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Understanding how struggling students approach math is vital to designing effective math lessons. Many low achieving students rely on a weak knowledge of procedures and attempt calculations without adequate consideration of the problem. We investigated how enabling or preventing premature calculations affected learning math. Students were presented with explanations of math problems that either contained numbers, thus allowing for calculations, or contained variables, thus preventing the possibility of calculations. In Experiment 1, we asked students to learn from a conceptual explanation and found that preventing calculations was beneficial, especially for students with less prior experience in math. In Experiment 2, when the lesson was procedures-focused, we found that preventing calculations did not have the same beneficial effect. Students with less prior experience performed poorly compared to those with more experience. Given students' prior math experience and their usual approach to problem-solving, we can facilitate learning by blocking maladaptive approaches.
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Journal of Cognitive Education and Psychology
Volume 17, Number 2, 2018
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Can Preventing Calculations Help
Students Learn Math?
Alyssa P. Lawson
Arineh Mirinjian
Ji Y. Son
California State University, Los Angeles, California
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Understanding how struggling students approach math is vital to designing eective
math lessons. Many low achieving students rely on a weak knowledge of procedures
andattempt calculations without adequate consideration of the problem. We inves-
tigated how enabling or preventing premature calculations aected learning math.
Students were presented with explanations of math problems that either contained
numbers, thus allowing for calculations, or contained variables, thus preventing
the possibility of calculations. In Experiment 1, we asked students to learn from a
conceptual explanation and found that preventing calculations was benecial,
especially for students with less prior experience in math. In Experiment 2, when the
lesson was procedures-focused, we found that preventing calculations did not have
the same benecial eect. Students with less prior experience performed poorly com-
pared to those with more experience. Given students’ prior math experience and their
usual approach to problem-solving, we can facilitate learning by blocking maladaptive
approaches.
Keywords:
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learning; preventing procedures; concept; transfer; instruction; prior
math experience
Students vary considerably in their ability to comprehend mathematical concepts
and eectively use mathematical procedures. What do students who struggle in
mathematics think math is about? In the United States, a country that scored lower than
the Organisation for Economic Co-operation and Development’s (OECD) average on the most
recent Programme for International Student Assessment scores for mathematics achievement
(OECD, 2016), students largely view themselves as “passive consumers of others’ mathematics”
(Schoenfeld, 1988, p. 18) or as one student put it, “In math, you have to just accept that that’s the
way it is and there’s no reason behind it” (Givvin, Stigler, & ompson, 2011, p. 7). at is, they
view themselves as learning things that “math people” have gured out. U.S. students consis-
tently believe memorizing and deploying procedures correctly is a critical part of mathematics
Pdf_Folio:178
178 © 2018 International Association for Cognitive Education and Psychology
http://dx.doi.org/10.1891/1945-8959.17.2.178
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Can Preventing Calculations Help Students Learn Math? 179
(e.g., Lubienski, 2001; Schoenfeld, 1992). Even in cases where students are encouraged to make
connections between procedure and concept, students often either fail to notice the connection
or do not attempt to understand it (Richland, Stigler, & Holyoak, 2012).
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Relatively low performing students (e.g., students in remedial/developmental math) mis-
takenly dene “understanding concepts” as being able to recall procedures quickly and precisely
(Oaks, 1987) and generally de-emphasize the role of ideas and connections in mathematics
(Givvin et al., 2011; Grouws, Howald, & Colangelo, 1996). In contrast, researchers and experts
dene and value conceptual and procedural knowledge dierently. Conceptual knowledge is
characterized by the underlying principles and interconnections between problems, and pro-
cedural knowledge is focused on the sequence of actions necessary to arrive at a solution to a
problem (Rittle-Johnson & Alibali, 1999). Experts and stronger students in mathematics are
more likely to emphasize these connections over calculations (Schoenfeld & Herrmann, 1982).
For most U.S. students, particularly low performing students, they only recognize procedural
knowledge in their math experience.
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Students struggling with developmental (or remedial) math demonstrate these beliefs in
their approach to problem-solving with a tendency to launch into calculations without trying to
understand the situation (Givvin et al., 2011; Stigler, Givvin, & ompson, 2010). Students who
struggle in mathematics often produce incorrect answers because of buggy algorithms, proce-
dures that are only partially correct or appropriate for some other set of circumstances (Brown
& Burton, 1977). Although buggy algorithms could be the result of students struggling to fol-
low procedures successfully, researchers have argued that students follow procedures well, but
the selected procedures are incorrect, and thus ineective, for that problem (Brown & Burton,
1977; Cauley, 1988; McNeil & Alibali, 2005; Resnick, 1982, 1983). So, if the fault does not lie
with the carrying out of procedures, perhaps the problem is the lack of understanding about the
situation at hand.
