# Teaching the Conceptual Structure of Mathematics

**ABSTRACT** This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae, and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand, or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material.

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**ABSTRACT:**Inspite of many strengthens of storytelling mathematics education, some problems are expected: when math is taught in concrete contexts, students may have trouble to extract concepts, to transfer to noble and abstract contexts, and they may experience inert knowledge problem. Low achieving students are particularly prone to these issues. To solve these problems some suggestions are made by the author. These are analogous encoding and progressive formalism. Using analogous encoding method students can construct concepts and schema more easily and transfer knowledge which shares structural similarity. Progressive formalism is an effective way of introducing concepts progressively moving from concrete contexts to abstract context.The Mathematical Education. 01/2013; 52(4). - [Show abstract] [Hide abstract]

**ABSTRACT:**Through examining a representative Chinese textbook series’ presentation of the distributive property, this study explores how mathematics curriculum may structure representations in ways that facilitate the transition from concrete to abstract so as to support students’ learning of mathematical principles. A total of 319 instances of the distributive property were identified. The representational transition among these instances was analyzed at three tiers: within one worked example, from the worked example to practice problems within one topic, and across multiple topics over grades. Findings revealed four features that facilitate the transition process in the Chinese textbook series. First, it situates initial learning in a word problem context, which serves as a starting point of the transition process. Second, it sets up abstract representations as an ultimate goal of the multi-tier transition process. Third, it incorporates problem variations with connections in carefully designed tasks that embody the same targeted principles. Fourth, it engages students in constant sense making of the transition process through various pedagogical supports. Implementations and future research directions are also discussed.Educational Studies in Mathematics 05/2014; 87(1):103–121. · 0.55 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**This study explores, from both constructivist and cognitive perspectives, teacher guidance in student-centered classrooms when addressing a common learning difficulty with equivalent fractions—lines or pieces—based on number line models. Findings from three contrasting cases reveal differences in teachers' facilitating and direct guidance in terms of anticipating and responding to student difficulties, which leads to differences in students' exploration opportunity and quality. These findings demonstrate the plausibility and benefit of integrating facilitating and direct guidance in student-centered classrooms. Findings also suggest two key components of effective teacher guidance including (a) using pretraining through worked examples and (b) focusing on the relevant information and explanations of concepts. Implementations are discussed.Mathematics Education Research Journal 07/2014; 26(2):353-376.

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Teaching the Conceptual Structure of Mathematics

Lindsey E. Richland a , James W. Stigler b & Keith J. Holyoak b

a Department of Comparative Human Development, University of Chicago

b Department of Psychology, University of California, Los Angeles

Version of record first published: 25 Jul 2012

To cite this article: Lindsey E. Richland, James W. Stigler & Keith J. Holyoak (2012): Teaching the Conceptual Structure of

Mathematics, Educational Psychologist, 47:3, 189-203

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The publisher does not give any warranty express or implied or make any representation that the contents

will be complete or accurate or up to date. The accuracy of any instructions, formulae, and drug doses

should be independently verified with primary sources. The publisher shall not be liable for any loss, actions,

claims, proceedings, demand, or costs or damages whatsoever or howsoever caused arising directly or

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EDUCATIONAL PSYCHOLOGIST, 47(3), 189–203, 2012

Copyright C ?Division 15, American Psychological Association

ISSN: 0046-1520 print / 1532-6985 online

DOI: 10.1080/00461520.2012.667065

Teaching the Conceptual Structure

of Mathematics

Lindsey E. Richland

Department of Comparative Human Development

University of Chicago

James W. Stigler and Keith J. Holyoak

Department of Psychology

University of California, Los Angeles

ManystudentsgraduatefromK–12mathematicsprogramswithoutflexible,conceptualmathe-

matics knowledge. This article reviews psychological and educational research to propose that

refining K–12 classroom instruction such that students draw connections through relational

comparisons may enhance their long-term ability to transfer and engage with mathematics

as a meaningful system. We begin by examining the mathematical knowledge of students in

one community college, reviewing results that show even after completing a K–12 required

mathematics sequence, these students were unlikely to flexibly reason about mathematics.

Rather than drawing relationships between presented problems or inferences about the repre-

sentations, students preferred to attempt previously memorized (often incorrect) procedures

(Givvin, Stigler, & Thompson, 2011; Stigler, Givvin, & Thompson, 2010). We next describe

the relations between the cognition of flexible, comparative reasoning and experimentally de-

rived strategies for supporting students’ ability to make these connections. A cross-cultural

study found that U.S. teachers currently use these strategies much less frequently than their

international counterparts (Hiebert et al., 2003; Richland, Zur, & Holyoak, 2007), suggesting

that these practices may be correlated with high student performance. Finally, we articulate

a research agenda for improving and studying pedagogical practices for fostering students’

relational thinking about mathematics.

Many schools are failing to teach their students the concep-

tual basis for understanding mathematics that could support

flexible transfer and generalization. Nowhere is this lack of a

conceptual base for mathematical knowledge more apparent

than among the population of American students who have

successfullygraduatedfromhighschoolandenteredtheU.S.

community college system (Givvin, Stigler, & Thompson,

2011; Stigler, Givvin, & Thompson, 2010). These commu-

nity college students have completed the full requirements

of a K–12 education in the United States and made the moti-

vated choice to seek higher education, but typically without

the financial resources or academic scores to enter a 4-year

institution. Despite having completed high school success-

fully, based on entry measures the majority of these stu-

Correspondence should be addressed to Lindsey E. Richland, Depart-

ment of Comparative Human Development, University of Chicago, 5730 S.

Woodlawn Avenue, Chicago, IL 60637. E-mail: lrichland@uchicago.edu

dents place into “developmental” or “remedial” mathematics

courses (e.g., Adelman, 1985; Bailey, Jenkins, & Leinbach,

2005). Too often, these remedial courses then turn into barri-

ersthatimpedeprogresstowardahigherleveldegree(Bailey,

2009).

ThenumbersofcommunitycollegestudentsintheUnited

States who cannot perform adequately on basic mathemat-

ics assessments provide some insight into the questionable

efficacy of the U.S. school system. More broadly, detailed

measures of these students’ knowledge further elucidate the

ways in which K–12 educational systems (in any country)

have the potential to misdirect the mathematical thinking of

many students. We begin this article by describing the results

of detailed assessment and interview data from students in

a California community college to better understand some

longer term outcomes of a well-studied K–12 educational

system (Givvin et al., 2011; Stigler et al., 2010). To antici-

pate, the mathematics knowledge of these students appears

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RICHLAND, STIGLER, HOLYOAK

to be largely bound to specific procedures, leaving the stu-

dents ineffective at reasoning through a mathematics prob-

lem. They are apt to attempt procedures that are partially or

incorrectly recalled without regard to the reasonableness of

the solution.

We then consider what may be missing from typical

U.S. K–12 mathematics instruction, a gap that leads to such

impoverished knowledge representations. In particular, we

consider one key to developing flexible and conceptual

understanding: comparing representations and drawing

connections between them. This topic has been the focus of

a great deal of cognitive and educational research, enabling

us to forge relationships between these literatures to draw

implications for pedagogical practice. An integration of

these literatures leads us to posit the crucial roles of

developing causal structure in knowledge representations,

and in supporting students in learning to represent novel

problems as goal-oriented structured systems.

