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4MathAMATYC Educator ~ Vol. 2, No. 3 ~ May 2011
In a prior issue of MathAMATYC Educator, we
reported on our efforts to nd out what community
college developmental mathematics students understand
about mathematics (Stigler, Givvin, & Thompson, 2010).
Our work painted a distressing picture of students’
mathematical knowledge. No matter what kind of
mathematical question we asked, students tended to
respond with computational procedures, which they often
applied inappropriately and incorrectly. Their knowledge
of mathematical concepts appeared to be fragile and
weakly connected to their knowledge of procedures. But
we also found some reason for hope. First, we found
that when students were able to provide conceptual
explanations for procedures, they often produced correct
answers. Second, though students rarely used reasoning
on their own to solve problems, they could reason under
the right conditions.
To follow up on our previous article, we conducted
one-on-one interviews with a sample of community
college developmental math students. These interviews
were designed to further probe students’ mathematical
thinking, both correct and incorrect. The interviews not
What Community College
Developmental Mathematics
Students Understand about
Mathematics,
Part 2: The Interviews
only corroborate our earlier ndings, but also enable
us to piece together a richer picture of these students,
mathematically speaking. Before we describe the
methods and ndings of the interviews, we will present
a summary of the picture we are developing. We will
also speculate on how these students came to approach
math in the way they do. Though the latter was not the
direct object of our study, we nd such speculation
helpful for understanding how these students think about
mathematics, and what might be done to improve their
future prospects for mathematical prociency. We rely
on other research to inform this hypothetical account of
how students got where they are. After we present our
hypothetical account, we will present evidence from the
interviews to support it.
The Making of a Developmental Mathematics
Student: A Hypothetical Account
Once upon a time, the developmental mathematics
students of today were young children. Like all young
children, they had, no doubt, developed some measure
of mathematical competence and intuition before they
Karen B. Givvin
University of California, Los Angeles
James W. Stigler
University of California, Los Angeles
Belinda J. Thompson
University of California, Los Angeles
ACKNOWLEDGMENTS – This study was funded by grants from the Carnegie Foundation for the Advancement of
Teaching, Bill & Melinda Gates Foundation, William and Flora Hewlett Foundation, Kresge Foundation and Lumina
Foundation for Education. The views expressed in this article are part of ongoing research and analysis and do not
necessarily reect the views of the funding agencies. We wish to thank James Hiebert, Tony Bryk, and Catherine
Sophian for their constructive feedback on earlier drafts.
5www.amatyc.org
entered school. Research tells us that virtually all children
learn to count, develop basic concepts of quantity, and
develop understandings of basic mathematical operations
such as addition and division or sharing (Ginsburg, 1989;
NRC, 2001). Their concepts and procedures developed in
tandem. For example, they naturally learned to connect
rote counting procedures to concepts such as one-to-one
correspondence and cardinality (Gelman & Gallistel,
1978). Procedures such as counting were constrained by
concepts, and so made sense.
All this started
to change once our
students entered school.
There, few links were
constructed between the
understandings they had
and the symbols and
rules they were taught
(Hiebert, 1984). They
may have encountered
teachers with narrow views of what it means to
know and do mathematics. These teachers viewed
mathematics as primarily about computation
and applying rules (Battista, 1994), or otherwise
knew that mathematics should make sense but felt
that notion was implicit in the procedures they
presented, and therefore never made the underlying
concepts explicit. In their teaching, they emphasized
procedures and paid relatively little attention to
conceptual connections (Schoenfeld, 1988; Hiebert
et al., 2003). In the process, our students were
socialized to view mathematics as a bunch of rules,
procedures, and notations, all of which needed to be
remembered (Schoenfeld, 1989). At the same time,
they most likely gave up on the idea that mathematics
was supposed to make sense, learning that to do
mathematics well simply required following the steps
outlined by the teacher.
Students who were curious, or who tried to
understand why algorithms worked, were often
discouraged by the teacher either overtly or
inadvertently. Understanding procedures takes
time, and teachers have to “cover” the curriculum.
Many of these students slowly changed their view
of mathematics and came to view it as mainly just
steps to be remembered. Once this view started to
set in, and was reinforced by rewards such as high
quiz and test grades, if a teacher tried to get them to
slow down and understand how an algorithm worked,
the students would push back, usually by ignoring
the teachers’ explanations. Conceptual explanations,
these students felt, were just wasting their time,
time they needed for practicing and memorizing the
growing number of procedures for which they were
responsible and rewarded. The intuitive concepts that
supported their thinking and reasoning when they
were younger began to atrophy, serving no purpose in
the world of school mathematics.
Although most students in U.S. schools shared
these experiences, most did not end up needing to take
developmental mathematics courses in community
college. Some students
learned on their own,
or through exceptional
instruction, the value
of connecting rules and
procedures to concepts.
They discovered
that things that make
sense are more easily
remembered, and they
sought out sense-making strategies on their own. These
students went on to excel in mathematics. Still others,
though they experienced the conceptual atrophy we’ve
described, were able to rely on a strong memory. On
their college placement test they remembered what
to do without necessarily knowing why they were
doing it, and they managed to land themselves into a
transfer-level math course.
The community college students that are the
focus of our attention fall into neither group.
Without conceptual supports and without a strong
rote memory, the rules, procedures, and notations
they had been taught started to degrade and get
buggy over time. The process was exacerbated by
an ever-increasing collection of disconnected facts
to remember. With time, those facts became less
accurately applied and even more disconnected from
the problem solving situations in which they might
have been used.
The product of this series of events is a group of
students whose concepts have atrophied and whose
knowledge of rules and procedures has degraded.
These students lack an understanding of how
important (and seemingly obvious) concepts relate
(e.g., that
1
3
is the same as 1 divided by 3). They
also show a troubling lack of the disposition to figure
things out, and very poor skills for doing so when
they try. This leads them to call haphazardly upon
procedures (or parts of procedures) and leaves them
unbothered by inconsistencies in their solutions.
