text:
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.
- [Show abstract] [Hide abstract]
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.
Page 1
This article was downloaded by: [University of Chicago]
On: 29 August 2012, At: 11:01
Publisher: Routledge
Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer
House, 37-41 Mortimer Street, London W1T 3JH, UK
Educational Psychologist
Publication details, including instructions for authors and subscription information:
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
To link to this article: http://dx.doi.org/10.1080/00461520.2012.667065
Full terms and conditions of use: http://www.tandfonline.com/page/terms-and-conditions
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.
Page 2
Copyright C ?Division 15, American Psychological Association
ISSN: 0046-1520 print / 1532-6985 online
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
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
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: email@example.com
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-
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
Downloaded by [University of Chicago] at 11:01 29 August 2012
Page 3
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
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-
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.
Studying the U.S. mathematics instructional system pro-
vides insights into more general relationships between stu-
know from international research that American students fall
far behind their counterparts in other industrialized nations,
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
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-
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
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
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
Downloaded by [University of Chicago] at 11:01 29 August 2012
Page 4
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
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
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,"
(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."
do mathematics. If students don't believe that it is possible to
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,
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-
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.
views of mathematics come from, if not from their teachers?
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
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
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.
tion 4/5 on a number line. He carefully marked off a line, la-
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
Downloaded by [University of Chicago] at 11:01 29 August 2012
Page 5
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
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
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.
(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.
Downloaded by [University of Chicago] at 11:01 29 August 2012
Page 6
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
cesses. We first consider the cognition involved in students'
of teacher practices to examine alignment between pedagogy
and cognition.
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 &
analogical comparisons is closely linked to our tendency to
Downloaded by [University of Chicago] at 11:01 29 August 2012
Page 7
of causal models.
But by its very nature, mathematics is a formal system,
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
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
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
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
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
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).
Downloaded by [University of Chicago] at 11:01 29 August 2012
Page 8
How American Teachers Introduce Mathematical
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
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-
effectively requires teaching them to reason with mathemat-
ics. Further, there is agreement that this aim necessitates
drawing connections and fostering students' awareness that
using the student's broad repertoire of "sense making," in-
cluding causal and analogical thinking strategies. Approach-
remembered and used more flexibly in transfer contexts—to
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.
Hiebert and Grouws (2007) conducted a meta-analysis of all
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.
"treating mathematical connections in an explicit and public
way" (p. 384). According to Hiebert and Grouws, this could
asking questions about how different solution strategies are
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
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.
studies ever conducted in which mathematics classrooms
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.
Downloaded by [University of Chicago] at 11:01 29 August 2012
Page 9
One goal of these studies was to try to find features of
(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
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
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
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.
Downloaded by [University of Chicago] at 11:01 29 August 2012
Page 10
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
nections problems in a textbook) will not necessarily result
in improved learning.
A more compelling pattern emerges when we examine, in
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
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
the students push back! Teaching is a complex system, and
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
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-
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).
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
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
Downloaded by [University of Chicago] at 11:01 29 August 2012
Page 11
implemented as Making Connections problems versus those
that were transformed into Using Procedures problems.
Consider Hong Kong and Japan: Whereas they looked com-
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.)
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
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
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-
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
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
ing sciences what it takes to create conceptual coherence and
flexible knowledge representations that support transfer. But
Downloaded by [University of Chicago] at 11:01 29 August 2012
Page 12
Percent of Analogies
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?
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
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
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
teacher routines, particularly in this area of supporting stu-
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
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-
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
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
Downloaded by [University of Chicago] at 11:01 29 August 2012
Page 13
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
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
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
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
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
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
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
Downloaded by [University of Chicago] at 11:01 29 August 2012
Page 14
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-
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
and resource tools that include more connected, comparison-
based suggested instruction.
In sum, we posit that leveraging students' reasoning skills
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
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-
may become effectively integrated into classroom teaching.
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.
ington, DC: United States Department of Education.
Bailey, T. (2009). Challenge and opportunity: Rethinking the role and func-
tion of developmental education in community college. New Directions
for Community Colleges, 145, 11–30.
Bailey, T., Jenkins, D., & Leinbach, T. (2005). Community college low-
ers College, Community College Research Center.
Bartha, P. (2010). By parallel reasoning: The construction and evaluation
of analogical arguments. New York, NY: Oxford University Press.
Bassok, M., Pedigo, S. F., & Oskarsson, A. T. (2008). Priming addition facts
with semantic relations. Journal of Experimental Psychology: Learning,
Memory, and Cognition, 34, 343–352.