The term “conceptual understanding” has been given

many meanings, whichinturnhas contributedtodifficulty in

changing teacher practices (e.g., see Skemp, 1976). For our

purposes in this article, we rely on a framework proposed by

Hatano and Inagaki (1986), which characterizes conceptual

understandingasattainmentofanexpertlikefluencywiththe

conceptual structure of a domain. This level of understand-

ing allows learners to think generatively within that content

area, enabling them to select appropriate procedures for each

step when solving new problems, make predictions about

the structure of solutions, and construct new understandings

and problem-solving strategies. For the sake of clarity, rather

than discussing “conceptual understanding” throughout his

article, we primarily focus our review of the literature and re-

searchagendaonthegoaloffacilitatinglearners’acquisition

of the conceptual structure of mathematics.

We next turn from consideration of student knowledge to

studies of videotaped teacher practice, to examine the align-

ment between current teacher practice and the strategies we

hypothesize to be effective. We find that the practices of

American teachers often do not correspond at all well with

the strategies that we believe would promote deep learning

and acquisition of the conceptual structure of mathematics.

Finally, we consider the role that researchers can play in un-

derstanding how teachers might practicably engage students

in effective representational thinking. We lay out a research

agenda with the aim of developing strategies for facilitat-

ing students’ learning to reason about mathematics and to

generalize their mathematical knowledge.

WHAT COMMUNITY COLLEGE STUDENTS

KNOW ABOUT MATHEMATICS

Studying the U.S. mathematics instructional system pro-

vides insights into more general relationships between stu-

dentknowledge,studentcognition,andteacherpractices.We

know from international research that American students fall

far behind their counterparts in other industrialized nations,

bothonstandardizedtestsofmathematicsachievement(Gon-

zales et al., 2008) and on tests designed to measure students’

abilities to apply their knowledge to solving novel and chal-

lenging problems (Fleishman, Hopstock, Pelczar, & Shelley,

2010). We also know that the gap between U.S. students and

those in other countries grows wider as students progress

through school, from elementary school through graduation

from high school (Gonzales et al., 2008).

Many researchers have attributed this low performance in

large part to the mainly procedural nature of the instruction

American students are exposed to in school (e.g., Stigler &

Hiebert, 1999). By asking students to remember procedures

butnottounderstandwhenorwhytousethemorlinkthemto

core mathematical concepts, we may be leading our students

away from the ability to use mathematics in future careers.

Perhaps nowhere are the results of our K–12 education sys-

tem more visible than in community colleges. As previously

noted, the vast majority of students entering community col-

lege are not prepared to enroll in a college-level mathematics

class (Bailey, 2009). We know this, mostly, from their per-

formance on placement tests. But placement tests provide

only a specific type of information: They measure students’

ability to apply procedural skills to solving routine problems

but provide little insight into what students actually under-

standabout fundamental mathematicsconcepts orthedegree

to which their procedural skills are connected to understand-

ings of mathematics concepts.

American community college students are interesting be-

causetheyprovideawindowforexamininglong-termconse-

quencesofawell-studiedK–12instructionalsystem.Notev-

eryone goes to community college, of course. Some students

do not continue their education beyond the secondary level,

and some American students, through some combination of

good teaching, natural intelligence, and diligent study, learn

mathematicswellinhighschoolanddirectlyenter4-yearcol-

leges. Some community college students pass the placement

tests and go on to 4-year colleges, and some even become

mathematicians. However, we believe much can be learned

fromexaminingthemathematicalknowledgeofthatmajority

ofcommunitycollegestudentswhoplaceintodevelopmental

mathematics courses. Most of these students graduated high

school. They were able to remember mathematical proce-

dures well enough to pass the tests in middle school and high

school. But after they stop taking mathematics in school, we

can see what happens to their knowledge—how it degrades

over time, or perhaps was never fully acquired in the first

place. The level ofusable knowledge available tocommunity

college students may tell us something about the long-term

impact of the kinds of instructional experiences they were

offered in their prior schooling.

We begin by looking more closely at what developmen-

tal mathematics students in community college know and

understand about mathematics. Little is known about the

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CONCEPTUAL STRUCTURE

191

mathematical knowledge of these students. Most of what

we know was learned in two small studies—one a survey

of several hundred students at one Los Angeles area col-

lege (Stigler et al., 2010), and the other a more qualitative

interview study in which interviewers engaged students in

conversations about mathematics (Givvin et al., 2011). Both

of these studies steered clear of the typical, procedural ques-

tions asked on placement tests. Students were not asked to

multiply fractions, perform long division, or solve algebraic

equations. The questions focused instead on very basic con-

cepts: Could students, for example, place a proper fraction

on a number line, or use algebraic notation (e.g., a + b =

c) to reason about quantitative relationships? More general

questions were also asked, such as, what does it mean to do

mathematics?Webrieflysummarizesomeoftheconclusions

from these two studies.

Students View Mathematics as a Collection of

Rules and Procedures to Be Remembered

Consistent with the view that K–12 mathematics instruction

focusesprimarilyonpracticingprocedures,thesestudentsfor

the most part have come to believe that mathematics is not

a body of knowledge that makes sense and can be “figured

out.” Instead, they see mathematics as a collection of rules,

procedures, and facts that must be remembered—a task that

gets increasingly more difficult as students progress through

the curriculum.

When asked what it means to be “good at mathematics,”

77%ofstudentspresentedviewsconsistentwiththesebeliefs

(Givvin et al., 2011). Here is a sampling of what they said:

• “Math is just all these steps.”

• “In math, sometimes you have to just accept that that’s the

way it is and there’s no reason behind it.”

• “I don’t think [being good at math] has anything to do

with reasoning. It’s all memorization.”

Thisis,ofcourse,adysfunctionalviewofwhatitmeansto

do mathematics. If students don’t believe that it is possible to

reasonthroughamathematicsproblem,thentheyareunlikely

to try. And if they don’t try to reason, to connect problems

with concepts and procedures, then it is hard to imagine how

they would get very far in mathematics.

Mathematicians, naturally, see reasoning about relation-

ships as central to the mathematical enterprise (e.g., Hilbert,

1900;Polya,1954),aviewthatalsoiscommonamongmath-

ematics teachers at community colleges. When data on stu-

dents’ views of mathematics were presented to a community

college mathematics department, the faculty members were

astounded. One said, “The main reason I majored in mathe-

maticswasbecauseIdidn’thavetomemorizeit,itcouldallbe

figured out. I think I was too lazy to go into a field where you

had to remember everything.” Every one of the other faculty

members present immediately voiced their agreement.

Giventhisdisconnectbetweenthestudentsandtheircom-

munitycollegeprofessors,onemightaskwherethestudents’

views of mathematics come from, if not from their teachers?

First,itisimportanttopointoutthattheybringthisviewwith

them based on their K–12 experiences. But it also is quite

possible that students’ views of what it means to do mathe-

matics arise not from the beliefs of their teachers but from

the daily routines that define the practice of school mathe-

matics (see, e.g., Stigler & Hiebert, 1999). Unless teachers’

beliefs are somehow instantiated in daily instructional rou-

tines or made explicit in some other way, they are unlikely to

be communicated to students.