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Not fully understanding a mathematical situation is particularly problematic when students
are faced with the problem of transfer, applying their previous knowledge to a novel situation.
Students routinely fail to recognize situations that require similar calculations and similarly fail
to adapt appropriate procedures for calculations in novel situations (e.g., Carraher, Carraher, &
Schliemann, 1985; Givvin et al., 2011; Lave, 1988). In some cases, it may not be that students
are “ignoring the situation” but rather that they are misinterpreting it. One reason for this may
be that novices have diculty distinguishing core structural information from incidental details
(Son & Goldstone, 2009). is inability to be able to recognize isomorphic, structurally critical,
information from a lesson may make transferring to a novel situation much more dicult.
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Similarly, another reason for misinterpreting the situation is that students develop incom-
plete heuristics that might not be relevant in all situations. For example, students may focus on
trying to identify keywords in a situation in order to gure out a solution procedure. Unsuccess-
ful problem-solvers exhibit a tendency to xate on the keywords and numbers presented in the
problem rather than reading and comprehending the entire problem (Hegarty, Mayer, & Monk,
1995; Sherrill, 1983). Although this focus on a few details might lead to a successful solution
in some cases, it is mostly ineective as a general solution strategy. Moreover, repeated use of
this shortcut approach might reinforce the association between certain keywords and a specic
algorithm regardless of whether the association is appropriate for the specic situation. is
strategy can be exceedingly detrimental when the keywords students focus on do not match the
meaning prescribed to those words (Hegarty et al., 1995; Mevarech, Terkieltaub, Vinberger, &
Nevet, 2010).Pdf_Folio:179
180 Lawson et al.
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In general, a larger disconnect between understanding the mathematical concepts and the
procedures for solving problems might increase the likelihood of deploying “buggy algorithms.”
In other words, a less conceptually knowledgeable student of mathematics might be more likely
to perform inappropriate calculations with the presented numbers and keywords without rst
analyzing the problem. In many cases, from the student’s perspective, applying some algorithm
can feel like a step toward a solution, even when it is a highly maladaptive strategy.
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Traditionally, there have been two dierent responses to trying to resolve buggy algorithms.
e rst solution is to slow down and teach the procedures “better” (repeating the procedure
more slowly, more clearly, etc.). Unfortunately, this solution may not be practical if the problem
lies with the poor selection of appropriate procedures, rather than an inability to carry out the
procedures. e second type of response is to teach students the semantics of procedures, what
the procedures mean (Resnick, 1984). is type of instruction can help students by strength-
ening the connections between the procedure and the underlying concept. By introducing stu-
dents to the meaning of the procedure, students can improve their understanding of how the
notation and procedures connected to the problem and the ultimate solution. is relation-
ship is often bidirectional, with meaning helping to develop procedure and procedure helping to
develop meaning (Rittle-Johnson & Alibali, 1999; Rittle-Johnson, Schneider, & Star, 2015). Yet,
some research suggests that providing students with both procedural and conceptual instruc-
tion during the same instruction may not be the most eective way of teaching (Rittle-Johnson,
Fyfe, & Loeher, 2016). If a student already believes that learning procedures are the primary
goal of math instruction, they may discount the conceptual part of the lesson. For example,
receiving instruction on procedures before a lesson on the conceptual meaning led to worse test
performance than learning about the concepts alone (Pesek & Kirshner, 2000).
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is viewpoint informed the focus of this article on a third option. We take as a starting
point the possibility that there may be some benets to making concepts the whole focus of a
lesson. e solution that we propose is one where students are not allowed the possibility of
performing the procedure in order to shift the attention to understanding the situation. Prior
research has shown that preventing students from being able to perform procedures (by asking
them to explain problems without numbers in them) actually helped students perform better on
posttests that involved calculations (Givvin, Moroz, Loftus, & Stigler, under review). Compared
to a group given the same explanation task with numbers, the no numbers version of the task
led to better performance in solving subsequent transfer problems. By removing numbers from
word problems, students may have had more opportunity to focus on explaining the problem
and underlying mathematical structure rather than jumping into calculations.