Battista, M. T. (1994). Teacher beliefs and the reform movement in mathe-
matics education. Phi Delta Kappan, 75, 462–470.
Bernardo, A. B. I. (2001). Principle explanation and strategic schema ab-
straction in problem solving. Memory & Cognition, 29, 627–633.
Beveridge, M., & Parkins, E. (1987). Visual representation in analogical
problem solving. Memory & Cognition, 15, 230–237.
Blackwell, L. A., Trzesniewski, K. H., & Dweck, C. S. (2007). Theories
of intelligence and achievement across the junior high school transi-
tion: A longitudinal study and an intervention. Child Development, 78,
Brown, A. L., Kane, M. J., & Echols, C. H. (1986). Young children's mental
models determine analogical transfer across problems with a common
goal structure. Cognitive Development, 1, 103–121.
Brown, A. L., Kane, M. J., & Long, C. (1989). Analogical transfer in young
children: Analogies as tools for communication and exposition. Applied
Cognitive Psychology, 3, 275–293.
Catrambone, R., & Holyoak, K. J. (1989). Overcoming contextual limita-
tions on problem-solving transfer. Journal of Experimental Psychology:
Learning, Memory, and Cognition, 15, 1147–1156.
Chen, Z., & Daehler, M. W. (1989). Positive and negative transfer in ana-
logical problem solving by 6-year-old children. Cognitive Development,
4, 327–344.
Chen, Z., Sanchez, R. P., & Campbell, T. (1997). From beyond to within
their grasp: The rudiments of analogical problem solving in 10- and 13-
month-olds. Developmental Psychology, 33, 790–801.
Chi, M. T. H., & VanLehn, K. A. (2012/this issue). Seeing deep structure
from the interactions of surface features. Educational Psychologist, 47,
Doumas, L. A. A., Hummel, J. E., & Sandhofer, C. M. (2008). A theory
of the discovery and predication of relational concepts. Psychological
Review, 115, 1–43.
Dweck, C. S., & Leggett, E.L. (1988). A social-cognitive approach to moti-
vation and personality. Psychological Review, 95, 256–273.
Fleischman, H. L., Hopstock, P. J., Pelczar, M. P., & Shelley, B. E. (2010).
Highlights from PISA 2009: Performance of U.S. 15-year-old students
in reading, mathematics, and science literacy in an international con-
text (NCES 2011–004). Washington, DC: United States Department of
Education, National Center for Education Statistics.
A general role for analogical encoding. Journal of Educational Psychol-
ogy, 95, 393–408.
Gentner, D., & Namy, L. L. (2006). Analogical processes in language learn-
ing. Current Directions in Psychological Science, 15, 297–301.
Gentner, D., & Rattermann, M. (1991). Language and the career of similar-
ity. In S. A. Gelman & J. P. Byrnes (Eds.), Perspectives on thought and
language: Interrelations in development (pp. 225–277). Cambridge, UK:
Cambridge University Press.
Psychology, 12, 306–355.
Gick, M. L., & Holyoak, K. J. (1983). Schema induction and analogical
transfer. Cognitive Psychology, 15, 1–38.
lege developmental mathematics students understand about mathematics,
Part II: The interviews. The MathAMATYC Educator, 2(3), 4–18.
Gonzales, P., Williams, T., Jocelyn, L., Roey, S., Kastberg, D., & Bren-
wald, S. (2008). Highlights from TIMSS 2007:Mathematics and science
achievement of U.S. fourth- and eighth-grade students in an international
Downloaded by [University of Chicago] at 11:01 29 August 2012
Page 15
context (NCES 2009-001). Washington, DC: National Center for Educa-
tion Statistics, U.S. Department of Education.
Goswami, U. (1992). Analogical reasoning in children. Hillsdale, NJ: Erl-
models. Hillsdale, NJ: Erlbaum.
(pp. 262–272). New York, NY: Freeman.
Heine, S. J., Kitayama, S., Lehman, D. R., Takata, T., Ide, E., Leung, C., &
Matsumoto, H. (2001). Divergent consequences of success and failure in
and malleable selves. Journal of Personality and Social Psychology, 81,
Hiebert, J., Gallimore, R., Garnier, H., Givvin, K. B., Hollingsworth, H., Ja-
D.C.: National Center for Education Statistics, United States Department
of Education.
teachingonstudents' learning.InF.K. Lester, Jr. (Ed.),Secondhandbook
of research on mathematics teaching and learning (Vol. 1, pp. 371–404).
New York, NY: Information Age.
Hilbert, D. (1900). Mathematische probleme [Mathematical problems].
Nachnchten der Akademie der Wissenschaften Gottingen, 290–329.
