

Ms. Sus an

0 11 7 38 3

Fig. 6. Different student representations in the lesson.

4.3. Situation models and solution models

Following this, when students solved Problem 2 (Fig. 3) independently, some students used the representations in Fig. 7to show their thinking.

While Example 7.1 is the solution model to find the distance between two numbers, Example 7.2 is a situation model thatshows this particular problem situation more accurately by illustrating two kite strings and comparing the lengths of thestrings. After modeling the problem this way (Example 7.2), these students found the difference between 189 and 215 bychunking the distance, or used the subtraction algorithm directly. Thus, some students were modeling the situation beforefocusing on the mathematics of the problem.

Some students also showed Separate and Part-Part-Whole problems slightly differently, using number lines as a situationmodel (and not as a solution model). Fig. 8 shows these examples.

In Fig. 8, Example 8.2 shows the problem situation for Problem 1 more accurately (situation model) than Example 8.1(solution model): Andy gave 89 cards out of 275 to Alex. The solution is the remaining amount, shown in 8.2 as the distancefrom 0. Example 8.4 shows the entire trip distance (situation model), while 8.3 only shows the abbreviated section of thetrip that is yet to be taken (solution model). Considering all three problem types, number-line representations as solutionmodel are most suited with Part-Part-Whole problems, since the problems typically require taking the first part of the total
and finding the second part, thus asking students to find the difference between the two numbers in the problem. Thus,when students showed their thinking process as depicted in Example 8.3 (solution model), the representation was the sameas a part of the situation models (Example 8.4).
Fig. 7. Examples of student models for Problem 2.


Fig. 8. Examples of student models for Problems 1 and 3.

Table 6Number-line representation used as a solution model and as a situation model.


moi

5

mutwmmsm

mtac

d

Solution model 63% 80% 70% 33% 100% 33% 100%Situation model 37% 20% 30% 67% 0 67% 0

Table 6 shows the percentage of student number-line uses, as a solution model and as a situation model.For Separate problems (1 and 5) and Part-Part-Whole problems (3 and 7), some students used number lines as a situation

odel initially, but quickly moved to use it as a solution model. For Compare problems (2, 4, and 6), the data show anpposite trend. That is, students used number lines as a solution model at the beginning, as initially shown in classroomnstruction, but came to use it as a situation model later on.

. Summary

In this study, Grade 3 students modeled subtraction situation problems differently according to problem types, and theirodels changed. Although the general trajectory appears to be from number-line models to number-only models, and from

sing addition to using subtraction (appropriate to the problem contexts), there were different patterns for the three problemypes. For Separate problems, with which students were most familiar, the majority of students used number-only modelsith subtraction over two problems. For Part-Part-Whole problems, more students used number-line models first, and thenoved to use number-only models. For those number-line models, more students used situation models at first, and thenoved to using solution models for the second problem. Similar to Part-Part-Whole problems, on Compare problems, more

tudents used number-line models first, followed by number-only models. As for types of number-line models, though, theajority of the students used solution models initially then later switched to situation models.The students’ models showed students were focusing on problem situations before using a more-general and abstract

odel. Overall, they were more likely to use addition for the problems that were worded as additive situations, and sub-raction for the ones worded as subtractive situations, but gradually used subtraction for all problems. When they used

number-line representation to model the situation for Separate and Part-Part-Whole problems, their use also graduallyame to focus on computation rather than representing the situations.

Looking at the results by different problem types, it is interesting to note how students approached Compare problemsifferently from other problem types. In working to solve three Compare problems, they used a number line more as a


situation model than as a solution model, and used addition more than subtraction to solve the problems than Separate andPart-Part-Whole problems. It could be because Compare problems are typically harder to identify as subtraction situations,and they are less likely to be used in instruction (in most U.S. curricula, Separate problems are used exclusively; Stigler et al.,1986), thus students had less familiarity with them. Unlike other problem types, for Compare problems, the situation modeland the solution model are quite different (the situation model requires double number lines), thus students will need totake an additional step to make sense of the problem situation and devise a solution plan.