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Given that preventing students from engaging in premature procedures might be helpful for
students trying to create explanations themselves, could preventing procedures also help when
students are learning explanations generated by others? For example, could this be helpful when
they are learning about a novel concept in math? In two experiments, we examined the eect
of preventing premature procedures during learning. We asked students to read explanations of
math problems that they had little experience with. Using the experimental manipulation cre-
ated by Givvin et al. (under review), we either prevented procedures by providing explanations
that had no numbers or allowed for the possibility of using procedures by providing explana-
tions with numbers. For Experiment 1, we hypothesized that the presence of numbers would
hinder students from learning something other than calculations. Specically, we predicted that
including opportunities to use procedures would hinder students from eectively learning from
conceptual explanations. To test this, students were randomly assigned to receive conceptualPdf_Folio:180
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Can Preventing Calculations Help Students Learn Math? 181
explanations of math problems with or without numbers (Experiment 1). Given that students
with a weaker background in math are more prone to buggy algorithms and novices are the least
likely to transfer their learning, we also examined how students of dierent math preparation
would perform on transfer questions after conceptual explanations with and without numbers
(Experiment 1). We also hypothesized that if the focus of learning was about the procedures
involved in calculations, providing the opportunity for the use of procedures would not have a
detrimental eect and perhaps even a benecial eect. So in Experiment 2, students were given
procedural explanations in the context of math problems with and without numbers.
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EXPERIMENT 1
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Methods
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Participants.In the rst experiment, 122 participants (83 female) were recruited from the
psychology subject pool from a state university and given course credit for their participation.
e subject pool is comprised of introductory psycholog y students, which is comprised of mostly
freshmen students. is class is taken by both students majoring/minoring in psychology as well
as students fullling their general education units. is class requires that students participate
in psychological research, but does not dictate which experiments students participate in.
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e average age of the participants was 19.58 (SD = 2.52). e participants described
themselves as Latino American/Hispanic/Mexican American (66%), Asian/Asian American/
Pacic Islander (18%), African American/Black (2%), White/European American (2%), and 12%
reported they were something other than what was listed. About half (49%) of the participants
reported English as their rst language. ese demographics are similar to the university popu-
lation from which the sample was drawn.
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As a measure of prior math experience, students self-reported whether they had ever taken
a calculus course: 54 participants reported having taken calculus, and 68 participants reported
never having taken calculus.
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Research Design.A 2 × 2, number × prior math experience, between-subjects design was
used in this study. Students were randomly assigned to receive instruction with numbers or with
no numbers. We operationalized more prior math experience as having taken calculus and less
prior math experience as not having taken calculus.
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Materials.Participants completed two packets: the learning packet and the testing packet.
e learning packet included a consent document, the conceptual explanation of a problem,
and explanation practice problems. e testing packet included posttest questions and a demo-
graphic questionnaire. ere were two versions of the learning packet, with numbers and no
numbers, and participants were randomly assigned to one. e testing packets were the same
for all participants.
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Learning Packet. e learning packet consisted of two parts: the conceptual explanation
and the explanation practice problems. e conceptual explanation began by presenting a prob-
lem (about using the arithmetic mean as a model for a distribution) either with numbers or
without numbers. Underneath the problem, a conceptual explanation of how to solve the prob-
lem was presented (so we will refer to this as the “explained problem”). e explained prob-
lem described why, without knowing each individual number in a data set, the average can
serve as a substitute or model for each individual number. Participants in the number condi-
tion were shown the explained problem with numbers (see Table 1) and a conceptual expla-
nation that also referenced the numbers. For participants in the no number condition, thePdf_Folio:181
182 Lawson et al.
explained problem and conceptual explanation used variables in place of numbers. (Both the
number and no number explanations are provided in full in Appendix A.)
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In each explained problem provided to the students, there were four components that were
key to understanding and solving the problem: old average, new average, N, and new amount.
ese four components and how they were presented to the participants are included in Table 2.
In the explained problem, we provided the students with old average, new average, and N, and
asked the participant to nd the new amount added to the old average that created the new
average.
TABLE 1. Explanation Problems and Practice Problems for Both Experiment 1 and
Experiment 2
Number Condition No Number Condition
Explained
problem
Stephanie is collecting money
from her coworkers in order to
buy their boss, Andie, a birthday
gift. Each of Stephanie’s
coworkers can donate any
amount of money that they
would like. Aweek before
Andie’s birthday, Stephanie
collects an average of $10 from a
total of ve5ve coworkers.
A few days before Andie’s
birthday, Stephanie’s coworker
Jim, who just returned from
vacation, added money to the
birthday fund, which increased
the average donation amount to
$10.50. How much did Jim
donate to the birthday fund?