Hill, H., Ball, D. L., & Schilling, S. (2008). Unpacking "pedagogical con-
tent knowledge": Conceptualizing and measuring teachers' topic-specific
knowledge of students. Journal for Research in Mathematics Education,
39, 372–400.
Hill, H. C., Schilling, S. G., & Ball, D. L. (2004). Developing measures
of teachers' mathematics knowledge for teaching. Elementary School
Journal, 105, 11–30.
(Ed.), The psychology of learning and motivation (Vol. 19, pp. 59–87).
New York, NY: Academic Press.
Holyoak, K. J. (2012). Analogy and relational reasoning. In K. J. Holyoak &
R. G. Morrison (Eds.), The Oxford handbook of thinking and reasoning
(pp. 234–259). New York, NY: Oxford University Press.
Holyoak, K. J., & Cheng, P. W. (2011). Causal learning and inference as a
rational process: The new synthesis. Annual Review of Psychology, 62,
Holyoak, K. J., Junn, E. N., & Billman, D. O. (1984). Development of
analogical problem-solving skill. Child Development, 55, 2042–2055.
Holyoak, K. J., Lee, H. S., & Lu, H. (2010). Analogical and category-based
inference: A theoretical integrationwith Bayesian causal models. Journal
of Experimental Psychology: General, 139, 702–727.
ture: A theory of analogical access and mapping. Psychological Review,
104, 427–466.
Hummel, J. E., & Holyoak, K. J. (2003). A symbolic-connectionist theory
of relational inference and generalization. Psychological Review, 110,
Jonassen, D. (2010). Learning to solve problems: A handbook for designing
problem-solving learning environments. London, UK: Routledge.
Kalyuga, S. (2007). Expertise reversal effect and its implications for
learner-tailored instruction. Educational Psychology Review, 19, 509–
analyses of classroom video predicts student learning of mathematics:
Further explorations of a novel measure of teacher knowledge. Journal of
Teacher Education, 61, 172–181.
Koedinger, K. R., & Roll, I. (2012). Learning to think: Cognitive mecha-
nisms of knowledge transfer. In K. J. Holyoak & R. G. Morrison (Eds.),
The Oxford handbook of thinking and reasoning (pp. 789–806). New
York, NY: Oxford University Press.
Kotovsky, L., & Gentner, D. (1996). Comparison and categorization in
the development of relational similarity. Child Development, 67, 2797–
Lee, H. S., & Holyoak, K. J. (2008). The role of causal models in analogical
inference. Journal of Experimental Psychology: Learning, Memory, and
Cognition, 34, 1111–1122.
Legare, C. H., Gelman, S. A., & Wellman, H. M. (2010). Inconsistency with
prior knowledge triggers children's causal explanatory reasoning. Child
Development, 81, 929–944.
Loewenstein, J., & Gentner, D. (2001). Spatial mapping in preschoolers:
Close comparisons facilitate far mappings. Journal of Cognition and
Development, 2, 189–219.
Loewenstein, J., Thompson, L., & Gentner, D. (2003). Analogical learning
in negotiation teams: Comparing cases promotes learning and transfer.
Academy of Management Learning and Education, 2, 119–127.
Ma, L. (1999). Knowing and teaching elementary mathematics. Teachers'
understanding of fundamental mathematics in China and the United
States. Mahwah, NJ: Erlbaum.
National Council of Teachers of Mathematics. (2006). Curriculum focal
points for prekindergarten through grade 5 mathematics: a quest for
coherence. Reston, VA: Author.
National Council of Teachers of Mathematics. (2009). Focus in high school
mathematics: Reasoning and sense making. Reston, VA: Author.
Novick, L. R., & Holyoak, K. J. (1991). Mathematical problem solving by
analogy. Journal of Experimental Psychology: Learning, Memory, and
Cognition, 17, 398–415.
Penn, D. C., Holyoak, K. J., & Povinelli, D. J. (2008). Darwin's mistake:
Explaining the discontinuity between human and nonhuman minds. Be-
havioral and Brain Sciences, 31, 109–178.
Polya, G. (1954). Mathematics and plausible reasoning: Induction and
analogy in mathematics (Vol. 1). Princeton, NJ: Princeton University
Richland, L. E., Holyoak, K. J., & Stigler, J. W. (2004). The role of analogy
in teaching middle-school mathematics. Cognition and Instruction, 22,
Richland, L. E., & McDonough, I. M. (2010). Learning by analogy: Dis-
criminating between potential analogs. Contemporary Educational Psy-
chology, 35, 28–43.
