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BRIEF REPORT

The benefit of interleaved mathematics practice is not limited

to superficially similar kinds of problems

Doug Rohrer &Robert F. Dedrick &Kaleena Burgess

Published online: 28 February 2014

#Psychonomic Society, Inc. 2014

Abstract Most mathematics assignments consist of a group

of problems requiring the same strategy. For example, a lesson

on the quadratic formula is typically followed by a block of

problems requiring students to use that formula, which means

that students know the appropriate strategy before they read

each problem. In an alternative approach, different kinds of

problems appear in an interleaved order, which requires stu-

dents to choose the strategy on the basis of the problem itself.

In the classroom-based experiment reported here, grade 7

students (n= 140) received blocked or interleaved practice

over a nine-week period, followed two weeks later by an

unannounced test. The mean test scores were greater for

material learned by interleaved practicerather than by blocked

practice (72 % vs. 38 %, d= 1.05). This interleaving effect

was observed even though the different kinds of problems

were superficially dissimilar from each other, whereas previ-

ous interleaved mathematics studies had required students to

learn nearly identical kinds of problems. We conclude that

interleaving improves mathematics learning not only by im-

proving discrimination between different kinds of problems,

but also by strengthening the association between each kind of

problem and its corresponding strategy.

Keywords Learning .Mathematics .Interleaved .Spacing .

Practice

Learning techniques inspired by research in the laboratory can

improve learning in the classroom (for recent reviews, see

Dunlosky, Rawson, Marsh, Nathan, & Willingham, 2013;

Roediger & Pyc, 2012). In the study reported here, a simple

intervention designed to improve mathematics learning was

assessed in a classroom-based experiment. We first describe

the intervention and the relevant research.

Interleaved practice

The solution of a mathematics problem requires two steps, as

is illustrated by the following example:

A bug flies 48 m east and then flies 14 m north. How far

is the bug from where it started?

This problem is solved by using the Pythagorean theorem

to find the length of a hypotenuse (

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

48 þ14

p¼50 ). In other

words, students first choose a strategy (Pythagorean theorem)

and then execute the strategy. The term strategy is used

loosely here to refer to a theorem, formula, concept, or proce-

dure. Learning to choose an appropriate strategy is difficult,

partly because the superficial features of a problem do not

always point to an obvious strategy (e.g., Chi, Feltovich, &

Glaser, 1981; Siegler, 2003). For example, the word problem

about the bug does not explicitly refer to the Pythagorean

theorem, or even to a triangle or hypotenuse. Additional

examples are given in Fig. 1.

Although students must learn to choose an appropriate

strategy, they are denied the opportunity to do so if every

problem in an assignment requires the same strategy. For

example, if a lesson on the Pythagorean theorem is followed

by a group of problems requiring the Pythagorean theorem,

students know the appropriate strategy before they read each

problem. The grouping of problems by strategies is termed

blocked practice, and the large majority of practice problems

in most mathematics textbooks are blocked. Blocked practice

served as the control in the study reported here.

D. Rohrer :R. F. Dedrick :K. Burgess

University of South Florida, Tampa, FL, USA

D. Rohrer (*)

Psychology PCD4118G, University of South Florida, Tampa,

FL 33620, USA

e-mail: drohrer@usf.edu

Psychon Bull Rev (2014) 21:1323–1330

DOI 10.3758/s13423-014-0588-3

In an alternative approach that is evaluated in the present

study, a majority of the problems within each assignment are

drawn from previous lessons, so that no two consecutive prob-

lems require the same strategy—a technique known as inter-

leaved practice. With this approach, students must choose an

appropriate strategy and not only execute it, just as they must

choose an appropriate strategy when they encounter a problem

during a cumulative exam or high-stakes test. Put another way,

blocked practice provides a crutch that might be optimal when

students first encounter a new skill, but only interleaved practice

allows students to practice what they are expected to know. To

create assignments with interleaved practice, the problems

within a set of blocked assignments can be rearranged (Fig. 2).

In addition to providing opportunities to practice choosing

a strategy, interleaved mathematics assignments guarantee

that problems of the same kind are distributed, or spaced,

across different assignments (Fig. 2). Spacing typically

improves performance on delayed tests of learning (e.g., for

recent reviews, see Dunlosky et al., 2013; Roediger & Pyc,

2012), and several studies have shown that spacing can

improve the learning of mathematics, in particular (Rohrer &

Tay lor, 2006,2007; Yazdani & Zebrowski, 2006). To sum-

marize thus far, interleaved practice has two critical features:

Problems of different kinds are interleaved (which requires

students to choose a strategy), and problems of the same kind

are spaced (which usually improves retention).