Stephanie is collecting money
from her coworkers in order to
buy their boss, Andie, a birthday
gift. Each of Stephanie’s
coworkers can donate any
amount of money that they
would like. A week before Andie’s
birthday, Stephanie collects an
average amount of money from
ve coworkers. A few days before
Andie’s birthday, Stephanie’s
coworker Jim, who just returned
from vacation, added money to
the birthday fund, which
increased the average donation
amount. How much did Jim
donate to the birthday fund?
Practice
explanation
problem 1
Carlos is collecting money from
members of his basketball team
to pitch in and buy a bunch of
pizza. e players can donate
any amount of money that they
would like. Carlos gured out the
average amount of the donations
from vemembers of the team
is $6.50. en Carlos’ coach
decides to chip in as well and
this increased the average
donation amount to $7.75. How
much did Carlos’ coach donate?
Carlos is collecting money from
members of his basketball team
to pitch in and buy a bunch of
pizza. e players can donate
any amount of money that they
would like. Carlos gured out the
average amount of the
donations. en Carlos’ coach
decides to chip in as well and
this increased the average
donation amount. How much
did Carlos’ coach donate?
(Continued)
Pdf_Folio:182
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Can Preventing Calculations Help Students Learn Math? 183
TABLE 1. Explanation Problems and Practice Problems for Both Experiment 1 and
Experiment 2 (Continued)
Number Condition No Number Condition
Practice
explanation
problem 2
A group of sevenfriends loved
hiking and were comparing all
the dierent trails they had
hiked that year. Harold, one of
the friends, loved statistics so he
gured out an average number
of four hikes per person in the
group. When Harold’s father
joined the discussion, it turned
out that he had hiked so much as
to raise the mean number of
trails to ve. How many trails
did Harold’s dad hike this year?
A group of friends loved hiking
and were comparing all the
dierent trails they had hiked
that year. Harold, one of the
friends, loved statistics so he
gured out the average number
of trails each person hiked.
When Harold’s father joined the
discussion, it turned out that he
had hiked so much as to raise the
mean number of trails to a larger
amount. How many trails did
Harold’s dad hike this year?
TABLE 2. Breakdown of Questions in Learning Packet and Posttest Packet
Old Average New Average NNew Amount
Learning packet:
example problem
Given Given Given Missing
Posttest: two-
averages problem
Given Missing Given Given (in the
form of
another
average rather
than a single
new data
point)
Posttest: N
problem
Given Given Missing Given
Posttest: old
average problem
Missing Given Given Given
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e learning packet also included explanation practice problems. Participants were asked to
practice explaining with two more problems similar to the example with this prompt, “Write an
explanation that you think would help another student understand the problem.” Both practice
problems asked students to nd the new amount when given the old average, new average, and
N, like in the explained problem. ese explanation practice problems either had numbers or no
numbers consistent with the condition to which the participant had been assigned (see Table 1).Pdf_Folio:183
184 Lawson et al.
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Posttest. e posttest included three types of transfer problems. e transfer questions
were created by changing whichof the four components was missingin the average problem
(see Table 2). e three types of transfer problems were two-average, N, and old average. e two-
average problems were most similar to the explanation provided in the learning packet about
why an average can be used to stand in for individuals in a distribution.
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Procedure.Participants came into the laboratory in groups of up to 10 people at a time, and
each person was randomly assigned to receive the number or no number packet. Participants
worked on their own packets monitored by a research assistant. Participants had up to 35 min-
utes to complete the learning packet, then had the remaining time (25 minutes) to complete the
testing packet.
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Coding.e responses to the explanation practice problems were coded for two strategies:
(a) whether students attempted to solve the problem without any explanation (even though
they were asked to explain these problems), what we call “solve only,” and (b) whether those
explanations and/or solutions were correct, or what we call “correct.” To be considered “solve
only,” participants either needed to solve the problem with the numbers provided or solve by
manipulating variables or substituting made-up numbers. To be considered “correct,” partici-
pants could have provided a correct conceptual explanation or correctly demonstrated the steps
to solve the problem. If the response was “solve only,” the response was coded as “1” in that cat-
egory, and if not, the response was coded as “0.” is was the same for the “correct” responses
as well. Figure 1 depicts examples of responses that either contained some explanation or were
“solve only” and example responses that were correct and incorrect.
FIGURE 1.
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Explanation practice problems completed by students for Experiment 1.
is matrix shows examples of how explanations were coded for both
correct/incorrect explanations/solutions as well as explanation or solve only.