If an effective approach
to applying procedures is
“ready, aim, fire,” then it’s as
if those students fail to take
aim.
6MathAMATYC Educator ~ Vol. 2, No. 3 ~ May 2011
Our story is summarized in Figure 1.
In the sections that follow, we describe the
interviews we conducted. As we will show, these
interviews are consistent with the story we have created,
further enriching our view of what community college
developmental mathematics students understand
about mathematics. We conclude by laying out some
hypotheses for how we might get community college
developmental mathematics students to become
successful mathematics learners, which is what we argue
many have been capable of being all along.
The Interview and the Interviewees
We interviewed thirty students (15 female, 15 male)
enrolled in developmental mathematics courses at a
community college in the greater Los Angeles area.
Students were drawn equally from Arithmetic, Pre-
Algebra, and Elementary Algebra sections. (The only
developmental mathematics course not included in the
interviews was Intermediate Algebra.) Students ranged
in age from 17 to 51 (M = 21.6, SD = 6.5). Eleven were
Hispanic, nine were white, four were African-American,
and six were of mixed ethnicity. For ten students, English
was not the primary language spoken at home. Interviews
ranged in length from 54 to 128 minutes (M = 75.6,
SD = 14.2). Each student received $50 for participating.
The one-on-one interviews took place on the campus
at which students were enrolled and were scheduled at
students’ convenience, outside class time. Guided by a
protocol that was rened through extensive pilot testing
(see Appendix for questions relevant to this paper), the
interview set out to assess students’ understanding of
some key math concepts, from arithmetic through pre-
algebra. We opened each interview by asking students
to think about what it takes to be good at mathematics.
That was followed by several mathematics problems
(e.g., comparisons of and operations with decimals,
comparisons of fractions and placement on a number line,
solving equations with one variable, and equivalence).
For each line of questioning, we anticipated possible
responses and created structured follow-ups. The general
pattern was to begin each line of questioning at the
most abstract level and to become progressively more
concrete, especially when students struggled. Each of
the mathematical questions concluded with prompts
that pressed for reasoning. The interviewer told students
that she was more interested in their thinking about
mathematical problems than she was in their answers and
students were encouraged to talk through their thinking
aloud as they worked. The interviewer avoided pointing
out when students made mistakes, often refusing to
conrm an answer as correct or incorrect when a student
asked. The closing of the interview included a question
about advice students would give to their teachers
about how to teach math so that students would better
understand it. A Livescribe Pulse pen was used to record
Figure 1. The making of a community college developmental math student: A hypothetical account.
7www.amatyc.org
dialog and to capture written work.
Each interview question was coded by two (or
more) coders. If their coding was initially discrepant, the
relevant portion of the interview was reexamined and
discussed until consensus was reached.
Students’ Approach to Problems: Math as a
Collection of Procedures to Be Applied
When we asked what it means to be good at
mathematics, 77% of students spoke to the perceived
procedural nature of the subject. (Only 13% spoke of
being good at math at understanding concepts.) As
one prealgebra student put it, “Math is just all these
steps.” Other students responded in a way that not only
supported the role of rules and procedures, but also
discounted the role of conceptual understanding. For
instance, one Elementary Algebra student stated that, “In
math, sometimes you have to just accept that that’s the
way it is and there’s no reason behind it,” and another
Elementary Algebra student responded that, “I don't think
[being good at math] has anything to do with reasoning.
It's all memorization.” When we asked students to give
advice to a math teacher with respect to what might be
done to better promote learning, the dominant themes
were about presenting material more slowly and with
more repetition, and breaking down procedures into
smaller steps, all of which reect an acceptance that math
is about procedures.
Consistent with those beliefs, when our interview
questions asked students to solve problems, students
would quickly choose a procedure they remembered
from school, and then set about applying it to the
situation. (This was the case even when we deliberately
asked questions for which executing a procedure was
not necessary.) By itself, that approach might not have
been problematic. The problem lies in the fact that the
procedures called upon were often either inappropriate
for the situation or were executed with critical errors—
errors that would surely have been caught had students
understood the concept underlying the procedure or
noticed that the magnitude of the answer was not
reasonable. Without a conceptual understanding of the
procedures in their toolbox, students were left to rely
solely on a memory of which to use and how to use it.
It appeared that over the years and with an increasing
collection of procedures from which to draw, that
memory had eroded.
This behavior of drawing on a collection of
procedures is evident in students’ responses to an
interview item that asked them to place multiple fractions
(i.e.,
4
5
,
5
8
, −
3
4
,
5
4
) on a number line. Thirty percent of
students set about dividing, having some recollection that
it might help them with the task at hand. Although it is
true that one reasonable approach to this task would be to
divide numerators by denominators, convert to decimals,
and use decimals to place each fraction on a number line,
this is not what was done by students who chose to use
division. All of the students who used division did so to
create a new fraction. (The advantage of doing so was
unclear; if they could place the new fraction, then why
not the original?) More alarmingly, two-thirds of those
who used division created a new fraction that wasn’t
equivalent to the original. Specically, those students
divided the denominator of a fraction by the numerator.
As an illustration, Student #17 (enrolled in Prealgebra)
converted
4
5
to 1
1
4
, which he then converted to
5
4
without noticing (or without being bothered) that
5
4
was
different from the
4
5
with which he started. He did the
same in converting
5
8
to 1
3
5
and then to
8
5
. It appeared
that students’ primary goal was not to write a fraction in
an equivalent form, but rather to perform an operation.
With the procedure complete, they accepted the resulting
value as correct, with no apparent regard for whether it
made any sense.