Richland, L. E., Morrison, R. G., & Holyoak, K. J. (2006). Children's devel-
opment of analogical reasoning: Insights from scene analogy problems.
Journal of Experimental Child Psychology, 94, 249–271.
Richland, L. E., Zur, O., & Holyoak, K. J. (2007). Cognitive sup-
ports for analogy in the mathematics classroom. Science, 316,
Rittle-Johnson, B., & Star, J. R. (2007). Does comparing solution methods
facilitate conceptual and procedural knowledge? An experimental study
on learning to solve equations. Journal of Educational Psychology, 99,
Rittle-Johnson, B., & Star, J. (2009). Compared to what? The effects of
different comparisons on conceptual knowledge and procedural flex-
ibility for equation solving. Journal of Educational Psychology, 101,
iarity when comparing examples: Impact on conceptual and procedural
knowledge of equation solving. Journal of Educational Psychology, 101,
Ross, B. H., & Kennedy, P. T. (1990). Generalizing from the use of ear-
lier examples in problem solving. Journal of Experimental Psychology:
Learning, Memory, and Cognition, 16, 42–55.
Schoenfeld,A. H.(1988).Whengoodteaching leads tobad results: Thedis-
asters of "well-taught" mathematics courses. Educational Psychologist,
23, 145–166.
overzealous transfer: Coordinating previously successful routines with
needs for new learning. Educational Psychologist, 47, 204–214.
Downloaded by [University of Chicago] at 11:01 29 August 2012


text:
In The News
Increasingly, buyers say they'd choose a walkable, transit-oriented location over a larger home, if cost weren't an issue, says a study for the Pembina Institute and RBC.
Toronto Star,
Residents doubt Mississauga can afford massive transit plan
Forum poll finds 72% approve of new Mayor Bonnie Crombie, but only 34% think the city can afford her massive transit construction promises.
Toronto Star,
Portrait of typical TTC rider emerges from survey
She's probably richer than you think. She also tends to use tokens, ride the subway and avoid rush-hour travel.
Toronto Star,
Mayor John Tory wants to let kids ride free on TTC, raise fares by 10 cents
Tory and TTC chair Josh Colle announce $95 million in proposed changes to the transit system.
Toronto Star, January 19, 2015
Road to transit heaven will be paved with facts: James
Transportation Tomorrow Survey reveals surprising truths about who's going where in the GTA — and hints at where we'd really get the most bang for the buck.
Toronto Star,
TTC-GO fare pilot project launched on Lakeshore lines
A $60 sticker added to a Metropass allows unlimits rides to Exhibition or Danforth.
Toronto Star,
SmartTrack studies to cost another $1.65 million
Council has already approved $750,000 in study spending for Mayor John Tory's top policy priority.
Toronto Star,
Joint TTC-GO fare pilot project expected on Friday
The province and chair of the TTC will make a joint announcement Friday at the Danforth GO station.
The Globe and Mail,
TTC and GO Transit set to announce fare integration program
The goal is to offer a much quicker way to get to the city core, while allowing people to bypass the over-crowded subway exchange at Bloor-Yonge.
Toronto Star, January 7, 2015
What's on Toronto's transit horizon in 2015?
Transit riders can expect incremental boosts to service as they await the completion of new subways and other mega-projects.