Previous studies of interleaved practice

Four previously published studies compared the effects of

interleaved and blocked mathematics practice (Le Blanc &

Simon, 2008; Mayfield & Chase, 2002; Rohrer & Taylor,

2007; Taylor & Rohrer, 2010). In each of the studies, partici-

pants received interleaved or blocked practice of different kinds

of problems, and interleaving produced better scores on a

delayed test. However, in each of these studies, the different

kinds of problems (and the corresponding strategies) were

nearly identical in appearance (Fig. 3). In one study, for exam-

ple, every problem included a variable raised to an exponent,

and, in another, every problem referred to a prism. We refer to

problems with shared features as superficially similar problems,

and this similarity might hinder students’ability to distinguish or

discriminate between different kinds of problems. Indeed, the

benefit of interleaved practice is often attributed to improved

discrimination, as we will detail in the Discussion section.

Therefore, the superficial similarity of the problems used in

previous studies leaves open the possibility that the test benefit

of interleaving is limited to scenarios in which students learn to

solve kinds of problems that look alike, and such a boundary

condition would curtail the utility of interleaved practice in the

classroom, where students encounter problems that are often

easily distinguished from other kinds of problems.

Present study

We compared interleaved and blocked mathematics practice

in a classroom-based experiment with a counterbalanced,

crossover design. Students learned to solve different kinds of

problems drawn from their mathematics course, and they

received the lessons and assignments from their regular

teacher over a period of nine weeks. Two weeks after the last

assignment, students sat for an unannounced test. Unlike

previous studies of interleaved mathematics practice, the dif-

ferent kinds of problems were superficially dissimilar.

ygetartSetucexE.2ygetartSesoohC.1melborP

A A bug flies 48 m east and then 14 m north.

How far is the bug from where it started?

Pythagorean

Theorem

48

B A bug flies 48 m east and then 14 m west.

How far is the bug from where it started?

Number line

arithmetic

C Find the length of the line segment with

endpoints (1, 1) and (5, 4).

Pythagorean

Theorem

3

D Find the slope of the line that passes

throu

g

h the points (1, 1) and (5, 4). slope

3

4

Fig. 1 The two steps in the solution of a problem. To solve a problem,

students must choose a strategy and then execute it. Superficially similar

problems may require different strategies (A and B, or C and D), and

superficially dissimilar problems may require the same strategy (A and

C). Regardless of similarity, students know the strategy in advance when

working a block of problems requiring the same strategy

1324 Psychon Bull Rev (2014) 21:1323–1330

Method

Participants

The study took place at a public middle school in Tampa,

Florida. Three teachers and eight of their seventh-grade math-

ematics classes participated. Each teacher taught two or three

of the classes. Of the 175 students in the classes, 157 students

participated in the study. Of these, 140 students attended class

on the day of the unannounced test, and only these students’

data were analyzed. Nearly all of the students were 12 years of

age at the beginning of the school year.

Material

Students learned to solve four kinds of problems drawn from

their course (Fig. 4). To confirm that students could not solve

these kinds of problems before the experiment, we administered

a pretest with one of each kind of problem. Averaged across

problems, just 0.7 % of the students supplied both the correct

answer (e.g., x= 7) and the correct solution (the steps leading to

the answer). When scored solely on the basis of answers (which

presumably included guesses), the mean score was 3.2 %.

The four kinds of problems were not only superficially

different from each other, but also quite unlike other kinds

of problems that the students had seen prior to the completion

of the experiment. For example, although students ultimately

learn how to solve many kinds of equations, a linear equation

was the only kind of equation that these students had encoun-

tered previously in school (Fig. 4A). Likewise, a linear

equation was the only kind of equation that the students had

previously graphed (Fig. 4C). The slope problem (Fig. 4D)

was also moderately unique, because the term “slope”is used

only in limited contexts. However, the proportion word prob-

lem (Fig. 4B)doesresembleotherkindsofwordproblems.

Design

For the study, we used a counterbalanced crossover design.