One way to interpret why students divided the
denominator by the numerator is that they “forgot” the
order. That hypothesis would suggest that instructors
should “remind” students. Another interpretation, one
that is based on a more conceptual view of mathematics,
is that students aren’t appropriately connecting fractions
and division. If students thought about
1
3
as one whole
that has been divided into three parts, then how could
they “forget” what to divide by what? The other concept
that demands attention is that of equivalence. For a
student to be able to assess whether a result in this case
is correct, s/he must understand that using division to
rewrite a fraction as a decimal preserves the value of
the number. Perhaps Student #17 knows that
5
4
isn’t
equivalent to
4
5
but doesn’t know that an appropriate
result has to be equivalent.
Another problem provides further evidence of
students’ reliance on procedures. Students were asked
to select the larger value, given
a
5
and
a
8
. With a basic
8MathAMATYC Educator ~ Vol. 2, No. 3 ~ May 2011
understanding of fractions (or of division), a student
could answer this question with no procedure at all, yet
most students chose to apply one. When the problem
was posed to Student #24 (enrolled in Arithmetic), she
selected
a
8
as the larger of the two fractions, explaining
that 8 is larger than 5. When prompted to substitute a
value for a as a way to help her make a comparison, she
substituted a 1 in place of each a, and then multiplied
the resulting two fractions, obtaining
1
40
. When the
interviewer asked if that process helped her to compare
the two fractions, she responded, “In a way.” It seemed
almost as if she multiplied the two fractions because
she knew how to, regardless of whether it could help her
answer the question she had been asked.
If an effective approach to applying procedures
is “ready, aim, re,” then it’s as if those students fail
to take aim. Students’ tendency to apply procedures
without thinking of the concepts that underlie them
was by no means limited to fractions. A question on
decimal subtraction revealed the same inclination.
Students were given the problem 0.572 – 0.86, written
horizontally. One student (enrolled in Elementary
Algebra) wrote the problem vertically and then
simply treated each column as a separate problem,
subtracting the smaller value from the larger no matter
its placement. For her, the procedure of subtracting
smaller values from larger values applied to individual
digits rather than to each value as a whole. Another
student wrote the problem vertically with 0.572 above
0.86 (an approach we saw from 80% of students), and
when calculating, added “1” to 0.572 so that there
would be a value from which to borrow. He appeared
untroubled by the change. When their problem set-
up didn’t initially “work,” those students might have
taken it as sign to reconsider the set-up itself. Had
they stepped back and thought about the underlying
concepts, they might have selected an appropriate
procedure. Instead, students clung to a desire to
make their chosen procedures work, even when the
adjustments they made were not mathematically valid.
Students’ approach to the subtraction problems
appeared to be very mechanical. We might assume
that had they been given the problem 5 – 8, at least
some would have correctly responded –3, or at least
recognized that something about the order of the
numbers must be addressed. However, students failed
to connect the problem they had been asked to set up
to simpler problems (such as 5 – 8) that might have
helped them decide on a sensible course of action.
This might be because they don’t have the habit
of investigating possible procedures using simpler
problems, or maybe they don’t think the rules for
operating with integers will necessarily apply when
operating with decimals.
To further investigate the degree to which students
clung to familiar procedures (even when they were the
most cumbersome option) we presented students with a
series of multiplication problems and asked that they do
the calculations mentally. They were as follows:
10 × 3 =
10 × 13 =
20 × 13 =
30 × 13 =
31 × 13 =
29 × 13 =
22 × 13 =
The series was designed to see if students would make
use of decomposition and the distributive property,
or perhaps rely on answers to earlier problems
in the series to help them solve later ones. Either
method would have made it easier to perform the
calculation mentally by reducing the load on working
memory. Seventy-three percent of students never
used decomposition and the distributive property;
77% never relied on answers to earlier problems in
the series. (Sixty-three percent never used either of
the two approaches.) The standard algorithm was the
most frequent approach chosen, with 80% of students
using it at least once and 20% of students beginning
to use it as early as 10 × 3. Some students enacted the
algorithm with ngers in the air (or on the desk) and a
few even “erased” when necessary, thus demonstrating
their reliance not only on the algorithm itself but also
on the method of carrying it out.
In many cases, answers to adjacent mental
multiplication problems could have prompted students
to question their work, but it very often did not. Of the
23 students who made at least one error in the series
of questions, 74% made an error that was sufciently
inconsistent with their other answers that had they
compared their solutions, the mistake should have
been caught. Students #5 and #10 (both enrolled in
Elementary Algebra) are examples. The fact that even
the most egregiously incorrect answers did not alarm
students reinforces for us the commitment of students
to procedures they have been taught, even if the results
make no sense.
9www.amatyc.org
Student #5
10 × 3 = 30
10 × 13 = 113 130
20 × 13 = 86
30 × 13 = 120
31 × 13 = 123
29 × 13 = 116
22 × 13 = 92
Student #10
10 × 3 = 30
10 × 13 = 130
20 × 13 = 260
22 × 13 = 52
30 × 13 = 120
31 × 13 = 124
29 × 13 = 126
It’s unclear whether students considered that there
might be an easier way to solve the series of mental
multiplication problems. But it appeared as if students
relied not just on procedures, but on a single, familiar
procedure. Perhaps they thought that if some people nd
such problems easy or can calculate them quickly, it’s
because they can simply keep track of partial products
better, and not that they use a more efcient strategy. We
would argue that if instruction includes an exploration of
multiple solution methods and analysis of how they’re
related, students might come to think of procedures more
exibly. (In a later section we’ll include an example of
what can happen when this is done.)
It appeared in the interviews that procedures were
sometimes memorized without any meaning at all,
making it difcult for students to know when to use
them across situations. We asked two direct questions of
students aimed at assessing whether they knew why they
were doing what they were doing. The rst concerned the
standard algorithm for multiplication and why we put a
“0” (or a “*” or a blank) in the rightmost position of the
second partial product, as illustrated in gure 2.
22
× 13
66
220
286
Figure 2. Example of the standard algorithm for multiplication,
with a zero in the rightmost position in the second partial
product
Following are examples of student responses:
I guess I’m just used to it. My teacher always says to
write a zero….You know, I don’t really know the answer
to why we can’t, but I’m already programmed to do it
like that, so. I wish I knew. I know there’s a proper term
for it, but I don’t really know the term. But you can’t
[right align it]. Because you just get a different answer...