As we see later, the routines of K–12 school mathematics

emphasize repeated recall and performance of routine facts

and procedures, and these routines are supported by state

standards,assessments,andtextbooksinadditiontoteaching

practices. Although a small percentage of students do seek

meaning and do achieve an understanding that is grounded

in the conceptual structure of mathematics—and we assume

that community college mathematics faculty are among this

small percentage—the majority of students appear to exit

high school with a more limited view of what it means to do

mathematics.

Regardless of Placement, Students Are Lacking

Fundamental Concepts That Would Be Required

to Reason About Mathematics

Although the developmental mathematics students in the

studies were placed into three different levels of mathe-

matics courses—basic arithmetic, pre-algebra, or beginning

algebra—they differed very little in their understanding

of fundamental mathematics concepts. Their similarity

may not be that surprising given their procedural view of

mathematics: If mathematics is not supposed to make sense,

consisting mainly of rules and procedures that must be

memorized, then basic concepts may not be perceived as

useful. That said, the range of things these students did not

understand is surprising.

Onestudent,intheinterviews,wasaskedtoplacethefrac-

tion 4/5 on a number line. He carefully marked off a line, la-

beledthemarksfrom0to8,andthenput4/5between4and5.

Manystudentsappearedtohavefundamentalmisunderstand-

ings of fractions and decimals, not seeing them as numbers

that could be compared and ordered with whole numbers. In

the survey, students were shown the number line depicted in

Figure 1, which spanned a range from –2 to +2. They were

asked to place the numbers −0.7 and 13/8 on the number

line. Only 21% of the students could do so successfully.

Most young children know that if you add two quantities

together to get a third, the third quantity is then composed of

the original two quantities such that if you removed one you

would be left with the other. The students in these studies,

however, seemed happy to carry out their mathematics work

without connecting it to such basic ideas. In the interviews

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RICHLAND, STIGLER, HOLYOAK

FIGURE 1

13/8.From“WhatCommunityCollegeDevelopmentalMathematics

Students Understand About Mathematics,” by J. W. Stigler, K. B.

Givvin, and B. Thompson, 2010, The MathAMATYC Educator, 10,

p. 12. Copyright 2010 by American Mathematical Association of

Two-Year Colleges. Reprinted with permission.

Number line on which students were to place −.7 and

a student was asked if he could think of a way to check that

the sum of two three-digit numbers was correct. The spe-

cific example presented was 462 + 253 = 715. The student

proceeded to subtract 253 from 715 and ended up with 462.

The interviewer then asked him if he could have subtracted

the 462 instead. He did not think so; he had been told, he

said, that you subtract the second addend, not the first. But

would it make a difference, the interviewer asked? He wasn’t

sure. The interviewer told him he could try it, and he did.

He seemed genuinely surprised to find that, indeed, he could

subtract either addend to get the other.

In one final example, students were asked, “Which is

greater? a/5 or a/8. Only 53% correctly answered a/5, a

percentage that could have been obtained just by guessing.

The students were also asked to explain how they got their

answer. About one third (36%) could not come up with

an explanation (half of these had answered correctly, the

other half incorrectly). The ones who provided some sort

of explanation tended to summon some rule or procedure

from memory that they thought might do the trick. Many

of the students said that a/8 is larger because 8 is larger

than 5. Not surprisingly, another group claimed just the

opposite, having remembered that the lower the denominator

the larger the number. Some students tried to perform a

procedure: Some found common denominators, though

often they made mistakes and got the wrong answer anyway.

Others cross-multiplied (something they apparently believed

you can do whenever you have two fractions).

Only 15% of the students tried to reason it through. These

students said things such as, “If you have the same quantity

and divide it into five parts, then the parts would be larger

than if you divide it into eight parts. Assuming you have the

same number of these different-sized parts, then a/5 must

be larger.” Although it is discouraging that only 15% took

this approach, it is interesting to note that every one of these

students got the answer correct. If we could only figure out

howtoconnectsuchfundamentalideaswiththemathematics

procedures students are learning in school, the mathematical

knowledge the students acquire might be more robust.

Students Almost Always Apply Standard

Procedures, Regardless of Whether They Make

Sense or Are Necessary

Students were asked a number of questions in the interviews

that could have been answered just by thinking. As evident in

the preceding example, only a small percentage of students

tried to think their way to a solution. For some questions, just

a bit of thinking and reflection might have guided students

to use a more appropriate procedure, or to spot errors in the

procedures they did use. Rarely, however, did students take

the bait.

In one part of the interview students were presented with

a list of multiplication problems and asked to solve them

mentally:

10 × 3 =

10 × 13 =

20 × 13 =

30 × 13 =

31 × 13 =

29 × 13 =

22 × 13 =

Clearly there are many relationships across these prob-

lems, and results of previous problems could potentially be

used to derive the answers to subsequent problems. But this

was not the way in which students approached this task.

Most students just chugged through the list, struggling to

apply the standard multiplication algorithm to each problem.

Fully 77% of the students never noticed or used any rela-

tion among the different problems, preferring to work each

problem independently.

Hereisanexampleoftheanswersproducedbyonestudent

(Givvin et al., 2011):

10 × 3 = 30

10 × 13 = 130

20 × 13 = 86

30 × 13 = 120

31 × 13 = 123

29 × 13 = 116

22 × 13 = 92

In summary, most students answered most problems by

retrieving answers or procedures from memory. Many of the

procedures they used were not necessary or not appropriate

to the problem at hand. Rarely were the procedures linked

to concepts, which might have guided their use in more ap-

propriate ways. When students were asked to solve multiple

problems, they almost never made comparisons across the

problems, leading to more mistakes and fewer opportunities

to infer the principles and concepts that could make their

knowledge more stable, coherent, flexible, and usable.

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CONCEPTUAL STRUCTURE

193

Why might students have developed such an orientation

toward mathematics through their K–12 mathematics educa-

tion? Although teaching in the United States is multifaceted

and the reasons behind student success or failure are much

too complex to fully treat here, we consider in particular one

candidate explanation: Students do not view mathematics

as a system because their teachers do not capitalize on op-

portunities to draw connections between mathematical rep-

resentations. In the following sections we first expand on

what kinds of processes might be required for the develop-

ment of deep and flexible mathematical knowledge. We then

consider, based on classroom observational studies, whether

Americanstudentshaveopportunitiestoengageinthesepro-

cesses. We first consider the cognition involved in students’

comparativethinkingandtransfer,andthenweturntostudies

of teacher practices to examine alignment between pedagogy

and cognition.

LEARNING RELATIONAL STRUCTURE

THROUGH COMPARISON

It seems a safe conjecture that the very same students who

apparently found no interesting patterns within a series of

juxtaposed multiplications by the age of 13 are quite capable

of noticing other sorts of potential comparisons and learn-

ing from them. They might compare the plots of movies, the

sources of difficulty in different video games, the reasons

why various romantic relationships have succeeded or failed.