subset: commoncrawl


text:
Popular Subjects
Perhaps Licensed to Ill was inevitable -- a white group blending rock and rap, giving them the first number one album in hip-hop history. But that reading of the album's history gives short shrift to the Beastie Boys; producer Rick Rubin, and his label, Def Jam, and this remarkable record, since mixing metal and hip-hop isn't necessarily an easy ... Read More Perhaps Licensed to Ill was inevitable -- a white group blending rock and rap, giving them the first number one album in hip-hop history. But that reading of the album's history gives short shrift to the Beastie Boys; producer Rick Rubin, and his label, Def Jam, and this remarkable record, since mixing metal and hip-hop isn't necessarily an easy thing to do. Just sampling and scratching Sabbath and Zeppelin to hip-hop beats does not make for an automatically good record, though there is a visceral thrill to hearing those muscular riffs put into overdrive with scratching. But, much of that is due to the producing skills of Rick Rubin, a metalhead who formed Def Jam Records with Russell Simmons and had previously flirted with this sound on Run-D.M.C.'s Raising Hell, not to mention a few singles and one-offs with the Beasties prior to this record. He made rap rock, but to give him lone credit for Licensed to Ill (as some have) is misleading, since that very same combination would not have been as powerful, nor would it have aged so well -- aged into a rock classic -- if it weren't for the Beastie Boys, who fuel this record through their passion for subcultures, pop culture, jokes, and the intoxicating power of wordplay. At the time, it wasn't immediately apparent that their obnoxious patter was part of a persona (a fate that would later plague Eminem), but the years have clarified that this was a joke -- although, listening to the cajoling rhymes, filled with clear parodies and absurdities, it's hard to imagine the offense that some took at the time. Which, naturally, is the credit of not just the music -- they don't call it the devil's music for nothing -- but the wild imagination of the Beasties, whose rhymes sear into consciousness through their gonzo humor and gleeful delivery. There hasn't been a funnier, more infectious record in pop music than this, and it's not because the group is mocking rappers (in all honesty, the truly twisted barbs are hurled at frat boys and lager lads), but because they've already created their own universe and points of reference, where it's as funny to spit out absurdist rhymes and pound out "Fight for Your Right (To Party)" as it is to send up street corner doo wop with "Girls." Then, there is the overpowering loudness of the record -- operating from the axis of where metal, punk, and rap meet, there never has been a record this heavy and nimble, drunk on its own power yet giddy with what they're getting away with. There is a sense of genuine discovery, of creating new music, that remains years later, after countless plays, countless misinterpretations, countless rip-off acts, even countless apologies from the Beasties, who seemed guilty by how intoxicating the sound of it is, how it makes beer-soaked hedonism sound like the apogee of human experience. And maybe it is, maybe it isn't, but in either case, Licensed to Ill reigns tall among the greatest records of its time. ~ Stephen Thomas Erlewine, Rovi Read Less
10/25/1990, Sony
CD, Fine/Like New
MirMedia Movies and Music
Reading, PA, USA
$0.99 $11.98
1995, Def Jam
CD, Good
Last Page Media-Columbus
Columbus, OH, USA
$0.99 $5.98
Goodwill Industries of Denver
Denver, CO, USA
Last Page Media-Dallas
Dallas, TX, USA
Worcester, MA, USA
Better World Books
Mishawaka, IN, USA
Sony Music Entertain
Good Books Will Follow
Blue Springs, MO, USA
CD, Very Good
Milwaukee, WI, USA
$1.06 $5.98
$1.91 $11.98
+ $3.99 shipping
Shipping Options:
Standard Shipping: $3.99
Trackable Expedited: $7.19
*You will be able to select a shipping option during Checkout. Shipping costs can vary based on destination.
Sold by
Ships from Reading, PA, USA
Questions? Contact the seller.
Seller's Description
Fine. CD IS LIKE NEW DIRECT FROM OUR WAREHOUSE...BEFORE SHIPPING USED ITEMS ARE INSPECTED AND POLISHED AS REQUIRED USING OUR PROFESSIONAL POLISHING EQUIPMENT...FAST SHIPPING AFTER YOUR ORDER IS RECEIVED ALL PURCHASES 100% GUARANTEED! ! ! BUY WITH CONFIDENCE! ! !
Ships from Columbus, OH, USA
Good. Customer service is our top priority! The item or packaging may have identifying markings from its owner or show limited signs of wear. Digital copies may or may not be present.
Ships from Denver, CO, USA
Ships from Dallas, TX, USA
Good. Connecting viewers with great music since 1972. All used discs are inspected and guaranteed. Customer service is our top priority!
Ships from Worcester, MA, USA
Good. Audio CDs-Complete in Jewel Case-with artwork All CD's/Records as represented Check the item notes to formats listed for complete matches..
Ships from Mishawaka, IN, USA
Good. Used CD. Shipped to over one million happy customers. Your purchase benefits World Literacy!
Ships from Blue Springs, MO, USA
Good. Disc(s) in good condition.
Ships from Milwaukee, WI, USA
Very Good. 731452735126 FREE UPGRADE TO FIRST CLASS. Immediate shipping. Satisfaction guaranteed. mc104.
Ships from MARLTON, NJ, USA
Fine. Books, CDs, DVDs, Videogames, LPs & more! Fast shipping! All items guaranteed!
Lucky you. Be the first to review this item!