You get the wrong answer. (Student #4, enrolled in
Prealgebra)
It’s just being taught, you know? Each time you go
down for the next number you put a zero. So if there’s
a third number, I’d put two zeros and you keep going…
[If you don’t put the 0] you get the wrong answer. But I
don’t know why. It’s just something from, you know, I
guess from 4th grade. They just teach you and go with it.
(Student #8, enrolled in Elementary Algebra)
Um, it might be correct, it might not. To me it’s
correct because that’s how I, you know, that’s how I got
used to it. So, I don’t know how other people might view
it. (Student #24, enrolled in Arithmetic)
I really don’t know. I don’t know why it’s done like
that but that’s the way I was taught to do it and I always
just did it like that. I don’t know the answer to that,
though. (Student #27, enrolled in Elementary Algebra)
The second direct question we asked was about why
we align the decimal points when we nd the difference
between two values. The following are example
responses:
I don’t know. You kinda just learn to do it with the
decimals. I guess you’re just programmed when you
learn something. I mean, I don’t really fully remember
decimals. But I know you have to know the decimals,
unless it’s multiplication or something. But I don’t
remember. But I guess you’re just programmed. (Student
#4, enrolled in Prealgebra)
I guess it just looks more organized and it looks
easier to approach and I think I somewhere along the line
I think I was just instructed to keep the decimals lined up.
I don’t remember where I heard that but I just, my logic
just says keep them lined up to each other. (Student #11,
enrolled in Elementary Algebra)
I think there is a reason for it, but I just can’t recall
right now. But I think I was supposed to line up the
decimals. I mean, but then again, I got 2 [different]
answers, so I don’t know. I’m not sure. [Experimenter: So
maybe you don’t line up the decimals?] Maybe I don’t.
(Student #17, enrolled in Prealgebra)
To the rst question, fewer than half of students
referred to place value in their response and to the second
10 MathAMATYC Educator ~ Vol. 2, No. 3 ~ May 2011
question only a third did. We might have hoped that a
student would respond that when we multiply 22 × 13,
a 0 is inserted in the second partial product because that
product represents the multiplication of 22 by 10, and not
22 by 1. At best, students stated the term “place value” or
“placeholder” without being able to explain its relevance
to the question. Just because students can recite the
place value names does not mean they attach a “powers
of ten” meaning to the places. The persistent neglect of
magnitude of numbers leads us to conclude that students
either do not understand the magnitude of numbers or
that they do not use this knowledge to reason about
values when they’re unsure of how to proceed.
The collection of student errors we’ve described does
more than illustrate the confused algorithmic thinking
in which many students engage. It demonstrates the
role that procedures play in the minds of students when
they are presented math problems to solve. Given a
problem, students think of a memorized procedure that
might be applied to it. They don’t think through the
appropriateness of the procedure to the situation but
proceed with a mechanical application of it. When they
arrive at an answer, they’re done. They don’t evaluate
the result’s appropriateness, nor do they nd reason
to reconsider the procedure they chose. A desire to
reconsider might have been fruitless anyway. Their lack
of understanding of the meaning underlying procedures
would leave them little clue as to what might be an tting
alternative.
Conceptual Atrophy and the Failure to Reason
“Conceptual atrophy” is a phrase we coined in
our prior article (Stigler et al., 2010) and we use it to
refer to what happens to developmental mathematics
students as a result of their many years’ experience of
school mathematics. When students enter school, they
bring with them intuitive ideas about quantity. Those
intuitive ideas are often incorporated into mathematics
lessons in lower elementary grades, and in later grades
they frequently nd their way into the introduction to
a new topic. However, when it comes time to become
procient at a procedure, that conceptual basis falls
away. The math instruction students then encounter
frequently fails to capitalize upon students’ intuitive
ideas and instead emphasizes steps disconnected from
meaning. Even efforts to capitalize on students’ intuitions
(as with estimating) often quickly turn to rules and
procedures (as in “rounding to the nearest”). The result
is that the potential for students to develop a stronger,
more powerful conceptual grasp—a strong muscle, if
you will—goes unrealized. Whatever sense of number
and willingness to reason that students once had withers,
and the conceptual basis that would keep procedures
under control goes undeveloped. In the section above
we provided examples of developmental math students’
heavy reliance on procedures. Now we ask the question,
do any of students’ intuitive ideas about quantity remain
and, if so, what are their limits?
Not surprisingly, we saw a continuum in students’
sense of number and ability to reason. For some students,
basic concepts of number appeared lost. One would
expect, for instance, that an upper-elementary student
would be able to state that a proper fraction—say
1
2
or
1
3
—is greater than 0, but less than one. Those young
students could likely draw a number line containing 0
and 1, and then partition that interval into the appropriate
number of segments of equal length based on the
denominator. We found students, though, who struggled
to place common fractions correctly on a number line.
Student #21 (enrolled in Arithmetic) produced the
following diagram when asked to place the numbers −
3
4
and
5
4
on the number line.
Student #21
Student #10 (enrolled in Elementary Algebra) started
her drawing like this, and then the following discussion
ensued.
Student #10
Interviewer: Can you put
4
5
5
8
and on a number line?
Student: Maybe. Oh my goodness! I don’t know if this
is right. The number line is not my friend. Oh my
goodness! I don’t think this is right. Gosh! They’ve
shown this a million times but it never processes.
I: Well, tell me what you’re thinking about.
S: I don’t know. I’m thinking of going to 4 but then it’s
just like, then between 5 but I don’t know if that’s
right. I was thinking it was just here but I feel like
that’s 4
1
2
and not
4
5
. But I don’t know. That’s the
only way that makes sense to me.