Pdf_Folio:184
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Can Preventing Calculations Help Students Learn Math? 185
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Two coders read through each answer and decided independently if the response was “solve
only” or not and “correct” or “incorrect.” ese coders agreed on all but ve of the responses. For
those ve responses, the two coders discussed the reasoning behind each of their responses and
then came to an agreement. After coding the explanations, each participant had a “solve only”
score and a “correct” score.
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Results
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Explanations Elicited During the Learning Phase.Table 3 provides the means and standard
deviations for the “solve only” and “correct” explanation scores. A 2 × 2, number × prior math
experience, analysis of variance (ANOVA) was performed for the “solve only” scores and revealed
a signicant main eect of both number, F(1, 107) = 6.17, p= .015, partial 𝜂²= .06, and prior
math experience, F(1, 107) = 5.38, p= .022, partial 𝜂²= .05. e interaction was not signi-
cant, F(1, 107) = 1.104, p= .296. Students given numbers in the learning phase were more likely
to solve without explaining even when the task was explicitly about explaining the problem.
Students with less prior math experience were also more likely to solve without explaining in
comparison to students with more prior experience. A 2 × 2, number × prior math experience,
ANOVA was also performed for the “correct” scores and revealed no signicant eects nor inter-
actions, Fs < 2.3, ps > .14.
ID:p0210
Problem-Solving in the Testing Phase.Preliminary analysis revealed that of the three
types of transfer problems, the two-average category was the only type where the mode of ques-
tions answered correctly was not 0. For the other two types (Nand old average), the majority
of participants could not answer any of these problems correctly. We focused our subsequent
analysis on the two-average problems.
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A 2 × 2, number × prior math experience, ANOVA was performed on performance on the
two-average transfer problems from the posttest. ere was a signicant interaction, F(1, 118) =
5.27, p= .02, partial 𝜂²= .04, as depicted in Figure 2. ere was no signicant main eect of num-
ber, F(1, 118) = .91, p= .34, nor prior math experience, F(1, 118) = 1.08, p= .30. Post hoc com-
parisons revealed that for students with less prior experience, explanations without numbers
led to signicantly better performance than explanations with numbers, t(66) = 2.62, p= .01.
For the students with more prior experience, there was no signicant dierence between these
conditions, t(52) = .83, p= .41. Furthermore, there was no signicant dierence between the
students with less prior experience and students with more prior experience in the no number
TABLE 3.
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Means (Standard Deviations) of Responses to Practice Explanations in
Experiment 1
Proportion of “Solve Only” Responses Proportion of “Correct” Responses
ID:t0150
ID:t0155
Less prior
experience
ID:t0160
More prior
experience
ID:t0165
Less prior
experience
ID:t0170
More prior
experience
ID:t0175
Number
ID:t0180
0.27 (0.416)
ID:t0185
0.10 (0.239)
ID:t0190
0.63 (0.369)
ID:t0195
0.82 (0.330)
ID:t0200
No number
ID:t0205
0.09 (0.229)
ID:t0210
0.02 (0.330)
ID:t0215
0.62 (0.390)
ID:t0220
0.75 (0.380)
Pdf_Folio:185
186 Lawson et al.
FIGURE 2.
ID:p0220
Means and standard errors of proportion correct in Experiment 1.
condition, t(60) = .89, p= .38, but there was a signicant dierence in the number condition,
t(58) = 2.36, p= .02.
ID:t
i0030
Discussion
ID:p0225
ere are two surprising aspects of this result: (a) here we nd that the students with less prior
math experience performed better when shown variables in explanations than when shown
numbers and (b) that in the case of no numbers, there is no dierence between the students
with less prior experience and students with more prior experience. ese results are surpris-
ing because one would expect that participants with more math preparation would perform
better than participants with less math preparation, but this was not the case for those given
variables in the explanation. One can argue that variables are more abstract than numbers and
thus should be more dicult (particularly for students with less math experience). However,
the benet of variables is that they do not allow students to focus solely on applying procedures
thus freeing up their attentional resources for understanding the underlying mechanisms of the
problem.
ID:p0230
When we examined the “explanations” written by the participants, those given numbers
in the problems were more likely to solve the problem without actually providing any explana-
tion. Students in the number condition, at least implicitly, believe that solving is explaining.