Starting at $0.99
Starting at $2.95
The Eminem Show
Buena Vista Social Club


text:
geographical name \sär-ˈdi-nē-ə, -ˈdi-nyə\Definition of SARDINIA
Variants of SARDINIA
(Concise Encyclopedia)Island and autonomous region (pop., 2001 prelim.: 1,599,511), Italy. Off the southern Italian coast, Sardinia is the second largest island in the western Mediterranean Sea. It measures 9,194 sq mi (23,813 sq km); its capital is Cagliari. Thousands of structures made of basalt blocks, called nuraghi, are a dominating feature of the island. These truncated conic structures of blocks taken from extinct volcanoes were built in prehistoric times without any mortar. Some date to c. 1500–400 BC. Phoenicians were Sardinia's first recorded settlers c. 800 BC. Greeks and Carthaginians followed; Roman rule began in 238 BC. In the early Middle Ages Pisa and Genoa struggled over its domination. The kingdom of Sardinia, centred on the lands of Piedmont in northwestern Italy and the island of Sardinia, was ruled by the house of Savoy from 1720 until the unification of Italy in 1861. Agriculture, fishing, and mining are economic mainstays of the island.
Variants of SARDINIA
Learn More About SARDINIA
Ask The Editor Videos
Seen & Heard
What made you want to look up Sardinia? Please tell us where you read or heard it (including the quote, if possible).


text:
Latest Dearborn Stories
FordDirect launches DealerConnection Elite Plus; Announces its General Counsel and Chief Compliance Officer, Beth Hill, will speak on data privacy at annual conference DEARBORN, Mich.,
- Spelman College in Atlanta takes home top prize in Ford's 2014 Historically Black Colleges & Universities Community Challenge with its plan for showing families in the West End community where it
New surgery center features eight surgical rooms designed to house a variety of medical procedures. Dearborn, Michigan (PRWEB) December 11, 2014 The
DEARBORN, Mich., Oct. 14, 2014 /PRNewswire-USNewswire/ -- Among the select few physicians honored with the prestigious Patients' Choice Award this year is Dr.
- Ford Motor Company and Ford Motor Company Fund have joined forces with the "Tom Joyner Morning Show" and "Rickey Smiley Morning Show" for the second year to launch the 2014 HBCU Community Challenge
- Ford is supporting the airing of the second season of Finding Your Roots with Henry Louis Gates, Jr., which debuts Tuesday, Sept. 23 at 8 p.m. ET DEARBORN, Mich., Sept.
Masri Clinic for Laser Cosmetic Surgery Set to Be a Valuable, Online Resource for Patients Dearborn, Michigan (PRWEB) August 15, 2014 Dr.
Latest Dearborn Reference Libraries
The Ford Motor Company is an American automobile manufacturer with its headquarters located in Dearborn, Michigan. It was founded by Henry Ford at 39 years old and incorporated on June 16, 1903. It is described by Forbes as "the most important industrial company in the history of the United States." Ford would go on to become one of the world's largest and most profitable companies and is one of the largest family-controlled companies in the world. Henry Ford made an attempt at an...
- totally perplexed and mixed up.


text:
A great revival of an old-school classic
If you experienced the slightest enjoyment from playing the HoMM series then this presses all the right buttons and plenty of others besides. The guys at Katauri deserve huge credit for preserving and actually enhancing a piece of classic gameplay. This is clearly a labour of love from a team who totally know what they're doing. Well done guys, I'm a fan!
What do I like most? I guess it must be the fact that they've created an utterly fascinating landscape packed with layers of content and unexpected discoveries. Not that any of this is particularly original although, oddly enough, that is a key part of its charm.
I'd give it '10' but then I'd obviously be getting no sleep at all!!


text:
News: a residential development designed by Richard Rogers to adjoin two streets in London's Mayfair via a paved arcade has been granted approval. More »
Public and leisure
Plans approved for Rogers Stirk Harbour
Open House by
Matthew Mazzotta
This house in Alabama folds open to provide seating for an open-air performance space (+ movie). More »
Mineirão Stadium renovation
by BCMF Arquitetos
Here are some photographs of the renovated 1960s Mineirão Stadium in Belo Horizonte, Brazil, set to host matches during the 2014 FIFA World Cup (+ slideshow). More »
Unbalanced Hotel
by OOIIO Architecture
Madrid firm OOIIO has designed a lopsided photo frame-shaped hotel that appears to have crashed into a cliff face near Lima in Peru. More »
Restaurant Castillito by Mathias Klotz
and Lillian Allen
Chilean architects Mathias Klotz and Lillian Allen have renovated a castle-like residence in Santiago's Parque Forestal to create a restaurant, exhibition space and ice-cream parlour (+ slideshow). More »