We randomly divided the eight classes into two groupsof four,

with the constraint that each group included at least one of the

classes taught by each teacher. One group interleaved their

practice of problems kinds A and B and blocked their practice

of kinds C and D, and the other group did the reverse.

Procedure

During the nine-week practice phase, students received ten

assignments with 12 problems each. Across all assignments,

the students saw 12 problems of each of the four kinds (Fig. 4).

The remaining problems were based on entirely different topics.

Students received the ten assignments on Days 1, 15, 24, 30/31,

36, 37, 57, 58, 60, and 64. Every student received the same

problems, but we rearranged the problems to create two versions

of each assignment—one for each group. The first four prob-

lems of kinds A, B, C, and D were the first four problems of

Assignments 1, 2, 4, and 5, respectively. If a problem kind was

learned by blocked practice, the remaining eight problems

appeared in the same assignment as the first four, meaning that

the assignment included one block of 12 problems. If a problem

Lesson

50 51 52 53 54 55 60 70 90

4 problems

on the

current

lesson

1 50 51 52 53 54 55 60 70 90

2 50 51 52 53 54 55 60 70 90

3 50 51 52 53 54 55 60 70 90

4 50 51 52 53 54 55 60 70 90

1 problem

on each of

8 previous

lessons

5 49 50 51 52 53 54 59 69 89

6 48 49 50 51 52 53 58 68 88

7 47 48 49 50 51 52 57 67 87

8 46 47 48 49 50 51 56 66 86

9 45 46 47 48 49 50 55 65 85

10 40 41 46 47 48 49 50 60 84

11 30 31 32 33 34 35 40 50 70

12 10 11 12 13 14 15 20 30 50

Fig. 2 A hypothetical set of assignments providing interleaved practice.

Each column represents an assignment, and each table entry indicates the

lesson number on which the problem is based. For example, if Lesson 50

is on ratios, the corresponding assignment includes four ratio problems

and one problem on each of eight lessons seen earlier in the school year

(or during the previous school year). Another eight ratio problems

(Lesson 50) are distributed across future assignments, with decreasing

frequency. In other words, problems of different kinds are interleaved

(which requires students to choose a strategy), and problems of the same

kind are spaced (which improves retention). Note that the arrangement

shown here is not the one that was used in the present study

Psychon Bull Rev (2014) 21:1323–1330 1325

kind was learned by interleaved practice, the remaining eight

problems of the same kind were distributed across the remaining

assignments. This meant that students saw the last problem of

each kind on a later date in the interleaved condition than in the

blocked condition, which is an intrinsic feature of assignments

with interleaved practice (Fig. 2). The effect of this difference in

“true test delay”is detailed in the Results.

Shortly before the scheduled date of each assignment,

teachers received paper copies for their students and a slide

presentation with solved examples and solutions to each prob-

lem. We asked teachers to present the examples before dis-

tributing the assignment. On the following school day,

teachers presented the solution to each problem while encour-

aging students to make any necessary corrections to their own

solutions. Teachers then collected the assignments. Within

two days, one or more of the authors visited the school, scored

each assignment (without marking it), and returned the as-

signments to the teachers. Although these scores do not mea-

sure students’mastery, because students could correct their

errors while the teacher presented the correct solutions, this

scoring of the assignments provided us with evidence of

teacher compliance with the experimental procedures.

Students were tested two weeks after the last assignment. We

asked teachers not to inform students of the test in advance,

because we did not want the final test to be affected by

cramming just prior to the test. Teachers did not see the test

before it was administered. The students were tested during their

regular class, and the teacher and one author proctored each test.

ygetartSetucexE.2ygetartSesoohC.1melborP

A Simplify. 8x ·4x Add exponents 32x

Simplify. Subtract exponents 2x

Simplify. 2x Multiply exponents 2 x

B Find the volume of a wedge

with radius 2 and height 3.

1

2πr h

1

2π2

Find the volume of a spheroid

with radius 2 and height 3.

4

3πr h

4

3π2

Find the volume of a spherical cone

with radius 2 and height 3.

2

3πr h

2

3π2

C The base of a prism has 5 sides.

How many faces does the prism have? base sides + 2 5 + 2 = 7

The base of a prism has 5 sides.

How many corners does the prism have? base sides x 2 5 x 2 = 10

The base of a prism has 5 sides.