Even when students are instructed to practice explanations, when those conceptual explana-
tions include numbers, students seem to extract and repeat the parts that are most related to
solving the problem. Students who did not have any numbers during learning are much less
likely to provide “solve only” responses. Perhaps they are also attending more to other parts
of the explanation besides the parts relevant to calculations. e absence of numbers seems to
prevent participants from jumping into number manipulation and allows more time to focus on
something besides the solution procedures.Pdf_Folio:186
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i0005
Can Preventing Calculations Help Students Learn Math? 187
ID:p0235
Furthermore, students with less prior experience were also more likely to provide “solve
only” responses during learning. Students with less pr ior experience may be more prone to want-
ing to start solving right away compared to students that have more experience with math. is
may suggest that students with less prior experience may rely more on calculations to solve
problems while their more experienced counterparts may use other sources of information to
make sense of the problem.
ID:p0240
e results support the idea that providing numbers drew students’ attention to calculating
solutions. By not providing numbers, students with less prior experience were more successful
after a presentation of concepts that underlie problem-solving strategies. Taking away the pos-
sibility of calculating the solution during learning may have encouraged other kinds of thinking.
Students with less math experience may be particularly prone to relying on procedures, perhaps
because they have diculty attending to the structure of the situation or they do not recognize
that analyzing the structure would help them determine which calculations would be useful.
us, the no number condition may have led students to more thoughtfully consider how each
part of the problem t with the explanation.
ID:p0245
For those in the number condition, students with less prior experience may have tried to
focus on learning (and practicing) the procedures to solve the problem or may have believed
they understood the situation fairly well. Either way, this may have taken away attention from
the conceptual explanation itself. For students with more math experience, there may be no
dierence between number conditions for a variety of reasons. Perhaps for these students, the
dependence on using a procedure to “solve” might not be as strong, they might be equally facile
with numbers and variables, or they might be more experienced at attending to structure in
math problems in general.
ID:p0250
In Experiment 1, preventing the use of procedures was critical for learning something other
than the calculations themselves. But what if the calculations were the focus of learning as in the
case of procedural instruction? Removing numbers might be expected to facilitate conceptual
understanding, but would it also facilitate learning of purely procedural skills? e next exper-
iment, presenting a straightforward procedural explanation in the context of numbers and no
numbers, was designed to address this question.
ID:T
I0035
EXPERIMENT 2
ID:t
i0040
Method
ID:p0255
Participants.In the second experiment, 129 participants (84 female, 1 declined to answer)
were recruited from the psychology subject pool of a state university. eir average age was
20.13 (SD = 3.06). e participants described themselves as Latino American/Hispanic/Mexican
American (68%), Asian/Asian American/Pacic Islander (19%), White/European (4%), African
American/Black (2%), Middle Eastern/Arab American (2%), and 5% reported they were some-
thing other than what was listed. Slightly less than half (41%) of the participants reported
English as their rst language.
ID:p0260
As in Experiment 1, students self-reported whether they had ever taken a calculus and 84
participants reported never having taken calculus.
ID:p0265
Research Design. e design was the same as the rst study (i.e., a 2 × 2, number × prior
math experience). We examined performance on two-average problems given participants’ dif-
culty in Experiment 1 with the other types of transfer problems.Pdf_Folio:187
188 Lawson et al.
ID:p0270
Materials. e only dierence with this study was that the explanations given in the
learning packet focused on procedurally solving problems. e procedural explanations showed
simply how to solve this problem, as opposed to why such a strategy can be deployed in this
circumstance. As in Experiment 1, there were number and no number versions of the learning
packet (both the number and no number explanations are provided in full in Appendix B).
ID:p0275
Procedure. e second study followed the same procedures as the rst study.
ID:p0280
Coding.e practice explanations for Experiment 2 were coded in the same way as Experi-
ment 1. Figure 3 depicts examples of responses that either contained some explanation or were
“solve only” and example responses that were correct and incorrect.
ID:t
i0045
Results
ID:p0290
Explanations Elicited During the Learning Phase.Table 4 provides the means and
standard deviations for “solve only” and “correct” explanation scores. A 2 × 2 ANOVA
was performed on the “solve only” scores in Experiment 2. Unlike Experiment 1,
there were no signicant main eects of number, F(1, 114) = .26, p=.61, nor prior
math experience, F(1, 114) = .01, p= .92. e interaction was also not signi-
cant, F(1, 114) = 3.175, p= .077. A 2 × 2 ANOVA was also performed on the “cor-
rect” scores and there were signicant main eects of number, F(1, 114) = 4.80,
p= .03, partial 𝜂²= .04, and prior math experience, F(1, 114) = 8.42, p= .004, partial 𝜂²= .07,
but no signicant interaction, F(1, 114) = .827, p= .37. Participants’ given numbers provided
more correct explanations and students with more prior math experience also gave more
correct explanations.
FIGURE 3.