11www.amatyc.org
“SOR RY, THAT’S NOT CORRECT.” “THAT’S CORRECT.”
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WA ad MathAMATYC Educator.indd 1 12/14/10 11:58:09 AM
12 MathAMATYC Educator ~ Vol. 2, No. 3 ~ May 2011
I: Okay.
S: I don’t see how you can go to 4 and then somehow
go to 5. Do you know what I’m saying? And then
like back-tracking there to 5.
I: Okay.
S: I just don’t see the connection.
I: Okay.
S: It’s one of those things that I just don’t understand.
My teacher would show me on the board and I still
won’t get it.
I: The number line?
S: Just like yeah how you can– Well just the fraction
or the number line. To me it’s like you either go to 4
or you either go to 5, you know? You don’t go [to 5]
and then like you back-track [to 4], like you know?
Like they should each be– like you graph 4 then you
graph 5. Not you go to 4 and you go over this much.
It just doesn’t process to me.
I: Okay.
S: And then do you want me to
5
8
?
I: Where would you think that might go?
S: Honestly, I don’t know.
5
8
. I was thinking of putting
it at– I don’t know why, just like putting it between
at like 6 or 7, only because it’s between 5 and 8.
When Student #16 (enrolled in Arithmetic) was asked
to compare
4
5
and
5
8
, he converted them to
32
40
and
25
40
,
respectively. When then asked to place the four original
fractions (i.e.,
4
5
,
5
8
, −
3
4
, and
5
4
) on a number line, he
placed only the numerators (see the following diagram).
He did not seem concerned by how the conversion
affected the values he placed or by the fact that he
converted only two of the four values. More importantly,
his understanding of fractions (like that of Students #21
and #10, above) didn’t take into account the relationship
between the numerator and denominator.
Student #16
Just as basic concepts of number appeared to have
atrophied, so too did understandings of basic operations.
It would be reasonable to expect that any young student
would be able to say that if you add two numbers to get
a third, subtracting either of the initial two from the sum
from would leave the other. This is often shown in the
“fact families” that students study in early elementary
grades. This understanding is important to establishing
the uniqueness of the sum of two numbers and the
inverse relationship of addition and subtraction. To assess
whether that understanding had withstood their years of
experience in math classes, interviewees were asked to
check 462 + 253 = 715 using subtraction. They nearly
always subtracted the second addend from the sum (i.e.,
715 – 253 = 462). However, when asked if they could
have subtracted the other addend instead, some didn’t
know, others were skeptical, and a few claimed it would
be incorrect to do so.
Student #26 (enrolled in Prealgebra) was among
those who believed it is possible to subtract only one of
the two addends to check the addition. The following
conversation ensued after she subtracted 462 from 715.
I: How do you know to subtract the 462?
S: [LONG PAUSE] I have no idea. [LAUGHS]
Because it’s the biggest number?
I: Okay. Could you have subtracted the 253, or can you
not do that because it’s smaller?
S: You could but [LONG PAUSE] I don’t think you
can. I think you have to subtract the top number.
I: Okay. Why is that, that you have to subtract the top
number?
S: [PAUSE] I have no idea.
Student #22 (enrolled in Arithmetic) was a skeptic.
She subtracted 253 from 715. The conversation after that
point was as follows:
I: How did you know to subtract 253?
S: Because it’s the smaller number. Well, I don’t know.
My teachers said to always subtract the bottom
number.
I: Okay. Could you have subtracted the top number
from 715?
S: No. Well, I mean I guess you can. I don’t think so,
though. I don’t know. ‘Cus all my teachers taught me
that way, so I don’t think so.
I: Okay. What if you try it and see what happens?
S: So 715 – 462 = 253. [STUDENT WORKS
PROBLEM ON PAPER] Oh, you can.
I: So what did you notice?
S: You could subtract any one and you’ll still get either
one as the answer.
Student #9 (enrolled in Elementary Algebra) had an
epiphany during the interview. In the following excerpt
he shares his excitement at discovering that either addend
can be subtracted from the sum to obtain the other
addend. It’s worth noting that it took little prompting to
13www.amatyc.org
set the student on the course of this discovery.
I: How did you know to choose 253 to subtract from
715?
S: I had a feeling you were going to ask me that right
when I pulled that in for some odd reason, and I
know I’m not going to give you an answer why. I
don’t remember why, but I just know that with some
odd reason when checking– I don’t know if it’s
always true that you pick the bottom one, ‘cus I was
thinking like why did I pick the bottom? How come
I didn’t pick the top one? I mean, what happens if
you pick the top one?
I: Well, you’re welcome to see, if you want.
S: Let me see that. Hold on. I never thought this until
now. Wow, this is very interesting. I never thought
about that. Let me see. So you’re going to get 3
that becomes a 6 and 6 or 5 will get you 11 32. Oh
wow, so it doesn’t even matter. Is that true? Does it
even matter which one you pick? ‘Cus 253 is in the
original equation so I guess you’re right. I guess it
doesn’t matter which one you pick. I don’t remember
that. I just always remember that picking the last one
for some odd reason, which is really interesting now
that I learned this right now. I guess all you have to
do is get your solution and then subtract it from any
of these numbers. Well, I don’t want to say “any”
because you could be left with a negative. I want to
say you could, but I’m not too sure. But I feel that
I’m safe going by the answer subtracted from the
bottom number and your answer should match that
one to make sure that it’s correct.
I: Okay.
S: So yeah.
I: Great.
S: I’m going to ask my math teacher today. It’s
interesting I just learned that.
We might speculate that Student #9, and others like
him, were encouraged by their K–12 teachers to check
their work by using inverse operations, and later in
algebra class students utilized inverse relationships to
solve equations. In both cases, addition and subtraction
are inextricably related. However, somewhere along
the line, the act became just one more procedure
disconnected from its meaning. The concept students
likely understood at one time had deteriorated, and was
never connected to new topics.
Interestingly, we did nd some instances in which
students could rely on their intuitive ideas about quantity.