Movie: Galaxy Soho by
Zaha Hadid Architects
Take a tour through the spaces of Zaha Hadid's Galaxy Soho complex in Beijing, China, in this movie by architectural photographer Dan Chung. More »
OMA selected for downtown
Santa Monica project
OMA lands Miami Beach
Convention Center commission
Movie: Campbell Sports Centre
by Steven Holl Architects
We take a tour through the staircases, gyms and study areas of Steven Holl's Campbell Sports Centre at Columbia University in the second of two movies by architectural filmmakers Spirit of Space. More »
"The concept was based on a football diagram"
- Steven Holl on Campbell Sports Centre
Architect Steven Holl describes how the design for his new athletics centre at Columbia University was based on a diagram for a football strategy, in the first of two movies by architectural filmmakers Spirit of Space. More »
Hut with the Arc Wall
by Tato Architects
These public toilets in Japan by Tato Architects comprise a single curved wall sheltered beneath a gabled roof (+ slideshow). More »
Gammel Hellerup Sports Hall
by BIG
Danish architecture studio BIG has completed a sunken sports hall where an arching wooden roof doubles up as a hilly courtyard (+ slideshow). More »
Frank Gehry reveals latest design
for trio of towers in Toronto
News: architect Frank Gehry has updated his design for a cluster of three towers in his home city of Toronto. More »
Harmonie Hall by
Takenaka Corporation
Huge clerestory windows reveal the exposed timber frame of this school sports hall in Kobe, Japan, by architecture firm Takenaka Corporation (+ slideshow). More »
Wimbledon masterplan unveiled
by Grimshaw
Ílhavo Maritime Museum Extension
by ARX Portugal
Ten years after completing the Ílhavo Maritime Museum in Portugal, Lisbon studio ARX Portugal has extended the building by adding an aquarium dedicated to codfish (+ slideshow). More »
Kontum Indochine Cafe by
Vo Trong Nghia Architects
Fifteen conical bamboo columns support the roof of this waterside cafe designed by Vo Trong Nghia Architects at a hotel in central Vietnam (+ slideshow). More »
Chungha in Gangnam
by MVRDV
Dutch studio MVRDV has transformed the facade of an ageing mixed-use building into a stack of shop windows in Gangnam, the trendy district of Seoul described in the world-famous Korean pop song. More »
Sign up for a daily roundup
of all our stories


text:
- News
- Video
- People
Find by writer
- Yasmin Alibhai-Brown
- Rebecca Armstrong
- Memphis Barker
- Max Benwell
- Chris Blackhurst
- Ian Burrell
- Andrew Buncombe
- Ben Chu
- Patrick Cockburn
- Mary Dejevsky
- Grace Dent
- Robert Fisk
- Andrew Grice
- Stefano Hatfield
- Lucy Hunter Johnston
- Howard Jacobson
- Alice Jones
- Ellen E Jones
- Comment
- Editorials
- Letters
- IV Drip
- Campaigns
- Archive
- Our Voices
- Commentators
- Columnists
- Democracy 2015
- IV Drip Archive
- Blogs
- Scottish independence
- Save the tiger
- The state of the NHS
Find by writer
- Sport
- Tech
- Life
- Property
- Arts + Ents
- Travel
- Money
- IndyBest
- Student
- Offers
Heads up: Top comment and controversy
Wills and Kate: Best baby tweets
03 December 2012 06:08 PM
Twitter is polarised over news of Kate Middleton's pregnancy announcement. Some tweeps are bubbling over with joy, while republicans bubble incandescently over the special treatment the foetus is currently receiving just for being a "#royalbaby".
Meanwhile, some commentators, like @amolrajan, are torn between simple joy at the news of new life and the opportunity to argue that the royals should immediately renounce their titles and go live in a hovel in Wales.
Here are some of the funniest, weirdest, loveliest and most controversial tweets so far.
@soukici: "In stark contrast to the rest of the economy, the bunting industry is having a booming start to the decade."
CarlaDobson: "Guardian is liveblogging the royal baby news wonder if they'll keep it up for six months"
@JeromeTaylor: The "worst rated" comment on MailOnline's royal baby report is hilarious http://twitpic.com/bisizs
@LisaBeesa: "The first sentence makes me giggle. Mainly because that's totally my reaction to the news! http://vnty.fr/XiDSBA"
@BrooklynDecker: "I hope Royal Baby and Blue Ivy Carter become best friends."
@TH3Consequence: "Already sick of the half blood prince royal baby jokes, snape wouldnt want to be associated with royals stop giving their baby his name"
@Joe_Whit: "Can someone save all the royal baby articles & then when it's grown up weird it out with how big the news of its conception was?"
@Commodity_Bull: "the royal baby has nothing on an American made iMac"
@Bourb0ns: "When the royal baby is born they better do the whole Lion King routine with it.
@YahooOMGUK: "The Jubilee, Olympics and NOW#royalbaby, has this been the best year EVER for Britain?!"