Across three problems, students came to use numerical models with Compare problems more often (although not asearly as with the other two types), but not comparable to other problem types. Since the instruction was short (2 weeks),it might not have been enough time for students to gain familiarity with this type of problems. With more experience overtime, they may come to generalize this problem type more. As for Part-Part-Whole problems, while these may also havebeen relatively new to students, solution models and situation models for this problem type were very similar, and in somecases, the same. Therefore, this aspect of the problem might have helped students to move along the learning path morequickly.

All of the Separate and Part-Part-Whole problem students solved had the total numbers given first in the problems. Thismight have been a factor in encouraging students to use more subtraction than addition. For Compare problems, too, Problem4 had the total given first, while Problems 2 and 6 had the totals second. Since the problems were written by Andrea, we didnot have control over the wording, but these word-order differences could be investigated in future research.

Classroom instruction that explicitly connected different student models helped students make sense of the differencesamong the models. Although uncertain about the differences herself, Andrea also learned how a solution model can be apart of a situation model, and when students showed different models, they are mathematically and situationally related.The time she spent planning the lessons and anticipating student responses in lesson study meetings with her colleagueshelped her learning while teaching, as she became more knowledgeable of the relations among different problem types,student solution methods, and models.

6. Conclusions and implications

While the distinctions between situation and solution models are typically not clear in textbook curricula or in teaching,this study showed that the students considered and used these models differently. When the problems were unfamiliar,students tended to model the situations by including more context-specific information, and as they gained more experiencewith the problems, they came to focus on the quantitative information in the model.

By teaching a general strategy only, it becomes a mere algorithm to follow without making connections to rich situationsthat problems can pose. In the present study, students were shown one general solution model with a number line, butwere able to invent their own situation models to bridge their understanding when necessary. They invented their ownrepresentation for the Compare problem, supporting their understanding of the situation and of the solution. In viewing theproblem-solving as a mathematization process with modeling, careful presentations and analysis of different models fordifferent problem types in classrooms should be helpful, because the developmental processes of local models may vary.Before students become fluent in using a general solution model, they must be assisted in making sense of the situationmodels of different problems in order to see the common features of the problems, and to develop the model for the contenttopic. Qualitative representations and mathematical models are both necessary for successful problem solving, and thesetwo formats need to be compared, contrasted, and explicitly discussed in classrooms for student understanding.

The difference between solution models and situation models need to be analyzed, because representing a problemsituation may not always lead to the answer directly; rather, representation is a necessary step in understanding the problemcontext in order to devise a solution plan. As Common Core State Standards (Council of Chief State School Officers, 2010)places more importance in mathematical modeling and drawings, it is becoming even more critical for teachers to clarifyearly on that a situation representation may only show quantitative relationships, but it may also show the quantities tobe related to problem solutions. If such practices are regularly seen in classrooms, students should be able to build betterunderstanding of the problem-solving processes based on contexts.

This study also found that students’ modeling approaches varied across problem types over time, even for the samecontent topic (multi-digit subtraction). The mathematization processes were not even and/or consistent, as different problemtypes posed different challenges for students. If students’ modeling trajectories differ across problem types, understandingdifferences and interactions among different trajectories will be important in designing curricula and in teaching. In thepresent study, students were exposed to three different problem types at the same time, and thus their understandingof Separate and Part-Part-Whole problems could have influenced the understanding of Compare problems. If we knowstudents are likely to need additional assistance with certain problems (due to particular complexity and/or challenges thatthe problems pose), we could anticipate the need and provide appropriate experiences within learning activities accordingly.For example, if developing a solution model for Compare problems indeed takes longer, we could take advantage of the fact
and engage students in activities to review how numbers relate in addition/subtraction problems by comparing and referringcompare problems to other familiar problem types. By doing so, we help strengthening their understanding of the contenttopic altogether. This could encourage the interactions among various emerging solution models to bring these modelstogether for a coherent learning of the content.