However, when the same concepts were transferred
into mathematical notation—when the math looked
more like a math class—students set their conceptual
understandings aside. For example, when we asked
students, “What would happen if you had a number and
added
1
3
to it? Would it be more than what you started
with, less than what you started with, the same as what
you started with, or can you not tell?” Eighty-seven
percent of students answered correctly that the resulting
number would be larger than the original. We followed up
immediately with the question, “If a x+ =
1
3
, is x more
than a, less than a, the same as a, or can you not tell?”
Now students weren’t so sure. Thirty percent thought the
second question was unanswerable unless a and/or x was
provided, in spite of the fact that 78% of those students
had just correctly answered the same question, albeit
without mathematical notation.
We then repeated the same two questions, but
replaced addition with multiplication. That is, “What
would happen if you had a number and multiplied it by
1
3
?” and, “If a x× =
1
3
, is x more than a, less than a, the
same as a, or can you not tell?” A student (enrolled in
Arithmetic) who was able to agree that
1
3
times a number
would result in a number less than the original number
used the idea of dividing by 3, a rare occurrence in this
sample. However, he took a very different approach when
asked the same question using the equation a x× =
1
3
. He proceeded to choose a number for a, and then to
multiply the number by 1 and by 3 to nd x. If a = 2, for
example, then 2 × 1 = 2, and 2 × 3 = 6. So, you end up
with x=
2
6
, which can be simplied to
1
3
. He went on
to show that this method works for every number, always
resulting in
1
3
. Whatever a is, x will always be equal to
1
3
! This particular student commented that he has been
“taught by like seven million teachers how to do this.”
While it is true that the student incorrectly used
2
2
(rather than
2
1
) as an equivalent form of 2, we would not
say that this was his most shocking error. It seems more
important to note that he failed to notice that multiplying
1
3
by a number other than one could not possibly result
in
1
3
. We would avoid saying “if he had used the correct
14 MathAMATYC Educator ~ Vol. 2, No. 3 ~ May 2011
representation of 2, he would have been able to state
that the result is larger.” Rather, we would argue that
more importantly, accepting his products shows that he
believes in his procedure, and so it is not necessary to
verify that his answer makes sense. It is interesting to
note that when multiplying, this student used
2
2
as an
equivalent form of 2, and when adding used
2
0
. As was
the case for other students, equivalence had been reduced
to a set of rules for writing equivalent values that came
to be far removed from the concept of equivalence.
“Equivalence” appeared to be thought of as something
you do or make, not as something you maintain.
When concepts atrophy, students are left with no
foundation upon which to reason. There were points
throughout the interviews where we saw students miss
opportunities to reason. One such place was when we
asked students to identify the larger of
a
5
and
a
8
. An
answer that involves reasoning might include that some
number of fths is larger than the same number of
eighths, if
a
5
and
a
8
are based on the same whole. Only
two interviewees reasoned in that way. Most students
relied on the application of an oft-practiced procedure of
creating common denominators, usually
8
40
a
and
5
40
a
.
Though not incorrect, it is evidence that students apply
known procedures rather than using reasoning, even
when reasoning is more efcient. (In our previous article
we showed that when students used only reasoning about
dividing a whole into pieces, the value they identied as
larger was always
a
5
.)
In the prior section, we concluded that students
rotely apply procedures. The failure we see among
them to draw on concepts is the other side of the
same coin. For some students we interviewed, basic
concepts of number and numeric operations were
severely lacking. Whether the concepts were once
there and atrophied, or whether never sufciently
developed in the rst place, we cannot be certain.
What we do know is that these students’ lack of
conceptual understanding has, by the time they entered
developmental math classes, signicantly impeded the
effectiveness of their application of procedures. And
application of procedures is, without concepts and a
disposition to reason with them, all that students have
left to go on.
What Might We Do to Remedy the Problem?
The goal of much of developmental math education
appears to be to get students to try harder to remember
the rules, procedures, and notations they’ve repeatedly
been taught. We are thinking about a different solution,
one motivated by the picture we’ve painted of
developmental math students. We propose a solution
with three elements, each of which is necessary for
success. Though in this paper we don’t take on the ne
details of how to successfully implement the elements in
developmental classrooms, we believe that guring out
how to do so might lead to dramatic improvements in
student outcomes.
Element One: We must nd a way to reawaken
students’ natural disposition to gure things out and re-
socialize them to believe that this is a critical element
of what it means to do mathematics. What might such
a class look like? Tasks presented to students would be
crafted to reveal intuitive understandings of quantity and
operations and build number sense, and teachers would
create conversations with students to elicit and enhance
these understandings. Students’ work would highlight
the value of mathematical reasoning. They would be
pressed to generalize and to consider the limits of their
generalizations, sometimes solving problems wholly
without the use of procedures. In fact, it would probably
be best to pose problems that cannot be answered by
applying standard procedures, in effect forcing students
to think to nd a solution. If we want students to
strengthen their ability to reason productively, we must
convince them that such an approach to mathematics can
yield them the answers they seek.
Element Two: Necessary to convincing students to
think is providing them with productive things to think
about. Specically, rather than asking students to call
to memory what they’ve learned about procedures, ask
them to consider the implications of concepts that seem
obvious and make those concepts explicit. A teacher
might, for instance, connect fractions and division,
discussing that a fraction is a division in which you
divide a unit into n number of pieces of equal size.
Alternatively, the teacher might initiate a discussion of
the equal sign, pointing out that it means “is the same as”
and not “here comes the answer.” Teachers may think that
it’s implicit in math that concepts, objects, and notation
are connected and to point out the obvious would be
superuous, or even demeaning. We argue instead that
making big, obvious concepts and connections explicit
helps students to organize the domain. The real challenge
here, and a place in which further investigation is needed,
is to gure out which are the most powerful concepts for
students to work with. Which will help connect together
15www.amatyc.org
the largest portion of the domain?