@Hel_374: "With all this excitement I have to keep reminding myself that it's not actually me that's pregnant"
@sophielelawrie: "And now a royal baby!!! This is a serious contender for the best Monday award"
@SeanWHughes: "What if it's twins? They should keep one under wraps - would come in very handy."
@George_E: "what we all want to know is, will she keep it? It could really interfere with her career."
Follow @EmilyJupp


text:
If Karnes is elected, he will be part of a growing credit union industry presence at the National Cooperative Business Association.
Technology problems in general outstrip other factors, new Guy Carpenter report says.
Company says more than 100 clients for new service. Massachusetts credit union stays with Portico.
Attracting new members is a big challenge but keeping them may be a bigger obstacle.
Former El Paso Employees FCU adds new service to new name in west Texas town.
Big Wisconsin credit union stays in 20-year core processing relationship with Fiserv, adds solutions.
They improve member service, speed up the loan process and comply with open-ended lending regulations.
Formerly Fynanz, company says it now has 250 lending partners on its platform.
Many credit union members around the country have been enjoying the ease of mobile banking for the past year or so, but until recently, some entrepreneurs still had to schlep to the branch to make a deposit.
Just over a year since it purchased noted mortgage CUSO Prime Alliance Solutions, Mortgage Cadence has been purchased by Accenture, an international financial services consulting and software firm.


text:
The average exerciser may not know Nia, but Sarah and Amy of Somatic Alchemy live and breathe the holistic workout. They teach the blend of dance, martial arts, and healing arts in an upbeat, yet relaxing environment. Set to music, Nia's steps are carefully guided, and they burn calories without high impact. So Sarah and Amy welcome students at all levels of fitness to their classes, held atop The Reach Center's sprung floors (which are easy on the joints).
Led by founder Sandy Adler, certified Anusara instructor, Freedom Yoga's instructors lead classes throughout the week for both beginners and more advanced students. Instructors tap into their experience, skill, and dedication to help students develop a safe and meaningful yoga practice, searching within themselves and their pupils to align spiritual centers, increase strength, and boost flexibility. Rather than using intense exercises or giant marionette strings to affect the mind and body from the outside, Anusara's poses originate from within the body, using deliberate breath and movement as a form of meditation to hone the mind while helping the body increase its strength and balance.
Just Breathe co-owners and instructors Quincy O'Toole and Mandy Baucum spread their love of movement and wellness through a variety of fitness classes. Though their diverse team of instructors comes from many different backgrounds—including massage therapy, academia, and traditional Chinese medicine—they all share a desire to help guests of all ages and abilities understand their own bodies. During yoga, Pilates, and tai chi classes, they not only work to boost strength, flexibility, and endurance, but to also relieve stress via meditation. They also host Nia aerobic classes, which blend movements from dance and martial arts.
At Everyday People Yoga, students unroll their mats onto bamboo floors in a spacious and tranquil studio. Skilled instructors lead a variety of classes, ranging from Vinyasa and zen yoga to gentle yoga, meditation, and Ashtanga yoga. While the classes work to boost strength and flexibility, they can also help students build self-awareness with the body, both internally and externally.
Length, grace, and a healthy body. These are the focuses of barre3 founder Sadie Lincoln. To achieve these results collectively, however, Sadie knew she needed to come up with a completely new routine. Working with yogis, professional dancers, doctors, and athletes, she created the barre3 system. It's a three-step sequence that consists of isometric holds, low-impact movements, and recovery stretching. This combination strengthens cores and aligns postures while sculpting long, lean muscles and burning calories. While upbeat music fills the studio, instructors lead small groups through a heart-racing sequence of yoga-inspired poses and pilates-based exercises. The ballet barre comes into use when striking muscle-building dance poses and stretches.
It's a system that has had proven results, which has spurred the successful growth of barre3 locations throughout the country. Today, fitness enthusiasts can find a barre3 studio in 16 states. An easy-to-follow routine, barre3 can also be performed at home with online workouts. These workouts are designed to fit busy schedules with routines that range from 10?60 minutes. All that's needed for online workouts is a barre3 core ball, weights, an exercise mat, and a sturdy, waist-high surface such as the top of Danny DeVito's head.
At Sweaty Ganesh Yoga studio, yoga serves as a transformative journey for every person who enters. All students, no matter their experience or skill level, are recognized for their current ability and empowered to test limitations of strength and flexibility. Along with beginners' hot yoga classes, held throughout the week, the studio also holds classes for kids 6–12. Workshops help students perfect bow poses, backward bends, forward folds, and stretching postures. The fundamental approach teaches strength and flexibility while the postures test balance.