How many ed

g

es does the prism have? base sides x 3 5 x 3 = 15

Fig. 3 Problems learned in previous studies of interleaved mathematics:

Students learned to solve several kinds of problems relating to (A)

exponent rules (Mayfield & Chase, 2002), (B) the volume of obscure

solids (Le Blanc & Simon, 2008; Rohrer & Taylor, 2007) or (C) prisms

(Taylor & Rohrer, 2010) [EE2] In each study, the different kinds of

problems (as well as the corresponding strategies) were nearly identical.

Note that each of the studies included four or five kinds of problems, but

only three are shown here

1326 Psychon Bull Rev (2014) 21:1323–1330

All of the test problems were novel. The test included three

problems of each of the four kinds, and each of the four pages

included a block of three problems of the same kind. We created

three versions by reordering problems within each block, and

students in adjacent chairs received different versions. Students

were allotted 36 min and allowed to use their school-supplied

basic calculator. Each test was scored on site that day by two

raters who were blind to each student’s group assignment. The

two raters scored each answer as correct or not correct and later

resolved the few discrepancies (17 of 1,680). Test score

reliability was moderately good (Cronbach’s alpha = .78).

Results

A repeated measures comparison of the two halves of the test

showed that interleaved practice was nearly twice as effective as

blocked practice, t(139) = 10.49, p< .001 (Table 1). The effect

size was large, d= 1.05, 95 % CI = [0.80, 1.30]. This benefit of

interleaving was observed for each of the four kinds of prob-

lems, ps < .01. The effect sizes for the four kinds (A, B, C, and

D) exhibited a positive trend (0.72, 0.45, 1.00, and 1.27, respec-

tively). This means that the interleaving benefit was larger for

problem kinds introduced later in the practice phase. In other

Problem 1. Choose Strategy 2. Execute Strategy

A

Solve the equation.

3(x + 1) = x + 17

Isolate x terms on one

side of the equation

3x + 3 = x + 17

2x + 3 = 17

2x = 14

x = 7

B

Penelope’s new tractor

requires 14 gallons of gas to

plow 6 acres. How many

gallons of gas will she need

to plow 21 acres?

Create a proportion

C

Graph the equation.

y = 2x + 1

Choose at least two

values of x and find

the corresponding

values of y.

x y

0 1

1 3

D

Find the slope of the line that

passes through the points

(3, 5) and (6, 7).

slope

Fig. 4 Examples of the four kinds of problems used in the present studies.

(A) Solve a linear equation requiring four steps. (B) Solve a word problem

using a proportion. (C) Graph an equation of the form, y=mx +b,wherem

and bare integers. (D) Determine the slope of the line defined by two given

points with integer coordinates

Psychon Bull Rev (2014) 21:1323–1330 1327

words, although the true test delay (the interval between the last

practice problem and the test) was larger in the blocked condi-

tion than in the interleaved condition (see the Procedure section),

the problem kinds with larger test delay differences (i.e., that

were seen earlier in the practice phase) were associated with

smaller effect sizes.Although this negative association might

reflect order effects—that is, all participants saw the four prob-

lem kinds in the same order—we cannot think of a reason why

order would matter. In brief, the effect sizes for problem kinds

introduced later in the practice phase were larger than the effects

for the earlier ones, and this trend was in the opposite direction

from what would be expected if the difference in test delays

contributed to the observed effect. Furthermore, if this difference

did play a role, it might be seen not as a confound, but as an

intrinsic feature of interleaved assignments (Fig. 2).

Discussion

Whereas previous studies of interleaved mathematics practice

had required students to learn kinds of problems that were nearly

identical in appearance (Fig. 3), the results reported here demon-

strate that this benefit also holds for problems that do not look

alike (Fig. 4). That is, the benefit of interleaved mathematics

practice is not limited to the ecologically invalid scenario in

which students encounter only superficially similar kinds of

problems. Although it might seem surprising that a mere

reordering of problems can nearly double test scores, it must be

remembered that interleaving alters the pedagogical demand of a

mathematics problem. As was detailed in the introduction, inter-

leaved practice requires that students choose an appropriate strat-

egy for each problem and not only execute the strategy, whereas

blocked practice allows students to safely assume that each

problem will require the same strategy as the previous problem.