Element Three: Finally, once students begin to
appreciate the value of guring things out and have
begun to lay the foundation of powerful concepts, we
can reintroduce procedures into the curriculum. In
many cases, standard algorithms map directly onto how
students solve problems without them. When this is the
case, it should be pointed out. The result will be that
procedures are no longer seen as arbitrary, sometimes
magical, series of steps, but rather as logical ways
to organize effort, connected to core concepts that
organize the mathematical landscape. We thus advocate
for teachers developing procedures while consistently
maintaining connections to the concepts that underlie
them. Procedures should be seen by students not as the
primary resource available for problem solving, nor as
a replacement for thinking. Instead, procedures should
be seen as efcient mechanisms for solving problems,
supporting and being supported by sense-making.
That we advocate for reasoning and building
knowledge of concepts shouldn’t be taken to imply that
we oppose practice (more commonly associated with
learning procedures). Mathematical thinking is a skill
and, as such, requires deliberate practice. We suggest that
teachers give students repeated opportunities to think
and reason, linking core concepts to rules, procedures,
and notations. The practice we envision is not one of
large numbers of problems similar to each other and
to the problem demonstrated by the teacher, but rather
small numbers of rich problems carefully selected to
highlight and develop concepts and build students’ skills
in applying them.
Glimmers of Hope
In spite of some of the distressing ndings we
reported on here and in our prior article, we also see in
the interviews some glimmers of hope that the remedies
we’ve suggested may prove effective. It’s a nontrivial
nding that students were eager to share with us their
mathematical thinking and that they required little
prompting to do so. Though students are rarely asked
to think aloud as they solve problems and to share the
rationale for their actions, they quickly fell into that
routine. When given an opportunity and limited guidance,
(often only the prompt “Why did you do that?”) they
were able to reason. It took little for us to set those
moments in motion: a few, well-crafted prompts, a
focus on understanding “why,” and a lot of listening
to students’ thinking. Students also reported having
learned from the interview experience (though that
wasn’t necessarily our intent!). They made connections
and saw value in them. Importantly, those experiences
of discovering connections were perceived positively—
indeed, sometimes joyfully—by students. The key
seemed to be to give tasks that presented opportunities
to reason and to press students to reason when the
opportunity was present.
Student #28 (enrolled in Prealgebra) provides an
example of how students sometimes came to reason
when encouraged to do so. This conversation shows not
only the student’s adeptness with procedures, but also
willingness to reason directly once he had exhausted the
two procedures he knew:
I: If I have these two numbers—I have
a
5
and
a
8
—
which one of those two is larger?
S: Let’s see,
a
5
.
I: Why do you say that?
S: I’m just guessing here, but I got the, I think it’s
called, the greatest common factor or something, a 5
and 8. So what I just did was I turned the 5 and the 8
into 40, both of them, and I multiplied 5 × 8, so it’d
be 8 over 40… 8a over 40, and this one would be 5a
over 40. So I gured that 8 a’s is greater than 5 a’s.
I: So what you did is you got common denominators,
and then you compared the numerators.
S: Yes.
I: Is there a way that you can think about this problem
simply by comparing the common numerators as
they are?
S: As they are? I guess I could’ve done
1
5
and
1
8
. So
1
8
would be– we could just probably turn them into
decimals. So like 5 divided into .10. No. It’ll be .2,
I guess. And then we turn this one into a decimal,
and, well, I think 2, 6, .125 if we turn this one into a
decimal.
I: Okay. Do you mind if I write down what you did?
S: Yeah.
I: You turned that
1
5
into this [.2], and you turned
1
8
[into .125].
S: Yeah, I guess you can do that. I’m not sure. But it
would seem like .2 would still be greater.
I: Yeah. So you’ve, so far, found two ways to compare
them. So you got common denominators and
compared the numerators, and then you substituted
for a and turned it into a decimal. But I’m still
curious if you can think about– Is there a way to
compare them without doing anything to them?
S: I guess we can, let’s say, I don’t know. Okay. So let’s
16 MathAMATYC Educator ~ Vol. 2, No. 3 ~ May 2011
say
a
5
and
a
8
are two pies and they’re the same size.
If you cut one into 5 pieces and you grab a slice, it’ll
be bigger than the other pie if you cut it to 8 pieces,
and you grab a slice.
I: Great. So three different ways to compare.
Other glimmers of hope came when we explained to
students how to use decomposition and the distributive
property to do mental multiplication. We demonstrated
that 22 × 13 can be thought of as (20 × 13) + (2 × 13) and
then asked students to compare that to what they had done
when they had solved 13 × 22 using the standard algorithm
(for which they normally placed 13 above 22 before they
multiplied). Student #8 (enrolled in Elementary Algebra)
discovered at this point why there’s a “0” in the second
partial product of the algorithm, and wondered aloud why
he had never noticed it before. By presenting a second
method for solving, he gained not only another tool in
his repertoire, but also gained a deeper understanding of
the standard algorithm. The algorithm was no longer a
set of random steps. From this we see that there are many
benets to exploring multiple solution methods and to
examining how they relate to one another and to their
underlying mathematical ideas. In the interviews, we set
up the potential for discovery and even without making
a further effort to teach, a student learned. Imagine then,
what can happen in a classroom when an effort is made to
surface and explore connections.
Directions for Future Work
Our interviews with community college
developmental math students cannot be used to fully
substantiate our image of where they are and how they
might have gotten there. Nor can we know that our
suggestions will prove effective. We are, after all, only
at the beginning of our journey toward understanding
what’s going on with these students, mathematically
speaking, and how we might be able to change it. We
hope that future work will seek to address questions
such as whether community college is too late to draw
upon students' intuitive concepts about math. Do those
concepts still exist? Is community college too late to
change students' conceptions of what math is? To what
degree will students resist a different approach to math
teaching and how difcult a task will re-socialization
be? (Furthermore, how difcult will be the task of re-
socialiazing instructors?) If we can change students'
beliefs about the nature of mathematics, will it have an
effect on their disposition to reason? To what degree does
better conceptual understanding impact the successful
application of procedures? When students have a
disposition and the tools with which to reason, do they
apply procedures more appropriately? Do they become
bothered by inconsistencies in the results produced by the
procedures they apply?