moWpccnit

rnn1iitdp

titpiog

A

tpp

R

AB

B

C

C

CC

C

CC

DF

FFF

F

F

F


Number line representations naturally lead students to use sequential-chunking method instead of decompositionethod. The sequential-chunking methods are less generalizable when numbers used in the problems become bigger, and in

ur student sample, too, the chunks were often added using standard algorithm after they were identified on number lines.ith the sequential-chunking methods, place value becomes invisible, while decomposition methods take advantages of

lace value (regrouping). When students used the standard algorithm as a main method at the end of the instructional period,lassroom instruction could have taken up the learning movement and helped make connections between the sequential-hunking method and standard algorithm, to support students’ sense-making process. For example, most students chunkedumbers in terms of tens and ones on number lines. Andrea could focus students’ attention on these chunks and help them

dentify them in the computation process with the standard algorithm. In these ways, students could be guided graduallyo make sense of the conceptual background of the algorithms.

The study also invites the question of the appropriate time to introduce and use number-line representations in class-ooms. While Grade 3 students are working with numbers up to 1000 (Council of Chief State School Officers, 2010), using ofumber lines may not be most ideal, considering the lack of transparency of place value as discussed above. In Netherlands,umber-line representations are used frequently starting early elementary grades (Beishuizen, 1993; Gravemeijer, 1994,999; Treffers & De Moor, 1990). In Japan, a family of linear representations (tape diagrams, number-line representations)

s used across grade levels over different content topics from early on (Murata, 2008). In these countries, number lines arentroduced early and used frequently over time, thus by Grade 3, students are able to switch among different representationso show situations and solve problems. With the new emphasis on math drawings and modeling in Common Core State Stan-ards, U.S. students are also expected to become fluent with using multiple representations early on, as their internationaleers, and able to represent problem situations with ease by mid-elementary grades.

The findings of this study also imply that students are taking their own steps to make sense of mathematics, whenhe instruction may only focus on developing skill base, therefore not allowing space for the sense-making process. Thenstructional focus on speed and right answers may be creating the need for students to learn on their own, since they haveo manage their learning without the support of curricula and teaching. If multiple students can share their mathematizationrocesses in classrooms, they will support and stimulate each other’s thinking and can ultimately speed up the process. It

s crucial to invite students’ prior knowledge into problem-solving situations and help them develop and refine a networkf knowledge. This will help students piece together the old and the new, and find a space for the new knowledge that willuide more students through their learning paths in a meaningful way and prepare them for further mathematical learning.

cknowledgements

This study was supported by the Wallenberg Global Learning Network II Preparation and Full grants. The authors wisho thank the teachers and students who participated in the study. We also would like to acknowledge the valuable feedbackrovided by close and distant colleagues on previous versions of the manuscript as we went through multiple revisionrocesses.

eferences

nderson, J. R. (2002). Problem solving and learning. American Psychologist, 48, 35–44.eishuizen, M. (1993). Mental strategies and materials or models for addition and subtraction up to 100 in Dutch second grades. Journal for Research in

Mathematics Education, 24(4), 294–323.oaler, J. (2008). Promoting ‘Relational Equity’ and High Mathematics Achievement Through an Innovative Mixed Ability Approach. British Educational

Research Journal, 34(2), 167–194.ai, J. & Lester, F. K. (2005). Solution representations and pedagogical representations in Chinese and U.S. classrooms. Journal of Mathematical Behavior, 24,

221–237.arpenter, T. P., Carey, D. A. & Couba, V. L. (1990). A problem-solving approach to the operations. In J. N. Payne (Ed.), Mathematics for the young child (pp.