ID:p0285
Explanation practice problems completed by students for Experiment 2.
Pdf_Folio:188
ID:t
i0005
Can Preventing Calculations Help Students Learn Math? 189
TABLE 4.
ID:p0295
Means and (Standard Deviations) of Responses to “Solve Only” Practice
Explanations in Experiment 2
Proportion of “Solve Only” Responses Proportion of “Correct” Responses
ID:t0225
ID:t0230
Less prior
experience
ID:t0235
More prior
experience
ID:t0240
Less prior
experience
ID:t0245
More prior
experience
ID:t0250
Number
ID:t0255
0.15 (0.334)
ID:t0260
0.05 (0.154)
ID:t0265
0.69 (0.452)
ID:t0270
0.85 (0.328)
ID:t0275
No number
ID:t0280
0.08 (0.219)
ID:t0285
0.18 (0.373)
ID:t0290
0.45 (0.395)
ID:t0295
0.75 (0.380)
FIGURE 4.
ID:p0305
Means and standard errors of proportion correct in Experiment 2.
ID:p0300
Problem-Solving in the Testing Phase.A 2 × 2 ANOVA was also performed on the posttest
scores of the two-average transfer problems. ere was no signicant eect of number, F(1,
125) = 1.03, p= .31, but there was a signicant eect of prior math experience, F(1, 125) =
5.53, p= .02, partial 𝜂²= .042, with the students with more prior experience outperforming the
students with less prior experience. Unlike Experiment 1, the interaction between number and
prior math experience was not signicant, F(1, 125) = 1.16, p= .28, as depicted in Figure 4.
ID:t
i0050
Discussion
ID:p0310
In this study, with procedural explanations, there was no discernable benet to preventing the
use of procedures. Regardless of whether students received numbers or no numbers in their
explanation and practice problems, this did not change how the students performed in solving
subsequent transfer problems. e presence of numbers in procedural instruction did support
Pdf_Folio:189
190 Lawson et al.
more correct explanations during the practice explanation problems. ese results show that
preventing the use of procedures may not always be benecial to students (e.g., in situations
when procedures are the focus of instruction). When practicing an explanation of a procedure,
having the numbers might help students do so correctly.
ID:T
I0055
GENERAL DISCUSSION
ID:p0315
Experiment 1 and 2 indicate that preventing premature procedures during learning may be most
useful (a) for students with less prior math experience and (b) when those students are trying
to learn about concepts.
ID:p0320
It is quite possible that the students with less math experience might be prone to relying on
procedures. For students with more math experience, it may be that a reliance on procedures
may not be as strong, they may be equally facile with numbers and variables, or they might
already be pretty good at attending to structure in math problems in general. In the number/
conceptual condition (Experiment 1), we found a very typical result: participants with more
prior experience outperformed participants with less prior experience. However, the no num-
ber/conceptual explanations (Experiment 1) were able to close the expected learning gap
between these two groups. Preventing premature procedures for the students with less prior
experience was so benecial in Experiment 1 that there was no discernible eect of prior math
experience. is is surprising as prior experience is one of the consistently best predictors of
learning (e.g., see Dochy, Segers, & Buehl, 1999 and Tobias, 1994, for relevant reviews). Yet,
with procedural explanations (Experiment 2), both in the number and no number conditions,
we return to the typical pattern: the students with more prior experience outperformed those
with less prior experience. e benet of preventing procedures is that it did not allow the stu-
dents with less prior experience to engage in the more commonly used procedural explanations.
ID:p0325
e results of this study are particularly interesting due to the dierences between Exper-
iment 1 and 2. With one type of instruction, there is no gap between students with diering
levels of prior experience and in another type of instruction, there is a gap between the groups.
is is important for instructors and researchers to understand when trying to help students,
especially those with less prior experience, perform better on tests that require them to have
some conceptual understanding of material, such as with transfer tests.
ID:p0330
is study is limited in that posttest performance was often very low. e type of problem
may have seemed quite novel for most students in the psychology subject pool (many of whom
were freshmen and likely, had not yet taken a college statistics course). We thought these prob-
lems would be appropriate because solving them has less to do with statistics than basic algebra.
But even in the most facile category of the transfer problems, performance ranged from 20%–
50% correct.
ID:p0335
Another limitation to this study was the way we presented mean to students in
Experiment 1. Given the time limitations, we had to quickly present what “mean” is in a clear,
but also a concise way. Consequently, it could be argued that the way in which we presented the
mean may have mirrored students’ beliefs about mean being a magic number, which might have
led students to try to focus on memorizing the procedure to nd the mean. In future studies, it
is important to understand how preventing calculations would benet students in a more rich
context, such as a realistic learning setting, to truly understand students’ math learning.