In such everyday situations people of all ages, including the

very young, spontaneously seek explanations for why things

happen, especially when faced with surprising events (e.g.,

Legare, Gelman, & Wellman, 2010). The answer to a “why”

question inevitably hinges on relational representations, par-

ticularly cause–effect relations (for a review, see Holyoak &

Cheng, 2011), or more generally (and especially in mathe-

matics), functional relations that govern whether inferences

are justified (Bartha, 2010)

Learning Schemas Via Analogical Reasoning

A causal model is a kind of schema, or mental representation

of the relational structure that characterizes a class of

situations. The acquisition of schemas is closely related to

the ability to compare situations and draw analogies based

in part on corresponding relations. Analogical reasoning

is the process of identifying goal-relevant similarities

between what is typically a familiar source analog and a

novel, less understood target, and then using the set of

correspondences, or mapping, between the two analogs to

generate plausible inferences about the latter (see Holyoak,

2012, for a review). The source may be a concrete object

(e.g., a balance scale), a set of multiple cases (e.g., multiple

problems involving balancing equations), or a more abstract

schema (e.g., balancing equations in general). The target

may be a relatively similar problem context (e.g., a balancing

equations problem with additional steps), or a more remote

analog (e.g., solving a proportion).

It has been argued that analogical reasoning is at the core

of what is unique about human intelligence (Penn, Holyoak,

& Povinelli, 2008). The rudiments of analogical reasoning

with causal relations appear in infancy (Chen, Sanchez, &

Campbell, 1997), and children’s analogical ability becomes

more sophisticated over the course of cognitive development

(Brown, Kane, & Echols, 1986; Holyoak, Junn, & Billman,

1984; Richland, Morrison, & Holyoak, 2006). Whereas very

young children focus on global similarities of objects, older

children attend to specific dimensions of variation (Smith,

1989) and to relations between objects (Gentner & Ratter-

mann, 1991).

Analogical reasoning is closely related to transfer. Cru-

cially, comparison of multiple analogs can result not only

in transfer of knowledge from a specific source analog to a

target (Gick & Holyoak, 1980) but also in the induction of

a more general schema that can in turn facilitate subsequent

transfer to additional cases (Gick & Holyoak, 1983). Peo-

ple often form schemas simply as a side effect of applying

one solved source problem to an unsolved target problem

(Novick & Holyoak, 1991; Ross & Kennedy, 1990). The

induction of such schemas has been demonstrated both in

adults and in young children (e.g., Brown et al., 1986; Chen

& Daehler, 1989; Holyoak et al., 1984; Loewenstein & Gen-

tner, 2001). Comparison may play a key role in children’s

learning of basic relations (e.g., comparative adjectives such

as “bigger than”) from nonrelational inputs (Doumas, Hum-

mel, & Sandhofer, 2008), and in language learning more

generally (Gentner & Namy, 2006). Although two examples

can suffice to establish a useful schema, people are able to

incrementally develop increasingly abstract schemas as ad-

ditional examples are provided (Brown et al., 1986; Brown,

Kane, & Long, 1989; Catrambone & Holyoak, 1989).

Why Schema Learning Can Be Hard (Especially

in Mathematics)

If humans have a propensity to use analogical reasoning to

compare situations and induce more general schemas, why

did the community college students described earlier ap-

pear not to have acquired flexible schemas for mathematical

concepts? Several issues deserve to be highlighted. As we

have emphasized, most everyday thinking focuses on un-

derstanding the physical and social environment, for which

causal relations are central (Holyoak, Lee, & Lu, 2010;

Lee & Holyoak, 2008). Not all relational correspondences

are viewed as equally important. Rather, correspondences

between elements causally related to a reasoning goal are

typically considered central (Holyoak, 1985; Spellman &

Holyoak,1996).Itseemsthatthehumanabilitytolearnfrom

analogical comparisons is closely linked to our tendency to

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RICHLAND, STIGLER, HOLYOAK

focusoncause–effectrelations,whicharethebuildingblocks

of causal models.

But by its very nature, mathematics is a formal system,

withinwhichthekeyrelationsarenot“causal”inanystraight-

forward sense. (Note that this observation applies not only

to mathematics but to other domains as well. For example, a

similar pattern of teaching and learning has been identified

in the domain of physics by Jonassen, 2010.) Worse, unless

mathematical procedures are given a meaningful interpreta-

tion, students may assume (as we have seen) that there are

no real “reasons” why the procedures work. In some sense,

the community college students we interviewed probably did

have a “schema” for multiplication, consisting of roles for

multiplicands and a product. However, lacking any mean-

ingful model of what multiplication “means” outside of the

procedure itself, the students lacked a reliable basis for find-

ing “interesting” relationships between juxtaposed problems

such as “10 × 13 = ” and “20 × 13 = ”.

In contrast, their community college professors clearly

viewed mathematics as a meaningful system, governed by an

interconnected set of relations. Though not “causal” per se,

these relations are seen as having relevance to mathematical

goals(Bartha, 2010).AsBarthaargued, thegeneral notionof

functional relevance (of which causal relations are a special

case) governs inference based on mathematics. Just as causal

relations determine the consequences of actions in the phys-

ical world, mathematical relations determine the validity of

procedures in a formal world. For example, multiplication

can be defined as repeated addition, which can be defined

in turn as the concatenation of two quantities, and quantities

can in turn be decomposed (e.g., the quantity 20 is equal to

two quantities of 10). This is the type of relational knowl-

edge required to notice, for example, that the value of 20 ×

13 has a special relationship to the value of 10 × 13. Sim-

ilarly, the professors, but not their students, understand that

numbersindecimalnotationlike–0.7andimproperfractions

like 13/8, along with integers, can all be placed on a num-

ber line because all of them are real numbers, representing

quantities along a continuum. One might say, then, that the

students and their professors have incommensurate schemas

for mathematics, in that only the latter place emphasis on

functional relations that serve to explain why various math-

ematical inferences are valid.

Clearly, simply solving sequences of math problems is no

guarantee that the student will end up deeply understanding

the conceptual structure of mathematics. Even in nonmath-

ematical domains, simply providing multiple examples does

not ensure formation of a useful schema. If two examples are

juxtaposed but processed independently, without relational

comparison, learning is severely limited (Gentner, Loewen-

stein, & Thompson, 2003; Loewenstein, Thompson, & Gen-

tner, 2003). Even when comparison is strongly encouraged,

some people will fail to focus on the goal-relevant functional

relations and subsequently fail on transfer tasks (Gick &

Holyoak, 1983). When mathematics problems are embedded

in specific contexts, details shared by different contexts are

likelytoendupattachedtothelearnedprocedure,potentially

limiting its generality. For example, people tend to view ad-

dition as an operation that is used to combine categories at

the same level in a semantic hierarchy (e.g., apples and or-

anges, not apples and baskets; Bassok, Pedigo, & Oskarsson,

2008), because word problems given in textbooks always re-

spect this constraint. At an even more basic level, analogical

transfer is ultimately limited by the reasoner’s understanding

of the source analog (Bartha, 2010; Holyoak et al., 2010). If

every solution to a math problem is viewed as “just all these

steps”with“noreasonbehindit,”simplycomparingmultiple

examples of problems (that to the student are meaningless)

will not suffice to generate a deep schema.

Thus, although relational comparisons can in principle

foster induction of flexible mathematical knowledge, many

pitfalls loom large. The teacher needs to introduce source

analogs that “ground out” formal mathematical operations

in domains that provide a clear semantic interpretation (e.g.,

introducing the number line as a basic model for concepts

and operations involving continuous quantities). Moreover,

even if a good source analog is provided, relational compar-

isons tax limited working memory (Halford, 1993; Hummel

& Holyoak, 1997, 2003; Waltz, Lau, Grewal, & Holyoak,

2000). In general, any kind of intervention that reduces

working-memory demands and helps people focus on goal-

relevant relations will aid learning of effective problem

schemas and thereby improve subsequent transfer to new

problems.