111–131). Reston, VA: National Council of Teachers of Mathematics.arpenter, T. P., Fennema, E., Franke, M. L., Levi, L. & Empson, S. B. (1999). Children’s mathematics: Cognitively guided instruction. Portsmouth, HN: Heinemann.arpenter, T. P., Franke, M. L., Jacobs, V. R., Fennema, E. & Empson, S. B. (1998). A longitudinal study of invention and understanding in children’s multidigit

addition and subtraction. Journal for Research in Mathematics Education, 29(1), 3–20.arpenter, T. P., Hiebert, J. & Moser, J. M. (1983). Problem structure and first-grade children’s initial solution processes for simple addition and subtraction

problems. Journal for Research in Mathematics Education, 12, 27–39.harles, R. (1987). How to evaluate progress in problem solving. Reston, VA: National Council of Teachers of Mathematics.ouncil of Chief State School Officers & National Governor’s Association. (2010, June). Common core state standards for mathematics. Retrieved from

http://www.corestandards.org/assets/CCSSI Math%20Standards.pdfsuval, R. (2006). A cognitive analysis of problems of comprehension in a learning of mathematics. Educational Studies in Mathematics, 61, 103–131.ranke, M. & Casey, D. (1997). Young children’s perceptions of mathematics in problem-solving environments. Journal for Research in Mathematics Education,

28(1), 8–25.reudenthal, H. F. (1978). Weeding and sowing. Preface to a science of mathematical education. Dordrecht: Reidel.reudenthal, H. F. (1991). Revisiting mathematics education. China lectures. Dordrecht: Kluwer.uson, K. (1992). Research on whole number addition and subtraction. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning

(pp. 243–275). NY: Macmillan.uson, K. (2003). Developing mathematical power in whole number operations. In J. Kilpatrick, W. G. Martin, & D. Shifter (Eds.), A research companion to

principles and standards for school mathematics (pp. 68–94). Reston, VA: National Council of Teachers of Mathematics.uson, K. & Willis, G. (1988). First and second graders’ performances on compare and equalize word problems. In Proceedings of the Tenth International

Conference on the Psychology of Mathematics Education (pp. 19–24). London: University of London Institute of Education.uson, K. C., Wearne, D., Hiebert, J. C., Murray, H. G., Human, P. G., Olivier, Al. I., et al. (1997). Children’s conceptual structures for multidigit numbers and

methods of multidigit addition and subtraction. Journal for Research in Mathematics Education, 28(2), 130–162.

http://www.corestandards.org/assets/CCSSI_Math%2520Standards.pdfs


Gravemeijer, K. (1994). Educational development and developmental research in mathematics education. Journal for Research in Mathematics Education, 25,443–471.

Gravemeijer, K. (1999). How emergent models may foster the constitution of formal mathematics. Mathematical Thinking and Learning, 1(2), 155–177.Gustein, E. & Romberg, T. A. (1995). Teaching children to add and subtract. Journal of Mathematics Behavior, 14(3), 283–324.Hershkowitz, R., Schwarz, B. B. & Dreyfus, T. (2001). Abstraction in context? Epistemic actions. Journal for Research in Mathematics Education, 32(2), 195–222.Hiebert, J., Carpenter, T. P., Fennema, E., Fuson, K., Human, P., Murray, H., et al. (1996). Problem solving as a basis for reform in curriculum and instruction:

The case of mathematics. Educational Researcher, 25(4), 12–21.Ishida, J. (1998). An analysis of first grade children’s using number sentences in solving word problems. Budapest: Paper given at the International Congress of

Mathematics Education.Kilpatrick, J., Martin, W. G., & Shifter, D. (Eds.). (2003). A research companion to principles and standards for school mathematics.. Reston, VA: National Council

of Teachers of Mathematics.Klein, A. S., Breshuizen, M. & Treffers, A. (1998). The empty number line in Dutch second grades: Realistic versis gradual program design. Journal for Research

in Mathematics Education, 29(4), 443–464.Lesh, R. & Doerr, H. (2003). Beyond constructivism: Models and modeling perspectives on mathematics problem solving, learning and teaching. Mahwah, NJ:

Erlbaum.Lesh, R. & Harel, G. (2003). Problem solving, modeling, and local conceptual development. Mathematical Thinking and Learning, 5(2 & 3), 157–190.Lesh, R. & Zawojewski, J. (2008). Problem solving and modeling. In F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning: A

project of the National Council of Teachers of Mathematics (pp. 763–804). Reston, VA: NCTM.Lewis, C., Perry, R. & Murata, A. (2006). How should research contribute to instructional improvement: The case of lesson study. Educational researcher,