ID:p0340
Furthermore, a potential alternative explanation for why students with less prior experi-
ence in math performed especially well in the conceptual/no numbers condition may be thatPdf_Folio:190
ID:t
i0005
Can Preventing Calculations Help Students Learn Math? 191
not having numbers in the explanation may have decreased the likelihood and the severity of
students experiencing math anxiety. By having less anxiety, students may have beneted from
the explanation more than students that saw the same explanation with numbers. It would be
interesting for future research to investigate how math anxiety may play a key factor in learning
math with and without numbers introduced.
ID:p0345
is study warrants further exploration into how teaching the procedures of other concepts,
both within and outside of statistics, may be aecting student understanding and thus student
performance. Many college students may already have some knowledge in the concept of mean,
either due to intuitive knowledge or previous formal instruction. If the results of this study were
dulled by previous knowledge of mean, results from areas students have less experience with or
less intuitive knowledge of may produce even more compelling results. us, to further under-
stand how preventing procedures can benet students, there should be further investigation on
how these eects play out in other groups of people as well as with other topics of instruction.
ID:p0350
Our method of “instruction” was very much like learning how to solve problems based on a
textbook explanation. Would preventing the use of procedures in other common instructional
forms such as lectures and active learning exercises have a similar eect? What would a class
culture that prevents students from rushing into the use of procedures look like? Given that stu-
dents have a preference for calculation and view math as being mainly about calculation, would
instructional routines that block this compulsion lead to a dierent view of mathematics? Our
own results and the few documented cases of preventing students from manipulating numbers
(e.g., Givvin et al., under review) have mostly shown evidence in one-time experimental settings.
But there is much work that needs to be done to translate these ideas into a longer term, more
authentic learning.
ID:p0355
ese experiments provide further evidence for reliance of procedures, which has eects
on student learning and understanding. As eorts such as the Common Core State Standards
(National Governors Association Center for Best Practices, Council of Chief State School O-
cers, 2010) shifts learning away from procedures and more toward a conceptual underpinning of
math practices, nding eective methods for circumventing ineective problem-solving strate-
gies might be key. e goal of helping students appropriately construe a situation is a bigger
goal than merely correcting the use of buggy algorithms. Several of the Common Core Math-
ematical Practices (National Governors Association Center for Best Practices, Council of Chief
State School Ocers, 2010) privilege a greater understanding of the situation over procedures
to solutions: make sense of problems, model with mathematics, use appropriate tools strate-
gically, and look for and make use of structure. Simply giving students instructions to engage
in these practices may not be enough. Removing numbers is one small way to adjust the con-
text so that students shift their attention from calculations to nding meaning in the situation.
By paying careful attention to how students view and approach math, we can gain insight into
what might help students, especially those with little math preparation, engage in consequen-
tial mathematical practices.
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Disclosure. e authors have no relevant nancial interest or aliations with any commercial interests
related to the subjects discussed within this article.
Correspondence regarding this article should be directed to Alyssa P. Lawson, California State Univer-
sity, Los Angeles, Psychology Department, 5151 State University Drive, Los Angeles, CA 90032. E-mail:
a.p.lawson14@gmail.com
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ID:T
I0065
APPENDIX A
ID:T
I0070
NUMBERS CONDITION
Pdf_Folio:194
ID:t
i0005
Can Preventing Calculations Help Students Learn Math? 195
ID:T
I0075
NO NUMBERS CONDITION
Pdf_Folio:195
196 Lawson et al.
ID:T
I0080
APPENDIX B
ID:T
I0085
NUMBERS CONDITION
Pdf_Folio:196
ID:t
i0005
Can Preventing Calculations Help Students Learn Math? 197
ID:T
I0090
NO NUMBERS CONDITION
Pdf_Folio:197
... We want students to expend their energy grappling with key concepts and the connections between them, but because US college students tend to equate doing math with doing calculations, they often will, if given the chance, start calculating before they have had a chance to think about a problem, how it relates to core concepts, and even which calculations might be most appropriate given the situation. One way to stave off premature calculations and reserve attention for conceptual connections is by giving students problems that do not have any numbers in them, making calculations impossible (Givvin et al. 2019;Lawson et al. 2019b). In our book, we hold off on presenting any formulas or calculations until chapter 5, asking students instead to work on developing their intuitive ideas about models and model comparison by examining and discussing graphical representations of data (e.g., histograms or box plots). ...
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