Future work needs also to focus on existing practice
in community college math classrooms which, to date,
very little work has sought to describe. If there are select
places where instructors are emphasizing concepts
in their teaching, in what ways and to what degree is
it effective? Finally, can some of our suggestions be
implemented at the K-12 level and, if so, can we prevent
students from having to enroll in community college
developmental math classes in the rst place?
References
Battista, M.T. (1994). Teacher beliefs and the reform
movement in mathematics education. Phi Delta
Kappan, 75(6), 462-470.
Gelman, R., & Gallistel, C.R. (1978). The child’s
understanding of number. Cambridge, MA:
Harvard University Press.
Ginsburg, H.P. (1989). Children’s Arithmetic. (2nd Ed.)
Austin, TX: Pro-Ed.
Hiebert, J. (1984). Children’s mathematics learning: The
struggle to link form and understanding. The
Elementary School Journal, 84(5), 496-513.
Hiebert, J., Gallimore, R., Garnier, H., Givvin, K.B.,
Hollingsworth, H., Jacobs, J., Chui, A. M-Y.,
Wearne, D., Smith, M., Kersting, N., Manaster,
A., Tseng, E., Etterbeek, W., Manaster, C.,
Gonzales, P., & Stigler, J.W. (2003). Teaching
mathematics in seven countries: Results from the
TIMSS 1999 Video Study (NCES 2003-013).
Washington, DC: U.S. Department of Education,
National Center for Education Statistics.
Knuth, E.J., Stephens, A.C., McNeil, N.M., & Alibali, M.W.
(2006). Does understanding the equal sign matter?
Evidence from solving equations. Journal for
Research in Mathematics Education, 37, 297–312.
National Research Council. (2001). Adding it up.
Washington, DC: National Academy Press.
Schoenfeld, A. H. (1988). When good teaching leads
to bad results: The disasters of “well taught”
mathematics classes. Educational Psychologist,
23, 145-166.
Schoenfeld, A.H. (1989). Explorations of students’
mathematical beliefs and behavior. Journal for
Research in Mathematics Education, 20(4), 338-355.
Stigler, J.W., Givvin, K.B., & Thompson, B.J. (2010).
What community college developmental
mathematics student understand about mathematics.
MathAMATYC Educator, 1(3), 4-16.
17www.amatyc.org
Appendix: Questions from the Interview Protocol
Opening
1. If someone is good at math, what exactly are they good at? For example, some people say math is about
remembering rules and procedures. Other people say it’s about understanding and reasoning. What do you
think?
Mental Multiplication
2. 10 × 3 = ______
3. 10 × 13 = ______
4. 20 × 13 = ______
5. 22 × 13 = ______
6. 30 × 13 = ______
7. 31 × 13 = ______
8. 29 × 3 = ______
9. How would you do 22 × 13 it if you weren’t asked to do it mentally?
10. Why did you put a “*” [or ‘0” or blank] here?
Reverse Operations
11. How would you check to see if the answer here is correct?
• If reworks problem: Is there another way to check?
• If no other way: Is there a way you can use subtraction to check?
12. Of these two numbers, 572 and 86 [written horizontally], which is larger? How do you know?
• If ‘has more digits’: Can you always apply that rule? What about 572 and 367?
13. Of these two numbers, 0.572 and 0.86 [written horizontally], which is larger? How do you know?
• If ‘has more digits’: Can you always apply that rule? What about 0.9 and 0.1111?
• If incorrect: correct student and ask if s/he can see why 0.9 > 0.1111.
14. Can you show me how you would set up 572 – 86? [written vertically]
15. Can you show me how you would set up 0.86 – 0.572? [written horizontally]
If incorrectly lined up: Is the placement of the decimals important? How did you decide where the decimals go?
22
× 13
66
22*
286
462
+ 253
715
18 MathAMATYC Educator ~ Vol. 2, No. 3 ~ May 2011
16. Can you show me how you would set up 0.572 – 0.86? [written horizontally]
• If incorrectly lined up: Is the placement of the decimals important? How did you decide where the decimals go?
17. Here [in whole number subtraction] you lined up the 8 and the 7 and lined up the 6 and the 2. Here [in
subtraction of decimals] you lined up the 5 and the 8 and lined up the 7 and the 6. Why is that?
Comparing Fractions and Placing Them on a Number Line
18. Which of these two numbers is larger
a a
5 8
and ? How do you know?
• If student struggles: Could you try substituting a number for a? Would that be a way to think about it?
• Will that work no matter what number you choose for a?
19. Which of these two numbers is larger
4
5
5
8
and ? How do you know?
• If 4 and 5 are closer together than 5 and 8: What about these two numbers
4
5
2
3
or ?
• If
4
5
is closer to 1: Tell me a little more about why that strategy works.
20. Can you draw a number line and place
4
5
and
5
8
on it?
If student is not able to draw a number line, draw just this much:
21. Can you now add these numbers to it −
3
4
5
4
and ?
aand
1
3
22. What happens if you take a number and add
1
3
to it?
23. In this equation a x+ =
1
3, do you think x is bigger than a, smaller than a, equal to a, or can you not tell.
24. What happens if you take a number and multiply it by
1
3
?
25. In this equation a x× =
1
3
, we’ll say that a is a positive, whole number. [Make sure that student understands
that ‘×’ means to multiply.] Do you think x is bigger than a, smaller than a, equal to a, or can you not tell?
• If student struggles: What if you had 6
1
2
×?
• If student gets it, return to a x× =
1
3
.
Closing
26. If you could give math teachers advice about how to teach in a way that would better help you understand math,
what would you tell them?
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