For example, Gick and Holyoak (1983) found that induc-

tion of a schema from two disparate analogs was facilitated

when each analog included a clear statement of the under-

lying solution principle. In some circumstances, transfer can

also be improved by having the reasoner generate a prob-

lem analogous to an initial example (Bernardo, 2001). Other

work has shown that abstract diagrams that highlight the ba-

sic solution principle can aid in schema induction and subse-

quent transfer (Beveridge & Parkins, 1987; Gick & Holyoak,

1983). Schema induction can also be encouraged by a pro-

cedure termed “progressive alignment”: providing a series

of comparisons ordered “easy to hard,” where the early pairs

share salient similarities between mapped objects as well as

lesssalientrelationalcorrespondences(Kotovsky&Gentner,

1996).Moregenerally,tounderstandthepotentialroleofana-

logical reasoning in education, it is essential to consider ped-

agogical strategies for supporting relational representations

and comparative thinking. Next we consider several such

pedagogical strategies, including highlighting goal-relevant

relations in the source analog, introducing multiple source

representations, and using explicit verbal and gestural cues

to draw attention to relational commonalities and differences

(see also Schwartz, Chase, & Bransford, 2012/this issue; and

Chi & VanLehn, 2012/this issue).

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CONCEPTUAL STRUCTURE

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How American Teachers Introduce Mathematical

Relations

Are teachers invoking these and related strategies in U.S.

mathematics instruction, either explicitly or implicitly? Al-

though the list of potential “best practices” in mathematics

is long and varied, there is general agreement about the im-

portance of drawing connections and supporting student rea-

soning. The National Council of Teachers of Mathematics

(NCTM) has issued strong recommendations in this vein,

publishing a new series of books for high school mathemat-

ics under the titled theme of Reasoning and Sense-Making

(NCTM, 2009). They define “reasoning” broadly, including

any circumstance in which logical conclusions are drawn

on the basis of evidence or stated assumptions, from infor-

mal explanations to deductive and inductive conclusions and

formal proofs (p. 19). Sense making is characterized as the

interrelated but more informal process of developing under-

standing of a situation, context, or concept by connecting it

with existing knowledge (p. 19). Based on reviews of educa-

tional research in mathematics and mathematics education,

the authors explore the following theme throughout the main

volume in this series as well as in books with specific cur-

riculum foci:

Reasoning and sense making are the cornerstones of mathe-

matics. Restructuring the high school mathematics program

aroundthemenhancesstudents’developmentofboththecon-

tentandprocessknowledgetheyneedtobesuccessfulintheir

continuing study of mathematics and in their lives. (p. 19)

These themes, though under the different title of “Focal

Topics in Mathematics,” are also central to their description

of high quality elementary instruction (NCTM, 2006).

Thus, there is growing consensus in both the psychologi-

calandeducationalresearchliteraturesthatteachingstudents

effectively requires teaching them to reason with mathemat-

ics. Further, there is agreement that this aim necessitates

drawing connections and fostering students’ awareness that

mathematicsisasensiblesystem,onethatcanbeapproached

using the student’s broad repertoire of “sense making,” in-

cluding causal and analogical thinking strategies. Approach-

ingmathematicsinthiswayenablesstudentstodevelopbetter

structuredknowledgerepresentationsthatmaybemoreeasily

remembered and used more flexibly in transfer contexts—to

solvenovelproblems,tonoticemathematicallyrelevantcom-

monalities and differences between representations, and to

reason through mathematics problems when one cannot re-

member a procedure.

Although drawing connections and sense making do not

guarantee transfer, these are cognitive routines that lead to

schema acquisition and knowledge representations that sup-

port transfer. Positive transfer will be facilitated by noticing

similarities between two or more representations or objects.

INTERNATIONAL VARIATIONS IN STUDENTS’

OPPORTUNITIES FOR LEARNING TO DRAW

CONNECTIONS IN MATHEMATICS

Hiebert and Grouws (2007) conducted a meta-analysis of all

studiesinwhichfeaturesofteachingwereempiricallyrelated

to measures of students’ learning. They found that two broad

features of instruction have been shown to promote students’

understanding of the conceptual structure of mathematics.

First,teachersandstudentsmustattendexplicitlytoconcepts,

“treating mathematical connections in an explicit and public

way” (p. 384). According to Hiebert and Grouws, this could

include

discussingthemathematicalmeaningunderlyingprocedures,

asking questions about how different solution strategies are

similartoanddifferentfromeachother,consideringtheways

in which mathematical problems build on each other or are

special (or general) cases of each other, attending to the

relationships among mathematical ideas, and reminding stu-

dents about the main point of the lesson and how this point

fits within the current sequence of lessons and ideas. (p. 384)

The second feature associated with students’ understand-

ing of mathematics’ conceptual structure is struggle: Stu-

dents must spend part of each lesson struggling to make

sense of important mathematics. Hiebert and Grouws de-

fined “struggle” to mean “students expend effort to make

sense of mathematics, to figure out something that is not

immediately apparent” (p. 387). Thus, students must expend

effort to make connections between mathematical problems

and the concepts and procedures that can be marshaled to

solve them. Note that Hiebert and Grouws did not identify

any single strategy for achieving these learning experiences

inclassrooms,pointingoutthattherearemanywaysofdoing

so. And clearly, not all struggle is good struggle. The point

they made is simply that connections must be made by the

student (i.e., they cannot be made by the teacher for the stu-

dent) and the making of these connections will require effort

on the student’s part.

Corroborationoftheseconclusionscomesfromthelargest

studies ever conducted in which mathematics classrooms

havebeenvideotapedindifferentcountries,theTIMSSvideo

studies. Two studies were conducted: the first in 1995 in

Germany, Japan, and the United States (Stigler & Hiebert,

1999), and the second in 1999 in seven countries: Australia,

the Czech Republic, Hong Kong, Japan, the Netherlands,

Switzerland, and the United States (Gonzales et al., 2008;

Hiebert et al., 2003). In each country, a national probability

sample of approximately 100 teachers was videotaped teach-

ing a single classroom mathematics lesson. An international

team of researchers collaboratively developed and reliably

coded all lessons to gather data about average teaching prac-

tices across and within countries.

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RICHLAND, STIGLER, HOLYOAK

One goal of these studies was to try to find features of

teachingthatmightdifferentiatethehigh-achievingcountries

(in general, all except the United States and Australia in the

preceding list) from the low-achieving countries. Of interest,

the findings fit nicely with the conclusions of Hiebert and

Grouws (2007) and also help to explain why we see the kind

ofoutcomesjustreportedinthestudiesofcommunitycollege

developmental mathematics students. Many surface features

of teaching did not appear associated with cross-national dif-

ferences in student achievement. For example, among the

high-achieving countries, there were countries that empha-

sized teacher lecture as the primary mode of instruction, and

countries that tended to have students work independently or

in groups on learning assignments. There were countries that

used many real-world problems in their mathematics classes,

and countries that dealt almost completely with symbolic

mathematics. None of these simple variations could explain

differences in student outcomes.