However, the interleaved practice effect observed here might

reflect the benefit of spaced practice rather than the benefit of

interleaving per se. As we explained in the introduction, the

creation of interleaved mathematics assignments guarantees not

only that problems of different kinds will be interleaved, but also

that problems of the same kind will be spaced across assign-

ments, and spacing ordinarily has large, robust effects on delayed

tests of retention. We therefore believe that spacing contributed to

the large effect observed here (d= 1.05). Still, we have reason to

suspect that interleaving, per se, contributed as well. In one

previous interleaved mathematics study, students in both the

interleaved and blocked conditions relied on spaced practice to

the same degree, and interleaving nevertheless produced a large

positive effect (d= 1.23; Taylor & Rohrer, 2010). In the present

study, though, we chose to compare interleaved practice to the

kinds of assignment used in most textbooks, which is a massed

block of problems.

Theoretical accounts of the interleaved mathematics effect

How does interleaving improve mathematics learning? The

standard account holds that the interleaving of different kinds

of mathematics problems improves students’ability to distin-

guish or discriminate between different kinds of problems (e.g.,

Rohrer, 2012). Put another way, each kind of problem is a

category, and students are better able to identify the category

to which a problem belongs if consecutive problems belong to

different categories. This ability to discriminate is a critical skill,

because students cannot learn to pair a particular kind of prob-

lem with an appropriate strategy unless they can first distinguish

that kind of problem from other kinds, just as Spanish-language

learners cannot learn the pairs PERRO–DOG and PERO–BUT

unless they can discriminate between PERRO and PERO.

This discriminability account parsimoniously explains the

interleaving effects observed in previous mathematics inter-

leaving studies, because participants in these studies were

required to discriminate between nearly identical kinds of

problems (Fig. 3). For instance, one of these previous studies

included an error analysis, and it showed that the majority of

test errors in the blocked condition, but not in the interleaved

condition, occurred because students chose a strategy

corresponding to one of the other kinds of problems that they

had learned—for example, using the formula for prism edges

rather than the formula for prism faces (Taylor & Rohrer,

2010). Furthermore, the students in this study were given a

second final test in which they were given the appropriate

strategy for each test problem and asked only to execute the

strategy, and the scores on this test were near ceiling in both

conditions. In sum, the data from this earlier experiment are

consistent with the possibility that interleaving improves stu-

dents’ability to discriminate one kind of problem from another

(or discriminate one kind of strategy from another).

However, in the present study, discrimination errors

appeared to be rare. In a post-hoc error analysis, three raters

(two of the authors and a research assistant, all blind to

conditions) examined the written solution accompanying each

incorrect answer and could not find any solutions in which

students “used the wrong strategy but one that solves another

kind of problem.”The raters then expanded the definition of

discrimination error to include solutions with at least one step

of a strategy that might be used to solve any kind of problem

other than the kind of problem that the student should have

solved. With this lowered threshold, discrimination errors still

accounted for only 33 of the 756 incorrect answers (4.4 %),

with no reliable difference between conditions (5.1 % for

Tabl e 1 Proportions correct on test

Mean SD

Interleaved practice .72 .30

Blocked practice .38 .35

1328 Psychon Bull Rev (2014) 21:1323–1330

interleaved, 4.0 % for blocked). For the other incorrect

answers, students chose the correct strategy but incorrectly

executed it (45.9 %), or they relied on a strategy we could not

decipher, often because they did not show their work (49.7 %).

The virtual absence of discrimination errors is arguably not

surprising, partly because the different kinds of problems did

not look alike, and partly because some strategies were obvi-

ously an inappropriate choice for some kinds of problems

(e.g., trying to graph a line by creating a proportion). The

rarity of discrimination errors in the present study raises the

possibility that improved discrimination cannot by itself

explain the benefits of interleaved mathematics practice.

We suggest that, aside from improved discrimination, inter-

leaving might strengthen the association between a particular

kind of problem and its corresponding strategy. In other words,

solving a mathematics problem requires students not only to

discriminate between different kinds of problems, but also to

associate each kind of problem with an appropriate strategy, and

interleaving might improve both skills (Fig. 5). In the present

study, for example, students were asked to learn to distinguish a

slope problem from a graph problem (a seemingly trivial dis-

crimination) and to associate each kind of problem with an

appropriate strategy (e.g., for a slope problem, use the strategy

“slope = rise/run”), and the latter skill might have benefited from

interleaved practice. Yet why would interleaving, more so than

blocking, strengthen the association between a problem and an

appropriate strategy? One possibility is that blocked assignments

often allow students to ignore the features of a problem that

indicate which strategy is appropriate, which precludes the

learning of the association between the problem and the strategy.