35(3), 3–14.Lewis, C. & Tsuchida, I. (1998). A lesson is like a swiftly flowing river: Research lessons and the improvement of Japanese education. American Educator,

Winter, 14–17, 50–52.Murata, A. (2008). Mathematics teaching and learning as a mediating process: The case of tape diagrams. Mathematical Thinking and Learning, 10(4), 374–406.Murata, A. (2010). Teacher learning with lesson study. In P. Peterson, E. Baker, & B. McGaw (Eds.), International encyclopedia of education (pp. 575–581).

Oxford: Elsevier.Murata, A. (2011). Conceptual overview of lesson study: Introduction. In L. Hart, A. Alston, & A. Murata (Eds.), Lesson study research and practice: Learning

together. Norwell, MA: Springer.Murata, A., Bofferding, L., Pothen, B., Taylor, M., & Wischnia, S. (in press). Making connections among student learning, content, and teaching: Teacher talk

paths in elementary mathematics lesson study, Journal for Research in Mathematics Education.Murata, A., & Takahashi, A. (2002). District-level lesson study: How Japanese teachers improve their teaching of elementary mathematics. Paper presented at

the research presession of National Council of Teachers of Mathematics annual meeting. Las Vegas, NV.National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: Author.National Council of Teachers of Mathematics. (2000). Principles and standards of school mathematics. Reston, VA: Author.Ng, S. F. & Lee, K. (2009). The model method: Singapore children’s tool for representing and solving algebraic word problems. Journal for Research in

Mathematics Education, 40(3), 282–313.Nunokawa, K. (2000). Heuristic strategies and problem solving situation. In J. Carrillo, & L. C. Conteras (Eds.), Problem solving in the beginning of the 21st

century: An international overview from multiple perspectives and educational level (pp. 81–117). Huelva: Hergue.Nunokawa, K. (2005). Mathematical problem solving and learning mathematics: What we expect students to obtain. Journal of Mathematical Behavior, 24,

325–340.Pape, S. (2004). Middle school children’s problem-solving behavior: A cognitive analysis from a reading comprehension perspective. Journal for research in

Mathematics Education, 35(3), 187–219.Pearson Education. (2008). Investigations in number, data, and space. Glenview, IL: Author.Selter, C. (1998). Building on children’s mathematics – A teaching experiment in Grade three. Educational Studies in Mathematics, 36(1), 1–27.Schoenfeld, A. (1992). Learning to think mathematically: Problem solving, metacognition, and sense making in mathematics. In D. A. Grouws (Ed.), Handbook

of research on mathematics teaching and learning (pp. 334–371). New York: MacMillan.Stigler, J. W., Fuson, K. C., Ham, M. & Kim, M. S. (1986). An analysis of addition and subtraction word problems in American and Soviet elementary mathematics

textbooks. Cognition and Instruction, 3, 153–171.Thompson, C. & Hendrickson, A. D. (1983). Verbal addition and subtraction problems: New research focusing on levels of difficulty of the problems and of

the related number sentences. Focus on Learning Problems in Mathematics, 5, 33–45.Treffers, A. & De Moor, E. (1990). Specimen of a national program for primary mathematics teaching. Part 2. Basic mental strategies and written computation.

Zwijsen: Tilburg, The Netherlands.Willis, G. B. & Fuson, K. (1988). Teaching children to use schematic drawings to solve addition and subtraction word problems. Journal of Educational

Psychology, 80, 192–201.Wittmann, E. (1981). The complementary roles of intuitive and reflective thinking in mathematics teaching. Educational Studies in Mathematics, 12(3),

389–397.

Grade 3 students’ mathematization through modeling: Situation models and solution models with...

Documents

Transcript of Grade 3 students’ mathematization through modeling: Situation models and solution models with...