Finding common features among the high-achieving

countries required looking more closely at what was hap-

pening in the lessons. It was neither the kinds of problems

presented nor teaching style employed that differentiated the

high-achieving countries from the others, but the kinds of

learning opportunities teachers created for students, namely,

making explicit connections in the lesson among mathemat-

ics procedures, problems, and concepts and finding ways

to engage students in the kind of productive struggle that

is required to understand these connections in a deep way.

The ways that teachers went about creating these learning

opportunities differed from country to country. Indeed, an

instructional move that inspires a Japanese student to engage

might not have the same effect on a Czech student, and vice

versa, due to the different motivational beliefs, attitudes, in-

terests, and expectations students in different cultures bring

to the task at hand. But the quality of the learning opportu-

nities teachers were able to create did seem to be common

across the high-achieving countries.

This conclusion was based on an analysis of the types of

problems that were presented, and how they were worked on,

in different countries. Across all countries, students spend

about 80% of their time in mathematics class working on

problems, whether independently, as part of a small group,

or as part of the whole class. The beginning and end of each

problem was identified as it was presented and worked on in

the videos. The types of problems presented were character-

ized, as was how each was worked on during the lesson.

The two most common types of problems presented were

categorized as Using Procedures and Making Connections.

Using Procedures problems, by far the most common across

allcountries,involvedaskingastudenttosolveaproblemthat

they already had been taught to solve, applying a procedure

they had been taught to perform. This is what is typically

regarded as “practice.” Take, for example, a lesson to teach

students how to calculate the interior angles of a polygon.

If the teacher has presented the formula [180 × (number of

sides – 2)], and then asks students to apply the formula to

calculatethesumsoftheinterioranglesoffivepolygons,that

would be coded as Using Procedures. If, however, a teacher

asks students to figure out why the formula works, to derive

the formula on their own, or to prove that the formula would

work for any polygon, that would be coded as a Making

Connections problem. A problem like this has the potential

for both struggle and for connecting students with explicit

mathematical concepts.

The percentage of problems presented in each country

that were coded as Using Procedures versus Making Con-

nections is presented in Figure 2. As is evident in the figure,

there was great variability across countries in the percentage

FIGURE 2 Percentage of problems that were coded Using Procedures and Making Connections.

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CONCEPTUAL STRUCTURE

197

of problems of each type. All countries have some Making

Connections problems, though only Japan has more Making

Connections problems than Using Procedures problems.

Clearly, just presenting more Making Connections problems

does not appear to be related to student achievement. Two

of the highest achieving countries in the group are Hong

Kong and Japan. Hong Kong has the lowest percentage of

Making Connections problems, and Japan has the highest.

The United States, it is interesting to note, falls in between

Hong Kong and Japan. This pattern suggests that curriculum

changealone(e.g.,increasingthepercentageofMakingCon-

nections problems in a textbook) will not necessarily result

in improved learning.

A more compelling pattern emerges when we examine, in

thevideos,howthepresentedproblemswereactuallyworked

on in the lesson. Although “struggle” per se was not coded,

each problem was coded a second time to determine whether

the teacher and students engaged with the problem in a way

that required them to grapple with concepts or draw connec-

tions, or whether the teacher or students changed the activ-

ity to reduce the conceptual demand. As evidenced by the

data, once a Making Connections problem was presented, it

was often changed, by the teacher into something else, most

commonly a Using Procedures problem. In other words, just

becauseaproblemhasthepotentialtoengagestudentsinpro-

ductive struggle with mathematics concepts, it will not nec-

essarily achieve that potential. For example, a teacher might

give additional instruction or a worked example to aid the

students in solving the Making Connections problem, which

means that the activity becomes only practice for students.

In the United States, one of the reasons that problems do

notsucceedinengagingstudentsinproductivestruggleisthat

the students push back! Teaching is a complex system, and

teachingroutinesaremultiplydetermined.Ateachermayask

students why, for example, the equation for finding the sum

of the interior angles of a convex polygon works. But stu-

dents may disengage at this point, knowing that the reasons

why will not be on the final exam. Reasons why also may be

misalignedwiththestudents’emergingsenseofwhatmathe-

matics is all about: a bunch of procedures to be remembered.

Cultural and individual views of the nature of intelligence

and learning, specifically as they relate to mathematics, and

related processes such as stereotype threat, sense of belong-

ing,andself-efficacy,mayunderminestudents’motivationto

engage in persistent effort toward achieving a mathematics

learning goal (see, e.g., Blackwell, Trzesniewski, & Dweck,

2007; Dweck & Leggett, 1988; Heine et al., 2001; Walton &

Cohen, 2007, 2011).

Butteachingpracticealsomaybelimitedbyteachers’own

epistemological beliefs about mathematics and how to learn

it. Although many K–12 teachers espouse the importance

of teaching for “conceptual understanding,” the meaning of

this phrase has quite variable interpretations (see Skemp,

1976). Because the ability to successfully complete math-

ematics problems requires both conceptual and procedural

skills, teachers regularly find these difficult to distinguish,

and may define conceptual understanding and successful

learning as comfort with procedures. For this reason, again

we find it useful to articulate our hypothesis that students

will be best served by learning to represent mathematics as

a system of conceptual relationships in which problems and

concepts must be connected.

Figure 3 presents just the Making Connections problems,

showing the percentage of problems that were actually

FIGURE 3

up to 100% because there were other transformations that sometimes occurred.

Percentage of making-connections problems that were implemented as Using Procedures and Making Connections. Note: Bars do not add

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RICHLAND, STIGLER, HOLYOAK

implemented as Making Connections problems versus those

that were transformed into Using Procedures problems.

Consider Hong Kong and Japan: Whereas they looked com-

pletelydifferentwhencomparingthepercentageofproblems

presented, they look very similar when we look just at how

the Making Connections problems are implemented. Both

Hong Kong and Japan, and most of the other countries too,

are able to realize the full potential of Making Connections

problems approximately half the time. The United States,

now, is the outlier. Virtually all of the Making Connections

problems presented in the United States were transformed

into Using Procedures problems, or something requiring

even less student conceptual participation. (The reason the

percentages do not add up to 100 is that teachers sometimes

did other things with Making Connections problems, e.g.,

just giving the students the answer without allowing them

the opportunity to figure it out.)

Thekindsofcomparisonprocessesthatwouldberequired

forconceptuallearningofmathematicswouldtendtohappen

duringtheseMakingConnectionsproblems.Butforavariety

of reasons, such processes do not occur, at least with much

frequency, in the U.S. classrooms.

Similar patterns were revealed in smaller scale, more de-

tailed analyses of subsets of the TIMSS video data (Rich-

land, Holyoak, & Stigler, 2004; Richland, Zur, & Holyoak,

2007). Richland et al. (2007) focused specifically on struc-

tured analogies, or opportunities for drawing connections

and comparative reasoning. These investigators examined a

subset of the United States, Hong Kong, and Japanese video-

taped lessons to identify teacher practices in using and sup-

porting students in making comparisons between problems,

representations, or concepts. These included opportunities

for comparisons between problems (e.g., “These are both

division problems but notice this one has a remainder”) be-

tween mathematical concepts (e.g.,between convex andcon-

cave polygons), between mathematics and nonmathematics

contexts (e.g., “an equation is like a balancing scale”), or

between multiple student solutions to a single problem.