In the present study, for example, students who worked 12 slope

problems in immediate succession (i.e., used blocked practice)

could solve the problems without noticing the feature of the

problem (the word “slope”) that indicated the appropriate

strategy (slope = rise/run). In other words, these students could

repeatedly execute the strategy (y

2

–y

1

)/(x

2

–x

1

) without any

awareness that they were solving problems related to slope. In

brief, blocked practice allowed students to focus only on the

execution of the strategy, without having to associate the prob-

lem with its strategy, much like a Spanish-language learner who

misguidedly attempts to learn the association between PERRO

and DOG by repeatedly writing DOG.

It might be possible to experimentally tease apart the effects of

interleaving on discrimination and association. In one such ex-

periment, participants would receive either blocked or interleaved

mathematics practice during the learning phase, as they typically

do, and then take two tests. The first test would assess only

discrimination. For example, students might be shown a random

mixture of five problems—four problems of one kind (e.g., word

problems requiring a proportion) and one problem of a different

kind (e.g., a word problem requiring the Pythagorean theorem)—

and then be asked to identify the problem that does not fit with

the others (the Pythagorean theorem problem). Students would

repeat this task many times with different kinds of problems. On

a second test measuring both discrimination and association,

students would see problems one a time and, for each problem,

choose the correct strategy, but not execute it. Scores on the first

test (discrimination only) should be greater than scores on the

more challenging second test (discrimination and association),

with larger differences between the two test scores reflecting a

poorer ability to associate a kind of problem and its strategy.

Therefore, if interleaving improves association, the difference

between the two test scores should be smaller for students who

interleaved rather than blocked.

Category learning

Finally, although we focused here on mathematics learning,

several studies have examined the effect of interleaved prac-

tice on category learning. For example, participants might see

Problem Strategy

Easy

Discrimination

Hard

Discrimination

A bug flies 48 m east and then 14 m west.

How far is the bug from where it started?

Association

Number line

subtraction

Find the length of the line segment with

endpoints (1, 1) and (5, 4).

Association

Pythagorean

Theorem

Find the slope of the line that passes

through the points (1, 1) and (5, 4).

Association

slope

Fig. 5 Discrimination and association. The solution of a mathematics problem requires that students discriminate one kind of problem from another and

associate each kind of problem with an appropriate strategy. Interleaving might improve both skills

Psychon Bull Rev (2014) 21:1323–1330 1329

photographs of different kinds of birds (jays, finches, swallow,

etc.) one at a time, in an order that was either blocked (each of

the jays, then each of the finches, etc.) or interleaved (jay,

finch, swallow, etc.), and interleaving would produce greater

scores on a subsequent test requiring participants to identify

previously unseen birds (e.g., Birnbaum, Kornell, Bjork, &

Bjork, 2013; Kang & Pashler, 2012; Kornell & Bjork, 2008;

Wahlheim, Dunlosky, & Jacoby, 2011; but see Carpenter &

Mueller, 2013). As with the results of previous interleaved

mathematics tasks, the positive effect of interleaving on cate-

gory learning could also be attributed to an improved ability to

discriminate between, say, a jay and a finch. To our knowl-

edge, though, it remains an untested possibility that this effect

might also reflect a strengthened association between each

category (e.g., finches) and the category name (“finch”). The

relative contributions of enhanced discrimination and stronger

associations to interleaving effects could be disentangled by

an experiment analogous to the mathematics experiment pro-

posed in the previous section: Participants would receive two

tests: a discrimination-only test requiring them to sort birds (or

identify the one bird that is different from others), and the

usual test requiring them to name novel birds, which would

require both discrimination and association. In summary, al-

though strong evidence exists showing that interleaved prac-

tice can improve both mathematics learning and category

learning, it seems unclear why either of these effects occur.

Author Note This work was supported by the Institute of Education

Sciences, U.S. Department of Education, through Grant No.

R305A110517 to theUniversity of South Florida (PI: D.R.). The opinions

expressed are those of the authors and do not necessarily represent the

views of the U.S. Department of Education. We thank Sandra Stershic for

her help with the data analysis, and we thank Jennifer DeMik, Brendan

Paul, Nancy Self, Liberty Middle School, and Hillsborough County

Public Schools for their participation.

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