Every instance identified as a comparison was coded to

reveal teachers’ strategies for supporting students in draw-

ing the connections intended by the teacher. An international

team coded the videos, with native speakers from each coun-

try, yielding high reliability across all codes. Because (as

previouslydiscussed)thecognitivescienceliteratureoncom-

parative reasoning indicates that novices in a domain often

fail to notice or engage in transfer and comparative thinking

without explicit cues or support, the codes were designed to

determine the extent to which teachers were providing such

aids. The codes were developed based on the cognitive sci-

ence literature and on teacher practices observed in other

TIMSS videotaped lessons, in an iterative fashion.

Specifically, the codes measured teacher instructional

practices that could be expected to encourage learners to

draw on prior causal knowledge structures and reduce work-

ing memory processing load. The codes assessed the pres-

ence or absence of the following teacher practices during

comparisons: (a) using source analogs likely to have a fa-

miliar causal structure to learners (vs. comparing two new

types of problems or concepts), (b) producing a visual rep-

resentation of a source analog versus only a verbal one (e.g.,

writing a solution strategy on the board), (c) making a visual

representation of the source analog visible during compari-

son with the target (e.g., leaving the solution to one problem

on the board while teaching the second, related problem), (d)

spatially aligning written representations of the source and

targetanalogstohighlightstructuralcommonalities(e.g.,us-

ingspatialorganizationoftwoproblemsolutionsontheboard

to identify related and unrelated problem elements), (e) us-

ing gestures that moved comparatively between the source

and target analogs, and (f) constructing visual imagery (e.g.

drawing while saying “consider a balancing scale”).

Teachers in all countries invoked a statistically similar

number of relational comparisons (means of 14–20 per les-

son). (These are different from the numbers of Making Con-

nections problems identified in the analysis described previ-

ously,astheseincludedalsoadditionaltypesofopportunities

for drawing relationships.) Of interest, the data revealed that

the U.S. teachers were least likely to support their students

in reasoning comparatively during these learning opportuni-

ties. These findings were highly similar qualitatively to those

from the overall TIMSS results, suggesting that U.S. teach-

ers are not currently capitalizing on learning opportunities

(i.e., opportunities for comparison) that they regularly evoke

within classroom lessons. Both teachers in Hong Kong and

Japanusedallofthecodedsupportstrategiesmoreoftenthan

did the U.S. teachers. As shown in Figure 4, some strategies

were used frequently, others less often, but the Asian teach-

ers were always more likely to include one or more support

strategies with their comparisons than were teachers in the

United States.

Overall, these data suggest that although the U.S. teachers

are introducing opportunities for their students to draw con-

nections and reason analogically, there is a high likelihood

that the students are not taking advantage of these opportu-

nities and are failing to notice or draw the relevant structural

connections. At this point, we have come full circle in our

discussion and return back to the students with whom we

started. Community college developmental mathematics stu-

dents don’t see mathematics as something they can reason

their way through. For this reason, and no doubt other rea-

sons as well, they do not expend effort trying to connect the

procedures they are taught with the fundamental concepts

that could help them understand mathematics as a coherent,

meaningful system. The roots of their approach to mathe-

matics can be seen in K–12 classrooms, where, it appears,

teachers and students conspire together to create a mathe-

matics practice that focuses mostly on memorizing facts and

step-by-stepprocedures.Weknowfromresearchinthelearn-

ing sciences what it takes to create conceptual coherence and

flexible knowledge representations that support transfer. But

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CONCEPTUAL STRUCTURE

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020406080100

F

E

D

C

B

A

Percent of Analogies

FIGURE 4

supports: (A) visual and mental imagery, (B) comparative gesture,

(C) visual alignment, (D) use of a familiar source, (E) source visible

concurrently with target, (F) source presented visually. Note. White

denotes U.S. teachers, gray denotes Chinese teachers, and black

denotes Japanese teachers. From “Cognitive Supports for Analogy

in the Mathematics Classroom,” by L. E. Richland, O. Zur, and K.

J. Holyoak, 2007, Science, 316, p. 1128. Copyright 2007 by the

American Association for the Advancement of Science. Reprinted

with permission.

Percentage of analogies by region containing cognitive

we have found it difficult to implement such ideas in class-

rooms. Where do we go from here?

HOW MIGHT TEACHERS BETTER SUPPORT

STUDENTS IN SYSTEMATIC MATHEMATICAL

THINKING? A RESEARCH AGENDA

Thewell-establishedcognitiveliteratureonlearningbystruc-

tured comparisons, together with our analysis of current

teacher practice and student outcomes in mathematics in-

struction, provides insight into strategies that might better

leverage students’ reasoning capacities to lead to meaningful

understandingofmathematics.Weproposeseveraldirections

for research that would develop a foundation for integrating

these ideas into classroom teaching. The first are experimen-

tal studies directly investigating the relationship between the

pedagogicalpracticesofcomparisonandanalogyandstudent

learning for flexible, transferable mathematics. Second, we

outline the importance of understanding teacher and student

epistemologies of mathematics in general, and epistemolo-

gies of comparison and cognition more specifically. This is

essential to understanding the origin of the current problem

and to develop recommendations that are likely to have an

impactonpractice.Third,wecallforresearchonprofessional

developmentstrategiesbecausetheproblemofhowtoimpact

teacher routines, particularly in this area of supporting stu-

dents’connected,transferablethinking,haveprovendifficult.

Classroom Efficacy Tests of Strategies for

Supporting Comparisons

Although the TIMSS 1999 video data results just reviewed

are provocative, they do not allow us to make causal infer-

ences about the relationship between teacher practices and

student learning. Several projects have begun to experimen-

tally test the strategies for supporting comparisons that were

identified as more frequent in the high-achieving countries

(e.g., Richland & McDonough, 2010). So far, this work has

found that using a combination of the most common sup-

port cues invoked by teachers in Japan and Hong Kong was

not necessary to teach basic memorization and use of an

instructed strategy, but these cues did increase students’ flex-

ibility and ability to identify relevant similarities and differ-

ences between instructed problems and transfer problems.

We and other research groups are addressing the question

of how to best design and support instructional comparisons.

Our team is using controlled videotaped presentation of var-

ied instruction, whereas other research groups are designing

tools that aid teachers in leading instruction by comparisons

as well as studying comparisons made by peers (see Rittle-

Johnson & Star, 2007, 2009; Star & Rittle-Johnson, 2009).

More work is needed to investigate strategies for optimiz-

ing teachers’ current use of problems and comparisons that

could be used to encourage students to draw connections and

reason meaningfully about mathematics.

Specifically, one of the strategies that needs further re-

search is to better understand how students’ prior knowl-

edge structures are related to the types of representations and

comparisons that are of most use in supporting sense making

andrelationalreasoning.Adequatepriorknowledgeisessen-

tial for reasoning by comparison, primarily because without

awareness of the fundamental elements of a representation,

one cannot hope to discern the important structural corre-

spondences and draw inferences on that basis (e.g., Gentner

& Rattermann, 1991; Goswami, 2002). Yet surprisingly, us-

ing very familiar source analogs was the comparison support

strategy identified in the TIMSS video studies that was em-

ployedproportionallyleastfrequentlybyteachersinallcoun-

tries. As reviewed earlier, the lack of well-structured knowl-

edge about the source will limit students’ schema formation

and generalization from the target, as they are simultane-

ously acquiring and reasoning about the causal structure of

both the source and target analogs. At minimum, the practice

ofusingunfamiliarsourceanalogswillimposehighcognitive

demands on the learner, making the additional supports for

cognitive load even more important to ensure that students

have sufficient resources to grapple with the relationships

between the two problems.

Despite the challenges of drawing inferences from a rela-

tively unfamiliar source analog, the literature is not clear as

to whether generalization from two less well-known analogs

can be as effective as between a known and less well-known

analog, assuming the learner has access to optimal supports

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for causal thinking and sense making. Providing multiple

representations certainly can be helpful, even when the do-

main is fairly novel, through a kind of analogical scaffolding

(Gick & Holyoak, 1983).

However, this may vary depending on the background

knowledge of the learner. Rittle-Johnson, Star, and Durkin

(2009) found that general algebraic knowledge about manip-

ulating equations predicted whether students benefited at all

from being taught through comparison between two solution

strategies. Those who began instruction with better initial

algebra intuitions about procedures for balancing equations

(even if the procedures were not executed properly) bene-

fited from this type of comparison, whereas those who were

less prepared benefited more from serial instruction about

two problems without explicit support for comparison, or

from comparisons between two problem types. These stu-

dents were working in collaborative pairs of peers, so those

who began the lesson without adequate knowledge may not

have had the level of support necessary to surmount the dif-

ficulty of aligning and mapping the representations, but it is

not clear what types of supports would have been sufficient.

Kalyuga (2007) proposed an “expertise reversal effect”

for the role of cognitive load. This could be interpreted to

imply that until students have adequate knowledge, they will

benefit from all possible efforts to reduce cognitive load, in-

cluding reducing the instructional objective to have students

encode the structure of a new representation. Once students

have more expertise, however, they will gradually be able to

handle more cognitive load and may actually benefit from

more effortful work to align and map between source and

target analogs. Thus, the optimal level and role of teacher

supports for relational thinking and sense making may shift

over the course of students’ learning (cf. Koedinger &

Roll, 2012).

Overall, research is necessary to better understand the

role of individual differences in prior knowledge and opti-

mal relational learning conditions. Relational thinking and

alignment between prior conceptual knowledge and new

representations may be a way to characterize an impor-

tant element of the more general construct “struggle” as

described by Hiebert and Grouws (2007). According to

this construct, the level of struggle must be attenuated

based on students’ level of prior knowledge so that the re-

quirement to reason causes struggle, yet the challenge is

surmountable.

Although theoretically a very powerful framework for

understanding the relationship between student learning

needs and instructional content, one can imagine that

this level of flexible instruction may be very challenging

for teachers. In particular, learning to use such strategies

is difficult for novice teachers (Stein, Engle, Smith, &

Hughes, 2008), and much more research is needed to

better understand teachers’ beliefs about comparisons and

students’ analogical reasoning.

Teacher Knowledge and Professional

Development

The instructional strategies we have discussed to this point

will be heavily reliant on a teacher who orients to mathemat-

ics as a meaningful system and is able to flexibly vary his or

herinstructionbasedondiagnosisofstudents’currentknowl-

edgestates.Thereareseveralpartstothisdescriptionofahy-

pothesized ideal teacher that may be important to understand

before we can know how to realistically integrate cognitive

principles of comparison into classroom instruction.

The first pertains to the structured organization of teacher

knowledge and beliefs about the role of connections in math-

ematics learning. In the community college sample, there

was a clear distinction between the professors’ and students’

orientations to mathematics, with the professors viewing

mathematics as more of a meaningful system than their stu-

dents. K–12 mathematics teachers may be more similar to

their students in their stored knowledge systems of mathe-

matics, however, appearing more focused on rules (Battista,

1994; Schoenfeld, 1988). Several measures have been de-

signed to assess teacher knowledge about mathematics con-

tent and about students’ mathematical thinking (e.g., Hill,

Ball, & Shilling, 2008; Hill, Schilling, & Ball, 2004; Kerst-

ing, Givvin, Sotelo, & Stigler, 2010), yet we need to learn

moreaboutteachers’beliefsandknowledgeaboutmathemat-

ics as a system, specifically with respect to the roles of multi-

ple representations and drawing connections among content.

In particular, it will be important to try to discover where

students acquire their belief that mathematics is a series of

memorized rules.

International studies suggest that despite variability in

teacher expertise in the domain within the United States,

there may still be differences in the ways that the mathe-

matical knowledge of American teachers is organized when

compared with either mathematics domain experts or with

teachersinothercounties,particularlywithrespecttotherole

of interconnections within the content. Ma (1999) found that

the U.S. mathematics teachers had taken more mathemat-

ics courses than the average Chinese mathematics teachers

in her sample, but the Chinese teachers’ representations of

mathematics were far more systematic, interconnected, and

structurally organized. The U.S. teachers tended to represent

the mathematics curriculum as linearly organized, whereas

theChineseteachers’representationsofthecurriculummore

closely resembled a web of connections. Further research in-

ternationally as well as within the United States may better

reveal teachers’ underlying conceptualizations of mathemat-

ics, with particular attention to the role of interconnections

and meaningful systems of relationships, in much the same

way we have gathered information from the community col-

lege students (Givvin et al., 2011). Greater understanding

of teachers’ knowledge of mathematics may aid in develop-

ing procedures or tools to better facilitate teacher practices

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Page 14

CONCEPTUAL STRUCTURE

201

and to optimize the effectiveness of comparison strategies as

pedagogical tools.

Finally, we join the NCTM and other mathematics teacher

educators in calling for further research on professional de-

velopment strategies for promoting a conceptual shift for

teachers from teaching mathematics as memorization of pro-

cedurestoastructuredsystemofgoal-orientedproblemsolv-

ing. We in particular emphasize the need for professional

development tosupport teachers inlearning how torepresent

problems as goal-oriented systems that can be connected

meaningfully to other problems, representations, and con-

cepts. As we have identified in the TIMSS analyses, U.S.

teachers are not currently leveraging opportunities for draw-

ing connections and thereby encouraging students to orga-

nize their knowledge around mathematical relationships. We

require research to better understand how to provide such

knowledge to teachers in a way that is usable. In addition, it

mayproveusefultosupportteachersthroughbettertextbooks

and resource tools that include more connected, comparison-

based suggested instruction.

In sum, we posit that leveraging students’ reasoning skills

duringK–12mathematicsinstructionmaybeacrucialwayto

enhance their ability to develop usable, flexible mathematics

knowledge that can transfer to out-of-school environments.

U.S. teachers are not currently providing most students with

opportunities to develop meaningful knowledge structures

formathematics,asrevealedbystudiesofcommunitycollege

students’ mathematical skills and video-based observations

of teacher practices. Cognitive scientific research on chil-

dren’s causal and relational thinking skills provides insights

into strategies for supporting students in gaining more sen-

sible, meaning-driven representations of mathematics. How-

ever,moreresearchisnecessarytodeterminehowtheseideas

may become effectively integrated into classroom teaching.

ACKNOWLEDGMENTS

Preparation of this article was supported by the Insti-

tute of Education Sciences (Award R305C080015), the

Spencer Foundation, and the Office of Naval Research

(N000140810186). This article is also based on work sup-

ported by National Science Foundation grant 0954222. We

thank Belinda Thompson and Karen Givvin for helpful con-

versations. Robert Goldstone and Dan Schwartz provided

valuable comments on an earlier draft.

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